Welcome back, my fellow adventurers! Last post, we explored warm-up routines that spark curiosity and community in the first 30 minutes of my supported calculus class. In this blog post, we will examine what happens during the rest of a typical class day by thin slicing through a lesson inspired by the research of Peter Lilejdahl and his book Building Thinking Classrooms (BTC).

Lilejdahl has conducted extensive studies of math classrooms, interviews with students, and professional development with math teachers. From this work, he has developed clearly defined principles for executing an active learning pedagogy that is taking the math education world by storm. (He was the keynote at AMATYC 2022.) These principles are designed to be easy-ish to implement and to intentionally disrupt traditional classroom norms that may undermine student engagement, persistence in solving problems, and understanding. I have found that BTC works well in my supported calculus classroom where students have varying levels of precalculus proficiency and confidence in their math abilities.

**Overview of the key components of the BTC approach**

There are 14 BTC principles, which are summarized in these fantastic sketch notes by Laura Wheeler. If you want to read more, but not an entire book, this Edutopia article also has a succinct summary: Building a Thinking Classroom in Math

If I had to narrow this list down to two, the big game changers in my supported calculus class are Visibly Randomized Groups (VRG) and Vertical Non-Permanent Surfaces (VNPS)– Principles 2 and 3. If you are new to BTC, I recommend starting with these. I’ll discuss these two principles next.

**Visibly Randomized Groups (VRG)**

Liljedahl’s classroom observations and his interviews with students suggest that the random assignment of students to groups fosters a more inclusive classroom environment where all students are more likely to participate actively and engage deeply with the material. Random assignment also breaks down social barriers, as students cannot choose partners based on ability, social connections, or other biases.

In my experience, separating students into groups gives them a support system and allows me to check in with most students over the course of a lesson. By randomly assigning students to groups, they can’t fall into old routines – such as taking a backseat or relying on a friend to carry to the load. I’ve found that since using randomized groups, students are more likely to seek help from each other than come to me with questions, which gives me more time to intentionally check in with students who are struggling.

**Tip:** I use a deck of cards for my random assignment after removing a single suit to match my class cap of 36 students. This is quick and easy to do, and the excluded suit can be used to designate where in the room the groups will work.

**Vertical Non-Permanent Surfaces (VNPS)**

When students stand and work in groups on whiteboards or other vertical non-permanent surfaces, Liljedahl’s research found that engagement and mathematical thinking increased. The time that it took students to start a problem decreased and time they spent working on a challenging problem increased. In addition, all students in the group were more likely to participate in meaningful ways. This is exactly what has happened in my supported calculus class. What is it about this simple change that produces such dramatic effects on student engagement and learning? Here are some thoughts on that:

- Visibility: Work on vertical surfaces is easily visible to the instructor and other groups. This facilitates timely feedback and promotes a culture of shared learning. I can now see exactly which students need that intervention by glancing around the room! In an hour-long activity. I can ask extension questions to those students who are getting the material and provide hints to the students who need them.
- Mobility: Students can move around. This increases their engagement and energy levels. There is also the mobility of ideas and thinking – students now have more support than just the support system of their randomly assigned group because they can look around the room and engage with the work from other groups to gain ideas for how to approach a problem.
- Rough Draft Thinking: The non-permanent surface encourages students to try solutions without fear of making mistakes. We’ve all seen students not start a problem on their paper because they are worried about making a mistake, but they are much more likely to write down an idea on a whiteboard when it can be easily erased.
- Accelerated learning and individualized support: Most faculty would agree that students need to practice in order to learn math – it’s why we assign homework. Before I incorporated active learning, my students weren’t able to get feedback until at least 48 hours after they submitted work in the form of a classwork assignment, homework, or quiz. Even with computer-generated homework sets, the feedback is thin – students may know immediately if they got an answer right or wrong, but they are not getting feedback on their thinking. Are they mimicking procedures or reasoning through the problem? What did they understand? Where are they getting stuck and why? I can answer these questions now about each of my students because I stand in the middle of my class and watch them actively solving problems at the whiteboard for at least an hour every class period.

**Tip:** If you don’t have surfaces around your room that could be used for the BTC approach, WipeBook makes temporary whiteboards that I’ve used painters tape to adhere to my walls. I’m now on my second semester using them and they are still in great shape!

**A Sample Lesson**

This lesson on analytical limits occurred on day 5 of my supported calculus class. My students spend at least an hour each class doing what I refer to as “board work” in my class summaries. As you read through this lesson, remember that my supported calculus class is open access; however, almost every student has passed a course at the level of intermediate algebra though few have had a precalculus course.

My learning goal was to introduce students to three algebraic techniques for evaluating limits at a removable discontinuity. Previously, the class had worked with polynomial and rational functions and their graphs, including using limit notation to describe end behavior and asymptotic behavior.

I began with a short lecture in which I presented a single example:

In the context of this example, I began with evaluating the function at x=3 and noting that the result of 0/0 was undefined, which means that this strategy produced an inconclusive result. I then factored both quadratic expressions and reduced by the common factor to produce a limit that could now be evaluated. I made conceptual connections to our previous work by graphing this function and noting that there was a vertical asymptote at x = -2 but not at x = 3, even though the function was not defined at either point. This strategy helped us identify a removable discontinuity in the original function.

(In hindsight, I think it would have been helpful to discuss why

Conceptually what is happening is that the algebraic transformation produces a nearly identical, but not equivalent, function with the same local behavior at x = 3. Since this new function is continuous at x=3, we can determine the limit by evaluating the function at that point.)

You may have noticed that “mini lecture” is not the first step in the BTC lesson cycle, so why did I do it in this example lesson? I feel that starting with a short demonstration is reassuring to students (and me) because it is familiar and allows us to get focused at the start of the lesson, but I don’t always do them. I skip the mini-lecture if I feel that our previous work has set the stage for productive thinking about today’s problems or that students will naturally try things that will be productive or there are multiple solution strategies.

After the mini lecture, here is the blow-by-blow of what happened next organized by some of the BTC principles:

**Begin with a problem**

I created a series of problems on a slide deck and projected the first problem:

The entire problem sequence is shown below. It includes both limits at points of continuity and removable discontinuities, and well as one limit associated with end behavior. Problems are ordered with increasing levels of challenge and intentionally require students to grapple with foundational precalculus concepts, such as factoring, factoring out a -1, adding fractions and simplifying complex fractions, and multiplying by a conjugate. Note that the problems differ in form from the problem I presented and I did not front-load precalculus skill practice, with the exception of the short mini lecture where I factored.

**Randomly assign students to groups of three**

I use a deck of cards and remove one suit. I distribute the cards and project a classroom map that shows where the 1’s meet, where the 2’s meet, etc. Groups are randomly assigned at least once during each class, sometimes more.

**Use vertical non-permanent surfaces**

**Students stand and work in groups on the same problem on vertical whiteboards, one marker per group**. The student with the marker does the writing. This encourages discussion, particularly when the writer needs assistance from other group members. The marker is passed around the group on each subsequent problem. Most groups produced work similar to that shown below for the first problem.

**Keep students thinking**

**To keep students thinking, only answer “keep thinking” questions; give judicious and timely hints; encourage groups to interact with other groups frequently.** With this pedagogy, the goal is to keep students thinking. Lilijdahl’s adage is, “problem solving is what we do when we don’t know what to do.” If students ask questions like, “is this right?”, answer with another question that prompts more thinking, such as “how could you check your answer?”

As shown above, most groups dropped the limit as they worked through the algebra. Lieljdahl cautions teachers about correcting form too early in the problem-solving process. Thinking is often non-linear, messy and non-routine and students’ white board work should reflect this if they are grappling. Get the mathematical ideas down, then work on the formal notation. A “keep thinking” question to focus them on this issue of the limit might be, “What does the 7/2 tell us about this function?” “Can you write that in words and in mathematical notation?”

Per BTC principles, I also encourage students to look or walk around the room to see what other groups are doing, and I might strategically direct them to a group that has a different answer or approach or presentation. This often results in groups correcting their work or helping others to do so.

If I see a recurring or interesting error, I will at some point discuss this with the class in my “You killed a puppy!” routine. After an algebra error makes the puppy death list, I am known to yell across the room during whiteboard time, “8’s, you killed a puppy!” This gets everyone scrambling to recognize the deathly error.

I also have an embedded tutor, Allison, who is a helpful, nurturing presence. With each question, she moves to a new group that struggled on the previous question and acts as an additional group member. She also holds supplemental instruction sessions after class that are increasingly well attended. You will hear more about her role in the next blog post which Allison and I are co-authoring.

**And repeat**!

** I projected the sequence of problems one at a time without additional demonstration. **Each problem surfaced a new issue. For example, the second problem in this set involves a continuous function, so the limit can be evaluated by substitution. This generated some initial confusion as some groups looked for an algebraic manipulation that could be performed and others realized that the limit could be simply calculated. Some groups used their calculators and got an approximate answer; other groups wrote their answer in exact form. Students naturally gravitated to other groups with different answers and returned to their group with new ideas. Knowledge visibly spread around the room. My sense was that groups were not just copying from other groups. They were talking, explaining, questioning– in short, growing their understanding. Most students were able to answer my questions about their work as I moved about the room, even if they had borrowed an idea from another group. The third problem required factoring, but for the fourth problem factoring was a waste of time. This kept groups actively thinking and engaged.

The next set of problems involved complex rational functions. With the first problem, there was a long pause as students considered how to proceed. Some groups tried direct substitution and got 0/0 and asked me what to do next. I answered with a question, “What does this tell us?” “What makes this problem hard? Can you think of a way to simplify it?” As soon as a couple of groups had come up with the idea of finding a common denominator and rewriting the numerator, I give that as a hint to the rest of the class: “Oh! I like what the jacks and fives are doing with creating common denominators!” Frankly, that problem was probably more challenging than necessary because not only was it a new problem type, but it requires students to factor out a -1. Some groups did this successfully. Because of the complexity of this problem, I decided to debrief after a few groups finished this problem.

**Debrief some problems as a class**

Debrief as a class when every group has reached a minimum threshold, reference students’ whiteboard work to build solution strategies and investigate common missteps. For this debrief, I went to one group’s solution where finding a common denominator was the strategy and had students from that group explain their work. I asked questions, “why is rewriting the numerator as a fraction instead of a sum a good move?” or added clarifications, “So now you are dividing because every fraction can be viewed as a division problem.” Another group had great work showing how they factored out a -1 in order to reduce common factors, so I moved to their board next to feature their work. I emphasized that any time we saw (a-b)/(b-a) we could factor out a negative one and reduce (as this group had done) or just recognize this expression as -1. Last, I returned to the front of the room to extend the discussion by showing an alternative method: multiplying by x/x.

**Back to work with some intentional hints**

After this example, I asked the groups to work through the second problem in this set using BOTH methods, passing the marker. All groups realized that the simplification required multiplying by 5x/5x. For the next two problems in that set, there was some trial and error on identifying a useful multiplier. Allison and I intervened with a few groups where a puppy was in danger, but groups remained engaged and made progress. On the last problem in this set, some groups used direct substitution and others went the arduous route of simplifying the complex fraction, but every group came to the realization that direct substitution was always a good first step.

We then moved onto the limit problems that required rationalizing the numerator or denominator. Again, there was a pause as students considered how to proceed. Since I couldn’t be sure all of my students (or even one per group of 3) had seen rationalizing a denominator/numerator, and in the first 3 minutes no group came up with the idea, I provided a hint. For a handful of students, I did a quick example on the magic of conjugates. The technique spread quickly and all groups were able to get through the first two problems. Allison and I worked with groups who were struggling on the remainder of the problems, and there was noticeable cross-pollination happening between groups. I also saw some students from groups who finished early walk around and help other groups!

**Note-Taking**

I don’t do the true BTC consolidation-style notes, where students have autonomy to create their own representation of the day’s important ideas. Instead, I tend to consolidate the “check your understanding” and the note-taking parts of the BTC pedagogy. After the white board work, I guide the class as they fill out a set of skeletal notes that give some information and have space for us to work through the different types of problems we encountered.

During this phase of the lesson, I had an interesting experience that gave me pause. I asked students to “try this problem” in their skeletal notes, and I noticed that many students were stalling or faking it. They were waiting for me to work the problem on the board. It may have been a lack of space on the skeletal notes I provided – a fear that they wouldn’t have space to add my solution after their attempt. (I’m making a mental note to edit those notes to have a “your attempt” v. “class attempt” section in the future.) But it was a shocking juxtaposition with the work they had been doing 30 minutes prior at the whiteboards.

I later realized that the behaviors I observed mirrored Liljedahl’s observations of what students do in the traditional “I do, we do, you do” pedagogy. I had unintentionally recreated that pedagogy–and the associated disengagement, that I was trying to transition away from! This is an area of the BTC pedagogy that I am still troubleshooting. Students tend to believe that they need to leave class with a perfect set of worked problems, but I am still grappling with whether this in some way short changes the work I am doing to “build thinking” and my students’ self-efficacy.

For more tips on how to execute the different phases of the BTC pedagogy, check out these incredible sketch notes by A Klassen.

If you’re ready to take the first step towards a more engaging classroom with BTC, curious to know more, or have experience with BTC, let’s collaborate! Join me during community hours on Thursdays from 12-1pm in a Coreq Calc Discussions Zoom Room every week until May 23rd.

Hope to see you on Zoom soon!

Kelly