How can I help an eighth grader learn to write mathematically?

Question

I'm tutoring an eighth grader in math, and while he is particularly bright he is not very good at taking the time to show his work and write things down. He likes to try to solve a problem in his head, write down the answer, and move on. Oftentimes he's correct, but I'm afraid that down the road if he keeps this up he will eventually land in a class that he can't approach this way and won't have anything to fall back on. He just finished algebra.

Specifically, I'd like to help him learn to solve problems on paper, and also to help him learn to explain his solutions clearly. I have two thoughts I'd appreciate feedback on, and if anyone has any suggestions of other directions I could go I would appreciate that as well.

My two ideas at the moment are:

1. Make some worksheets of problems similar to what he saw in algebra, in which he is asked to solve/explain the problems in stages, e.g. 'What are we trying to accomplish?', 'What information is known?', 'Try X solution method and show all work', 'Explain your solution', etc.
2. Introduce him to some basic parity or divisibility proofs, and emphasize being able to explain things clearly and completely.

Thoughts?

Answer 1

Based on your description of the student (particularly bright, not showing his work, often answers correctly) my guess is that he feels the problems he's doing are too easy in general. In addition, he feels that showing he can do things quickly in the head is the best way to show his skill/understanding of the material (I think that way when I was at his age, and I feel like many kids around this age think the same way.).

I'll continue by assuming both your observation and my guess are correct.

Your ideas are fine. However, if you're going to do 1., I'll recommend making the problem complicated enough such that these things actually help/are necessary, rather than forcing him to explain the detail for a simple problem. For example,

$2x+1=5x-11\Rightarrow x=4$

Don't force him to explain adding 11 on both sides, subtracting $2x$ on both sides, and dividing 3 on both sides if he is able to get $x=4$ in his head quickly. He might feel you're insulting his intelligent.

Instead, make something like

$8x+9\{6x+7[5x+4(2x+3)]\}=1700x-63$

where I doubt he can compute everything in his head. In this case, writing down the detail steps is probably valuable if not necessary to get the final answer.

The bottom line is, don't force him to do the work that he feels unnecessary, instead give examples that shows such a work (of writing things in detail) is valuable.

I think 2. is a great idea, and probably more important than 1. if you want to compare.

Answer 2

Good for you for emphasizing this. I would recommend partly role-playing all the future situations in which he'll need to convince a skeptical colleague. How will he explain it to: a fellow student, someone he was tutoring, a confused fellow tutor, a mistaken teacher, a professional colleague, a boss on the job, a student when he becomes the teacher?

Look for exercises in books (usually near the end of a section) that ask for explanations/justifications and work on those (as evidence that it's not something you've made up). Possibly sell it as an extension that only advanced students are expected to perform (not to say that's strictly true).

Start working on proofs like establishing known identities in an algebra/trigonometry course.