I teach at Bard College at Simon’s Rock, where the Writing and Thinking workshop is the first week of each student’s Rocker experience and first year seminars are the extended year-long shared experience of Writing and Thinking. Despite this, mathematics and science courses always felt separate from Writing and Thinking. Indeed, my courses were always writing-lite. That is, I used writing activities as space fillers. I felt for the longest time that math class was a place to do math; but, when writing was framed for me as a means for thinking and learning, I became more purposeful with the writing activities I designed and integrated into my lessons. I plan writing-to-learn activities ahead of time so they make sense in a lesson, while previously I would do them on a whim. In some instances, the lesson itself can only come out in the writing. My goal is not to fill up time but, rather, to create different entry points for students to think more critically about math than they had before. Having some fun is also a bonus!
I’ll speak, more specifically, to my design for a particular Elementary Functions class. I always assign homework, though I don’t always collect it. There is a cycle I follow: prep work, class time, and then review work. I assume that the student should understand about 33% of the material after doing prep work, 66% of the material after class time, and 86% of the material after the review work. A B-level understanding is exactly where I want students who are learning. This assumption is made, though, with the additional assumption that students give each portion the appropriate amount of time they need for themselves as individuals. An average time is specified for students.
- Review (25 min)
- Read+Research+Write Prompt. Read p339. Summarize each section/example in your own words. What would you like to know more about compound interest? Do research on your bank account. What sweet deal do you have?
- Practice problems sec 4.1 # 9, 11, 13, 14, 17, 23, 31, 34, 41, 49, 61, 66; sec 4.2 # 3, 7, 9, 13-18, 33, 37, 41, 51
- Prep (10 min)
- Tell the story of the relationship between bx and logbx.
Figure 1: An example homework demonstrating the review stage for current day’s lesson and the prep stage for next day’s class.
There is some pushback in mathematics against the writing-based teaching practices. This resistance mostly comes from students who already believe themselves to be good at mathematics as it has always been taught. Based on feedback I’ve received, the students loathe doing the extra work when they find success (in terms of grades) doing math the way they’ve always done it. I find this interesting because of the fact that Simon’s Rock students are folded into a Writing and Thinking culture beginning their first week here. The students believe mathematics and writing are separate, like I once did. I hold the ship steady by reframing writing as it once was reframed for me, explaining the benefits of writing to students in a general metacognitive sense. Thus, I challenge these students to relearn how to learn math. In general, students end up enjoying a lot of the activities I’ve created.
I expect students to read and to write and to do math. This is a lot for students who are bogged down by 5 other courses. Many students don’t do homework that is not graded. They obviously have to prioritize graded work, even if graded work doesn’t necessarily benefit their learning. I have to organize class time with multiple entry points so all students can learn in a classroom that attempts to have a flipped structure.
In the example script below in Figure 2, notice that the first prompt connects back to the homework in Figure 1. While I would very much like for students to share what they wrote for homework, I quickly find out that students didn’t do the homework. The sharing usually happens in groups of 2-3 students, but walking around I hear students creating their stories in those moments together. So, I adjust, and ask the groups to share their stories to the whole classroom. One might expect a story about logbx being the inverse function of bx, found by reflecting bx over the y=x line. The first time I implemented this lesson, in Spring 2021, that is exactly what I got. The students did the reading *relief*. But in Fall 2021, I got very different stories XD
Script for Exponential and Logarithmic Functions
- To find the inverse of an exponential function, the logarithmic function
- To use properties of exponential/logarithmic functions to solve equations
- PFW (3 min).
- Share (2 min): Tell the story of the relationship between bx and logbx.
- (5 min) Using only the fact that logbx and bx are inverse functions, attempt to evaluate logb1 and logbb. Where do you get stuck? And why do you think you get stuck there?
- (5 min) Using only the fact that logbx and bx are inverse functions, attempt to rewrite the expression ln(x) + 2ln(x+1) as a single logarithm.
- (10 min) Properties of logarithmic functions with 2log4x – (1/3)log4(x+1).
- (10 min) Go over f(x) + ln(x), the “natural logarithm function”.
- Problem set (remainder of time).
- Reflect (5 min). What still confuses you about logarithmic functions?
Figure 2: An example script, for day 1 of logarithmic functions, one that follows the homework in Figure 1.
I find this to be the beauty of writing-based activities. This is a simple focused free write. I don’t specify the type of story I am looking for (though I am expecting something specific), so students use their imaginations. Normally, in mathematics, when a student hasn’t prepared for class time, they sit quietly, hoping to avoid the instructor’s attention. For students who do prepare but didn’t understand the assignment, they also sit quietly hoping to avoid the instructor’s attention. I cannot tell the difference between those groups of students. This FFW provides a way for me to determine a difference.
One set of students told the story of bx and how it kicked a log (as in a tree log) but injured itself, and so bx ended up on the ground lying next to the log. Another set of students told a very intricate, long story about the friendship between and the break-up of bx and logbx. Another group of students went up to the board and illustrated their story. A final group of students did tell the relationship between these two functions as expected. It was clear which group had at least one student who came to class prepared and understanding the material. I really appreciate the humor and creativity of all the stories though! It provides great fun for all students.
Highlighting the one “true” story, I’m able to segue into other parts of the lesson, which are more mathematical. However, I provide a sufficient amount of thinking time with additional reflective activities. The second prompt listed in the sample lesson (Figure 2) is another focused free write, but it is a metacognitive prompt, guiding students to think more critically about the mathematics but also about their own thinking. Some students might not address the two questions at the end because they think they aren’t stuck, but when sharing answers (and I also share my answer when it comes to answers that do have a correct/wrong response), these students demonstrate they do not fully understand the concepts and then they can learn! without me blatantly saying so.
By time I get to the problem set, students have been exposed, through the writing activities, to material they should have already seen in their reading. For students who did the reading, class time helps to clarify for them what they were confused about and the problem set is a place for synthesis during practice. For students who did not do the reading, the writing activities give students a place to, hopefully, use prior knowledge to build new knowledge in a focused way (for this particular lesson, students should already be comfortable with exponential functions and the idea of inverse functions). The goal of the problem set is to have students work together so that they can teach and learn and teach each other.
Ideally, I would like to end class with a reflection. Unfortunately, I never have time for it, despite having it scheduled in each script. Luckily, there is a review assignment, like in Figure 1, so students have a chance to reflect on what they learn outside of class time.
I do, however, start each class with a private free write. I explain during the very first class that PFWs help clear the mind and increase focus during class time. I try not to limit what students do during PFWs. I make suggestions about all the ways they can use this time: draw, write lyrics to a song, make a list of what they remember from the previous class, create an assignment list for the week, etc. When students know they can write about (or draw about) what is on their minds, they do not complain about it at all. Of course, there are students who sit there and do nothing on their paper. I cannot force them to participate in PFWs, but I hope they are participating in some other mindfulness practice, like meditation, to clear their minds for class time. I hear from students that they appreciate this practice, like in Figure 3.
Figure 3: Student feedback about the writing-based activities.
I’ve really appreciated the structure and flexibility that scripts provide me and my class. All students seem to benefit from the writing-based activities, even with the reduced amount of practice on mathematics problems. It seems that thinking more critically about mathematics prepares them for many types of problems.