Student explaining math

[Sarah0000]

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?

[daijobu]

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 October 30, 2019 by daijobu

[Gil]

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.

[Eatysleepy]

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. 

[daijobu]

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: 

 

 

 

[Sarah0000]

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.