kitchen table math, the sequel: Making tens for subtraction, PM standards ed.

Sunday, March 30, 2014

Making tens for subtraction, PM standards ed.

As per post below, here are some small examples of how frequent and explicit the making ten strategies are in addition and subtraction in Primary Math, standards ed. This not not meant to be enough to explain these methods, just to show what an actual Teacher's Guide says.

This is a subtraction lesson, the hardest, where you can't just decrement the ones by the subtrahend.


GoogleMaster said...

This wording is awful: "Guide students to say: 12 minus 5 is 1 ten minus 5 ones and 2 ones."

Recall that first graders have not yet studied order of operations.

When I read that to myself, I hear "and" binding tighter than "minus". Any of the above would have been clearer: "1 ten minus 5 ones, and 2 more ones" with a comma and the word "more", "1 ten minus 5 ones plus 2 ones" without punctuation, "1 ten minus 5 ones, plus 2 ones" with a comma, or even better, "2 ones and 1 ten minus 5 ones", which is actually what is illustrated by the number bond thing.

Cassandra Turner said...

Yep, English can be different in Singapore. No teachers I know stand in front of the class and read the teacher's manual.

allison said...

Google master:

10 -5 +2 is correct mathematically. The kids are not confused by this because there is no reason for them to interpret this any way other than what is in front of them.

You have learned to hear non existent parentheses. They haven't. Taught this way from the beginning, there isn't any confusion.

Anonymous said...

"Taught this way from the beginning, there isn't any confusion."

As long as they never venture outside a classroom where people talk this way.

If they ever learn to think clearly, they'll recognize this as ambiguous and ask what it means.


Allison said...

I'm sorry--what are you talking about? It's math. Math is unambiguous.

10 - 5 + 2 is unambiguous. It has exactly one correct answer.

English isn't math. This is why we need to teach children that math is precise, using precision from the beginning. We use symbols for this reason,rather than relying on english statements in math. They have to learn to follow the rules precisely to get the right answer, and they will.

Children successfully learn all the time that different domains have different rules. You can run around the playground but not the dining room; you can bark like a dog outside but not in class; you can speak slang with friends but not grandparents; you can roll your eyes in some places but not at your mother (or a cop).

That math is specific is not the problem. That other domains have different rules is not a problem either. That English is not math is not a problem.

Thinking 10 - 2 + 5 is ambiguous is a problem. It isn't, and it shouldn't be taught as if it is.

Anonymous said...

"10 - 5 + 2 is unambiguous. It has exactly one correct answer"

True enough.

But "ten minus five ones and two ones" isn't unambiguous.

It could be 10 - (5 + 2).

That was the point of GoogleMaster's comment.


Anonymous said...

"10 minus five ones plus two ones" feels like a math expression to me.

"10 minus five ones AND two ones" feels like an English expression that has to be translated into math and it does have that ambiguity.

If you are going to say it in English, why not say "Take five ones away from 10 and then add two ones"?

But to be fair, this was a little box in a guide. Think of it as a telegraphic reminder of an approach that is spelled out in greater length and clarity elsewhere (I hope).


Allison said...

The problem here is might simply be one of someone walking into a 50 minute lecture 45 mins in and complaining it doesn't make sense.

But I think it is something else deeper.

The complaint is that the words are ambiguous, and the math is learned from the words, therefore, the math is ambiguous.

But this is wrong from beginning to end: we do NOT learn math from words.
And worse, children in US schools whose teachers and books try to teach math from words are incapable of addressing the necessary precision to teach math accurately.

Math is learned from the math. The pedagogy for math in Singapore math is defined:
they go from Concrete to Pictorial to Abstract.
note the lack of a "verbal" or "written" stage.

The children have spent months at this point learning how to add and subtract, making parts and wholes with concrete things. They have physically performed the subtracting of 5 from 12. There is absolutely no ambiguity when doing this that you might somehow be supposed to take away two more than 5 when you take away 5.

Then, after succeeding concretely, you move on to pictorially: again, performing the algorithm by subtracting 5 from 12 the same way (breaking the ten into a part of size 5, having 5 remain, and your original two.)

Then you move onto the abstraction of the equation.

The words used are to convey what has already been done and taught countless times. They aren't teaching the math; the meaning in the words IS INFORMED BY the knowledge of the operation.

This could have been seen by reading the notes carefully, I think. But it requires looking at it differently than through the kens most of us were taught with.

Crimson Wife said...

I actually dislike the way that Singapore teaches this type of subtraction problem because I find it confusing. I much prefer the way Right Start Math teaches it. Right Start would have the student first decompose 5 into 2 + 3. Then the student would do 12 - 2 = 10, followed by 10 - 3 = 7. Much more straightforward IMHO.

Allison said...

Judging a curriculum by how much we adults like it is not judging for mathematical accuracy or coherence.

CW,that method you use isn't how the manipulatives work.The Right Start method isn't building up from concrete the same way.

Here, you decompose the ten because those are the facts learned to automatic recall, and there is only one fact needed to be remembered, the number bond for the ten.

Your way you need to remember what you have taken away already, and what is left after you take that away, which you can't count up, because it isn't there. This is compensation, and it is taught eventually, but it is a much more abstract and difficult concept for students to gradsp.

I get adults find this unnatural, but that isn't a reason to claim the book shouldn't do it. The pedagogy is coherent in Primary Math. All of these alternatives are less so, because they violate the C-P-A.

Jen said...

"CW,that method you use isn't how the manipulatives work.The Right Start method isn't building up from concrete the same way."

What? The manipulatives only work one way? That doesn't make any sense! It's math, of course the manipulatives could show either way!

I don't think that anyone is arguing that this way "doesn't work" but are instead saying that there are other ways which work just as well for children (not for adults, I agree that we can't use our understanding to understand children's thought processes.)

You can certainly and easily use manipulatives to do it the CW way. And in CW's model, a subtraction problem doesn't involve taking away twice and then adding: 12-2 and 10-5 and then remembering to add back that 2 you "took away" first -- but not adding back the 5 you "took away." That sort of switching IS something that can be confusing to kids.

I think you are confusing a curriculum you feel works well (and may work well) with a deeper understanding that can't be gained by another means. In reality, this curriculum may work because it teaches a series of steps in great detail, over and over again, not because of the way they approach tens.

It's the practice part that is likely the effective part. Often it takes a long time of doing things by learned steps before comprehension dawns. If they hear it in class for two years and have tutors who teach it the same way, they are hearing those steps a lot of times!

Crimson Wife said...

Right Start does start with using manipulatives. The student has previously spent a lot of time working with what Singapore calls "number bonds" and what RS calls "part-whole circles". So the student would know 5 = 2 + 3 as an automatic fact by this stage.

The student would first build 12 on the abacus by having 10 beads on one strand and 2 beads on the strand next to it. The student first subtracts the 2 to get 10. Then he/she subtracts 3 more out of the 10 to arrive at the final answer of 7. Easy-peasy and no need to subtract then add back like the convoluted Singapore method.

After the student has mastered the procedure using the abacus, then he/she switches to mentally visualizing the abacus.

Jen said...

CW -- I realize I didn't make it clear that I was quoting Allison, having forgotten to add "Allison said:" before that quote!

She seemed to be saying that because of manipulatives, the other method was clearly the better one.

My point was yours, that *any* manipulatives for early math will emphasize making tens, and adding on to tens.

Allison said...

Again, that method Right Start uses isn't how *the manipulatives USED IN THIS LESSON* work. The Right Start method isn't building up from concrete the same way.

I didn't say RS doesn't use manipulatives; I said doesn't use *the* manipulatives we were using in *this exact lesson*.

RS has an abacus as you explain. They show 12 as 12 beads on two strands, as you explain.

So it is really EASY to do the decomposition because the "ten" is ALREADY TEN ONES. A total of 12 ones on two strands.

This is *different than* a ten rod which cannot be broken immediately. Instead of exchanging, though, the algorithm IN THIS LESSON just subtracts from the ten.

It's all well and good to say "RS's method is easier", but it isn't using the same manipulatives here so it is solving the problem a different way--not easier, because it sin;t relating to a ten rod, which is the purpose of THIS lesson; that way works with an abacus, but doesn't work with a ten rod. PM still wants a method that works from the ten rod because we need to build up place value in small steps--and the base 10 block counts different than the unary counting of an abacus.

These may all seem like pointless trivialities, but they aren't. No, not all manipulatives teach making tens. (cuisenaire rods don't.) Not all ways to make ten teach place value--the abacus is still a unary system, not a base 10 system. The base 10 blocks aren't a full place value system yet but are moving along a line of abstraction up to that.

So given a certain kind of manipulatives, yes, I am saying the step to the procedure is defined differently. Use a different manipulative, and the steps get expressed differently.

Why does all of this matter? Let me give an imperfect analogy. Some kids just figure out how to read. They crack the code. But not all kids do. And different kids have trouble in the tiniest of things the readers may take for granted, and may not even see all of the steps required to get from text to knowledge.

Some kids break down at proper phoneme production, some can't distinguish some phonemes from others, some get some phonemes but can't blend, others have limited working memory for remembering the words after they've decoded, etc.
Detailed instruction in reading and literacy has helped teachers to teach these specific skills using explicit methods, certain scripts, etc. Not all methods teach the vital component skill a given learner needs. Saying "this other way is easier" misses the point that some students can't "skip over" a given method and make progress.

Science hasn't gotten as far in knowledge of how math is learned as they have in literacy. But we do know by watching classrooms of kids that some kids do not make the connections others do--e.g. they do not successfully understand the steps of written algorithms without specific concrete manipulatives. Some can't infer the math is the same on a grouping of ten ones to a ten rod without explicit practice. The tiniest leap from concrete to abstract may be too much, so specifically engineering the manipulatives to teach a given lesson one way and use others for another way matters. Not for all kids, but for some, and in a classroom setting, then it is worth explicitly using the tools and lessons to get these details right.

Lsquared said...

I'm reading Allison's and CW's battling algorithms with interest. I've tried teaching both of those ways and several more both to my students (college pre-teachers) and students I volunteer with (grade 2 at the time). I quite like the SM way, and though I've often used CW's way when doing mental arithmetic, I've gotten frustrated with the college students who prefer it (they generally seem to have a harder time with the relationship between addition and subtraction. My current favorite way is counting or adding up: 12-5 would be done: 5+5=10 and 2 more is 12, so the difference between 5 and 12 is 5+2=7 (the counting up to version of this, where you count on your fingers from 5 up to 12, is recommended in Ronit Bird's book on Dyscalculia).

One thing I notice is I think that CW may be misrepresenting (misunderstanding?) SM's strategy. In the SM strategy, 12 is seen as 10+2, and the subtraction strategy is to either take away from the 10 or take away from the 2--whichever is easier (or possible). To use the strategy you need to be thinking of 12 as 10 and 2, so you're not doing a mental step of 12-2=10 to solve the problem.

So far as I can tell, students who have a reasonable facility with numbers in terms of splitting them into parts and joining them back together can learn and use either strategy effectively and easily. I can also share that students who don't have that facility can't learn either strategy so far as I can tell--it's a fairly rare problem, but I have encountered some students who just can't seem to hold number relationships in their heads in any reliable way: if you don't know whether 8=5+2 or 5+3, it's going to trip you up no matter what.

The one thing that is a great advantage to both strategies is that they are both part of a curriculum that builds in a coherent way across grades and uses manipulatives in a way that fits the processes and algorithms they are teaching. That is, so far as I can tell, incredibly rare. Most curricula I see bounce back and forth between lots of different processes and materials, and there isn't enough attention to detail. I had 2 children go through a Montessori elementary sequence, and am really impressed with the consistency both across grades and tasks and consistency with how well the manipulatives and processes build on each other. Singapore Math and Right Start both have a lot of those advantages. I think Singapore math has an edge, but it's not that this specific process is better than the other, it's that it has more resources developed for it and it is designed and tested to feed into an algebra/geometry program in the next grades in a coherent way.

Allison--thanks for sharing those teacher's guide pages. I really like the attention to detail that shows how the process and the teaching fits together. Very nice stuff.

Allison said...

LSquared, to this specific point:

"I have encountered some students who just can't seem to hold number relationships in their heads in any reliable way: if you don't know whether 8=5+2 or 5+3, it's going to trip you up no matter what. '

Every time I've encountered a child in grade 1 or 2 like this, I have gone back to watching them count, and their one-to-one correspondence is a massive problem (can't say if it is the problem, or just a correlated variable to whatever the problem is.)

Their act of counting is usually deeply compromised: they cannot reliably count 6 things, EVEN when I start by saying "There are 6 cubes. Count them." They have all known the sequence of whole numbers, but their verbal cadence for numbers doesn't match their touching, or they can't reliably tell which object they've touched/counted already.

I find I can make some headway by re-teaching how to count:

The student must touch one cube, and move it, definitively separating it from the others, while saying the number during the motion. (Grab block. Start sliding block. Say '1' while moving. End word and motion. Let go. Repeat.)

Ime, until counting 6 yields 6 99 times out of 100, it doesn't seem to work that number bonds are meaningful.

I agree that the strength of Singapore's program is the careful building up of these manipulatives and processes to perfectly match each other, and keep moving along these algebraic methods. But part of the strength is in writing it down for the teacher to learn.

By now, every US textbook has stolen number bonds, yet to no avail. They use them for expressing the four equations, but not for mental math, or they use them for mental math, but haven't preferenced them by how their mznipulatives actually work. Fundamentally, the teacher's guides don't explain how the manipulatives are tightly bound to the methods. They just sorta throw them all into the mix, making incoherent arithmetic yet again.