# What Makes A Strong Mathematical Explanation?

I recently had the pleasure of working with some Algebra 2 students while teaching a public re-engagement lesson. Students had worked on the following problem and we were focused on the quality of their written explanations.

We first got reacquainted with the problem by having students work through it again on their own. We talked about the difference between an answer between 5 and 6 weeks and an answer between 12 and 13 weeks. We accepted both. Can you guess why?

After building consensus about the correct answer(s) we talked about the strengths and potential revisions for three different pieces of student writing. Note: These work samples came from this particular class.

Explain how you figured out which week the total views first exceeds 1,000,000.

• Student A: 10(2.5)^13 = 1490116.119
• Student B: I kept multiplying the previous week’s number of views by 2.5 until I got to over 1 million views.
• Student C: I plugged numbers into the equation until I found that 13 equaled 1,490,116 and the term before that is not 1000000.

Here is what students came up with:

When we analyze student explanations at Math ANEX, we look for three particular components:

• Mathematics (M): Did the student describe their mathematical thinking and/or the mathematical model that they applied?
• Connection to Context (C): Did the student include units and important information connecting their calculations back to the situation/story?
• Logic (L): Is their explanation logically clear and complete? Did they answer the question that was posed?

As you can see from the student’s table in the picture above, they didn’t call out M, C, and L specifically, but they were certainly tracking on each.

• M: Correct growth factor, correct equation, guess and check (clear and correct), provide an equation (feedback)
• C: 13 weeks, term before was not equal to 1 million (clear and correct), how you found your values (feedback)
• L: Explain their thinking (clear and correct), how you found your values (feedback)

By eliciting these responses from students (what is clear and correct and what feedback would you give to each student) I was able to build upon the things that they found valuable and then provide more formal names for each component (Math, Context, Logic).

Lastly, students were given their original written explanations back to revise. Check out some of the before and after responses we collected:

What growth do you see in the students’ explanations? How could we continue to build on this growth?

What strategies are you using to support your students in their mathematical writing? In what ways have you found success in using student work to inform your instruction?