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n many AoPS classes, students write a detailed solution to one of the homework problems each week. In these writing problems, the explanation is more important than the answer. We expect students to communicate their mathematical reasoning clearly and justify every step in their solution. An AoPS staff member reads each submission carefully and provides the student with detailed feedback.

When I joined the AoPS team in 2015, I was excited to share my love of math and solving challenging problems with talented and motivated students. I continue to be inspired by the outstanding students and rich curriculum, but I also discovered another joy: helping students develop their mathematical writing skills.

When reviewing students’ work, I first reinforce key mathematical concepts and problem-solving strategies discussed in class, and correct any misunderstandings of the class material. Then I look for ways in which a student can articulate an idea or organize their work more effectively. This can also include formatting advice (presentation counts!), and, if necessary, correcting spelling and grammatical errors. One of my goals is to give each student something to consider, even if they already have a strong solution.  As an instructor, I see the writing problem feedback as my opportunity to check in individually with every student on a weekly basis.

I also start every class with a tip of the week, most of which focus on the writing problems. Some of my favorite tips include: using precise mathematical language to express ideas clearly, ending every solution with a clear concluding sentence, and organizing longer solutions into several short paragraphs. I am elated when I see students implement these suggestions, and impressed by their substantial improvement in their mathematical writing from the beginning of the course.

Here’s an example of a writing problem (adapted from Introduction to Algebra A), a mock student solution, and mock feedback.

Problem:

If Anna gives Brendan 15 dollars, they will have the same amount of money. If instead Brendan gives Anna 20 dollars, then Anna will have three times as much money as Brendan. How much money does Brendan currently have (in dollars)?

Mock Student Solution:

From the first sentence, we see that Anna has 30 dollars more than Brendan. So, Anna has 𝑥+30 dollars.

Now we look at the second sentence. If Brendan gives Anna 20 dollars, after the exchange Brendan will have 𝑥−20 dollars and Anna will have (𝑥+30)+20=𝑥+50 dollars. We are told that this is three times as much as Brendan. Thus, we have the equation 3(𝑥−20)=𝑥+50.

3𝑥−60=𝑥+50.

3𝑥−𝑥=60+50.

2𝑥=110.

𝑥=55.

Brendan currently has 55 dollars.

Mock Feedback:

You have the right idea using the information in the problem to set up and solve an equation, and you are correct that Brendan currently has 55 dollars. However, there are a few ways you can strengthen your explanation.

You wrote:

“From the first sentence, we see that Anna has 30 dollars more than Brendan. So, Anna has 𝑥+30 dollars.”

Whenever you introduce a variable it is important to be precise about what the variable represents. You could say “Let 𝑥 be the amount of money Brendan currently has.”

Also, it would be good to explain in more detail how you determined that Anna currently has 30 dollars more than Brendan, since this might not be obvious to the reader. You could elaborate as follows:

“If Anna gives Brendan 15 dollars, she loses 15 dollars and he gains 15 dollars; that’s a 30 dollar difference. We’re told they would have the same amount of money after this exchange, so Anna must currently have 30 dollars more than Brendan. As an expression, this is 𝑥+30 dollars.”

When solving the equation, you did a good job showing your intermediate steps. However, it’s a bit hard to follow what’s going on. You could make your work easier to understand by placing each equation on its own line and explaining in words what you are doing in each step. For example, you might write:

“Thus, we have the equation:

3(𝑥−20)=𝑥+50.

We distribute the left side:

3𝑥−60=𝑥+50.

Next, we isolate the variable:

3𝑥−𝑥=60+50.

Then, we combine like terms:

2𝑥=110.

Finally, we divide by 2:

𝑥=55.”

Nice job organizing your work into several small paragraphs. It is also good that you ended your solution with a clear concluding sentence, as we discussed in the tip of the week.

Keep working hard!

I see the long-term impact of students’ experience doing the writing problems in my work running the USA Mathematical Talent Search (USAMTS). The USAMTS is a free proof-based math contest for advanced middle school and high school students. Whereas most math contests place a premium on speed, the USAMTS gives students over a month to solve each set of fun and challenging math problems. Students are required to write full problem solutions, and their work is scored on both the mathematical accuracy and quality of the explanations. Many students, especially those who have taken AoPS courses, submit well-written, detailed, and beautifully formatted solutions. (Shameless advertising: Year 31 of the USAMTS started in September, and we would love to see you participate! You can learn more and register for the contest at the USAMTS website. Our Round 3 problems will be available on December 5th.)

My motivation for helping students improve their mathematical writing is also personal. I grew up attending New York City public schools, and the class sizes were too large for me to get intensive feedback on my writing; it wasn’t until college that I became a strong writer. I know that some of our students at AoPS might not have other opportunities to get detailed feedback, so I am especially motivated to give these students the kind of writing feedback I wish I had received when growing up.

I hope you are even more excited for your next writing problem!