Uncovering the Perception of Difficulty: Personalized Learning in Pre-Calculus – Part 2

David Buller

David Buller has been teaching high school mathematics and computer science in Madison, CT for 18 years. Prior to teaching, David spent 15 years working in the information technology arena, where he utilized many of the skills and strategies he now teaches. David and his wife are the proud parents of three grown children, and currently reside in CT with their black lab, “Solo.”


In Part 1 of this three-part series, I spoke about coming up with the idea to look at perception of difficulty with my pre-calculus students. This post will dig into what happened after I implemented that idea.

When class started the next day, I really had no idea what to expect, and what I found was immediately interesting.

First, virtually all of my students had followed through on my request, and none of them expressed any displeasure with the added task. Second, I was surprised at the variety of ways students had carried out such a simple task:

  • Some students had circled the number of those problems they found difficult, while leaving the others unmarked.
  • Other students traced over the problem number in red for hard problems, green for easy.
  • Several others chose to use two highlighters to distinguish between the problems.

As I walked around the room checking homework, I would often comment on their novel approach, and occasionally asked if they’d permit me to share their technique with the class (they all did). While I don’t check homework every day, I made a point of checking every day for the next week to reinforce the habit and also to provide feedback on their process.

Every few days, I’d do a quick check-in to ensure they were still following through with the instructions, and on occasion would comment on their recordings.

A New Testing Experience

As our first chapter test approached, one of my students mentioned that she planned to spend extra time studying the concepts associated with the problems she marked as “hard.” Allison and I discussed how we could connect the students’ insights regarding their homework to their assessment, and we arrived at the following:

Students were instructed to mark each question on their test in the same manner as they had marked their homework: easy or hard. When their tests were returned, students used a department Chromebook or their own personal computer to record their results, indicating how they had rated the question, and how they performed on it.

An example of a student’s entries is shown below:

Please complete the table – this will not be shared with other students. Consider an answer correct only if you received more than half of the points possible.
Question Perceived Difficulty (Easy/Hard) Correct (Y/N)
1 Easy Yes
2a Easy Yes
2b Easy Yes
3a Easy Yes
3b Easy Yes
4 Easy Yes
5 Easy Yes
6 Easy No
7 Easy Yes
8 Easy Yes
9 Easy Yes
10a (center) Easy Yes
10b (radius) Easy Yes
11a (x-int) Easy No (copying + multiplicity mistake)
11b (y-int) Easy Yes
11c (symmetry) Easy No (labels in notes wrong)
12 Easy Yes
13 Easy Yes
14 Easy Yes
15 Easy Yes
16 Easy Yes
17 Easy Yes
18 Easy Yes
19b (equation) Easy Yes
19c (slope interpretation) Easy Yes
19d (projected sales) Easy Yes

Reflecting on the Test Experience

For homework that night, students used the same document to reflect on their test experience. As the first reflections started to trickle in, I began to get a sense of just how valuable this simple activity had been.

While some of the questions seemed to provoke more introspection than others, all of the reflections opened up a window into my students’ perceptions about their practices and abilities. It also had the unintended consequence of initiating a conversation between my students and me, a conversation that would typically have only occurred with a very small subset of the class.

I took a few minutes to read each reflection, and to add a few comments and/or questions specific to the student’s responses. In my computer science classes, I often provide written feedback that isn’t dissimilar to what I found myself writing.

But this was the first time in my teaching career that I was reading and commenting on my students’ views of themselves as mathematicians and math learners. Students were so incredibly honest, thoughtful, and introspective in their comments that I couldn’t help but be moved by their responses.

As you might expect, students’ responses varied greatly across all of the questions.

Here are the questions and sample student responses. [These are the questions that were so revealing that I continued to use them in Chapter 2 Test Reflection.]

Reflect overall on where you are right now as a mathematician, and see if you can uncover how you work right now based on the first unit. What behaviors/strategies did you uncover that are working for you? What behaviors/strategies did you uncover that are areas of concern?

  • I like to do my homework after school in Mr. Buller’s room in case I have questions, and I don’t have to take the textbook home with me
  • I actually try to pay attention and understand what we are learning.
  • That I tend to focus on other things instead of math when we are learning
  • I am concerned for my lack of checking work, especially because I know how beneficial it can just looking at the pre-course quiz or from computer science that it is so easy to forget one little thing which makes the solution not work.
  • I think I need to put more effort and time into my studying. I am concerned for my lack of checking work, especially because I know how beneficial it can just looking at the pre-course quiz or from computer science that it is so easy to forget one little thing which makes the solution not work.
  • I need more help with circles.
  • I need to keep practice on testing for symmetry because I did not really understand it until the night before the test and at that point it was more of a memorization than a conceptual learning point.
  • I will also once again revisit the homework problems that I marked as hard the first time and try more of those problems in order to identify my weaknesses.

What is the one single thing you can do next to improve your performance in Chapter 2. Be specific (NOT… I should study more …)?

  • I should try to take better notes and do out the example problems from class so that I can review it to better understand the homework. I should also start coming after class for help if I don’t understand something.
  • I can ask questions on the homework the day after completing it instead of waiting a few days to do so. I also think that I should keep coming in for extra help if I need reassurance on how to do a specific problem.
  • I need to check my work and not rush to go do the next thing whether it’s the next problem or start a different subjects homework because most of my errors are due to not checking my work and rushing.
  • I think I should come into the classroom and ask for help. I did use the videos online which were helpful, however coming into the classroom at least once for extra help would definitely have helped.

Here are a few of the insights that came from the reflection:

  • Most students have at least an informal strategy for taking a test and most work through the test in order. Many pay some attention to the difficulty of a question, often marking the more challenging ones for further review once they’ve completed the test.
  • Some students tended to accurately identify problems they got wrong as difficult, while others justified their mistakes as “dumb” or due to external forces. Several students identified problems as difficult, but in reading their reflections, it was clear that the difficulty was due to a lack of studying the relevant material. That allowed me to suggest possible strategies such as coming in for extra help in the future.
  • Many students identified a particular type of problem as being difficult. I will provide students with additional resources and extra help on the relevant concepts, and will allow students to demonstrate mastery of the concept on subsequent unit tests.
  • Several students felt the homework tended to be easier than the questions asked on the test. I will ensure that future questions are sufficiently challenging and also try to understand if students are being honest with themselves about whether or not they answered a homework question completely correctly. [I read an article ten years ago by a math professor who described this as “irrational confidence,” which was one of the inspirations for why I went down this personalized learning path in the first place.]

Overall, I couldn’t have been more pleased with how my first foray into personalized learning unfolded. For the first time in my career, students revealed their feelings regarding how they approached math, reflected on their strengths and areas needing development, identified habits that they’d like to change and habits that were working, and described obstacles to learning they’d like to overcome.

The reflection document created a conversation where I could validate and/or question their beliefs, offer suggestions, or just “listen” to their concerns.

Part 3: Moving the Chains: Updating What Works and Continuing to Take Things Out for a Test Drive

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