# Do you understand my first-grade child's homework?

My six-year-old told me she doesn't understand her homework. After studying it for 15 minutes, I *think* I understand what she's supposed to do, but I'd like a second opinion.

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1. Vorn says:

The task here is to “make ten” – add counters to the top bin (right bit), overflowing to the bottom bin, according to the bottom number in the addition. Then: the box below the 10 is the number of counters in the bottom bin, and the sum box in both will be the same.

1. goldengalaxia says:

Vorn is 100% correct, that’s how the problems need to be done. The two ending problems will equal the same number. ^_^

2. AirPillo says:

This looks like the same lessons my 9-year-old brother is forced to do. Whatever workbook they’re from, it’s honestly chock full of ambiguous nonsense that’s far worse than this, where the largest challenge is deciphering what the heck is being requested of the student.

Whoever wrote this damned workbook is a mountebank and a fool.

Agreed (at least about being a Mountebank). This might (I emphasize, *might*) make sense within the context of a small lecture on place value (as in, this is how addition *works*), but as a stand-alone homework assignment it suffers from insufferable vagueness, plus incommensurability with any standard mathematical terminology.

3. Caroline says:

The layout of the counters tipped me off. They’re in groups of ten. She’s supposed to draw the appropriate number of counters, count up the total — and then figure out how many counters plus ten give the same total.

Like in the first one, there are 8 counters already. She draws 3 more. Count them up and get 11. Then, look at it again to see that 11 counters makes one group of 10, plus one extra.

I think this is supposed to teach them about base-10 numbers — that 11 means one 10 and one 1 — but it’s awfully unclear what to do without instructions or an example.

4. Anonymous says:

8+3 = 11; 10 + 1 = 11; put three pennies in the bottom ten-box.

9+3 = 12; 10 + 2 = 12; put three pennies in the bottom ten box

7+ 4 = 11; 10 + 1 = 11; put 4 pennies in the bottom ten box.

I hope.

1. Anonymous says:

No. The spillover to the bottom grid is 1 for the first one, 2 for the second one, and 1 for the third one.

WHile the directions aren’t crystal clear, I think it only takes about 30 seconds to figure out.

This is a very good and useful math exercise for children to learn the kind of visual and physical insight into mathematics that comes naturally to some people. I highly approve.

2. Anonymous says:

OMg those are PENNIES!!! Could not tell in the pic they just look like spots in shades of brown, kind of like a makeup pallet lol.

3. Anonymous says:

I’m with you… We had a eerily similar homework assignment recently (my son is in 1st grade, also.) HE explained it to me in the way that you outlined. I would go with that.

What “Making 10” has to do with anything – I just don’t know.

5. Micah says:

I agree with Vorn. But it took me several minutes of studying this to come to that conclusion. And I don’t think I ever would have figured it out with just one problem (it wasn’t until I compared problem #1 to the other two that I figured it out).

Presumably this was demonstrated by the teacher in class?

6. Anonymous says:

It’s two ideas-first adding 8+3=11, the second is that the number “11” is 1 unit of “10” plus “1”

In the first problem, she’s supposed to draw three more counters, to show 11.
For the second part, it should be 10+1=11

Disclaimer: I have a 15 year old, so I’ve seen things like this before

7. Vorn says:

The actual /point/ is that you add just enough to the top number to make it 10, then subtract the same amount from the bottom number, and the resulting bottom number is the ones digit of the result.

Oh. And I’m a professional math tutor (though I work at college level), and I had to stare at it for a bit to get it.

1. Anonymous says:

Vorn, in #7, I can’t understand your answer. Subtract? I don’t mean to make fun of you, but what are you saying? The kid has to subtract in order to do simple addition?

8. Anonymous says:

I think, and I teach 6th grade so the math is very different, that you solve the first one — 8 + 3 is 11 — then you solve the next one to 11 — 10 + ? = 11.
I think.
Once you fill in the top counters, you have 10, so the bottom counters would be the second number in the 10 + problem.
Yes, it’s a long hard struggle to make heads or tails of elementary math these days.
Luckily, I’m multiplying fractions.

Good luck!

9. Anonymous says:

Draw three more counters, which will require spilling one over to the second box. Then consider the first box as one unit of ten, and count the tokens in the second box to derive 10 + 1 = 11.

Which I guess is one way to start talking about the place-value system, but not necessarily the one I would choose.

10. Tdawwg says:

The task is to become numbed by the soul-crushing mindlessness of the American educational industry.

11. Anonymous says:

Dang it, I was wrong . . . fill the extra pennies (3, 3, 4) all the way through and overflow like the folks above said.

And on your free time, teach the child to count on fingers like normal first-graders.

1. bwcbwc says:

“…teach your kid to count on her fingers like normal first graders”

Better yet, teach your kid to count on her fingers in binary, like Isaac Asimov.

Bart: Cool, I can count up to 10 on my fingers!

Nigel: That’s nothing. My fingers go up to 11!

Isaac: What a wimp. Mine go up to 1023!

1. funkyderek says:

@bwcbwc:
Counting to 1023 is technically possible but quite difficult due to having to use binary and the considerable dexterity required. An easier way to get more than 10 from your digits is to use the fingers of one hand to count from 1-4 and use the thumb as a 5. That way you can easily count to 9 on one hand. Use the finger of the other hand as tens (and the thumb as 50). This brings you up to 99. Should be enough for most tasks and it won’t make you look like an arthritic Metallica fan!

12. Myrcurial says:

And this is exactly why I’m worried we’re headed for another dark age — we can’t teach our kids to do arithmetic without trying to fit it with a new pair of fancy pants and prove to those beknighted predecessors that we’re smarter.

No wonder I get a dirty pile of crumpled bills, a receipt and some coins piled on top for good measure, that kid working the checkout couldn’t do the math necessary to give change if it was the “skill testing question” on a \$50MM lottery prize.

13. Anonymous says:

MMath here: took me more than a minute. I weep when I see this kind of homework. Who wants to do that? It’s like putting an Ikea bed together. Teach in class, homework is reinforcement.

1. JackThompson says:

I actually just put an Ikea bed together.

Ikea beds have clearly stated instructions laid out as relatively clear, more-or-less universally understandable, uncomplicated diagrams. This homework assignment is the manual for a Volvo in comparison.

14. Anonymous says:

Heh, these directions are poorly written.

15. Anonymous says:

i think maybe, its if 9+3=12, then 10+2=12 ? if 8+3=11, then 10+1=11… and so on..

1. Anonymous says:

That’s what I got out of it. This is hilarious!

16. AirPillo says:

Oh and Mark, could you please let us know the source of these assignments if you happen to come across it? I’d like to confront the local school district about its use, it’s extremely unfair to kids for their homework grades to be hamstrung by the incompetent design of their assignments, and this is the _only_ math homework my poor brother receives.

17. RikF says:

I care not for this mathematical frippery, but shall celebrate AirPillo’s use of mountebank. An underused word if ever there was one!

18. Antinous / Moderator says:

Basically, they’re encouraging her to count on her fingers instead of memorizing simple math equations.

19. KWillets says:

“Everyday Math”?

Draw in the number of additional counters specified by the second operand. Decompose into 10+x.

This is fairly common mental computation task, eg for any addition which shifts the 10’s digit. These days the tricks are taught formally, but the vocabulary to describe the process is lacking, so parents can’t decipher what’s supposed to happen.

20. Anonymous says:

That’s like a really hard question from an iq test. Wow! Caroline is right, in each problem the total should be the same. Like for 1. 8+3=11 and 10+1=11
2. 9+3=12 and 10+2=12

There has to be more instruction for this. This can’t be the whole thing can it? Crazy. That’s terrible.

21. Anonymous says:

So Vorn, are you saying the arrow functions as an equals sign? Why didn’t they just use an actual equals sign?

1. bwcbwc says:

The arrow functions as a transformation indicator. 8+3 becomes 10+1 when you write it in positional decimal notation.

2. Anonymous says:

Exactly!!! or the equivalent sign (< =>) ???!!!

22. optuser says:

OOH! Mr. Kotter! I know!

I think the exercise is to teach kids “base 10” and help them visualize adding numbers across the great abyss of double digits. So to visualize adding 8 + 3 in the first one, I would write the numbers 1 and 2 in the top block of the grid to complete the first 10 block, then the number 3 in the first (upper left) block of the empty set below. The equivalent problem to the right of the arrow should be 10 + 1 = 11. Your child visualizes that when you count up from the number eight three units, it’s the same thing as counting up from ten by one unit.

Problem 2’s grid would have number 1 in the almost-full grid, then 2 and 3 in the empty grid. The equivalent problem should be 10 + 2 = 12.

Can I have a gold star on the daily achievement poster board?

23. Anonymous says:

I agree with what other people are saying. It took me just a minute to figure out.

It seems like the kind of assignment that would make sense if they’d been doing some examples as a group in class. I know I had some assignments as a kid that only made sense because we’d done some together.

Of course, that doesn’t seem to be the case. It’s a stupid assignment.

24. Aloisius says:

I recognize this ploy. You pretend that this is your child’s homework but really you just want us to help you cheat!

:)

25. Anonymous says:

Okay, so I totally missed it. I assumed the bottom set of boxes was for charting the second half of the problem, the “10+” problem. Maybe they need a little graphic design help, too.

26. Anonymous says:

I agree that it is presented poorly. But if they had included a worked example, it is a fun little exercise and a clever angle at introducing different bases.

27. Church says:

I had to read Vorn’s explanation (twice) to make sense of it.

KWillets seems to decipher the intent, but it’s not really a situation where that is useful. So I have to wonder if this is helping anything.

28. Anonymous says:

My 6-year old gets the identical homework in North Carolina, and we’re equally baffled that they don’t give homework with clearer expectations. I’m always surprised how much I have to guess at to deduce what they’re looking for… and I’m a scientist with a PhD!

29. thequickbrownfox says:

Would be better to teach your daughter Roman numerals.

30. Anonymous says:

First line, the name that’s worth more than the answers. Second step, first box, the answer to the addition problem (tray 1 + the remaining amount of tokens given in the math problem). Third, next box over the box to the right of the first box, the answer to the addition minus 10 which will always equal the bottom rectangle ice cube tray . Fourth step, the answer to the addition: in the box on the bottom of the other box (the sum of tray 1 + tray 2). Fifth step, which should be done most likely during the second step, fill out the boxes with the correct amount of tokens. Sixth, repeat the second through fifth steps for the next problem (#2 and #3). Seventh step, hunt down anonymous commenter for either helping or ruining your child’s future.

31. SamSam says:

This is a GREAT exercise, and the research literature supports it.

The idea is that young students should become fluent in the mental re-arrangement that most people who are fluent in math do subconsciously.

When we add 8 + 3 in our heads, some people add 8 + 3 directly, while others see that you can add 2 to 8 to get 10, and 1 more “spills over” to get 11. It turns out this second way is a great way to think about numbers, and helps with much harder sums later on. The same kinds of skills are used when you multiply 8 x 16, and realize that you can simply add 8 x 10 and 8 x 6 together.

Many people, including those in this comment thread, look at things like this and “despair.” I despair when I read those comments, because it shows that, when it comes to learning, far to many people are stuck in the notion that “If what I was taught in 1973 was good enough for me, then it should be good enough for anyone. Why change how math is taught?”

No: if evidence and research show otherwise, embrace new teaching techniques.

(Full disclosure: I’ve worked for a NSA-funded non-profit educational software and research company for the past four years.)

1. AirPillo says:

I do appreciate the value of the actual concepts these lessons are communicating, but I am repeatedly outraged when I’m the third person my brother has to come to for help because nobody could decipher the abhorrent writing used to explain the problems.

Once he actually understands what the slothful instructions are actually instructing, he usually solves the question immediately.

It makes me furious when, in a math lesson, a child who understands the math is incapable of performing the task, because the lesson is written in incompetent prose. The real challenge presented to him by his homework is deciphering the awkward fumblings of someone who either glossed over their job, or has a poor command of the language they’re writing in.

2. SKR says:

I have to agree with SamSam. When I first saw this I was a little confused. This is why I think they need to have better directions for the parents that never were taught this way, so they can help their children like they ought. But this is the kind of fluency with numbers that people that are self-proclaimed, “bad at math” have a difficult time grasping. For me, it usually comes up regarding multiplication or division. Usually it is something like, “what is 8×8?” “Do you know what 8×4 is?” “Yeah, 32.” “Ok, 2 of those.”

3. dssstrkl says:

I’m sorry, but I have to completely disagree with what you said. Research and literature? Care to provide some, because without providing links to any relevant papers that have been published in recognized journals, then you’re full of it. If it takes people who work with advanced math (I need to use calculus in work fairly regularly) some effort to figure this problem out, then the problem is not with us, how we learned math, or with the child, but with the material and the way that its presented. I was able to figure the problem out (assuming that the first few commenters were correct), but explaining it in a way that made sense took even longer.

Problems like this will have the exact opposite of the effect that you seem to think. Instead of giving children a more effective way of dealing with math, problems like this will simply make them feel stupid and turn them off from school. I had this very problem when I was younger. Clearly it wasn’t me, since I somehow managed to make it through college with a science degree, but was clearly with the way that I was taught in primary school.

If my child came home with homework like this, I would make it my point in life to find out why trash like this was being passed off as educational material and to remove it in favor of useful tools taught by competent professionals.

By the way, you wrote “to” when you clearly meant “too.” Not to be the grammarian, but this is a question of proper education, after all.

4. mbruceevans says:

This definitely looks like my daughter’s Everyday Math curriculum homework. At first I tried to be open-minded about this “new”math, then, as my daughter’s self-esteem plummeted along with her math grades, I checked out the wikipedia entry for Everyday Math and found that I wasn’t alone in doubting the effectiveness of the curriculum. I have no objection to making research-validated improvements in the way we teach our children; however, I don’t think Everyday Math is a step in the right direction.

1. kostia says:

As part of my job I have done design and layout for a LOT of Department of Education evaluations of studies of elementary math curricula. The results for almost all of them are disappointing.

But what really gets me is that these reports are published so quietly that they are invisible to the people who would benefit most from them. By looking up the original studies in the reports’ references, a parent really could find out the strengths and weaknesses of a particular curriculum, leading to knowledge of how best to work with his or her child within that curriculum or even recommend a better one to the school.

The Department of Education report for Everyday Math is at http://ies.ed.gov/ncee/wwc/reports/elementary_math/eday_math/

The “potentially positive” result corresponds to a 4 on a 5-point scale. Most elementary math curricula I did reports for got scores lower than this.

5. alowishus says:

Thank you! I had absolutely no friggin idea what was going on with this assignment until I read your comment. I was never taught math this way and I’m math illiterate to this day, despite two college degrees and 10 years in my profession.

1. peterbruells says:

I didn’t get it either, but didn’t stop to think about it. (Well,actually i thought this was about algebra. )

Any way, part of the problem in judging these exercises is that math and even more so reading are overlearned (direct translation if the German term, don’t if there’s an English one). Both behaviours become quite ingrained, deciphering and solving are basically the same and done with little or no concsious efforts. Both techniques change how we think and it becomes near Impossible to get in the mindset of someone who hasn’t mastered these techniques as an adult.

6. lhopitalified says:

“When we add 8 + 3 in our heads, some people add 8 + 3 directly, while others see that you can add 2 to 8 to get 10, and 1 more “spills over” to get 11. It turns out this second way is a great way to think about numbers, and helps with much harder sums later on. The same kinds of skills are used when you multiply 8 x 16, and realize that you can simply add 8 x 10 and 8 x 6 together.”

I disagree – for me, adding one-digit numbers is pure memorization (or very close to it). Whether I use spill-over or not for adding larger numbers is case-specific, sometimes I figure out what needs to be added to get to the next power of 10, sometimes I just do “plain” addition and carry over one’s. This is very different from the distributive property for multiplication. And any “good” teacher who is explaining how to multiply 8 and 16 together would use the “standard” algorithm of multiplication to demonstrate how it all inter-related with place-value notation and distribution of multiplication over addition.

You can also go further in your “optimizations” and note that 8 x 16 = 8 x 8 x 2, and then you can exploit your knowledge of perfect squares. (or in the competitive math world, 8 x 16 = 2^3 x 2^4 = 2^7 = 128)

1. Anonymous says:

I agree that for one-digit numbers it’s just a memorization, 8+3 = 11. But for larger numbers this trick comes in handy all the time, e.g. to add 998 + 239 it’s much much faster and less error-prone to realize that it’s 1000 + 237 than to do the standard algorithm and all the borrowings. A lot of people do this kind of thing without even thinking about it, but for a first-grader I can see the value of teaching the shortcut directly, and explaining the concept with smaller numbers.

I have no problem with the curriculum (as others have pointed out, it’s basically the same principle as an abacus), but it seems like it may not have been communicated well. Of course we’d need to see more context: did the students see examples of this type of problem in class? Did the workbook include worked-out examples? What kind of instruction was given in class? etc. etc. It’s easy to pick a random page from a workbook and say it’s unclear without seeing the full curriculum.

7. Hools Verne says:

I use this trick for multiplications that I never managed to memorize. It gets me by but it can be tedious so, yes, it’s a nice skill to have but I still want to memorize my multiplication tables through to twelve by twelve one of theses days. Moreover, that it’s a good skill to have doesn’t make the presentation any less appalling.

8. RevEng says:

I don’t think it’s the idea that leaves them in despair; I’m left in despair and this is exactly the same way I think about math (and I’m an engineer).

What they find terrible is the lack of explanation. The title says “Making Ten”, which to me means, “find numbers that add to 10”. Then they show 8+3 — but that’s 11! Okay, they also say “draw counters to solve”. Well, unless you’re aware that they are being taught to count using pennies, it might not be obvious what the counters are. To make matters worse, they show 8 pennies (“Ah,” you say, “the 8 from the 8+3 problem”) but then there are only 10 spaces in that group (again you say, “but 8+3 is 11!”). And worst of all, the arrow from one problem to the next is completely unintuitive — what does an arrow mean in math? What does it mean in everyday life? I thought it meant, “Do the same thing for the next question,” since arrows are usually used to point to the next thing and to imply some sort of relationship between the two.

The despair others have expressed is that the instructions are too ambiguous for the parents to understand, let alone for the children. The children may understand better because their teacher has explained it to them in class, but the parents are left unable to help their children with basic arithmetic. Don’t we want to encourage parents to help their kids? The school system is already expected to do too much; we should be happy that parents are helping their kids. Yet, these worksheets (whose purpose may be sound, but whose execution is flawed) are getting in the way of that, and could well be getting in the way of the child’s learning.

Education is a difficult thing, but it seems we get it wrong more often than right.

The problem is that the research standard for education is so much lower than the research standard for the sciences, that the evidence that any of this is working is questionable. Combined with the fact that most of our elementary education teachers went into teaching in the lower grades because they didn’t like math, and that way they didn’t have to take anything past Algebra I, and they could get a D in that — explains why the math scores in my children’s school district are so abysmally low, and the children taught this way are still counting on their fingers in 6th grade.
Teaching “make 10” is a valid strategy, and there are programs out there using it that do it in a far more straight forward way. Try Singapore or Saxon Math.
Every week I have to straighten out the tripe my son learns at school. No — it’s not a number sentence, it’s an equation, and no you can’t really write a paragraph discussing all the three sided polygons that are NOT triangles, because there aren’t any.
I’m so tired of math programs that suck the joy out of the math, confuse the children, and create yet another generation of the confused.
Sorry SamSam, all of these math programs are ineffective, and it is time you all got over yourselves and admitted it.

1. Anonymous says:

Agree with this, 100%.

Thank god my Mom stepped in (a PhD in math) and taught me math. This was back in the late 70’s/early 80’s when this type of B.S. math started to creep in. It drove her (and still does) crazy. I can’t tell you how many times she’d re-teach the lesson to me, because the ‘new method’ was a bunch of nonsense.

I think if we would teach kids math, the real fundamentals of it, and the various applications of it (from music to engineering)… we would be pleasantly surprised at the amount they would grasp. We might even hear kids saying how much they like math. Seriously.

32. SamSam says:

One addition, though: if your child didn’t understand it, then it should have been explained better, or at least the homework should have included an explanation so that the parents could help. Great evidence-based teaching methods are only so good as their actual implementations.

1. thoris says:

33. lerasmus says:

Whatever this assignment is, I’m sure it passed a “learning outcomes assessment” that was administered by a school board.

34. Anonymous says:

The actual thought behind this, is to get the kids to think. My kids school district uses a similar (same?) curriculum called investigating math. It teaches the child how you got the answer, not just memorizing x’s tables, and not knowing why 2×2 is 4. this worksheet is part of a class room project, discussion, involving several consepts, arays, counting by 10’s, addition….
you should call the teacher, in our version there is a take home strategy guide for the parents who are stuck in the old system of math.

The real fun comes in a couple years when they start requiring more then one solution to the problems. I have a very literal and straight forward daughter, who when asked what 2×2 is will just write 4, and get annoyed that she has to show several ways of getting it. this system requires the student to fully understand the ways in which you get four. which seems tedious and stupid when they are young, but as they get older and learn more complicated math concepts this kind of thinking works much better.

35. Vorn says:

Anonymous #23: They didn’t use an equals sign because what they really mean is “transforms into”; I will often do the same thing when doing algebra, especially when on IRC when I’m constrained to one line.

36. Leslie says:

Maybe I’m a freak, but this was immediately obvious to me. Anon. #4 shows it perfectly. And the point is, if you know how many you need to get to the next 10, then add the remainder… So simple, a first grader could do it!

37. Anonymous says:

What’s the name of the book from which this lesson originated or is this just a scan of a print out?

38. Anonymous says:

I remember when I was in elementary school. Math questions consisted of:

1. 4+5=
2. 6+9=
3. 7-5=
4. Anne had three apples. George had six apples. George took Anne’s apples and ate them. Anne came at George with an axe and brutally caved in his chest. How fast is Train B going if it left the station at 6pm?

What reason is there to make it so confusing and backwards?

1. Maneki Nico says:

I’m with Anon. @ #40. That’s how I remember math too. The answers are:

1. 9
2. 15
3. 2
4. 65 years old.

I think…

39. Anonymous says:

face it. It’s better to be an English major.

40. Anonymous says:

what do the arrows mean? Couldn’t they have just used words?

And, seriously, some thing you just need to memorize. Addition and multiplication tables are 2 examples.

41. Vorn says:

I agree with SamSam on both counts; this exercise would be exactly right if they provided examples (and, imo, a “how-to” guide for parents), and I do hope they did more than just these three there. Ten or so, I think, would be appropriate. Enough to get the pattern figured out well enough that the next class session you can abstract it away.

42. Anonymous says:

@#32 – While the goal of the exercise is good (I personally do exactly what it’s trying to teach), the language used to instruct the kid on what they’re supposed to be doing is threadbare. It’s basically being presented with 10+x=y, define x, y.

43. Anonymous says:

Reminds me of Florida’s “butterfly ballot”

44. Anonymous says:

I also got it pretty quickly. However, I just don’t understand why you’d want to do it this way. I guess emphasizing the tens and the ones is important, but I’m not sure this exercise is really doing that.

SamSam – I’m glad that you are enthusiastic for this and glad that you are enthusiastic for trying new ways of teaching and definitely agree with you that we should not automatically reject stuff because it’s different than when we were kids. However, if the kid and his reasonably intelligent dad can’t figure it out, it’s not accomplishing what it’s supposed to. Am I right?

45. Laslo Paniflex says:

The learning assistance students at my school use a program that has very similar type of activities (An amazing little non-profit called JumpMath).

SamSam is completely correct. The idea to get kids to create mental shortcuts in their heads when they’re doing the adding. The problem is that people who do not need the shortcuts see directly teaching them as ridiculous. Most kids learn how to do this through repetition and their brains learn to do it automatically. But if I really try to deconstruct how I add 8+3, I really do think 8+2=10 plus 1 is 11.

In my school, there are dozens of students above grade give that would have a hard time answering 8+3 without using their fingers. The good news is, that schools today have resources have resources like this to help kids’ brain make those connections. Those kids use to just drop out.

1. alowishus says:

Jesus Christ, this is like a fucking revelation to me. So THIS is how you add. Nobody ever taught me this. Seriously, I’m one of the kids who had to count on his fingers all the damn time. So I just said screw it, I’ll read a book instead. Flunked Math, aced English.

See, this is why I hated school and most of my teachers. Instead of breaking down addition, subtraction, or any basic math into these little short cuts, they would just punish me for not doing my homework. Can I sue? I can sue, right? Lifelong damage!

46. slywy says:

Sorry, but this is the kind of approach that made me LOATHE math and miss out on all its potential fun.

47. Anonymous says:

It’s really the same algorithm an abacus uses for addition. I guess they figure it is easier to lay out counters in rows, than it is to slide beads back and forth on a wire.

48. C White says:

Seems like a Graphic Design problem.

49. Anonymous says:

Because of teachers not understanding this crazy stuff back when i was in elementary school, my entire graduating class from that school has a horrible time with basic arithmetic.

50. mdh says:

It does look like an IQ test. Maybe it’s in there to find the 2% of kids who will get it at that age.

51. Anonymous says:

What? Grids of pennys called “counters” and arrows pointing right that may or may not represent an equal sign? OK and we are “making 10.” How about at least using the proper word, “addition.”

52. Patrick Austin says:

I got the highest score in my Diff Eq’s class and aced my Quantum Mechanics class.

I have no idea what the fuck this is.

53. clowtilda says:

ok. I have it….

8+3… you have 8 pennies in the top box, so you draw three more… two fill up the first box and make 10. you put your last one in the second box, and you have 11.

this illustrates the fact that 10+1 also = 11. Because then you can clearly see that you have one box of 10 and one extra.

54. zizzybaloobah says:

I was high school math teacher once. Way back in the last century. If nothing else, I found one thing to be true: just as it takes a certain physical maturity to perform certain tasks, it takes a certain mental maturity to understand many math concepts. For some students the light comes on in time, for some it doesn’t (and for some I don’t think the bulb is broken).

In any case, I think its great to teach a child an alternate way(s) to do math, but to expect a 1st grader to understand the concepts presented by this exercise is ludicrous to me. Furthermore, having encountered stuff like this w/my own daughter, students won’t be give the choice of choosing which way to perform the task on test – they’ll have to do it both ways on the test regardless of whether or not they have a preference or understand one (or both) of the ways. In other words, they end up being tested on knowing both methods instead of the more practical behavior performing addition correctly.

This is why students grow up hating math.

1. Anonymous says:

Oh, I think you have it. I get that providing different ways of thinking about math is a good thing, but I recall totally hating this type of teaching. Because the focus is often on the methods, not the math.

55. David Bruce Murray says:

It appears to be an effort to stop parents from doing their kid’s homework for them.

Seems to be working.

56. takeshi says:

It’s perfectly simple. If you’re not getting your hair cut, you don’t have to put your brother’s clothes down on the lower peg, you just collect the note before you do your scripture prep after lunch when you’ve written your letter home before rest, move your clothes down a peg, greet the visitors.

57. Anonymous says:

This looks like the “every day math” curriculum that my daughter’s elementary school is using. Each new concept should be accompanied or preceded by a “home link” sheet that a) explains the concept and how its presented and b) assumes the parent will be involved and suggests questions that your child should be able to answer based on the material covered in class. It’s certainly different than the way I was taught math, but I’ve been impressed with it. The 1st grade unit on pattern recognition is pretty cool. And I’ll bet you’ll be impressed when you see your daughter doing elementary algebra (they don’t call it that, but that’s what it is) toward the end of the 1st grade.

58. Man On Pink Corner says:

Well, the problem is clearly isomorphic to all real solutions of the Riemann zeta function in a four-dimensional Hilbert space. If your kid doesn’t pay attention in class, you’re not doing her any favors by doing her homework for her.

59. Alex says:

@2 AirPillo: I earned an awesome new word today. Thanks!

@40 Laslo Paniflex: I don’t think that way for adding numbers. 8+3 is an instant 11 in my head. I don’t know how it ended up that way, but whenever I see an 8 and a 3 to be added, I instantly think that the last digit will be a 1 and the previous digit will need to be incremented. I see a 7 and 6, and I already know there’s a 3 coming.

I *can* think of it as 10+1, but I find that that adds several extra steps. First I have to recall that 8 + 2 is 10 (or perhaps I subtract 8 from 10), then I subtract 2 from 3 to get 1. I prefer to not have to use 10s complement and do a subtraction problem when I’m adding. Methods like that can be useful in some esoteric situations, and knowing that you CAN do it that way is cool and probably helpful, but for expediency I’ll take a bit of one-time memorization.

1. AirPillo says:

All credit for my use of that word goes to Shamus Young, for reminding me of how awesome it is.

2. Laslo Paniflex says:

That’s all well and good for you. But work like this is designed to get kids away from just counting with their figures and starting make abstractions. Those thoughts need to have some type of organization and this is a common one.

@71 I think the reason why math teachers (myself included) emphasize writing stuff down is that eventually (around grade 7 at my school) students reach a limit on how complicated a computation can be to perform in their head. Making kids show their work benefits them as it allows them to learn how to organize written math. This helps to prepare them for high school math. Cause there aren’t many people who can do a quadratic equation in their head.

3. Anonymous says:

Interesting. I almost always when doing addition make a ten then do the rest.
So 9 +24 I convert to 10 + 23 and add. I also multiply by doing them by places.127+4 is 400+80+28 which I can quickly turn into 508.Not sure how other people do it or how long it takes. But I find it works pretty fast and as a kid my mother used me instead of a calculator when shopping and I can shop without a calculator and calculate prices, including tax, to within 25 cents of the total.Even when buying a months worth of groceries for a family of 6.

Not bragging about my skills, just trying to provide some anecdotal evidence about the merits of non-traditional math instruction/techniques.

60. Anonymous says:

I taught First Grade for 5 Years. I felt I had to be constantly on the look-out for exactly this kind of thing. If I had gotten this, I would have instructed the students to put a BIG X over the second problem. What the people writing this curriculum think people should “intuitatively know” is that the answer is “11” in both the first and second problems. Jerks. This is confusing, and it most certainly is not your fault! There needs to be another homework where the idea is to find “everything that adds up to 11.”

It’s not your child’s teacher’s fault either. It’s that the teacher has had this curriculum approved by a committee. The members of that committee were “sold” this curriculum. (Shrug!)

The answer is 11 for all problems.

Hope this helps,

Leila

61. Anonymous says:

This incenses me!!! I am a special education teacher who is outraged by confusing Math programs like Growing with Math and Everyday Math. We are creating a generation of students who are Math illiterate!!!

62. krebscout says:

Wait, people don’t always think of numbers like this?

…do people really think in terms of memorization? I could never remember the charts.

63. mellowknees says:

wow…I totally understand the “math” part, but the “counters” part would have me stumped, too. Seems like an extremely arduous way to teach a kid how to count up stuff to add. Even when I was a kid, I probably would have gotten the math part, but I would have likely not understood the “counter” part because I was fortunate enough to have a math-inclined brain.

I don’t remember how we were taught basic addition in first grade, but I’m sure this was not it. I seem to recall a lot of story problems that related to stuff a kid could understand, like “Nancy has 8 pieces of candy. John gives her three more. How many pieces of candy does Nancy have now?”, along with a drawing of said candy.

Well, also, back then we didn’t get homework until 4th grade. Remember when kids actually got to come home from school and be kids for a little while? Yeah, those were the days…

64. Anonymous says:

@Laslo Paniflex, the funny thing to me is that I used all those shortcuts to break down problems and do them in my head in elementary school and would get in trouble for not showing my work on paper. I never understood why I had to do it on paper if I could do it all in my head. Tended to hate math class as a result.

Though doing things all in my head really caught up to me when I got to calculus…

65. ackpht says:

Physics and engineering degrees here. Took me about a minute to figure it out. Badly-worded problem statement, no question.

In English: “If you add these two numbers together, how many more than ten is it?”

66. AsteriskCGY says:

“mountebank”
Hey, I learned something.

67. Anonymous says:

The exercise is trying to illustrate the act of “Borrowing”. This shows that adding numbers up to 10 then adding the remaider to 10 is sometimes easier than adding the numbers directly

The single boxes on the left are always filled with 1’s to represent the 10’s column while adding. the student can observe the Coins block to determine the number of coins required to get to ten.

The box below the 10 gets the remainder of coins (2) and then the equation is 10 + 2 = 12.

The idea being that it is easier to add 10+2 than 9 + 3.

68. Haro! says:

I have tons of younger relatives and I’ve noticed that the math workbooks have gotten really ambiguous with their exercises over time. Not sure if they’re trying to dumb it down or what, but I don’t think they’re being taught these basic concepts effectively.
I won’t even start on how ineffective teaching kids how to read has become. But then, this may also be NYC’s lousy public school system at work.

69. Anonymous says:

Wow, seriously guys, it should immediately be apparent to anyone beyond 1st grade math:
root the first number in the equation, take the second, multiply it by its inverse, then take the absolute value of the two constants, assemble in a 2×3 matrix {{311}{111}}, add another row or 1s to make it square, find the determinant, then ^2. The value of that should either be -1^2=1,1^2=1, or 0^2=0, then you know you’ve got it, anything else is wrong. Take this solution (thus either 1 or 0) and place it in the funny box thing, then use it for the missing box in the equation to the right.

Haven’t figured out what to do with the pennies yet, but I’ll post if I figure it out.

70. Church says:

There’s a couple problems here. The first is that this is horribly laid out, and not at all explained. It really does read like an intelligence test (I assume the kids got instruction in school, but it doesn’t help the parents any, nor does it refresh the kids’ memories.)

The second is that this doesn’t really mimic the way numerate people ‘fit’ numbers together. In fact, it makes it harder by turning an addition problem into an addition and subtraction and addition problem. (Barring access to a series of helpfully grouped pennies, of course.) It doesn’t strike me as teaching them anything actually useful. Playing with differently scored blocks would be better.

The fact that there are 70+ posts about this explains exactly what is wrong with the american educational system. If you are some hapless first grader with out to lunch parents you are so SOL. Thanks though — it explains my cities’ inability to create one not “failing” school.

72. Anonymous says:

I see some people in the comments have figured it out but I wasn’t able to. 8+3=? so ?+?=?

yeeeeees

73. Jamie Sue says:

Stuff like this is why I hated math and why my parents could never help me with it. It’s also why I always got low marks in math, was frustrated to the point of crying at every homework paper, and why I stayed grounded for failure to bring my grade up.

74. twobeeshawn says:

After reviewing this with my 1st and 2nd grade kids, we’ve decided the best homework solution is crowd sourcing on boingboing and sending unresolved questions to SamSam, keep the thread open, we’ve got a spelling test tomorrow.

75. Anonymous says:

76. andyhavens says:

77. pjcamp says:

This may seem ambiguous to you, but no way they haven’t done this already in class with real pennies or buttons or something. Or the teacher is an idiot, one or the other.

dssstrkl: Read Robert Siegler’s books Children’s Thinking and Emerging Minds. Siegler has been investigating early childhood math, among other things, since the early 70’s. You can get any number of refereed publications off his vita here:

http://www.psy.cmu.edu/~siegler/index.html

Around 1970, Siegler demonstrated the phenomenon of U-shaped development in children’s understanding of single digit arithmetic. He showed that during the acquisition of that skill, children’s performance actually goes backwards for a time. It has to do with a skill composed of multiple subskills, each of which individually develops at a different rate. So first children learn to use a counting up strategy which is gradually replaced by a recognition strategy. But there is a time when, after they have demonstrably learn the recognition strategy, they retreat to a counting up strategy for a time.

The kind of exercise we’re looking at here is clearly intended as a bridge between the two strategies, supporting the learner when he is in this intermediate developmental state by continuing to expose the connection between the two strategies.

U-shaped development appears to be a general feature of human cognition. I recently discovered evidence of it in the acquisition of Newtonian reasoning skills in college students.

Now we have our choice. We can *use* cognitive research of this sort to design better learning materials (whether they are appropriately implemented is another question, but there is research on that too), or we can stick with whatever crap was good enough for you.

But that, basically, was a filter. Anyone who couldn’t do it simply fell out of the system.

I made it through college with a science degree too — a Ph. D. in theoretical physics, specializing in relativity, quantum gravity and cosmological applications thereof. But unlike you, apparently, I came out of the experience with a healthy respect for data and scientific knowledge. So now, when I teach my classes, I try to learn whatever is known about human cognition that is relevant to my material.

You say “If my child came home with homework like this, I would make it my point in life to find out why trash like this was being passed off as educational material and to remove it in favor of useful tools taught by competent professionals.” Ok. Start with Google Scholar. It is free. It would have led you quickly to Siegler’s work. You can also look up his colleague, David Klahr.

You can also try the FREE online book How People Learn, Bransford et. al, at the National Academies of Science.

Sometimes a useful tool cannot be recognized by someone who sees it third hand from a standpoint of blissful ignorance of how people learn.

But that problem resides in the ignorance, not in the tool.

78. Choppers says:

The answer is 11 for all problems. That must be why the worksheet is entitled, “Making Ten”

79. Anonymous says:

I agree the directions are not clear, but the concept behind this exercise has been very successfully used in Montessori programs for many years. I have seen kids as young as 4 understanding addition by means of using visualization techniques instead of memorization. Unfortunately, most schools rely too much on the latter, and the consequences are that most kids do not understand math.

80. SophiesChoice says:

The exercise is a way to help kids learn how to work with numbers. It will make mental math far easier for them in the future. That being said, this exercise may be a bit premature, or it should have clearer instructions!

81. Anonymous says:

What’s missing here is that this is a page torn out of an EveryDay math workbook. She would have had a lesson in class where she was using counters (manipulatives) to do simple math problems. They show pennies on the page, and probably she used pennies to solve similar problems in class. She is supposed to draw in the three extra pennies, and it will show her the answer to both problems. It’s a great curriculum. It gives kids a chance to do problems several different ways, which is good for different learning styles, and also gives them more facility. If you can not only solve a problem, but flip it around in your head, then you’ve really mastered it. Recently I had fun with my son taking a 5th grade level EveryDay Math lesson and turning into a 10th grade algebra problem. The way they had broken down the lesson made it easy to get an intuitive feel for how the math really works.

The fact that you have to stop and think about this problem is actually a good thing. That’s what it’s supposed to do.

82. Individual says:

My kindergartner gets these same sort of confusing homework packets every week and my wife and I are forced to decipher them as best we can though the teacher sends home a second set of instructions for each problem because she’s realizes out of context they make little sense. What I’ve come to realize is that in class they actually have the “counters” they speak of and use them as a visual aid, but here at home, we’re expected to imagine, and opposed to homework as I am I’d just assume not bother, though so far we have.

83. Anonymous says:

the title pitches the curveball, swing and a miss.

84. beameup says:

Of course this assignment would be an extension of what the students would have been using in class using hands-on counters. Making 10’s (filling 10’s) is an excellent mental math strategy. I teach grade 8 math, and about 1/2 my students still add by counting-on (using their fingers or in their head) when I first get them. For example, if they see 18 + 5, they will say “19, 20, 21, 22, 23.” When they are taught to use filling 10’s and other mental math strategies, light bulbs go on and they can work much more efficiently on their grade 8 level math. They were probably taught to just memorize their addition facts in the younger grades and never developed a proper number sense. This is not “theory” talking, this is 20 years of teaching in the trenches grades 5 to 12 math.

85. Anonymous says:

A simple reformat of the page makes the task simpler. Move the counter grids to the middle between the equations. It suddenly becomes obvious what you’re supposed to do.

86. Idle Tuesday says:

I see this kind of dreck coming home with my kids all of the time. This is what they’re learning instead of memorizing times tables. I hate to sound like an old fart, but I don’t see the point of this.

87. Caroline says:

Leila, I’m pretty sure 9+3=12, not 11.

beameup, I agree that it can be a good mental math strategy. But it sounds like this homework was given totally out of context. That, or Mark’s daughter was paying absolutely no attention in class. I rather suspect it was just assigned out of context. Sadly, I had several teachers who did things like that.

88. Anonymous says:

A lot of judgemental comments on this exercise. Lighten up. I have a kindergartener. For a while, I was stunned at the homework she was bringing home, the concepts seemed way beyond a kindergartener. More fool me. She was frustrated at the beginning. By the end of the page, she would brag about how easy it was. Don’t assume the child hasn’t gotten an explanation of the concept. When I asked my daughter’s teacher, it turned out that she had received explanations. Until you verify with the teacher, it is not wise to jump to conclusions.

I don’t have a degree in primary education or child development. If I don’t understand what is expected in my daughter’s homework, it means *I* don’t understand, it doesn’t mean there is any defect in the teaching.

1. Antinous / Moderator says:

If I don’t understand what is expected in my daughter’s homework, it means *I* don’t understand, it doesn’t mean there is any defect in the teaching.

You must not live in California.

1. lorenzofarris says:

We did live in California. We left a little over a year ago. Sending our kids to a California public school is definitely a last resort for us. Our older girl was in private schools until we moved, and the little one is in a Montessori school now. It is an interesting contrast. The private schools in Silicon Valley are academically excellent, filling a gap left by the public schools. The top private schools accessible to the Eastside area near Seattle are more focused on “well-rounded” students. We moved our older girl into a more academically challenging public school program after a year in an “elite” private school.

89. KurtMac says:

I’m still stuck on what “counters” are supposed to be and how drawing them would help me. Should the counters I draw be granite or Formica?

1. Anonymous says:

:D granite or formica counter – brilliant – although in true maker style I think we could add a reclaimed hardwood counter or a handmade tile counter, a mosaic counter maybe…. that would be fun to draw. The concept is great and I’m sure they were shown this in class, but the design really could be improved. Even arguing that it is a workbook and not a textbook doesn’t make it a good design. Given that this workbook will probably be around for a while, my personal solution would be to write some instructions together after playing around with the key concepts for a while. You can then use the instructions that they come up with as a means of checking how well the child has learnt the concepts and the child gets to be a knower as well as a learner.

90. Anonymous says:

Without additional instruction, it would have been easier to understand had the “drawing” counters been in the middle. I never even used the counters because I didn’t get what they were for yet I still got all the right answers. Only after reading the comments did I understand what purpose the counter spillover boxes served. And then only then did I realize how the spillover was created. Then, I was a bit of a math prodigy in that I just know the answer but really have no frippin’ clue why I always know the correct answer(s). And this “spillover” effect is something I’ve done since day one with base ten and base x. I just didn’t know that’s what all you English majors called it.

– DonkeyBreath28

@82

Exactly. The tool is great IN THE CLASSROOM I have no problem with this tool. I do however have a problem with the ignorance perpetuated by the ape like instructions. As a parent with a learning child I would absolutely request an explanation from the teacher and if one were given, as satisfactory as yours was provided , would accept it.

However– as the child of parents who had a “do your own homework, helping is cheating” attitude and bitter old ax WWII era teachers who just refused to die or retire. I am able to rocket myself back to the frustration of first grade bad homework I didn’t understand, and that my parents, lacking the internet, didn’t understand and put myself in the place of some poor kid with uncaring parents, or no resources.

So in this case there IS a problem with the tool, because without explanation, it is NOT a tool. In this example an explanation of the homework is not easily (ie on the same or an attached page) accessible which is why it should be done at school, not at home.

92. LYNDON says:

Perhaps the lack of instructions is a fiendish plot to stop parents helping their children with their homework.

I think the crotchety-old-man response here was best summed up by Tom Lehrer.

93. Alex says:

Anonymous wrote: “the funny thing to me is that I used all those shortcuts to break down problems and do them in my head in elementary school and would get in trouble for not showing my work on paper. I never understood why I had to do it on paper if I could do it all in my head. Tended to hate math class as a result.”

That happened to me in second grade with subtraction. I could breeze through a page in 3 minutes, but with all the borrow-from-the-tens busywork it would take 20. I turned in one with just the answers, which was returned with a red “Where is your work?” written across the top by the suspicious-of-calculators teacher. The concept of needing to enact and document some farcical scenario of someone borrowing (with no intention to return!!) numbers from their rich neighbor seemed an absurd way of solving 24-17.

It really didn’t help a few weeks later when my mom proceeded to show me an even easier shortcut to subtraction than what I’d been doing myself.

94. Anonymous says:

Idle Tuesday summed it up. I can do some of these manipulations in my head, but failed at advanced math because I spent all of my time trying to figure out multiplication I should have memorized.

1. Anonymous says:

So do you memorize whole words too, without learning what the letters mean?

95. Anonymous says:

They lost me at “Counters”.

96. efalk says:

Holy mother of god. No wonder our children is failing.

97. Dan Mac says:

My daughter is in grade two, and we had a limited amount of this last year, which had kind of put her off math. I did some research, and ordered all of the preschool, grade 1 and 2 Singapore Math curriculum books and activity books off of ebay. (cost approx \$40.00) We are still working our way through grade 1 stuff, but she has a far firmer grasp of basic math, and especially base 10 numbers.
(Those crafty Singaporeans are #1 in math in the world, they say, and they have a whole system).
Math is enjoyable and she does great work on her own. On another note, I was told by her teacher that the school sytem cannot assign homework until grade 4 here in British Columbia, so be happy they are getting some!

98. griffdn says:

“It’s so simple, so very simple, that only a child could do it!”

99. EscapingTheTrunk says:

Mark, take your kid out of class. Her school is clearly run by NERV, and she’s being tested for competency as a future Evangelion pilot. RUN!

100. Alex_M says:

Jesus. I’ve got a PhD in physics (meaning knowing enough math for at least an undergrad degree in the subject) and I couldn’t figure that thing out after staring at it for a minute.

Not the first time I’ve run into something like that. Intelligence is really an impediment to solving stupid (=ambiguous) problems, since you see the ambiguity.

Which kind of makes it even worse, considering that math is NOT a subject that allows for ambiguity.

They should really start teaching algebra earlier too. All these elementary school mathbooks have ‘fill in the missing number’-type problems. Which are naturally intended as a ramp towards algebra. But I think it gives kids way too little credit. After all, an empty square is hardly more or less difficult to understand than a variable.

101. endymion says:

I think this is actually a pretty cool way of teaching what doing addition with carry-over really “means”. Presumably the teacher explained how it works, and if there were some instructions, it wouldn’t have taken us a few minutes to figure it out. The fact that it did in no way denigrates the approach, IMO.

102. zenkat says:

It’s “New Math”, people.

Grown-ups have a hard time of it because they do math reflexively, using rules they learned in 2nd-4th grades.

But it gives the young’uns a very good intuitive grasp of what those rules are all about.

And, as a computer scientist, I *love* the fact that they are also teaching an intuitive understanding of functions in first grade. This generation will be wizard programmers, mark my words!

(Full disclosure — I have a child in first grade, we review homework like this all the time. I’ve found it very effective.)

103. lolbrandon says:

8+3 = 11, 10+3 = 13, draw 3 pennies in the bottom box.

9+3 = 12, 10+3 = 13, draw 3 pennies.

7+4 = 11, 10+4 = 14, draw 4 pennies.

Err, I think. Maybe you should send a note to class with a link to this comment thread and a message that reads, “well, we tried.”

104. Choppers says:

I find it interesting that the editorial remarks in favor of this worksheet seem to come from confessed math-types who litter their fanmail with obtuse praise written at the 4th grade level.

105. beameup says:

Alex_M, A PhD in physics does not mean you understand effective strategies for teaching elementary mathematics. In my experience people who are gifted in mathematics (as you probably are) often struggle with teaching students that are not also gifted. Spend a few days teaching addition to grade 1’s, and then to older students that were only taught to memorize facts and algorithms when they were younger, and then decide if this method is “stupid”.

106. Anonymous says:

This would be a fine exercise if it said “use counters” rather than “draw counters”. 8 in the top grid, 3 in the bottom grid, move some to fill the top. Eventually kids should know these things cold, but in first grade there’s a lot more to be gained by just playing around with ideas.

107. Anonymous says:

Check out the publisher of the new math program and find their website. May be sample lessons on “You tube” or “Teacher Tube” Amazing new Math adoptions in Ca that I know of. Foundations are different than what we learned but critical for success!

108. joe blough says:

this looks suspiciously like my kindergartener’s math book. they are the “fish” books, right? these are new, and they do seem to be quite a departure from the books my older daughter used in K. i have a postgraduate degree in computer science and like many others in this thread these assignments confuse the heck out of me.

i guess the question is: “is it possible to have new teaching methods and technologies for math which are superior to old methods and technologies?” the answer is probably “yes”… but i worry that certain ways of doing math come very naturally to the human mind, and these “new fangled” teaching methods often go counter to those natural ways.

also new methods/textbooks “gives educational psychologists jobs.” :)

109. Alex says:

@Laslo Paniflex

I think that’s a considerable stretch. Showing work for the sake of showing work because something some time in the future may require work to be shown, I find extremely bogus. You can justify any amount of mindless busywork by making some obscure connection of how you think it would help them in 10 years. And that’s what it was. Mindless busywork, that made kids despise math and cut down on their time to explore the world. It was not uncommon for me to do my homework in tears, not because of the difficulty, but because it took a LOT of time while often accomplishing nothing. Imagine if, today, your job consisted of printing out 1000 pages from a word processor and manually using white-out to remove the serifs from capital Ts and lowercase Ls, and that your pay would be docked tomorrow if it wasn’t finished. That’s what it felt like to me.

FWIW, in high school we glossed over what quadratic equations were for and why they were used, and were provided with a calculator program to do it for us. Only in my first semester of college math did we look at how it was derived and what imaginary numbers actually meant (or, at least, when we covered it in college, it wasn’t surrounded by a backstory of LaShonda’s Kwanzaa party, so I was awake for the lesson).

If you want a kid to learn how to write down and organize information, have them do it with problems that lend themselves to writing down and organizing to solve. This wasn’t even solving… it was doing the same exact thing on 30 problems every night for 2 weeks. All it instilled in me was a deep hatred of subtraction.

@Alex_M: I completely agree that kids aren’t given enough credit. I’ll bet that if you introduced 4th graders to Algebra, a good many of them would pick it up. If anything, I’d say to give them algebra problems, then see which steps they don’t yet grasp, and work your way back from there (heck, even tell them that they’re learning such and such bit so they can do this cool harder math), rather than assuming all related concepts are beyond them without your express guidance. There’s some bizarre “I learned this after years spent on that, so that’s how to do it” mentality that pervades our school system, made worse by a good portion of teachers who themselves can’t do math to save their lives, that really kills a possible interest in math for lots of kids.

Heck, I was probably effectively doing Algebra in 4th grade with non-math-class-related concepts.

But that would make it difficult to establish our precious “standards”, and we can’t have that!

110. jonasgerson says:

I think everyone here is missing the point about ‘figuring it out’ — there shouldn’t BE anything to figure out! It’s adding for chris’sakes!

It’s this type of drivel that forces to kids to feel ashamed (‘stupid’) at a young age, and onto thinking they are ‘bad at math’ (How many rounds of public embarrassment, such as having to answer this in class in front of your friends, would it take?)

This has nothing to do with mathematics, or being numerate for that matter. What this does have to do with, however, is following meaningless instructions which are ‘taught’ in substitute of simply teaching actual arithmetic.

111. Anonymous says:

What’s frustrating is that there is no stated constraint that the sums on each row have to be the same. I mean, yes, presumably the “correct” answer to the first one is to put 8+3=11 and then 10+1=11, but what if you put 10+2=12? Is there anything in the instructions that clearly suggests this would be wrong? I don’t see it.

112. Catsidhe says:

The people who are saying things like â€œOh yeah, the intent of this is fine, and it’s teaching wonderful things and blah blah blah…â€ guys, you’re really not getting it. The people who read here are not stupid, nor ignorant. There are PhDs who have professed having difficulty figuring out WTF the questions mean. Once you know that, then figuring out the answer is trivial… but you have to get to that point before you can start, and if the effort makes a middle-aged post-grad cross their eyes (and takes a maths tutor a couple of goes before they figure it out), then what goddamned hope does a 6yo have?

Really, this question displays a profound problem with the ability of the person who wrote/designed it to present information, and equally serious problems with the reflection and judgement of the person/committee who thought this was an appropriate thing to put in front of young children. (Note that this was not necessarily the teacher.)

Unless the point is to teach that the world is bewildering, confusing, incomprehensible and nonsensical, and you will be punished if you fail the test.

My own 1st grader has homework come home, mostly for spelling. Her latest list is words like â€˜gnomeâ€™, â€˜boughtâ€™, â€˜caughtâ€™, â€˜signpostâ€™, and â€˜ghostâ€™. That’s fine, and she can even spell them just fine. What’s the homework? To write a fscking play!!! Whiskey Tango Foxtrot, Over? Another time, it was to write a rap song!

How in the name of all that’s holy does that tangential and irrelevant make-work bullshit help her remember how to spell â€˜ghostâ€™?

Is it just me, or has it actually gotten worse since we were in school?

1. Matt McKnight says:

Or perhaps you are missing the point that this is a workbook, and not a textbook. This is a great exercise, but sending it as homework for a 6-year old without the instructions to be taught by incompetent parents, such as you appear to be pretending to be, is really a bad idea. It’s abdication of pedagogical responsibility by the teacher.

Your complaint primarily seems to be that you don’t know how to teach numeracy in this way and didn’t want to spend 15 minutes studying this proven, effective method. My complaint is that the teacher shouldn’t expect you to.

I still, vaguely recall being required to write a sentence for each of my spelling words in first grade, and getting in trouble for combining them into a single sentence, such as: “My teacher said the spelling words for this week were, ‘gnome’, ‘ghost’, etc.”

I like the idea of composing a play or a rap song. The concept is practicing two things at once- the spelling and the written form. While it may be make-work, to some extent, and there’s no need to do it at home, my kids would definitely enjoy it. Again, the problem comes with shoddy teaching, where they are asked to write a play at home, but perhaps never having read a play, are likely to be unfamiliar with the conventions with which they are written. This again puts the burden on the parent to teach, although in this case it is at least something the parent is likely to have done.

1. Catsidhe says:

The exercise itself, when explained, is fine, wonderful, praiseworthy. The problem is not the exercise itself, but its execution. The worksheet is not well constructed, in my opinion (and not just mine); if you have to figure out what the question means (and the point isn’t about exactly that), then there is a failure somewhere. That there might be a book somewhere where this is explained doesn’t help if this isn’t properly explained to the child. And if it’s that badly expressed on the worksheet, then the parent can’t help, unless they have a blinding flash of insight.

It’s not that the exercise is not a good one, but that there really needs to have been some more thought put into the takehome sheets

113. Alex says:

Speaking of people assuming kids can’t handle certain concepts, I just remembered a game we used to play (well, I used to play it, but I think I remember playing it with other kids). I don’t know where we picked it up, but I’m pretty sure it wasn’t in math class. We would count from 1 to 100, but leave out, say, every number with a 7 in it. If you could get all the way there, you won. The tricky bit was remembering to leave out the 70s and count 65 66 68 69 80 81, and so on. I think we’d even do it omitting more than one digit in some cases.

We effectively learned to count in non-base-10 number systems without realizing it. In, I’d say, 3rd grade or so. For fun, while we were, say, waiting in the car (ZOMG KIDNAPPERS) for mom to grab something from the store.

114. Anonymous says:

My daughter has the same homework.
I despise these worksheets. She does about two a night.

115. stuhfoo says:

i found this to be very intuitive

i think you have to look at it going in with the skill set of a 6 year old

as the directions state use the counter

so for the first problem 8 + 3, 8 pennies are drawn, you must draw three more, 2 in the upper box and then 1 in the lower

count the pennies to get your answer 11, the penny in the lower box then is the answer to the second blank spot (below 10), and the total is then 11.

the key to this worksheet is the directions given. USE THE COUNTER… if you looked at solving the addition by itself you would probably confused :)

116. Alex says:

@Catsidhe: Then wait 10 years, and repeat the process in Spanish class. How spending two hours at home by yourself writing an interaction between a waiter and customer, to present to people who also don’t know the language, often taught by someone who maybe spent a summer in Spain back in ’84 and spends 90% of the time talking about Spanish words in English, is supposed to be at all an effective way to learn a language is beyond me.

End tangent.

117. Anonymous says:

From what I can tell, the only correct answer is to throw the workbook in the trash.

Then you can teach your kid addition the way everyone learned it 15-2000 years ago.

Seriously, addition is simple enough that we don’t need to find fancy new methods of teaching it.

118. Matt McKnight says:

These are very good exercises. The knee-jerk commenters here who don’t like the problems are missing that this is a workbook- a set of problems that goes with a textbook. The textbook explains the method of solving these problems. It is a progressive step in learning to carry, so that carrying is not an abstract rule, but a logical process that makes sense to the learner. The teacher is to be faulted for sending “homework” with a 6 year old and not sending the relevant texts, as it is the parent who will be assuming the teacher’s responsibilities for failing to explain how to do the work.

For all of you displaying your ignorance in deploring the “American” or newfangled nature of this method, I came across it in the Singapore math series (from 1983 or so?), which I am using to supplement the teaching my children receive at school. You may be aware that students in Singapore are typically the most advanced in the world in mathematics. The use of their methods of creating a deeper sense of numeracy should be applauded. Revisit this example with a beginner’s mind, and the benefit of the text, and you may see how well it works.

1. Anonymous says:

Honestly, stuff like this is why we have so many people who “hate” math in America. Growing up, I hated math. It wasn’t because I actually hated math, or because I was bad at it; it was because showing every step of my work made doing my homework take four times as long.
When you force children who understand the concepts to do busywork (what I would have considered this when I was 5), you create children who hate what they’re learning. That is never a good thing.
When I was 5, I had bad fine motor control. My kindergarten teacher said “Oh, Anonymous! You like math! Why don’t you take these beans and count them while you move them from one cup to another.” I looked at her and said “That’s stupid.” I never put any beans in a cup, and I have decent handwriting. I’ve passed over 3 years worth of calculus classes (but I still loathe them, due to the time requirements and the forced graphing). But still, whenever I’m faced with a teacher that wants me to do Math, I shudder at the thought, simply because I’ll have to spend half of my time figuring out how *they* want me to show my work. Even in college, each professor wants a completely different style.

2. AirPillo says:

Matt, trust me, I’ve had to work with these workbooks for years now.

The problem isn’t in finding the answer to the problem. The problem is in deciphering what the question is.

You can’t even begin to teach the child the answer if nobody can make sense of what the homework is requesting of you.

Does that make sense?

The author of the workbook is consistently inept in communicating what they are actually requesting. It takes my brother and whoever is helping him 30 seconds to perform the actual mathematical operation or manipulation once identified, but frequently takes 15 minutes to translate the question into anything resembling comprehensible logic or language.

It’s like working with a tutor with a severe speech impediment. If you get lost, it may very likely not be because you don’t understand the concept. It’s because you can’t understand what the hell they’re saying.

119. geekage says:

Why do so many people have to declare having advanced degrees and they can’t even understand the homework? Is that part of the problem?

I’d think that anything over an elementary school education would be all that’s necessary. Heck, even a first grader could do it.

Probably the best help would come from a fourth grader.

120. BobbyMike says:

Seriously Mark, she’s supposed to ask you to be homeschooled so that you are free to pick a curriculum that makes sense to both of you. Heaven knows you’re bright enough to do an excellent job.

Our public education has very little to nothing to do with with bad teachers and everything to do with poorly thought out curriculums/textbooks.

121. Choppers says:

I feel for the kids out there who laboriously drew miniature pennies in the boxes because they comprehended the edict to “draw the counters to solve.”

FYI: Students in Singapore are typically the most advanced in the world in mathematics. Ask them to look both ways before crossing the street however, and you’ve got a BIG BIG problem on your hands.

122. Deidzoeb says:

Make all tens for great justice. Somebody set us up the bomb.

123. Anonymous says:

I think I understand it.
1) 8+3=11; take that 3 from that equation to represent 3 pennies and place those pennies in the lower “box set of ten”. You are then left with seven empty spaces. Subtract the empty spaces(7) from the 8 pennies represented in the top “box set of ten” above; 8-7=1, then add that one to ten; 1+10=11. Do the same for other problems.

2) 9+3=12, take that 3 and place three pennies into the lower “box set of ten”. Now subtract the empty spaces(7) from the 9 pennies in the top “box set of ten” above; 9-7=2, and then add that two to the ten; 10+2=12.

3)Same as the other two problems. 7+4=11 take that 4 and place four pennies into the lower “box set of ten”. Now subtract the 6 empty space left over from the 7 pennies in the upper “box set of ten”. 7-6=1, take that one and add it to the ten; 10+1=11.

1. Piers W says:

Hopefully if a child did that, the teacher would notice they’d drawn the coins in the ‘wrong’ place, ask them why, and set them a more advanced task next time.

It needs a better explanation, and an example, but it’s one way of demonstrating why 8+3=11 in base 10.

124. lerasmus says:

when I was in elementary school we had this cat something or another math learning thing, it involved colored cards, line-art pictures of cats, and progressively harder problems (you advanced levels). I can’t remember the specific mode of presentation, but those of us who used it learned multiplication tables, division, fractions, polynomials and other stuff very quickly. There were cats. I made it all the way to calculus in 6th grade with those cats. Many of us did. It’d be interesting to find out how problems and learning were graphically presented then.

125. lerasmus says:

it might have been called compucat…

126. Griffin says:

The concept is wonderful. The execution, though… gods, it certainly leaves a bit to be desired. The only explanation is that they didnt think they needed an explanation, because it was already explained to them in class.

This is a design problem and a communication problem, but the concept is fine (and is a good concept to hold onto later when you start getting into more complicated maths – remember the easier parts, and then figure out where you can break more difficult sections into component parts)

This is actually kind of the way I do division

8 * 13
becomes 8 * 10 + 8 * 3 becomes 80 + 8 * 2 + 8
(that last ones kind of a lie, but to illustrate next addition step its good.)
becomes 80 + 16 + 8 becomes 80 + 20 + 4
becomes 100 + 4 becomes 104

Though of course in my head it goes a lot faster than writing it out.

The explanation and presentation still totally suck, but the concept in my opinion is better than many people deal with.

127. Anonymous says:

It matters a lot more that you care about your daughter’s learning than that some scrap of it be immediately, systematically comprehensible to every adult without any context (how easy would it be to translate if you’d just sat through a five minute talk about carrying ones, then worked out the same exercise with the teacher?).

I wouldn’t take a worksheet that was designed poorly to be evidence education is going down the tubes nor an outrage in and of itself – again, it certainly wasn’t intended for isolated scrutiny, and I imagine Mark and his daughter talked about it. I would take an abdication of the parent/guardian’s involvement in their child’s life to be a strong indicator of what that specific child will get out of their educational experience. And I’d take a crappy worksheet and a loving dad over any alternative (except a great worksheet and a loving dad). So cheers, Mark, to eleven more years of deciphering the enigma of public education workbooks and textbooks along her happy mutant side (though maybe an enjoyable side game for you to play is guessing which policy requirements shaped such a worksheet. Aaand, go!).

128. themiddleofnowhere says:

If this is a sample of the “progressive” educational tools/system being taught these days from grade one, then the proof of its failure is in the fact that we now have an epidemic of young people with no clue how to do basic math in their heads and have to rely on calculators.

129. Anonymous says:

Somewhere out there, Richard Feynman is crying :(

130. izzy says:

OMFG is this Everyday Math, Mark?
If so, get your kid OUT OF THAT CLASS.
it is the worst bit of math ‘education’ I have every seen perpetuated on our society. They started it in my kids school when he was in 5th grade. He went from getting A’s and loving math to F’s and absolutely hating math and then school.
The twits who crack dreamed this up should have their PhDs revoked.
And I told them that to their faces.

Seriously – home school your kids as true makers.

131. ROSSINDETROIT says:

I work at a good elementary school. I’ll show this to a first grade teacher. If she pauses for more than a 3 count before explaining it I’ll know it’s junk.

132. ROSSINDETROIT says:

From what I understand one problem some elementary students have is not that the parents can’t do the kids’ homework but that they DO do the homework. Good grades on lessons, lousy grades on tests.

133. Kyle Armbruster says:

Math education in the US is a crime. I made it all the way to college before I found out I was good at math. I had to get materials to study for the GRE before I read actual, competent explanations of absolutely fundamental things, like factoring and doing calculations in one’s head. These are things that, if I had been taught in grade school, would have had a major impact on my life. I would not be a linguist now, I’d probably be an engineer, and I would probably enjoy that more.

I’ve lived in Japan for a long time and I’ve worked at every level of the Japanese education system. Do you know where they whoop us (meaning the US)? It isn’t in their high schools; those are club-activities and test cramming factories. It isn’t in their junior highs–those years are a wash no matter where you go.

No, it’s in primary school. That’s the only place I worked where I said, “Wow. This is… This is really good.” They actually study things there, while simultaneously nurturing the kids as they should. This was especially evident in the math and science education. The kids were studying things I didn’t get to until late junior high (because let’s be honest here, is Algebra I really that difficult?), and doing actual physics experiments. My math classes in grade school went like this:

“Here is how you take two numbers and do some stuff to give you another number, but it needs to be the same number as what is in my book. Now, do that 100 times before tomorrow, and then we’ll add a digit to the numbers and you can do 100 of those tomorrow.”

Is it any wonder that everyone hates it? It’s pure, meaningless drudgery. But physics? Physics is cool. I would have really gotten into that as a kid. Or basic chemistry (vinegar and baking-soda level). Or anything that would have made it seem like the point was something beyond making numbers out of other numbers.

The best math lesson I ever had was the time my dad showed me how to calculate gear ratios with my Lego Technics set, and that was about 5 minutes. That’s the kind of thing that should have happened every week in grade school. We might not have such an embarrassingly incompetent populace if we did such a thing.

And this homework here? This is worse than anything I remember. I’m not really in early education, but as a teacher, I have no idea what pedagogical objective that is even trying to achieve.

134. doubleoseven says:

I may have missed the comment, but I believe that some educators choose to build their own self esteem by creating puzzles as teaching tools. Many of the replies to this issue appeared to come from people who had many years of experience and training in their lives and still had to think hard to understand the question. My sympathy to the student who must learn in spite of the poor attitude of the teacher who seems to be intent on being clever at the expense of the student. I wonder if the teacher’s peers encourage this kind of teaching process or if the peers are blind to the problems it causes and thereby enable truly self absorbed people who enjoy a form of bullying that would be (should be) punished under different circumstances.

135. Anonymous says:

Comment number 33 explains the point of this exercise clearly. As a primary teacher I know that teaching children to think about numbers in this way is great. However as someone who did not grow up being taught maths this way I also see that there should be a short, clear explanation there to support parents.

136. ronton says:

More like a puzzle than a math problem!

137. Anonymous says:

Actually I’m shocked how many people would expect an equal sign instead of the arrow. There is no equal sign because there is none supposed to be, as you don’t put equal signs between equations.
You should definitely leave the teaching to the teachers. ^^

138. lhopitalified says:

I won’t claim to have read all the comments, because, well, there are a ton of them! However, I do think I have gotten the main gist of what people have said, and thought I’d throw in my own analysis, FWIW. (Again, given the # of comments, not sure if anyone will read this…)

I agree with some comments in that historical methods of teaching arithmetic have been flawed. People learn in different ways, and trying to force everyone into the same path leaves people behind. At the same time, though it is silly to teach students how to solve a problem in multiple ways and then test them on all the different ways! (which by any measure is worse than forcing everyone to use just a single method.) If student A understands how to carry-over tens using method X and student B does fine using method Y, let them use their individual methods on a test – as long as they can both do the problems correctly, who cares what method they use?

I would also caution against generalizing results from psychological studies. There is a tendency to believe that because a result is statistically significant, that it applies to everyone. This is simply not the case. A study on a new diet pill may show that people who take the pill lose more weight on average than people who take the placebo. The pill may even work properly – but it doesn’t have to work for everybody – as long as the mean in the experimental group is different from the mean in the control group, statistics can tell us that the pill had an effect. Similarly, individual variance in education often trumps research that shows how teaching in some new way results in better learning.

Education in the US is frustrating because it is a huge system with significant momentum in a certain direction. Many of us could write something better, but we are not billion-dollar publishing companies in a weird mutualistic symbiosis with the Texas and California state boards of education.

From what I have read about education, the individual teacher makes a huge difference (the best teachers may cover as much as twice the material covered by the worst teachers) As a parent, the best you can do is talk to the teachers at your local school. If the situation looks bad, you’ll either have to transfer to a different school, consider private / home schooling, or be heavily involved through extracurricular learning -type activities.

Then again, what do I know, I’m just a graduate student with no kids – but you’re a smart guy, Mark, I’m sure you’ll absorb these comments with the appropriate amount of salt.

A final parting note: if you’re interested in math acceleration, there is this great website/company run by some acquaintances of mine: http://www.artofproblemsolving.com/

which has books, links, and a massive community of young (math) problem solvers.

139. Sweetcarovan says:

One of the first things I learnt while training to be a teacher is ALWAYS GIVE AN EXAMPLE.
I’m amazed that a workbook would set task without giving an example!

140. Anonymous says:

At Christmas in 1st grade, my parents gave my daughter a chalkboard and my father taught her about “carrying” in addition. Within 15 minutes, she was adding up 5 10-digit numbers.

Thanks to Everyday Math’s “spiralling” concept, the first thing they did in 4th grade this year was 2 digit addition! WTH?

lol, my CAPTCHA word is “horrific”

141. Anonymous says:

Last year my nephew Drew brought home this problem and was stumped by the directions as was his mother. I took a stab at it and came up with the answers (which I now know are right thanks to this thread) in about 10-15 minutes.
Then I explained the ‘answers’ to Drew in a, “I think this is what they are trying to get across to you *if* these are indeed the correct answers. Ask your teacher for verification.” kind of way.
The conversation was a fun exercise for me but may have been moot for Drew who was just happy to have his homework done (by me) and over with.
I had him rewrite the answers before we put it away for the night for reinforcement, but it went back to school with the backside covered in my handwritten notes on the subject, and a note asking for a complete explanation and additional problems of the same nature from the teacher.
(I’ll have to ask my sister if the teacher responded.)

142. Anonymous says:

The answer is “I’m not good at math, screw homework!”

143. Anonymous says:

And we wonder why students hate school. When they don’t have instructions that explain what is happening it makes school frustrating and they stop trying to learn. Our children need instructions Junior High & High School is the time to challenge them not grade school. With too little information. If the parents can’t figure out the homework than it’s time to talk to the school/school board as to what is happening. Grade school homework should be easy to do for adults but challenging for the students.

Oh, and if the teacher gave instructions on the homework, which they will inevitably say, how many kids (after other sections being taught) will remember what said instructions are once they are home with Mom & Dad.

144. ndollak says:

I teach, and I often find these worksheets terribly confusing. It’s mostly because these workbooks are “simplified” (read: “dumbed-down”) with the misguided idea that kids are too easily confused by straight information. The over-simplification results in the omission of so much relevant data that it’s hard to tell if specific numbers are expected to go in the squares, or if anything goes, as long as the sum is correct. As it turns out, unless the direction “Make up your own problem” is given, it is implied that only certain numbers go into the squares. (That much I’ve figured out!) But which ones? Unless you were actually sitting in the classroom & paying attention, you can’t tell. Maybe it’s also to prevent parents from doing their kids’ h.w. for them.

My guess: Since 10 is not one of the numbers that appear in the first-column of sums, they’re not learning “fact families” (sets of problems involving the same three numbers to show their relation). In problem 1, the first square probably gets a 3 (as in the first column), and the answer would be 13. And so forth. It might be to show how much the answer changes when one of the numbers is larger, or to show the pattern that emerges when 10 is a number. All well & good, but it would be nice if somewhere on the sheet was an explanation of the concept or a proper example!

As a kid, I would have gotten correct answers but failed the assignment, because my partial nerve-deafness made it very hard to follow what the teacher was saying. I would have noticed there was insufficient direction on the worksheet, and in the absence of specifics I would have put any integers I wanted in the squares and worked out the sums correctly from there. I completely sympathize when my own students do the same. (And guess what? I have students who really enjoy maths, but have a terrible time in the class because of this very problem: poorly-written or nonexistent directions!)

145. Anonymous says:

What I find amazing is that this is just what we used to call “carrying”. When you add 8 and 3 and get 11, you know that you put a 1 down and carry the “1” that is really a 10. So, 8+3 = 10+1 = 11.

Nowadays, they call it “regrouping” instead of carrying. Not sure why this has all been made so complicated; carrying was easy enough for me.

146. Anonymous says:

Its kinda ingenious if they had cared to explain it to the kids IN THE CLASSROOM!
This basically teaches them that there are different solutions to one problem and how you might go about trying to figure them out.

Of course, it could easily have been done by teaching children to count by…I don’t know what you call them, but I call the finger divisions. You know, your fingers on the palm side has 3 divisions due to the joint creases.

147. Anonymous says:

The worksheet is attempting to teach the students the precursors of partial products…which is another algorithm for multiplying…

For example,

Instead of a student computing 19×6 the student would reason in their head…

9*6= 54

6*10= 60

60+54= 114

This requires students to KNOW PLACE VALUE, which is a forgotten art in the standard algorithms taught for adding, subtracting, multiplying, and dividing most of us were taught as kids.

For example, when most of us were taught to multiply, we were taught to multiply 6×9, carry the 5 and write the 4, then multiply 6×1, add the carried 5 and write 11 for an answer of 114. The problem is MOST students, and teachers for that matter don’t realize that they aren’t multiplying 6×1, they are multiplying 6×10 (the 1 is in the 10’s place). The standard algorithm works but shows no understand of number sense.

The problem on the worksheet is trying to get students to realize that 8+3=10+1.

I know that it is frustrating to parents…heck, it is frustrating to me as a teacher, to unlearn a flawed way of thinking, but trust me, students today, if taught, will have a much greater understanding of numbers than we did.

Hopefully I didn’t ramble too much!

Bryan McDonald

148. nigex says:

I’ve got a degree in electronics, and I thought this was really tricky. Probably says more about me than the homework, though.

When I was at primary school I developed a finger counting method that allows counting to 99, without realising no-one else did it. What you do is use the right hand fingers for 1’s, and the thumb for 5’s, which gives 1 – 9 on that hand. The left hand is the same, but counts 10’s.

My missus can’t get the hang of it.

149. Liassic says:

Are you Americans crazy?
First graders shouldn’t be having homework.
Especially when even the parents can’t do it!

150. ab3a says:

I have three children in elementary school. My oldest is having problems doing larger multiplication and division problems because she doesn’t know her multiplication tables well enough.

At some point, we have to dispense with the explanations and simply say memorize stuff like this “because it’s the only way to get things done.” That never seems to have happened. We practice separately at home with flash cards to help my children. All that pattern matching stuff, the counting stuff, and the rest of that nonsense does is slow them down.

I’ve seen enough of this crap to know what it looks and smells like. It may meet educational theories, but it doesn’t seem to help in the long run.

151. Gal_n_AL says:

Wow! this post drew A Lot of comments. My son is in kindergarten, and he brings in homework that baffles both me and my husband occasionally. This is saying a lot because my husband works for NASA for crying out loud.

152. Anonymous says:

I actually agree with the new math. While learning to read by using “sight words” is counter-productive (the idea here is to foster comprehension, not actually learning to “read” by sounding out words, etc.) this math teaches you a shortcut that I use regularly.

Sure adding 2+2 is rote memorization. How about 245+317? Why not break it down to 245+300+17? Or +12+5? Try it both ways and see which one is faster. Multiplication? 5X5 is simple. How about 72X4? Well – 70X4 is 280, and 4X2 is 8. Try doing it the old way and see which comes faster.

On the same hand – my son’s last homework had a statement that said, “Explain why two odd numbers when added equal an even number”. Explain in a 2″x2″ square. Ummmmm…. “Because it just is?”

1. Anonymous says:

“Explain why two odd numbers when added equal an even number”

I knew this when I was 7:
Even numbers are always of the form 2X (where X is an integer), and odd numbers are of the form 2X+1.
Adding two odd numbers gives (2X+1)+(2Y+1)=2X+2Y+2=2(X+Y+1).
As X and Y are both integers then X+Y+1 is also an integer, so the answer is even.

153. benher says:

Is this one of those damn puzzles from the “Nightmare” setting on Silent Hill?

154. All Children Left Behind.

155. Anonymous says:

156. Anonymous says:

Caroline: In section #1 there are three problems. In each case the workbook writers want the student to “understand” that the student must have 11 for an answer to 8 + 3, 10 + (the missing 1,) and “They draw eight counters and you draw three counters.” Three problems there. The answer to all is 11. I’m not talking about #2, and I didn’t even look at #3 yet. They work the same way though. Same answer to all three problems.

Leila

157. Brandon Heyer says:

I promised myself I wasn’t going to post. BUT I am getting such a kick out of everyone saying they have this many degrees and this many initials after their name and an IQ this big with a few extra inches where it counts. And then they get REALLY upset and throw a tantrum, because all of that comes to naught when solving this problem.

This is why college education classes have to teach COLLEGE students how to do FIRST GRADE math. You can spend your days solving calculus problems, or figuring out complex physics equations, or playing video games. The truth is you are SO FAR REMOVED that solving basic problems is TOO DIFFICULT for your initial post-fixed brain.

Yes their are no instructions, but maybe it was just me, and I go “hrmm no instructions, but ooh look, pictures” and then I saw two equations, one with a number that matches up to the numbers of items in a pair of ten-grouped boxes. Then I saw a ten, then I put two and two together, took the third derivative and questioned the stars.

It isn’t rocket science people, and maybe that is the problem. Tout your degrees after you prove your common sense…and for a twist of irony, maybe it is just the fact that I am lucky enough to use both sides of my brain, and this got me two lowly bachelor degrees in Computer Science and Graphic Design.

158. Oshkosh John says:

This is from number theory. What’s weird for me is that number theory is a college-level course, usually presented after a couple of semesters or more of calculus.

159. Anonymous says:

Hey, any way you look at it, it’s better than having a meth problem. . . .

160. Anonymous says:

So many responses… so this may be a duplicate idea.

The boxes in the right column are unknowns. The only place they can come from is the solutions on the left column.

1) 8 + 3 = 11 (copy the 11) 10 + 11 = 21.

2) 9 + 3 = 12 (copy the 12) 10 + 12 = 22.

3) 7 + 4 = 11 (copy the 11 [again]) 10 + 11 = 21.

161. Anonymous says:

All they are missing is an = on the bottom row of each problem and it would be clear. So sloppy.

162. Anonymous says:

I didn’t understand this AT ALL until I read all of the comments. Towards the tail end of the comments, I now get it.

The concept of this homework (teaching kids an easier way to do math by using more 10s in there) is fantastic.

On the other hand, the instructions with the homework and the layout of the problems are embarrasingly bad.

163. AGC says:

There are great hard cover math books costing lots of money sitting on selves in the classrooms across the world.

Kids get ugly, badly photocopied, worksheets, missing information, compiled quickly by teachers before classes start.

For every nice math textbook collected dust, thousands of pages in poor photocopies get used.

Do away with all copyright in education, do away with companies that sell text book. You need people that know how to compile nice workbooks for kids from freely available sources.

A photocopier in every classroom, with free toner would solve 50% of problems in schools.

164. Anonymous says:

This is SO wrong.

If I wanted a 6 year old computer, I would just buy a refurbished one.

1st grade = simple learning skills taught in a human friendly manner

165. KarenAScofield says:

Rule 1: Instructions should not be more difficult to the average ______ grader than the math being taught.

Practical: Young students’ already developed communication skills are an asset in learning new concepts and tasks. We should teach children to use their assets while they are in the lower grades.

Emotional: This not only makes incredible sense but there will be fewer negative associations and attitudes to hauntingly interfere with natural or guided exploration of aptitudes, interests, and careers.

Rule 2: Especially in the lower grades, instructions should be sufficient and clear enough that the average parent, caregiver, or older sibling can readily understand them and help the child with their homework. The reasons for that are obvious.

When those obvious reasons don’t have an impact on the institutionalized dysfunction seen in our schools, i.e. the questionable methods of teaching math, then their cries for increased parental involvement seem sensible and sincere on some fronts but disingenuous on a most crucial level.

Rule 3: Teaching students in the lower grades in what would be a more cryptic manner has a limited purpose and should be limited accordingly.

The purpose should encourage the habits and attitudes of learned empowerment (love of learning) rather inadvertently foster frustration and dislike for math or any other topic being taught.

The purpose should not, in any convoluted manner, continue to play a role in turning No Child Left Behind into No Child Left a Brain, not that they would actually ever call it that.

166. Anonymous says:

I was surprised reading the comments that people thought this was difficult or confusing. This was super easy and intuitive to me. Its just a pattern, but maybe it was really simple because I’m younger and have probably done a million of these worksheet things in elementary school. Maybe math was taught differently to older people.

You add the first two numbers together to get your sum. Then for the second number you add whatever number to make the same sum. You draw in the same amount of counters to equal the sum.

167. Anonymous says:

It would be nice if they’d explain it, but this looks like the kind of math that makes it easier for french fry vendors and bowlers – in a universe without computers.

Its fun with Associative Property.

8 + 3 = (8 + 2) + 1 = 10 + 1 = 11

9 + 3 = (9 + 1) + 1 = 10 + 2 = 12

7 + 4 = (7 + 3) + 1 = 10 + 1 = 11

168. Anonymous says:

OK, I know I am probably just repeating some older comments, sorry not to have read them before.
From the perspective of a Math teacher: “going over 10” is (supposedly) a big deal in primary Math. First they start adding “to ten” – easy to model on fingers. Then a way to show the learners how they can add when the result is more then ten is to point out a simple fact: you start with the bigger number, fill it up to ten and see how much is left. It is usually modelled with coins or other small objects, beans etc. It would actually really be useful to give the child the real objects to play with, modelling is a very powerful tool – that’s what “draw counters” is trying to suggest in the instructions. It is important for the child to understand this principle, because it will help them in adding larger numbers. Good luck!

169. Milton Ramirez says:

I wonder how such a simple post has generated deep conversation. But I am still thinking about the source.

Tweets from @nealchambers, @teacherspirit and @tonnet, keep asking about what textbook that worksheet is from? Would you, please?

170. Anonymous says:

I don’t even understand the explanations.

171. zandar says:

@Micah- no, this is the new math way. new concepts are introduced without warning in homework to encourage a parent-child dialogue about it.

No lie, this really is what I was told by my sons’ teachers, both of whom have suffered greatly with nonsensical homework just like this. It’s not like we wouldn’t have talked about it anyway, but with homework like the kind we get, most of the time is spent figuring out how to jump through the hoops and hardly any time is invested in developing new knowledge of the subject.

I have to agree with Antinous and TDawwg, this is designed to (1) make you dependent on counting on your fingers instead of having the concept of equasions drilled in you (which is a bad thing); and (2), to crush your soul.

172. Anonymous says:

I was taught this same kind of “new math” when it was actually new – the late 60s early 70s. Everything was set theory and different bases. At least we had cherries instead of pennies (that’s Capitalism for you).

Because math was so abstract and about making collections of things I developed a kind of synaesthesia about numbers: I thought there were numbers that didn’t get along. And because everything was spatial – like in the diagram above, I thought that solving problems was not about steps but about figuring out the shortcut to change the shape of things.

For this reason, as an adult in college I had to take remedial algebra. I was still concerned with finding quicker ways to solve problems – like I was Stephen Hawking or somehting – I would look for my own unifying theory that would let me do problems more quickly. Of course I never found it. But that’s because I had never just memorized the basic tables and learned the steps. Everything was still, somehow, about collections of cherries.

Now, PhD (humanities) when I have a dinner party and need to halve a large recipe, I still need to pour the full amount of X into a measuring cup, look at it to eye what “half” would be, and pour the extra back.

Don’t let this happen to your child.

173. Anonymous says:

The students should have a companion (hardcover?) book that contextualizes this, plus a take-home sheet from the start of this unit preparing the parents, and probably even an online companion.

We are often baffled by my kids’ (2nd & 5th grade) “Investigations” math homework here in Rhode Island. It incenses my wife, but I figured out these strategies on my own as a kid so most of the time I can figure out what the classroom portion of the lesson was.

Other parents in town, however, are confused and angry and feel condescended to when the school department just intones “Trust us” whenever they’re challenged. Few people have developed these tricks, and when kids ask parents for help, it generates really bad emotional responses.

I believe that many teachers now offer both methods of doing math, which is good for parents’ feelings, possibly good for kids (yay! more strategies!), but arguably a drag for the teachers.

174. crankypage says:

Time to teach your child Chisanbop.

175. Anonymous says:

I believe this is from a text book that was published when Mike Harris was Premier. School boards were given a load of money and told to make a purchase from one of two publishers. Both text books were ambiguous and are largely not used anymore for the very reason this person posted this. It seemed that publishers knew that there would be a cash injection and produced poor texts knowing they would be purchased because of the short review time. I teach grade one and if I were you I would send it back. Kids shouldn’t be sent home with work they haven’t done before or are not familiar with.

176. J. Jones says:

58 now but when I was growing up there was the emphasis on the “new math.” And what is in these problems is the introduction of the relationship between the flat construct of a named value i.e. 13 or 105 concept of notation and the actual size of the value. At the same time the emphasis of base 10 notation and arrangement helps with understanding how to estimate in a way that is align with that notation system. (As opposed to having to do this in several different bases which really was one of the banes of New Math.) The result is that the notation system has a more concrete representation as is needed for concrete thinking found in most young kids.

177. ethancoop says:

“Make all tens for great justice. Somebody set us up the bomb.”

That was by far the best response I’ve seen for this nonsense. Math isn’t the problem here, clear instruction is. How about an example problem for goodness sake.

178. BryE says:

OK, I’m going to defend the homework. I have a 2nd-grade son, and I’ve helped him do this type of problem before. The idea for the 8+3 problem is to add 2 to the 8 and remove 2 from the 3, so that the problem becomes 10+1, which is very easy to solve. Google for “making tens” for more examples. The problem is that the sheet has no instructions. I’m sure it was discussed in class, but every sheet brought home by my son has a completely-worked example that has been most helpful for me.

So why is it defensible? It encourages the child to think about the numbers, rather than just count on fingers. Also, this is a simple example of the associative property of addition, which will be discussed in pre-algebra courses. Does it work? Last night, my son was working on a paper that had to do with ordinal numbers. There was a row of 20 circles, and the problem was to make a mark on the Nth circle. The particular problem was “Write a check on the 13th circle”. Rather than just count all the circles up to the 13th as I expected him to do, he started at the 10th and counted up 3 – he’d done a mental “make ten” and added three. The next problem was “Write an X on the seventeenth circle.” He started at the 20th circle and counted backwards to the 17th. I was floored – he’d done the “make ten” with 20 and subtracted three, and there was no counting on fingers or anything at all – he’d done it all in his head. I could only assume that he’d been able to do this from all the “make tens” stuff.

1. Anonymous says:

why do you have to remove the 2 from the 3 then add the one? that makes it complicated

i guess there are some assumptions here being made, one is that all 3 graphics are a representation of the same value

the instructions say USE THE COUNTERS TO SOLVE

so you have 8 + 3. a quick count tells you that there are 8 pennies in a space for 10.USE THE COUNTERS TO SOLVE. so add 3 pennies

now you have 10+___=____ well, your counters show you 10 pennies and then a one. so you already have your 10, the missing piece is the solo penny and then count both to get the answer.

it takes longer to type than to do because once you recognize that the boxes contain 10 spaces, and fill it in on the first equation, you can just LOOK at the lower box in each problem to see the answer to the 2nd equation.
its so easy it should be cheating. the instant you recognize that there are 3 pennies, you have your answer.

btw, i do have a degree in science but i also have taught my kids math- one was autistic. so im pretty well versed on math for big grown ups, little kids and those who have language problems. i find it interesting that people think the instructions are hard. the problem is easily grasped visually. it seems like an ideal way of explaining the meaning of 11 etc to someone who doesnt speak your language and doesn’t comprehend our numerals.

179. warreno says:

This is beyond doubt the most shoddily designed homework I’ve ever seen. The title, “making ten”, is confusing and deceptive. It should be called “working with ten”.

I sincerely hope this is not representative of what US schoolkids are being faced with now under NCLB. If it is, we as a nation are doomed.

1. Anonymous says:

This pre-dates NCLB. By far. I was nearly ruined by “new math” in the 1960s… had to learn the old ways in college, and I’m still math-phobic.

180. dancentury says:

That’s just insane. No child left sane.

181. danlalan says:

I’m somewhat bemused by the reactions here. The exercise is simple if explained and slightly cryptic if not because the form is unfamiliar, so a couple of sentences of explanation or an example would have been helpful and wise, especially for the parents. Assuming the teacher explained what was expected (and shame on them if they didn’t) this isn’t such an off the wall introduction to how base 10 mathematics work. But to read some of the comments here, it’s the first grade equivalent of the “kobayashi maru”.

182. Anonymous says:

I’ve had one of my daughter’s 1st-grade math worksheets pinned to my bulletin board for years. The instructions for one of the problems: “Circle the correct estimate.” My daughter’s penciled-in answer: “Correct estimate makes no sense.” Made me so proud.

183. Anonymous says:

vorn’s got it right. you solve 1st part of problem 8+3 =11 which is same as 10+1=11 or a full top box with 1 left in bottom
I think it’s trying to teach associative property,
if 8+3=11
and 10+1=11
then 8+3=10+1

184. slida says:

People are over-complicating this. The problem is how do you get to eleven. What plus ten equals 11? How many more pennies do you need to have 11 pennies? What they should have done is included the sum ’11’ in the second example, that’s what makes this confusing, as the point isn’t to guess the sum because you are supposed to know that already.

185. Anonymous says:

The purpose behind activities like this is to teach the simple math facts in a meaningful way (so that students understand them rather than just memorizing them so that they can forget them later) and to start teaching young students to understand place value so that the standard addition algorithm will actually make sense when they learn it later on.

Most people who are reading this post probably learned the addition facts through rote memorization when they were in school and then learned the standard addition algorithm without really understanding the math going on behind it. These math activities look different from what most of us are used to, but they’re based on the principle that students should understand math concepts before they learn the algorithms. That way, the algorithms become shortcuts but don’t compromise understanding.

186. peter313 says:

Mark, my son has the same workbook and I also struggle with the overly succinct and cryptic instructions. I get calls from other parents trying to understand homework assignments that a 6-year-old would never be able to figure out on their own. I just keep promising my kids how much they will love college.

187. Anonymous says:

Yeah there are a lot of comments here but I wanted to add my two cents. My son has had math homework like this in the past. Many parents tried to get the math program kicked out of the school. I like the idea of formalizing some of the shortcuts many of us used and figured out on our own. But that is a strategy to get to the correct answer. Before you learn strategy you should always know the rules of the game. You still need to know your times tables and just understand basic arithmetic. Teaching shortcuts like this to students who don’t fully understand the rules is where it breaks down in my mind. It’s like trying to teach someone the finer points of chess before they even understand how a rook moves.

188. Anonymous says:

Not one comment on this yet:
The reason why you have to show working out in maths is to demonstrate you understand the concept and to show you didn’t just copy the answers from your neighbour/computer/parents.
I’m sure it would be easier for teachers to just mark your answer too, but there is a reason for it. And if it makes you hate maths then nobody explained that to you.

189. whiznat says:

The real purpose of this type of problem is to be a precursor to solving harder problems using distribution of multiplication over addition or subtraction. For instance,

78 * 8 = (80 – 2) * 8 = 640 – 16 = 624

Knowing the groupings of numbers which sum to 10 (in this case 8 + 2) makes doing these type of problems in your head quick and easy.

190. RobJ says:

Hey Mark, this is wild. My daughter, in first grade up here in San Jose, also had this same page of homework to do recently. Took me a few minutes to figure it out (like other commenters have done in this post), but it wasn’t too out of control. IIRC, there is another page that explains it a bit more, and you are showing the “Practice” page. Anyway, that’s how I interpret these workbook pages.

Our teacher isn’t too thrilled with them either, but I guess they’re mandatory.

BTW, did you guys have the same kind of standard workbook last year in kindergarten? Now *those*, I could not make heads or tails of! These first grade questions, at least I feel I can interpret them!

Good luck, BTW.

191. dtaylor404 says:

Vorn and Caroline, of course, have it right. And as several have pointed out, the original workbook has instructions that this teacher chose not to send home.

Now, what I don’t get is why they want you to “draw counters”. These are a lot easier to do if you actually use pennies. It defeats the purpose of using counters at all if you have to draw them.

192. Ito Kagehisa says:

There’s a basic dysfunction that is evidenced in many of these comments: In the Real World [tm] not every child heard the instructions given in class. In the Real World [tm] sometimes they were being distracted by other children, or the teacher was called into the hallway to help somebody who slipped and cracked their elbow and didn’t finish the lesson, or there was a fire drill, or any of ten thousand other problems. A textbook that does not have coherent instructions on the work pages is worse than no textbook at all.

Mark, you will have to teach your children multiplication, division, and simple algebra yourself if you want them to be good at math. The US publically-funded school system is horribly broken, due primarily to the textbook selection system. Read Feynman on the subject for more information; the problems he describes haven’t been fixed in the decades since he wrote.

193. Bryan3000000 says:

The problem here is that they are trying to *teach* these kind of microconcepts to children at all. These kinds of numerical relationships are the things that children naturally discover on their own, given the opportunity.

The truth is, a standard K-12 math curriculum consumes about 20 weeks for an older kid who is ready to learn it.

I frankly gave up on math as a child and stopped trying to learn it because of tripe like this. I always thought it was just because my teachers were idiots – they could never tell me what this was supposed to teach me. The real problem is that they *and* the people who designed it are mainly idiots. I’ve had more success in getting my kids to understand and do math by explaining high level concepts and their application (even when the kids truly are not ready for it cognitively). At least then they know that there is a point and that it can actually be a cool point.

I swear, even my calculus teacher was completely baffled and unable to explain when I would ask him the point of a particular exercise. Given the problem in the context of a project, I, like just about any child, would have taken it hook, line, and sinker. If only I had known that calculus had a connection to electrical engineering and making cool things, I might have gone a very different path.

194. Anonymous says:

+1 on the abacus comment. Get your kid an abacus.

195. Bloodboiler says:

Florida 2000 elections ballot designer strikes again.

196. ROSSINDETROIT says:

Elementary school kids are pretty smart and mentally flexible. I’ve seen science tests for 3rd graders that were logically difficult and they breezed through them. Kids have the advantage that they haven’t yet been taught that there’s one fixed way to learn. I’m 50 and in my parochial elementary school there was rote learning and memorization. If they’re now teaching more creative methods of problem solving it’s all to the good.

terrible, terrible instructions.

198. Anonymous says:

Agree with @optuser —

This exercise is showing equivalencies between different addition problems. I like it.

The first example:
8 + 3 = __
10 + __ = __

To figure it out, the child looks into the third column at the visual representation. There are two groups of TEN boxes each.

Count how many circles are already in the boxes. There are eight. Now draw in THREE more, corresponding to the first addition problem given. (8 + 3)

Now count the filled in circles. There are 11.
8 + 3 = 11

The second addition problem regroups those same numbers. Instead of 8 + 3, we’re working with 10 (amount of filled in circles in the first set of boxes) + __ (amount of circles filled in on the second set of boxes). In this case, the child counts the circles in the second set of boxes. There is ONE circle. Now we can fill in the first blank.

10 + 1 = __

And then the lightbulb goes off — We haven’t added any circles, but our original visual representation applies to both problems! 8 + 3 is the same as 10 + 1. Which we now know to be 11. Count off the circles to verify.

9 + 3 is the same as 10 + 2
7 + 4 is the same as 10 + 1

And by the way, this is EXACTLY how I add numbers in my head. Probably a wacky way of doing it, but it works!

Sidenote: Also teaches child the difference (subtraction) between 10 and __ (number less than 10). In the first problem, we add 2 circles to hit 10, thus 10-2=8. In the second, we add 1 (10-1=9) in the third, we add 3 (10-3=7). Planting the seeds for subtraction?

199. Amos Moses says:

Re: SamSam’s comments of feeling bad about remarks by those who are obviously stuck in the “old way” of doing math. First, it’s the typical arrogance of those who swear by the “new” math as being a reflection of their own “advanced” thinking. Second, they won’t admit that the “new way” has produced 70% enrollment in remedial math classes in community colleges and 40% in four-year institutions. Mathematics is one of two universal languages (music being the other) and we are creating students who can’t talk with other countries because of our “inventive” reform mathematics. As a retired math teacher and elementary principal, I can tell you this “new” math is a disaster. This is especially true if you are a six-year-old who does NOT have the cognitive abilities yet to sort out “higher-level” thinking skills. Third, SamSam is an example of those who already have the math skills and who don’t realize that learning simpler, basic skills used for 2000 years by a variety of cultures around the world THEN allow the production of higher-level thinking. Fourth, when a primary grade math assignment cannot be quickly understood by a fairly well-educated parent (even a strong high school graduate), there is something wrong with the assignment.

200. Anonymous says:

good practice for putting together ikea furniture.

201. scb711 says:

I was in first grade in 1959 so I’m not sure I’d call this ‘new’ math. We did the exact same exercise with logs and bundles of logs. A bundle was 10 logs. We had better workbooks though with better illustrations and examples. Although since we can see the whole book there’s no way to know if there were instructions we just didn’t get to see.

I used to love those workbooks. And I’m good at math too.

202. Anonymous says:

My kid has to suffer through this Everyday Math garbage too. I finally realized the problem with the entire curriculum: it focuses way too much on “mental calculation techniques” instead of just simply and clearly presenting the notion of equality. You know how this problem could be made utterly intuitive and clear? If an equal sign was present below the two boxed “answers”, e.g.

..8….10
+ 3. + [ ]
___ .. ___
[ ] = [ ]
(boinboing comment formatting sucks)

There are two very basic and fundamental truths that underlie all of arithmetic. Equality is the first. The uniqueness and existence of arithmetic operations is the second. A + B has one unique answer, C, that is guaranteed to exist for any A and B, and the problem becomes utterly clear if you think about it in those terms. There is only one possible answer to put in all three boxes if you understand those two basic concepts (and the problem is stated correctly)

203. michaelamechanic says:

My boy, now in second grade, has come home with stuff like this. It’s not that hard for an adult to figure out, but I’d say it’s beyond the reasoning ability of most first graders to know what they want, unless someone explains it. Occasionally he comes home with something whose instructions totally baffle me.

204. ROSSINDETROIT says:

I showed this to D.R., a third grade teacher in Birmingham MI public schools. After staring at it for 7 minutes she declared “Clear as mud but it covers the ground.”

205. Anonymous says:

I’m sure this will get lost in all the comments, but the point of this worksheet is not to teach the kids that 8+3 = 11. The point is to visualize the carrying procedure that we were all taught so abstractly as kids.

8+3 =11 , so I’m supposed to put down one of the 1’s, but put the other up above the next column and add it to those numbers, so…um…what? If you sit and think about it, that algorithm makes no sense at all. And have you ever tried to explain long division to anyone? That’s utterly nonsensical.

The standard algorithms for addition (carrying, etc.) are just algorithms – they abstract what’s really going on in favor of being able to do things quickly. But kids who are just learning math need to focus more on what things means rather than trying to work quickly. This helps the kids learn that that 1 in the next column means 1 bundle of ten pennies, just like those blocks that had long rows of ten or big squares of one hundred blocks. Conceptual understanding leads to computational facility down the road much more easily.

206. desprez says:

This is an interesting discussion.
I learned to mentally count numbers completely different.

For addition I would start with the higher number and mentally count up, arranging an image of dots in my mind until I had formed the arrangement that represents the other number. When I got that, whatever number my count had reached was the answer.

For subtraction, I did the same thing except that I would count until I got to the other number, and look at the mental dot arrangement for the answer.

207. tingfod says:

Sorry, but worksheets like this kill any creativity! Mathmatics can be fun, but these worksheets are not!

208. lhopitalified says:

To everyone who believes that this worksheet teaches place value and/or serves a precursor to later mathematical properties like distribution of multiplication over addition, I have the following remarks:

I believe that you are wrong, and here is why: an actual worksheet that discusses place value would explain how 13 = 10 + 3. Instead, it seems that this worksheet is showing how to add numbers when there is “carrying” or “regrouping” or whatever you want to call it. In doing so, it demonstrates how when adding 3 to 9, first adding 1 gets you to 10, which leaves you with 2 “counters” remaining. Perhaps when you are first learning to add numbers together, this is a good way to demonstrate what happens when you need to “carry”, but otherwise, I fail to see how it explains place value or “higher” math concepts. Please don’t generalize the teaching methodology – I think most of us would agree that teaching why place value matters and how it is related to multiplying together “19×6” is important, but that is not the issue here. I think there are two main problems with this worksheet:

1) It may be inappropriate for teaching the lesson at hand (whatever it happens to be)
2) The “directions” are unclear. Unless you are teaching kids how to decipher badly worded and incomplete instructions, which IS a vital life skill, this is a *bad* thing.

To the person who commented that the reason you show work is to demonstrate that “you didn’t just copy the answers from your neighbour/computer/parents” – that philosophy assumes that there *are* people who are just copying the answers and not actually doing the work. Yes, if you didn’t require work to be shown, there no doubt will be people who “cheat”, but forcing people to show work assumes (however realistically) that there would have been “cheaters” and that those people need to be “caught”. You can argue that it identifies kids who don’t understand the concept and would need additional help, but I imagine few teachers have the time to personally address the educational gaps of all kids who don’t know the concepts. At the same time, forcing people show work can turn off some people from math who otherwise would be interested (because they know it so well, showing work is boring to them).

I dislike the fact that your response to why work needs to be shown implies that teachers distrust students. Students rise to meet expectations – if you tell them you don’t trust them and that’s why you force them to show work, there’s no expectation that they would have done the work anyway. It just leads to students who care just about grades and will do whatever they can get away with (and some things they can’t) for a better grade.

209. Anonymous says:

What a wonderful series of comments that range from frustration at the problems to frustration at the system to frustration at each others’ abilities to see the purpose and design behind the test. I have particularly enjoyed the Monty Python reference, the links to scholarly studies on the nature of learning, the humour and the fact that people far more qualified than me had the same difficulties figuring out the problem as I did.

What does this all say? It says that people learn and think in different ways and this implies that any system of teaching should be flexible enough to take this into account. Ideally, it should be individual learner driven and paced. The real problem is to set tasks that motivate the individual to learn the content by whatever means they find useful and this again is different for each of us. The challenge is how to do this in classes of 20 to 40 children. I believe that until now this has been an impossible task. However, with the introduction of the internet into the classroom we may have a tool that allows teachers and parents to allow the learner to drive their own education.

210. Anonymous says:

What a horribly frustrating way to learn math!

211. Amos Moses says:

Reply to Kostia: About the research showing Everyday Math had “potentially positive” effects…Further checking will help you learn the following: EM did well when compared to earlier (1st generation) versions of NCTM based “Reform” textbooks, but failed to show any improvement at all when compared to traditional textbooks in three other What Works Clearinghouse-accepted studies. Also, only affluent white and Asian kids were tested, telling us nothing about the major education problem of today, at-risk students. The complete explanation of this study can be found in the only peer-reviewed and published report in math education research today at http://web.uvic.ca/~whook/Hook-Bishop-Hook,AuthorVersion,1-3-07.doc.

212. Sara Bennett says:

The real question is: why are first graders getting homework. As the co-author of The Case Against Homework (Crown 2006) and the founder of Stop Homework, I’m appalled that, despite the research showing no correlation between homework and academic achievement in elementary school, a first grader would get homework at all. If all parents would stop trying to figure out their children’s homework and helping them, and if all parents would stop asking their kids whether they had homework and reminding them to do it, homework for young children would be abolished immediately. Our children would be better off–they’d have more time to play, daydream, read for pleasure, have fun–and they’d be better educated as a result. And then we could turn our attention to abolishing (or, at the very least, severely limiting) homework for older students, most of whom spend hours on work which most educators, if they took the time to examine it, would consider busywork.

213. Anonymous says:

My 6 year old has these exact worksheets she can do them in 1 minute flat. They’re annoying but they do work.

The answer is 8+3=11, 10+1=11, etc.

214. Lt DirtyFreq says:

So why are they making them do problems in tens & why are they making a little grid on the side? Why can’t they just make the kids do only the first equations? What will this solve in real life?

215. lzygenius says:

I have to agree with the regrouping theory already posited. I’m currently taking Teaching Math for Elementary School and we covered this concept earlier, basically they are trying to teach some of the mental strategies for regrouping.

216. Inconsequentiallogic says:

In a word “ridiculous”. My son is a 3rd grader and some of his stuff is way more far out than that. I recently got a kick out of some wise proverbs given with a first grader’s point of view

217. Gilbert Wham says:

I remember in the late ’70s/early ’80s, we head a Speak & Math delivered to us by mistake by some catalogue company, which we cheerfully kept (early parental influences in my Chaotic Neutral character alignment…). It used rather odd mathematics, too. F’rinstance: how many tens are there in a hundred and eleven? Eleven, right? Apparently not. There’s one ten, one hundred and one one. That orange & yellow box inculcated a dislike of machines telling me what I should do that came to fruition with that fucking talking paperclip…

218. beameup says:

There’s a lot of very strongly worded criticism here. I believe everyone should have an opinion, but maybe the strength of your opinion should be tempered by the amount of experience you have in teaching young children mathematics using a variety of methods. You know how your child learns, and you know how you learned many years ago. Look into it a bit deeper and you may be surprised. Speaking from many years of teaching experience (as I said before, not from theory), the method that this worksheet is practicing is a fantastic tool for most students for building strong number sense and mental math skills.

1. AirPillo says:

Still, the method is useless if the vagueness and cryptic nature of the instructions are making the children believe they are stupid because they can’t understand what to do.

This is what is happening with my brother. Despite the fact that he performs well with the actual math once he figures out what the assignment is, he believes he is dumb because it’s so hard for him to understand the instructions in the assignments.

Having arguably good theory behind one’s workbooks does not excuse them from actually making the attempt to educate (communicate) with those workbooks.

I honestly could care less if they wish to teach math differently from how it was taught to myself and my parents, but I do expect them to actually teach something. Stranding children in the dark with vague and nonspecific directives laid out amidst an unclear presentation of information is cruel at best and deliberately negligent at worst.

219. beameup says:

Is the “they” and “them” you are referring to all math teachers, or your brother’s math teacher?

1. AirPillo says:

Several people. Firstly this seems to be consistent between different teachers over the years, who actually seem to be decent, engaged teachers… which means someone at the school or district level has either encouraged the wrong approaches to homework or failed to provide the teachers much means of compensating for this workbooks’ failures.

Secondly, the author of the workbook. Honestly, when compiling something of this size there is little disincentive not to go the additional inch and include 1-2 sentences of clear explanation per question, detailing what the actual task is. This much effort was extended for my education and that of my parents.

The lack of explanation and the significant ambiguity present do not present any benefit to anyone except the author and publisher, and they actively hinder the materials’ stated purpose.

My impression is that there are multiple levels of failure at work here, but that seems to be the case in several different districts, in many different states, where this sort of workbook is assigned.

Minimal if any explanation of questions, ambiguity of language in assignments, lack of communication to the parents whose job it is to provide assistance and tutoring when necessary… there is a lot wrong here and I can’t pin it all on the author, but then again a lot of it would be significantly less necessary if they had seen fit to explain themselves clearly just once a page. That is not a very pressing demand, is it?

220. Pipenta says:

Lots of emotional responses here.

Me, I’m not a math wizard. When I was younger, I had such math anxiety that I simply could not do math. I didn’t know if it was because of the new math or old math or bad teachers or crazy parents or just innate baggage, but I felt freaked out and helpless. Presented with anything remotely mathematical, I was like a deer in the headlights. It sucked. I felt defective. And it limited my options. I wanted to be a scientist. Instead, I went to art school.

Nearly thirty years later, I decide I am going to do science, damn it all. I go back to school. I have to face down my math demons. As an adult, I still panic. But I can peel apart my fear, see the shame in it. I realize that, when I freeze up, much of what is going through my head is that people are watching me and thinking I’m stupid and lazy and what all. This points to my abusive parents being the root of my particular issues with math. Oh. Okay. Whatever. Not like I can do anything about that.

So I had to deal with this, right? And I realized, in order to do math, I had to let go of the negative emotion. I needed to stop pouring energy into all the negative looping I was doing when confronted with a math problem. And wouldn’t you know it? I had to breath and let go and approach each problem calmly from the most basic level.

Then, I could do math. I’m not stellar. I had to study and study and study and I couldn’t seem to crack an A. But I could always muster a B. I wasn’t fast. I wasn’t confident. I needed to chew my pencil. But I could do it.

More importantly, I often enjoyed it. There’s something comforting about it. When the rest of the world doesn’t make sense, math always does. Even this homework assignment makes sense. It just wasn’t what a lot of you expected. Well, maybe it was. You expected something incomprehensible. It is not like it was presented in a neutral context. The “Do you understand my first-grade child’s homework?” set it up such that I expected to be baffled by it. I wasn’t. The exercise, to my surprise, seemed clear and simple.

Maybe it didn’t bother me because I’ve never been good enough with math to have a regular fall back position with it. I have no “math comfort zone” whatsoever. So tend to approach anything that even has the faintest whiff of math as a puzzle.

Try looking it as a puzzle. As puzzles go, this is an easy one. You just have to be open-minded. So what’s wrong with a little mental flexibility? And especially, what is wrong with wanting our kids to have that kind of intellectual versatility.

So, all you sputtering BB-ers, if your kid doesn’t get it and you don’t get it, you are going to have to call the teacher. Is that such a bad thing? Are you simply angry because it makes you feel stupid? Are you bogging down like a mastodon in that tarry pit of shame? Let it go. Or do you feel your child has to learn math the same way that you learned math, if indeed you did? Let it go. That is not helping anyone. If you have issues or anxieties about math, or feel like math is an odious chore, for your kid’s sake, now would be a good time fake some enthusiasm. You lied to your child about Santa and that served no purpose whatsoever. This really is for the kid’s good.

This is a puzzle. This is fun.

221. Anonymous says:

The grids on the right are the only part that really confused me. Despite the use of an arrow rather than an equality sign, I determined that the student was supposed to write, “1,” under the 10 to make the two equations equivalent. I could not, however, divine the purpose of the grids and despite having read explanations, I am still not sure of their purpose. The instructions are rather unenlightening, but I think that this method disagrees with me. I have never liked or gained from the use of counters or such representations; they only ever confused me. I recall that when I was in sixth grade our teachers attempted to instruct us in simple algebra, but did so with some ridiculous contrivance of plastic pawns and balancing scales that confused me immensely. When I was years later instructed in solving more conventional simple equations,I learned how to do so. Learning to solve required some struggle, but succeeded in the end, whereas the nonsense with the pawns was just a fruitless, painful ordeal.

222. beameup says:

I would estimate that at least 90% percent of the math worksheets I’ve seen do not include detailed instructions, if any. Think back: multiplying two-digit numbers, multiplying fractions, subtracting integers, using the quadratic formula, factoring polynomials… Why no instructions? The instructions and lots of practice should have been done in class and the child should be clear on how to complete the worksheet. If that isn’t the case, then there is a concern.

223. JoshP says:

‘So, we’ll go no more a roving
So late into the night,
Though the heart be still as loving,
And the moon be still as bright.’

ah, a little couplet makes it all better.

224. Anonymous says:

It is the weekend now and I’ve had time to actually look at the homework problem in depth: I have figured out mainly that you, the Dad, were posting this and not posting a request from somebody else, so, of course you are not a Head-Scratching-Mom but a Head-Scratching-Dad. Whoops! My bad! I’m scratching my own head at my mis-reading error!

That being said, I still stand by what I said, although the posters with whom I agree most are those posting that First Graders should not be doing homework. When I taught First Grade I gave as little homework as I thought I could get away with giving. My Mom said it best: “Play is the work of children.”

I teach Art now. I see that the Homework lesson we are discussing is supposed to prepare students to work in tens, the basic idea of Place Value. I have done this Art Lesson with Fifth Graders twice. You could try it when your daughter gets a little older because it is fun.

Size Value Pictures (to 4 places):

Optional: Look at work by Mondrian. Mondrian liked very precise lines and forms. His work is “balanced” in that there seems to be the same amount of carefully made shapes and colors in every area of his pictures.

Make squares of 4 different sizes and at least 3 colors of paper. Suggested inch size of papers: 1/2×1/2, 1×1, 1&1/2×1&1/2, and 2×2 inches. Make sure there are are at least 9 of each size, but have more for an adequate color choice.

Discuss that the 4 sizes of squares represent 1s, 10s, 100s, and 1000s. Note that the squares can’t show this exactly the same way as numbers can because, if so, the smallest is 1/2 x1/2 inch,and then the next size would be papers 5 x 5 inches, but then…oh my! … the 100 size papers could cover over a bulletin board and the 1000 size papers would be like billboards!

On a 9″ x 12″ sheet of colored construction paper, arrange and glue from 0-9 of each of the four sizes. Try to make the arrangement look balanced and artistic. On the back, write what number was made. Let others look at the picture and see if they can figure out the number also!

Leila

225. Anonymous says:

Everybody was once in first grade, so everybody has some (usually strongly held) opinion about the best way to teach first graders.

Scanning through the comments, I see lots of “this is a great way to teach,” and also plenty of “this is why I hate math / this is why american students are so bad at math.” And while plenty of the posters are quick to point out all their advanced degrees in math/science/etc., or their own personal experience being really good/bad/ugly at math, very few of the comments are from either first grade teachers or researchers with expertise in early math education. For that matter, I don’t see any posts written by actual first graders :)

So, admitted I have no expertise whatsoever, a number of things seem obvious:

(1) The instructions on this worksheet are too sparse for a parent to be expected to immediately recognize what their child is supposed to do.

(2) The students probably have done similar work in class, under their teacher’s supervision and instruction, and are thus MORE LIKELY than their parents to know what to do.

(3) Some students probably weren’t paying attention in class and/or don’t remember exactly what to do, so the homework NEEDS TO HAVE BETTER INSTRUCTIONS so the parents can understand it.

(4) The issue of whether this is a good way to teach math is a COMPLETELY DIFFERENT ISSUE from the issue of whether the instructions are adequate on the worksheet in question.

(5) The method of teaching math used in this worksheet was probably designed by someone with MORE EXPERTISE in teaching math to first graders than the vast majority of posters, BUT…

(6) …there’s no guarantee that such expertise actually means the method is any better/worse than many other methods in use, and even if the method is good…

(7) …the worksheet was probably not created by the same people who devised the teaching method, and may in fact be quite flawed even if the method it’s attempting to use is sound.

(8) All of this is ridiculous because FIRST GRADERS SHOULDN’T HAVE HOMEWORK in the first place.

If school systems and/or the teachers who work in them want to use such methods to teach math in school, fine by me. Perhaps there’s good research out there that shows that this stuff works better than the way I learned math when I was a kid, and I am certainly not remotely up on the scientific literature in this area. But if you’re going to insist on sending worksheets home to be done as homework, there’s certainly some room for improvement in the implementation.

And, for whatever little it is worth, I personally (a) have a first grader who brings home exactly this type of worksheet, (b) didn’t have more than a few moments of trouble figuring out what was supposed to be done, (c) have a Ph.D. in engineering and teach math-heavy classes to undergraduate and graduate students, and (d) don’t think any of (a)-(c) give me much better insight than anyone else in to what is a good way to teach math to first graders.

226. sugaku says:

Oddly enough, this is how I taught myself to quickly do math on pairs I couldn’t remember well. If I wanted to add 8+6, I’d do 8+(2+4) then (8+2)+4 then (10)+4 then 14.

Of course, this was the “wrong” way to do it, and I needed to “memorize” more…but it’s mathematically sound and *if* it works for you, that’s great. As other posters have said, it’s probably an extension of a lesson, with a matching section in the textbook.

Since this is the newest flavor for teaching that subject, the best thing to do might be to make sure that your kid knows how to actually add numbers for real, and how to “play the game” and do whatever song-and-dance the teacher wants…

227. Anonymous says:

Don’t know but my first grader came home with a 69 on a math test that I would have flunked because the directions made no sense and were written in words he couldnt understand.

And seriously when did first grade start to cover full on high school geometry? My kids think its exciting I can multiply and add numbers well into the thousands in my head, and I fear they won’t ever be able to because this is so confusing they will give up.
Steff

228. Anonymous says:

To those complaining that the work is valuable but that the worksheet construction is flawed has no idea how this type of homework is supposed to work. I am familiar with the math curriculum that this comes from. In it, homework is never about introducing new topics and is always about reinforcement. You can even see this stated in the top right, where it is titled a “Practice” exercise. If utilized properly, this assignment should exactly mirror work done in the classroom already, and the student should be familiar with it. Obviously, that does not guarantee it has happened in this way. But that is how it is SUPPOSED to happen, which is why it is designed as such.

229. Anonymous says:

Got it wrong, I thunk the arrow meant ‘maps to’ so if 8 maps to 10 (+2) then 3 maps to 5 and the answer is 15.

230. Blaatann says:

I agree that teaching kids to view 8+3 as 10 and 1, the one “spilling over” is a good idea, this is exactly how I visualize adding large numbers. But they could have taken the time to line up the numbers, text and graphic. I think the reason I had to check the comments to understand what was going on was that I just couldn’t decipher the meaning due to the wonky text alignment.

231. TomChicago says:

This is one of many different strategies to solve basic facts problems and it’s a very good one. The ability to break apart numbers here will serve kids very well down the road when they add larger numbers (302 + 153 is easy to add mentally when you think of it as 300 + 150 + 2 + 3) or when you estimate.

The number 10 is taught to kids as early as grade K as a benchmark number as our number system is based on tens. So, they first learn in grade K different combinations of numbers that make ten — 8 and 2, 7 and 3, 9 and 1, etc… This provides the basis for this lesson. So, when you have 8 + 4, it’s really 8 + 2 to make 10 and then count 2 more to make 12.

As for the Practice page worksheet and instructions, typically the student’s textbook will have more description of this strategy and the teacher will, of course, go through this strategy in class. By the time they get to the Practice page, they are just practicing a strategy they have been taught. So, take a look at the student’s actual textbook for this page or ask the teacher about the strategy.

If your student has a textbook that uses these powerful strategies of breaking apart numbers to solve problems, you should be very happy. By teaching this ability to work with numbers and not just memorize facts, your child will be able to handle and memorize the basic facts today as well as be prepared for harder problems later on in math.

232. Anonymous says:

Might you have asked to see the lesson for which this was practice?

233. Anonymous says:

Not having read the other 100 posts, this may have already been said.

It’s to build number sense… something greatly lacking from curriculum today. Many of my 4th grade students can’t tell you want 10 + 8 is with 2 second turn around. They seriously count on their fingers. (Sigh.)

234. Anonymous says:

With all due respect to those who like this method (I’m not picking on you TomChicago), this is NOT a good method to use to teach math.

It’s currently in use in the school system where I live and the downside effects are seen on a daily basis especially when the kids move up to the middle school grades.

There are several problems with teaching this kind of math.

One, children don’t learn just one or two methods for solving problems, they end up learning several. While this may seem to be a good thing, the problem becomes apparent when children are unable to solve problems because they haven’t mastered one method, they are so-so at a number.

Second, this kind of math does NOT enforce simple math solutions or the relationships between math operations. Having children who have either been forced to learn these methods or are currently being taught these methods, I have seen a great weakness in the relationship between math operations.

Consider the following problem: A person needs to get 770 apples but only has 625. How many more apples does that person need? The easy way to do this is 770-625 = 145. However, when taught using methods as shown above (since that is one of many problems), kids start to default to:

625+100 = 725
725+40 = 765
765+5 = 5
100+40+5 = 145

Instead of learning subtraction, the student is doing four addition operations and providing 4 times the opportunity to get the wrong answer.

In addition, without full understand of the math operational relationships, there is no facility for checking one’s work. As I’ve watched my kids and their friends do this, they are completely lost at using the opposite math operation to see if they got the right answer. It’s not that they can’t, it’s that they aren’t confident in their use of it.

Third, I understand the arguments of relating the problems to ’10s’ and other arguments presented here, however, the issue becomes that detrimental when the math advances beyond basic math, such as algebra.

Consider the example: there are 3281 ft in a kilometer.
If I have 3.5km to travel, how many feet is that. Most people would simply do 3281 x 3.5 = 11483.5. However, the kids in middle school who have learned the type of math outlined here will do 3218 x 3, then 3281/2, then add them. That’s more room for error and doesn’t work when you get to algebra or makes algebra harder to learn.

In the schools in my town, they have a program in the 5th and 6th grade level which is supposed to ‘connect’ this math (referred to as TERC or Methods) to high school math. Ignoring the question of why we are teaching children math that isn’t applicable in the high schools, they have found that almost all students have to be retaught math principles.

This type of math teaching has some advantages. In our town, the major advantage has been to tutors who have done very well helping our 7th and 8th graders out while they have to be re-taught the proper skills in basic math to succeed higher order math.

1. beameup says:

770-625. Your way: 0-5, can’t do, borrow from the 7, add 10 to the 0, 15-5=5, 6-2=4, 7-6=1, read off the answer left to right: 145. This requires no understanding of place value, and for most people requires paper and pencil.

Students that have learned the filling 10s method can move on to filling 100’s. For this problem, I need 75 more to get to 700, add that to 70 = 145. Very quick, very simple, deepens understanding of place value and number. I think learning the mental math techniques is very valuable.

BUT, like you, I do think it is also useful to learn the traditional algorithms for most students.

235. Yooper says:

Homework in first grade, why? I am quite sure I have read that research as found no benefit to HW this young. Instead for your child, read a book, walk in the woods, play with hand puppets – they’ll learn a lot more.

This practice sheet was surely closely tied to the lesson. I suspect it was meant to be in-class practice. Please don’t complain about the lack of clear insturctions – they are FIRST graders!

For any innovative curriculum, parent-teacher/school communication via PTA meetings and/or parent conference days are critical but very hard to put in practice. But parents and teachers are very busy and stressed these days.

We NEED flexible thinking in math (by students, parents, and teachers)- but that does not preclude ‘know the facts’. Both are important; both take a lot of work.

236. Anonymous says:

I think the goal must be to create a new generation of math workbook writers. The old generation seems to have gone senile.

237. Anonymous says:

is it just me?
i think the problem is very intuitive and good to teach the concept of addition and 10-based numbers

238. Anonymous says:

But – i saw an 8 then a gap and a 3+? then an arrow and a 10 gap ?+? not to mention that the line between numbers usually means fraction. The more you know about maths the more this layout is flawed. The object grids on the side either suggest that the items should be taken out or totally filled in. The Ten thing at the top creates another question. Without doubt the worst cognitive layout I have ever seen. I am a professional designer.

239. Anonymous says:

Granted, i’m a physics student in college, but it only took me a few seconds to realize what they wanted. Unless you jump to some convoluted solution such as using negative numbers, to equate the two sides you simply have to find a partner for 10 to make the same sum.

I think its a good exercise for building mathematical logic in a smart kid. Maybe its not for everyone, but you have a public education tasked with providing a base for future scientists and mathematicians at a level (age) where they’re unable to discriminate. It’s a necessary evil under this system.

240. Anonymous says:

Not to be a pain, but the problems with this type of worksheet are due to the student not listening in class. It also should never be sent home as homework for that very reason. It should be an in class reinforcement of the concepts taught. This is the teacher’s fault for sending it as homework.

241. Anonymous says:

All this worksheet would have needed was the first question to be filled in as an example. As a teacher (albeit not an elementary teacher) I would have made one copy of the page, filled in the first one as an example, then copied a set for the class to take home. Voila.

(PS – The validation words I have to type to submit this are rather serendipitous- “The fiascos”!)

242. Anonymous says:

I know, I know, this topic is well and truly dead. But for some poor sap like me doing a search through Google, it’ll still come up. The idea behind the worksheet is called “bridging to ten”, “bridging through ten”, “make ten” and so on. It’s a basic strategy for mental computation – it may be overkill for the questions on that worksheet, but it provides foundations for later questions like 18+5 or 29+6. The idea is when one number is close to a multiple of ten, then you use part of the second number to get to the multiple of ten, then add on the amount that’s left. The picture is there to demonstrate the idea – get to ten, add on the remaining amount. It works because it is easier to count on from a multiple of ten than any other number. In all likelihood, the procedure would (or should) have been taught in class, with similar diagrams, so the expectation was that the student would know what to do at home. Blame the teacher, blame the worksheet, maybe even blame the child for not knowing what to do but that’s a separate issue to what is trying to be taught.