Math Papers Introduction

Writing in Math - How to communicate like a mathematician

Introduction

One thing that I think about a lot is the distinction between math in high school and math in the world. Generally students ask ‘when am I ever going to use this?’ and my reply is often ‘something exactly like this? Never, this is a problem bred in captivity. It’s a domesticated math problem, to be nice and obvious. The ones in the wild are more complex.’ I tell my students that we learn math formally so that they can use math informally. 

However, one thing that we do in learning math that I think is a disservice, is that we allow the solving process to be the totality of the mathematics, rather than how it is often utilized in real life - which is the explanation. To this end, I started working on ‘math papers’ with my students. I would like to share the barebones idea below to invite feedback and improvements to the formula. 

1st - The Problem

For the math, at least to start, begin softly. I simply pick a word or application problem from whichever textbook my school is using. For example, here is a problem adapted from McGraw-Hill’s Algebra 1 Textbook:

‘A rectangle has a width of 4 in and a length of x+7 in. It has the same area as a triangle which has a base of 6 in and a height of 4x in. What is the value of x?’

The problem does a couple of things that we are looking for. First, It gives context for the math problem which is critical to help students get ready to solve problems in real life. Second, it does not give the equations, but rather asks the students to create the equations they will use to solve.  This is one of the major skills we are working on. At this point, students should be proficient with the basic math the problem is asking - in this instance, solving linear equations with variables on both sides. Instead, what we are asking for is the explanation. THAT is our focus. 

2nd - The Ask

The rubric of the paper needs to focus on three major parts

Part 1  -  Introduction

Students should start the paper by stating the problem as it is written, and then interpreting it for the reader. I always tell them to imagine they are helping a younger sibling understand what’s happening. Speak to them like they almost understand - but don’t quite. Explain what the problem is asking and then write out their equation, explaining clearly why the equation is written the way that it is. Make sure the students write in full sentences and explain each part of their equation. In the example above, they should have 4(x+7) and explain that this is because of how you calculate the area of a rectangle.

Part 2 - The ‘Math’

Next, have students write out their work meticulously. If you are seeing what I am in a classroom, there has been a lack of focus on solving work being written clearly and completely. I like having the students write out every step. Also, for the paper, I ask that they write a description of what is happening in each line. This is (sneakily or not sneakily) a mathematical proof. In an Algebra class, we can think of it as preparing them for Geometry. Otherwise, it’s just good math practice to be able to justify each mathematical step you take. If you can’t - you probably did something incorrect (or, worse, got right by luck).  Make sure students give an explanation for each step. 

Part 3 - The Conclusion

Now that students have their answer, they should take their answer and put it back into context. Make sure they write a small paragraph explaining the answer and, critically, why it makes sense in the context of the problem. This is a good time to have them plug their answer back in to check that it works and show their reader what the answer they got means. 

3rd - Follow-up

It’s important that the mathematical conversation doesn’t end when the paper is submitted. Remember that math is a dialogue, a way of communicating, and it’s important that we treat it as such. Generally, for the first paper you are not going to get what you hope to from your students. This is because they have not ever had to communicate in this way. We need to discuss it and improve upon it.

One way that I do this is to take math papers, black out the names, and have students grade each other’s work in groups utilizing the rubric. We can discuss each part, how effective were the explanations? What could we improve? 

Generally, it’s better to pick some good examples, so that no student feels called out by their work, but sometimes I will write my own papers to use as ‘bad’ examples so that they can see what ineffective communication looks like. The grading process needs to be open, transparent, and done as a full class. It’s best to have students work in groups and then have a full class conversation afterwards to make sure the class is all on the same page and, critically, aligning with your explanation.

This process generally does two things: It helps students see that your expectations are not way out of their reach, and it helps them see what they can do to improve their own work. 

4th - Next Steps

  For the next paper, there will be added complexity simply because the math material will get more complex. However, at the same time, it’s worth trying to increase the complexity of the explanation. Consider a few of the following options:

• The answer needs to be interpreted further. An example of this is a problem like a trig triangle problem where what you are solving for is not directly the answer, but can then be used to find the answer

• The problem requires two answers that are compared. Consider the price of crushed gravel vs. pavement for the area of a driveway

• The problem requires students to research something, such as the price of grass seed, the gas mileage of a car, etc.

I will likely spend further posts discussing these options and providing examples that could be used to help differentiate for students.

5th - Choices

• Sometimes I provide a number of different prompts, often differentiated by level, some may require more complex problem solving, some more complex discussion about the meaning of the answer, and some that might require research. The benefit of differentiation is that you can engage all students. The goal is for everyone to find it somewhat difficult, but not impossibly so. 

• You could do the grading rubric class discussion before having students engage in writing their first papers, you would just need examples from a previous year or, to start, example papers you write yourself for the students to grade.

Conclusion

The hope is to push students towards authentic math communication by having them engage in the entirety of the math writing process including synthesizing the problem, explaining their mathematical process, and interpreting the solution in the context of what they were trying to solve. By having students take time and really focus on this process they can build a more intuitive understanding of what it means to utilize mathematics, it gives them a starting point for math conversation rather than the do-it-and-forget-it approach that often comes with word problems in class. I’ve found students truly engage in this, and it’s something that I highly recommend!

That said - and of course - I am always looking for feedback on my ideas! Let me know if you have ways to improve the process or if you try it and it works!

Thank you for reading. Be Excellent!