# Student explaining math

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Would you expect a second grader to be able to verbally explain why they did what they did on a math problem beyond reciting the steps they took? Or to answer an open ended question such as "But do you understand what you're doing with equivalent fractions?"

I know my son knows what he's doing because I can hear and see what he's doing while he's actually doing it, but sometimes our supervising teacher will ask what I think are too open ended questions for a seven year old. But maybe I'm expecting too little?

For example, yesterday I watched my son do his math for a bit. He was calculating 4^3 as part of a larger problem, and was talking through it to himself. He goes "4×4 is 16 and 16x4 is...hmmm...ummm....8x8 is easier so...64." I did not teach him to do that and as far as I know Beast Academy didn't either. When I asked him to explain why he did that he could only say it was easier to calculate 8x8 than 16x4. When I prompted him "So you divided one term by 2 and then..." he said "I multiplied the other by 2." But he could not explain why other than to say to keep everything equal/balanced and the calculation was easier that way.

There is, of course, no written work of this and if I were to ask him how or what he did days later he would probably simply recite the steps at best or just say something really general like "because that's what it equals."

So, does this sound totally fine at this age/stage or should we start working more on explaining the whys and wherefores of math? When does it become more of a critical issue?

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I think it's fine at this age.  Being able to communicate through math is a good skill to have, but not necessary at this age.

With my math olympiad students (4th-8th grade), I will scribe at the whiteboard as they describe their solutions.  If they forget some vocabulary, I will suggest it for them.  I think helps with their public speaking and helps reinforce vocabulary.

So if they are trying to calculate 4^3, I'll ask them how they did it.  First I write

4^3 =  on the white board.  If the student says "4x4 = 16" then I will write

4^3 = 4 x 4 x 4 = 16 x 4 =

and then he says, "and 16 x 4 is...hmmm...8x8 is easier so ... 64"

4^3 = 4 x 4 x 4 = 16 x 4 = 8 x 2 x 4 = 8 x 8 = 64  and I'll make a point of saying "So you rewrote 16 as 8x2?  And now we replace 2x4 with 8?"

ostensibly so the other students will understand how he solved it.

I often underline numbers that are being replaced.

I think it's good practice, but I don't expect my students to be perfect at it, especially orally.  It really helps to see it on paper, because the student can say, "Replace the 16 with 8x2" and it makes sense.

Edited by daijobu
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I do think that's reasonable, but I taught The Boys verbiage and terminology in mathematics and coached them to "speak Math correctly" so  yes, I did expect and require that they be able to explain their solutions. I didn't expect that they do it spontaneously or through osmosis.

Talking through solutions and "math speak" was something that we did focused practice of right along with writing numerals correctly and math facts.
Right around 7 is when they started really having to be able to justify and explain their work.
At 7, they should begin learning how to display their full solution for a problem.

One thing that really helped The Boys get used to it, is talking through/teaching from/explaining an example that I had written out.
Then writing out a full solution to an exercise that paralled almost exactly the one that I'd done and talking through their solution after they were done.
That evolved into explaining their full solution as they went.
Once they were comfortable with a solution format and type, I would intentionally make an error and have them find and explain what it should've been. Or cue them to listen in case I make a mistake and when I mis-explained something, they could correct it.

We have an "in house" style that I used very consistently so the notation and color scheme wasn't random. The way that a problem is worked is systematic. By watching me systematically work problems out, and teach/justify/explain the steps, by taking a few moments to focus on terminology and concepts outside of calculation exercises each day they learned to do the same thing. By now, they do it automatically and easily, but it wasn't something that they did spontaneously.

I think that 7 is old enough to learn to speak on and explain some process and concept that you are familiar with.
I would expect to be modeling and scaffolding in the beginning and I wouldn't expect them to be able to explain something that they aren't familiar with but for a skill/concept that they're confident with and understand? Yes.

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I do expect a second grader to explain his solution.  I would be looking for terms such as "I regrouped...." or a drawing to show the regrouping done in that step.

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I think it depends on what you mean by "expect".

I think that what you describe is really really typical for kids who are advanced in math.  Explaining your work is a hard skill.  So if you're asking if you should be alarmed, or concerned, then the answer is no.

On the other hand, if you're asking whether you should be asking him to explain his thinking regularly, and teaching him the vocabulary to do so, then yes, I do think so.  I think this would be a useful skill for him to develop, and one that a seven year old who is clearly bright should be able to start developing.

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So long as there's also reasonable expectation of some struggle and difficulty in being able to do it. Like CuriousMomof3 said, it's a difficult skill, but it's also one that should be cultivated and developed early on.

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I wouldn't hammer at it so much at the elementary level, simply because often these problems are rather self-evident and don't require a lot of explanation.  A "complete solution" often is in the eye of the beholder.

For example, the AMC solutions proffered by the Mathematical Association of America, are often terse to the point of inscrutability.

Here's a typical MAA solution:

And here's the solution to the same problem from AoPS:

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On 10/29/2019 at 9:30 PM, Sarah0000 said:

Would you expect a second grader to be able to verbally explain why they did what they did on a math problem beyond reciting the steps they took? Or to answer an open ended question such as "But do you understand what you're doing with equivalent fractions?"

I know my son knows what he's doing because I can hear and see what he's doing while he's actually doing it, but sometimes our supervising teacher will ask what I think are too open ended questions for a seven year old. But maybe I'm expecting too little?

For example, yesterday I watched my son do his math for a bit. He was calculating 4^3 as part of a larger problem, and was talking through it to himself. He goes "4×4 is 16 and 16x4 is...hmmm...ummm....8x8 is easier so...64." I did not teach him to do that and as far as I know Beast Academy didn't either. When I asked him to explain why he did that he could only say it was easier to calculate 8x8 than 16x4. When I prompted him "So you divided one term by 2 and then..." he said "I multiplied the other by 2." But he could not explain why other than to say to keep everything equal/balanced and the calculation was easier that way.

There is, of course, no written work of this and if I were to ask him how or what he did days later he would probably simply recite the steps at best or just say something really general like "because that's what it equals."

So, does this sound totally fine at this age/stage or should we start working more on explaining the whys and wherefores of math? When does it become more of a critical issue?

Yes, I absolutely expect explanations, and no, I wouldn't expect the explanation to involve multiplying one term by 2 and dividing the other one by 2. I think a good explanation at this age is "We had 16 fours, and I grouped them into pairs of fours, so then I had eight pairs (since 8*2 = 16), and each pair contained 8, so I did 8*8." I'd prompt an explanation along these lines, because I do think hooking up mathematical logic to verbal logic is valuable to most kids. But I wouldn't expect use of the associative property. I think doing this kind of shortcut sets him up for associative property later on :-).

Edited by square_25
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On 10/30/2019 at 12:02 AM, daijobu said:

I think it's fine at this age.  Being able to communicate through math is a good skill to have, but not necessary at this age.

With my math olympiad students (4th-8th grade), I will scribe at the whiteboard as they describe their solutions.  If they forget some vocabulary, I will suggest it for them.  I think helps with their public speaking and helps reinforce vocabulary.

So if they are trying to calculate 4^3, I'll ask them how they did it.  First I write

4^3 =  on the white board.  If the student says "4x4 = 16" then I will write

4^3 = 4 x 4 x 4 = 16 x 4 =

and then he says, "and 16 x 4 is...hmmm...8x8 is easier so ... 64"

4^3 = 4 x 4 x 4 = 16 x 4 = 8 x 2 x 4 = 8 x 8 = 64  and I'll make a point of saying "So you rewrote 16 as 8x2?  And now we replace 2x4 with 8?"

ostensibly so the other students will understand how he solved it.

I often underline numbers that are being replaced.

I think it's good practice, but I don't expect my students to be perfect at it, especially orally.  It really helps to see it on paper, because the student can say, "Replace the 16 with 8x2" and it makes sense.

Honestly, I prefer the kind of mental regrouping that kids do to using the associative property. Mental rearranging makes sense to kids. However, products of 3 numbers are basically not well-defined until you actually talk about the associative property, and when you calculate something like 8*2*4 you're taking it on faith that it's the same thing whether you do (8*2)*4 or 8*(2*4). Having spent some time talking about the associative property with my AoPS kids, it's not particularly well understood. Lots of kids think it comes from commutativity, for example.

So I guess I like the idea of working on this kind of regrouping as pure logic and as the kind of experience that'll later set you up for the associative property :-). I think it gives kids ownership.

Edited by square_25
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10 hours ago, square_25 said:

Honestly, I prefer the kind of mental regrouping that kids do to using the associative property. Mental rearranging makes sense to kids. However, products of 3 numbers are basically not well-defined until you actually talk about the associative property, and when you calculate something like 8*2*4 you're taking it on faith that it's the same thing whether you do (8*2)*4 or 8*(2*4). Having spent some time talking about the associative property with my AoPS kids, it's not particularly well understood. Lots of kids think it comes from commutativity, for example.

So I guess I like the idea of working on this kind of regrouping as pure logic and as the kind of experience that'll later set you up for the associative property :-). I think it gives kids ownership.

4th grade here is when the students are expected to use the property names.  For less advanced students, a 'friendlier' name may be given.

I think its reasonable; in science they are also learning series vs parallel in electrical circuits. So plenty of visualization and regrouping go on, and its helpful to have the correct terminology even if some are working on the spelling.

The OP asks when it becomes a critical issue...I'd say grade 4...the amount of terminology is going to increase dramatically in grade 4, so developing the expectation of a solution rather than an answer really needs to be done in the earlier years.  The concept of proof does start out simply in preschool with "how do you figure?" and develops further each year.

Edited by HeighHo
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17 minutes ago, HeighHo said:

4th grade here is when the students are expected to use the property names.  For less advanced students, a 'friendlier' name may be given.

I think its reasonable; in science they are also learning series vs parallel in elecrical circuits. So plenty of visualization and regrouping go on, and its helpful to have the correct terminology even if some are working on the spelling.

The OP asks when it becomes a critical issue...I'd say grade 4...the amount of terminology is going to increase dramatically in grade 4, so developing the expectation of a solution rather than an answer really needs to be done in the earlier years.  The concept of proof does start out simply in preschool with "how do you figure?" and develops further each year.

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I've been paying more attention the past few days and I feel like he's doing some arithmetic gymnastics sometimes without even thinking about it and that's when he seems to have a hard time explaining what he's doing. Or very basic concepts are so obvious it's hard to explain why it's so. He does ok explaining things he has to think about and find a way to solve it. I'm going to start having him practice explaining more when we're doing read aloud math. Thanks everyone.