Wednesday, October 4, 2017

Teaching Students to Write Mathematical Reasoning

My Context

I teach algebra 1 for freshman. With respect to reading/writing, my classes are heterogeneous. Some students read/write at grade level, while others are significantly below grade level--some as low as elementary school level reading/writing. Thus, my lessons on teaching how to write mathematical reasoning have a lot of scaffolding built in. 

Also, this is all new for me this year! I'm sharing my baby-step lessons on teaching how to write mathematical reasoning. I don't think what I'm sharing is the ultimate way to teach writing mathematical reasoning--but there have been a lot of questions about student work and activities I've posted on Twitter so I wanted to share what I've done so far--imperfections and all! 

Update 11/6/17

I recently gave the second unit test (linearity) in algebra. This was a chance for me to see what students can produce in terms of writing mathematical reasoning with no peer or teacher support. The results were all over the place, which is expected (student work is below). I have a wide range of students with a wide range of reading/writing readiness. Some students are reading/writing at a 4th grade level, while others are at grade level. However, I saw some growth in almost every student's work.

I'm also going to use the student work to do a test re-engagement activity. I still firmly believe that students need opportunities to revise their writing in order how to learn to write mathematical reasoning.  So we will do some type of activity to give students a chance to revise some of this work in order to get a feel of how to improve their writing.

Big Ideas on Teaching Students to Write Mathematical Reasoning 

There are a few big ideas I keep in mind when I'm designing lessons or activities to teach students how to write mathematical reasoning.

  • I've made it crystal clear to the students (and myself) that writing mathematical reasoning is not a goal they will accomplish this week or this month. This is something we will work on TOGETHER all year. So we all need to be patient.
  • I need to build space for revisions into my lessons. In English class, students don't write a perfect essay in one shot. They draft, revise, rewrite. I need to honor that process in my lessons for teaching how to write mathematical reasoning. 
  • I need to write an ideal response to a prompt BEFORE I teach/plan the lesson. I need to write the ideal response because it helps me see the patterns in language I need to teach and it helps me inspect what I expect students to produce in terms of writing. 
  • For the initial lessons, it was helpful that the math was done for them (or very accessible math with multiple access points) and they just needed to interpret the math and write about it. Having to do math AND write mathematical reasoning in one activity doubles the cognitive load. If I want to focus on writing, I need to lower the math demand and give space for students to focus on writing mathematical reasoning.
  • Claim, Evidence, Reasoning (CER) is a powerful structure for argumentative writing that students are familiar with (science and ELA use CER). I'm not sure I want to teach CER explicitly, but I use CER as a rough framework when I write my ideal response and I use CER to help me backward plan my lessons on writing mathematical reasoning. I also use CER as a scaffold. 
  • Teaching how to write mathematical reasoning is hard for me and hard for the students. I need to have a growth mindset for myself and for my students. The students have been told for a few years to "write in math" but I not sure anyone has taught them HOW to write in math. Similarly, I've been hearing for years I need to teach students how to write in math, but no one has taught me how to teach writing mathematical reasoning. Yes, this is hard!

Sample Lessons

Lesson 1 (week 1)

Lesson Goal: Students will understand the drafting, feedback and making revisions are an important part of writing mathematical reasoning. The math part of this lesson was 30 minutes. The writing part was about 40 minutes (I broke up the writing over two days).

I used this fantastic, open-ended math task from Jo Boaler's Week of Inspirational Math for the math portion of this lesson. I used this task because the math was open-ended and had multiple access points. Again, the focus was the writing more than the math.

In planning this lesson, the other algebra teachers and I wanted to scaffold the task to help students differentiate between the patterns they used to see how the shape grows and how they used those patterns to predict what case 20 would look like.

I also wanted to acknowledge that students might not have been taught how to write mathematical reasoning in previous math classes. Thus, we provided "reasoning" sentence starters and a bank of transition words.

For the lesson, after students had time to work on the math individually and with a partner, I gave students about 10 minutes to write their reasoning. As students worked, I circulated to look for student work that had examples of what I was hoping to see based on the ideal response I had written.

After about 10 minutes, I stopped all students and showed a few examples of exemplar student work (these students had transition words, or did a good job in explaining HOW they used the patterns to predict case 20, rather than just explaining what case 20 looked like.

Then, I gave students a few more minutes to write. The idea of showing the exemplar work was to help students see examples of what they could do if they hadn't been able to get started yet.

Next, I had students stop writing. As a class we discussed how to give warm and cool feedback.

Slide from my lesson on how to give warm and cool feedback.
After we discussed how to give warm and cool feedback, I had students trade papers with a partner and give their partner feedback on mini-post-its. Students then traded with another partner and did more warm and cool feedback. This way, students recieved feedback from two other students.

Students got their own work back, read the feedback, and then made revisions to their writing USING ANOTHER COLOR PEN. I love using a different color because I can see their original work versus the edits they made.


Lesson 2 (week 4)

Lesson Goal: Students will write mathematical reasoning to interpret the result of math work. 30 minutes.

Slide from this lesson. 

We are in the solving unit at this point. The previous lesson had been about solving equations with infinite and no solutions. The goal of this lesson was to interpret the result of math work that had already been done.

For this lesson, the math was done so that students could focus on the writing.

Again, I let students just write for about 5 minutes. I emphasized that their first attempt was not a finished product and we would come back to edit their work.

As students wrote, I looked for student work that had elements of the ideal responses that I wrote. For this lesson, I thought about the response as having a CER structure.  My ideal response: 

Claim: the claim should state their interpretation of the result: "this equation has no/infinite/one solution(s)"

Evidence: the evidence should reference the mathematical work. Something like "because when I solved, the variable disappeared and I was left with 5 = 5, which is true."

Reasoning: The reasoning should synthesize the claim and the evidence. "Therefore, I can substitute any value for x and my equation will balance."

After I showed exemplar student work, I let students go back and revise their original writing--in another color. 

I didn't explicitly teach this lesson using CER because I didn't want to get cookie-cutter responses. However, I explained that their writing should follow a rough CER structure. I don't know how I feel about that. Like I said, all of this is a work in progress for me. 

Sadly, I didn't save any student work from this lesson. 

Lesson 3 (week 6, Day 1) 

Lesson Goal: Students will review for the test. Students will explain their math thinking. 40 minutes. 

This was the Middle Bits activity (thanks Shira Helft for sharing this great idea!).  The concept of MIddle Bits is that the student is given a question AND the answer and they need to fill in the middle part (the middle bits--haha). 

How I run Middle Bits:

1. I intro the rubric. I let students know that the expectation is they work together (each partner using a different color pen) and they support each other. I made very clear that since I am not able to see or hear all the great conversations they are having, I want them to put a star on the paper at any point where they stop to discuss a point of confusion or mistake. After they get started, I usually circulate to look for students that are putting stars and show that work on the doc cam so they see how important that is to is also on the rubric. Each partner only gets one paper.

2. I hand out a rubric to partners. We also go over it as a class.

3. Partners need to switch off the work. So if I do the "math" for one problem, you do the writing. Then, we switch for the next problem and you do the math and I do the writing. 

4. I usually stop the class 3 or 4 minutes in to do a shout out to partners that have exemplar body language. They are leaned over the paper together, writing at the same time, no one has arms crossed or is daydreaming while the other works. I have all freshman, so I need to make very clear what I expect!

Lesson 4 (week 6, Day 2) (this is a repeat of Lesson 2 from above)

Lesson Goal: Students will write mathematical reasoning to interpret the result of math work. 15 minutes this time since it was a repeat. 

This lesson went well the first time and it was a good chance to do a quick writing activity in class again that had space for revisions--so why not do it again!

Unit Test Work

This was the summative assessment for the Solving Unit. Granted, writing mathematical reasoning wasn't the primary objective in this unit, but we did spend a lot of class time on writing lessons and practice. Overall, I was very happy as I graded the unit tests! Even students that had the wrong mathematical solutions had solid writing. I was especially happy to see the consistent use of transition words...something I hadn't seen in my algebra 1 students' work before. So I felt like this was evidence of my growth as a teacher.

I think this work also highlights the point that writing mathematical reasoning is difficult for students because there are multiple academic demands happening simultaneously: students have to know how to do the math, as well as how to clearly expresses their thinking using academic language.

This also raises the question, if the math is wrong, but the writing is solid, how do I grade that? Or, if the math is great, but the writing is lacking, how do I grade that? What am I assessing here? And how do I grade that?

More questions than answers so far, but I'm happy with the progress I've made. I'll keep you updated as I continue to work on how to teach students to write mathematical reasoning.

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