For the Love of Maths

Journey to Interleaved Practice #1

“Learning to pair problem types and procedures is especially challenging in mathematics because different problem types are often superficially similar.” -Doug Rohrer

This semester I decided to create a study on interleaved practice with my second-semester calculus class. By no means is the study empirical in nature – I am not using any controls, and haven’t thought much about confounding variables. The study is more observational in nature, with the goal of collecting student solutions to analyze how students are answering specific questions.

The idea came about through an email discussion with Yana Weinstein of the Learning Scientists (and University of Massachusetts, Lowell), and Doug Rohrer of the University of South Florida. I had been interested in using some of the interleaved practice tools that Yana had helped develop in our Slack team, and she thought it would be nice to touch-base with Doug, as he thinks a lot about how interleaved practice affects students’ learning in mathematics.

There were two specific papers that I remembered reading, this being one of them. I thought the discussion on discrimination rather appealing, and something that I tended to see each semester. Roughly speaking, we teach mathematics in a particular way, scaffolding from one idea to the next, with practice questions always coming from specific chapters. As students practice, they always know the strategy that they need to use in order to solve the question (ie. they think “the questions are at the end of the lesson on the Pythagorean Theorem, so I probably have to use the Pythagorean Theorem to answer the question”). They don’t, however, get much practice mixing the different strategies that they learn. Unfortunately, this means when they come to a summative test, extra effort has to be initially put in to determine what strategy to use to solve a question.

My main goal is to do a bit of observational research around discrimination on summative tests. I have developed a schedule of interleaved homework and interleaved lessons for my integral calculus class, and we are currently off to the races. When I check back in next time, I will share some of the tools I am using to create the interleaved homework.

A Calculus-Themed Amazing Race

“The world is waiting for you. Good luck. Travel safe. Go!” – Amazing Race

On Monday this week, I ran an Amazing Race event on my campus where my calculus students moved from location to location solving review questions and racing against the clock for extra credit.

The hardest part for me was thinking through the original logistics of the event. Luckily with the help of our campus recreation coordinator, Jo Ann, we developed a preliminary schedule with colour groups. The premise was the we would divide students into groups of three, assign a colour group, then have a series of four clues at various locations around campus. At some locations, the clues would be hung on the wall, and at other locations the group would have to interact with someone (eg. our receptionist) to receive their next question.


Schedule of colour groups and locations of clues.

Now that the schedule was created, it was up to me to design questions that gave an appropriate clue letting the team know where the next question was. This was a bit tricky, but after a few hours of hard work, I had a series of questions that gave clues:

  1. Room PE 10, clue: p'(e) = 10
  2. Room PE C14, clue: C = 14
  3. Room PL 117, clue L = 117
  4. Room PC 133, clue C = 133
  5. Room PC 146, clue C = 146
  6. Room PC 108, clue C = 108

So you can see for the most part, the parameter C was involved and students were somehow solving for C to receive their hint as to what room to go to. My favourite was designing a slope question in which when x = e was substituted, the answer was 10. This not only gave the number of the room, but the building too (really proud of that one)!


Students solved a limit question for a parameter L to get their clue to the next location PL 117 (our library).

When the day finally arrived, I got to school early to distribute the envelopes containing the clues in their various locations. Each envelope was colour coordinated, and contained a coloured paper. This way students & staff helping would know which was the correct envelope.


Envelopes hung at the Student Union Office PE C14.


Colour coded paper containing the question, and a marking on the envelope telling students which clue number and what colour.

Overall, I believe the students had quite a bit of fun with the activity (I heard rumor that they were telling other students about how much fun this math class was as they ran around campus). Also, the questions were challenging, but not overly taxing, which allowed a bit of group work along the way (not one student could brute-force through all of the questions). Most teams completed in about 50-60 minutes, and I ended up giving out extra credit to all who participated (tiered so that the “winners” received more). I will absolutely use this activity again – and may put in some extra locations and questions to make it a bit longer. Thank you to all the staff who helped out!


Students working together on the first clue.

The PPT Template I used for locations where I hung envelopes on the door.

My List of Questions for each colour group.

The introductory PPT Template stating ground-rules and prizes for winners.
(of importance here is that I required full-solutions and all group members to be back in order to get their final question)

What I Learnt from #rEDWash

“I’m hopeful. I know there is a lot of ambition in Washington, obviously. But I hope the ambitious realize that they are more likely to succeed with success as opposed to failure.” George W. Bush

Well, it has been officially one week since I took flight to D.C. to attend and speak at researchED Washington. And what a wild ride it was. I returned to class this week on a high – completely abuzz from meeting with people who were willing to listen about what matters in education.

I decided that I wanted to write a post about researchED, but wanted it to be more reflective in nature. So here are a few lessons that I learnt from researchED Washington:

Setting up routines is important.

I thought David Didau hit on a few important ideas in his talk Poor Proxies for Learning. One that stuck out to me was the idea that anything that occupies working memory resources reduces our ability to think, and that we need to think about something in order to learn it. I was wondering how do I lessen or eliminate unnecessary distractions in my own teaching? To me, this feels tied to Tom Bennett’s discussion about the three Rs of classroom management – the first R being routines.

Should I be more mindful of the routines that I am setting up in my college classrooms? How do I feel about technology? I am mindful to tell students that they need to think deeply about something to be able to learn it, and that cell phone use in class often leads to decreased learning capability – however, I don’t generally enforce non-use of cell phones in my classes. Would it be weird to get into a routine of no cellphones in class? This would help eliminate some necessary distractions and allow students to focus more on what matters – thinking about the mathematics we are working on. I need to spend more time on this as I move forward in the future.

Teachers don’t seem to be ready.

Maybe all of us that got together in Washington are those pioneers harvesting good things to come. But I look back, thinking to myself, “Geeze, where are all the teachers?” I like that the event had a mixture of policy-makers and experts in different fields that were interested in the future of education, but it still felt lacking in teacher attendance. Maybe I am setting my hopes too high, and that I should buy into the “Build it and they will come” mentality. Or is it that teachers in their current state, see little utility in an even like researchED? The NCTM regionals in Phoenix and Philadelphia occurred at the same time as researchED Washington, and their teacher attendance and ticket prices were much higher (researchED is a non-profit, thus ticket prices can remain low). Either way, I left the event thinking about what might make the event more useful for teachers. How do we truly make researchED a space in which practitioners can seize control of their own professional development, rather than it becoming an echo-chamber of like-minded individuals?

Good teaching boils down to more than jumping on the latest bandwagon.

One thing I really enjoyed about Cassy Turner’s talk was that she gave some indication of the power of the Singaporean mathematical ideas. The big take-away for me was that the bar-modelling, while a very strong visual in of itself, is not the end goal. The bar model is meant to introduce mathematical ideas in a visual way to allow for pattern recognition – movement to the abstract is always the end-goal. That is, there is a clear destination: building fluency with numbers or with algebra.

A potential problem arises when we adapt a particular soup de jour, without critically thinking about how it is connected to present or future mathematical ideas. Eric Kalenze discussed this through the lens of over-correction: an adaptation to a current problem without thoughtful analysis of what the problem entails. I feel like there are also connections to my talk on teacher training programs needing to contain horizon content knowledge, or knowledge of connections between mathematical ideas, and how one mathematical idea progresses to the next.

One of my highlights last weekend was responding to the question “So what do you make of Dan Meyer?” In my personal journey as a mathematics teacher, I find myself returning back, time and again, to the practices of Dan Meyer. So my response was that there is something interesting happening there – I can’t quite put my finger on what it is that I like, but I think it holds promise. Over the next month or so, I am going to spend some time reading up on some of Dan’s educational interests and trying to formalize some of the aspects of Dan’s teaching practices that I particularly enjoy, and maybe even some of the practices that I don’t. Be on the lookout for that over the holidays – and many thanks to Dan for supplying a few articles to start my journey.

It is intimidating to party with the cool kids.

While I like to pretend that I am as cool as a cucumber – I actually care a fair bit about how others perceive me. On bad days, it can be particularly challenging, as it becomes too easy to get caught up in the exhausting hustle of “How do others perceive me?” Perhaps the stress of preparing and presenting at the conference got to me, but I walked away thinking that I had let my audience down. One of my audience members left before I was able to finish. Was it because we started a bit late? Did that person really need a coffee? These things nag at me, and make me feel a bit disheartened. Maybe I didn’t deserve my spot alongside the other speakers? If I ever get to hang out with the cool kids again, one thing that I can guarantee is that I will make sure I am ready to play.

For those of you interested in the slides from my presentation, you can find them here: bridging-mathematics-mathematics-education.

When is “Doing” Enough?

“Our focus should be on the relationships between conceptual and procedural knowledge.” -Jon R. Star

So I have come to that subtle realization that just because a student can algebraically work through an example, doesn’t necessarily mean that they possess the concepts I would like them to know. That is, I am slowly realizing why so many educators are concerned that students can not only show their work, but also explain their work; and why “conceptual understanding” and “big ideas” have moved to the forefront of mathematics education. This blog is a walk through my thoughts for limits in first-year Calculus and is somewhat based around the ideas that Barry Garelick & Katharine Beals discussed in the Atlantic  last year.

My Experiment and Hypothesis

In my first-year calculus class, I wanted to get a sense of whether or not my students understood conceptually what a limit was. We had discussed them somewhat informally using the definition that “as x gets close to a, f(x) get arbitrarily close to L” and not in an epsilon-delta kind of way. They came up naturally as a way to help us answer the question “How do we define 2^pi?” in which we observed two one-sided limits (over the rational numbers) converging to a decimal number somewhere around 8.82. That is, their first work with limits came as observing one-sided limits in chart form. From here, I connected limits to graphical representations using Desmos and sliders, and came up with a few introductory rules (such as when we are allowed to use the substitution rule). After this, we worked on more of the algebraic rules: factoring, conjugation, vertical & horizontal asymptotes.

By the end of all of this, I was curious as to how they were thinking about and visualizing limits. I came across this paper, borrowed a few of their questions, and created a Desmos lab to try to help me determine this. I hypothesized that my students would be very good at the algebraic portion – that is, given a limit, I figured they would be great at working through this kind of question and getting to a final answer. However, I didn’t think that they would be as keen conceptually – that is, I thought they would have some major misconceptions about how to think about a limit (for instance, they would think that a limit is an approximation). So, let’s dive in and compare how their answers on the first test (algebraic) compared/relate to their answers from the Desmos limit lab activity (conceptual).

Test Results

The test results weren’t as good as I had originally hoped. This may be to a few factors – not enough studying or poor study habits on behalf of the students, testing being a stressful situation for the students, not assigning or covering enough questions on my behalf, the questions being too challenging on my behalf. Overall, the limit question was out of ten marks and had five parts. The mean and median mark was 4.5/10, with the highest mark being 9/10 and lowest being 0.5/10. The first three parts were standard, (a) being a conjugation [2 marks], (b) being an analysis of a horizontal asymptote [3 marks], and (c) involving factoring and analysis of a vertical asymptote [3 marks]. The last two questions were particularly challenging, with students requiring some knowledge of the asymptotes of arctan(x) and ln(x) to be successful [1 mark each]. The first three parts were definitely done better than all five parts combined; however, I did not parse these results.

So, at best, I can say that algebraic simplification of limits was average from a whole-class perspective. With that said, it might be interesting to parse down to individual student responses and analyse these to make some comparisons/contrasts. I have picked out three students that did exceptionally well with algebraic simplification of limits (some responses shown below). Let’s label these students as Student A, Student B and Student C. I am curious if exceptional knowledge of the procedure for solving limits is equivalent to a excellent conceptual knowledge of limits.



Student responses to the limit questions on the test.

Lab Activity Responses

Question: How do you feel about the following statement “A limit is a number past which a function cannot go.”

Student A: Depending on the function, the limit is a number the function approaches at a certain value of x.
Student B: True.
Student C: A function can approach the limit from either side of the limit.

Interpretation: This statement is false. Student A seems to be on the right track – that this statement depends on the function and x-value we are looking at. Interestingly, he/she has drawn what looks like a horizontal asymptote at y = 5. Is the student trying to make a connection based on the wording of the question, or because the limit is being viewed as related to an asymptote? Perhaps they are showing an example of a function that is getting arbitrarily close to y = 5, but does not pass this number? For Student B, it is hard to tell what he/she is thinking, especially without a drawing. They may be thinking about how the limit is not concerned about what is happening at x = a (based on a previous response not shown). Student C has given a statement about one-sided limits, perhaps thinking that if the limit approaches from both sides that the function must necessarily go there. Further questioning would be needed for  all students to really understand their thinking.

Question: How do you feel about the following statement “A limit is a number that the function value gets closer to but never reaches.”

Student A: A limit is a number that the function value gets closer to, but may or may not reach, depending on the function and x value chosen.
stuplot2.pngStudent B: True if a cannot equal x such as with an asymptote.
Student C: True because the function is undefined at the limit.

Interpretation: This statement is true, as the limit is unconcerned about what is happening exactly at x = a. Student A is perhaps equating f(a) with the limit of f(x) as x approaches a, which is a common misconception. Student B thinks the statement is true, provided that f(a) does not exist. However, the statement is true in general, even if f(a) is defined and is equal to the limit of f(x) as x approaches a. Making this disconnect is perhaps the most challenging piece conceptually with regards to limits. Student C perhaps is thinking of a removable discontinuity (hole); however, is falling into a similar thinking compared to Student B.

Question: How do you feel about the following statement A limit is an approximation that we can make as accurate as we wish.”

Student A: A limit may either be an approximation or an exact value, depending on the function and x value chosen.
Student B: False.
Student C: True.

Interpretation: The statement is false. Without any more information, we cannot make any judgement on the answers of Students B or C – they may only be best guesses. As for Student A, there is no separation of the limit from the approximation procedure used to illustrate the limit (often we try to illustrate the limit as the upper/lower bound of a sequence of values). One common misconception is to view the limit as a process of approximation when the limit is actually the upper/lower bound of this sequence of approximation values.

Question: How do you feel about the following statement “A limit can fail to exist at a certain point.”
Student A: True
Student B: True
Student C: True, when there are two points for the same x value

Interpretation: The statement is true. Student A and Student C have insightful responses. I believe Student C is thinking about a piecewise function where the two one-sided limits are not equal (even if the wording is a bit off).

Analysis and Conclusion

Overall, there does not appear to be much correlation with students success algebraically with limits and students conceptual understanding of limits. However, when we parse down to a more individual level, selecting a few students who show great success algebraically, we note that a couple misconceptions began to arise: (1) viewing the limit as an approximation, and (2) believing that the limit is somehow connected to f(a). There may be some confounding factors at play here. I am teaching calculus for the first time and my own explanations may not be as clear as a seasoned teacher. Also, the wording of the lab activity questions were taken directly from the paper above, and could have been difficult for first-year students to comprehend – more translation into easier-to-work-with statements could have bee done.

In conclusion, we have to ask ourselves what is it that we want of our students? Do we want students who are only able to work algebraically through the problems without any sense of the conceptual underpinnings? Do we want students who are conceptually strong, but lack algebraic skills? It is obvious that we want neither of these. So how do we attain students who show both algebraic finesse and conceptual understanding? We have to be respectful that algebraic proficiency and conceptual understanding pave a bi-directional road. One helps build the other, and we should spend ample time in each neighborhood (that’s a calculus joke). In fact, our students may learn better if we are able to combine the best of both approaches, without falling into the trap of promoting one type of knowledge over the other.

Social Identity within the Math Wars

“I cannot tell which side you are on.” -Anon

When looking at the members of an ethnic group other than your own have you ever found yourself thinking “they all look the same to me.” This example illustrates the cognitive bias of out-group homogeneity, or the perception of individuals who do not fit into our identified in-group as less variable than they are. This way of thinking is particularly damaging in the context of the math wars for a couple reasons:

(1) Favorability within the in-group comes at the expense of individuals in the out-group.

One’s in-group may defined around a particular trait or dimension. In order to develop positive ties to the other group members, one must show that they too share the same trait or dimension. One way to accomplish this is to criticize an individual of comparable status within the out-group. The act of criticizing a member of the out-group reinforces one’s connection to the in-group, boosts a positive sense of self-esteem, and displays to other members that ideals are shared.

In the context of the math wars, let’s say we have two groups: a ‘traditional’ group and a ‘constructivist’ group. Whether or not these groups exist is the context of another discussion. For now, I will assume these rough boundaries, as I have certainly felt a divide in individuals over the past few years. As I entered my career in the math-education sphere, I recall (sadly) picking on individuals I deemed to be in the ‘constructivist’ group. At the time, I have to say it felt great. I was defining myself within my in-group at their expense. I remember I even tried calling out Boaler on one occasion (I was just as ambitious then apparently!). These call-outs did boost my favorability within my group. However, I failed to understand that those on the receiving end likely felt stupid, hurt and angry. And the main problem about arguing this way, is that both sides psychologically end up feeling stronger about the original differences, strengthening the divide between the two groups. I would like to take the time to own my previous interactions, and apologize for creating a space that was uncomfortable, unwarranted and unfair.

(2) Group-think. 

It becomes very easy to call-out an individual from the out-group when we perceive him or her to believe an ideal that our in-group disagrees with. As the divide between the groups grows stronger, we begin to define ourselves by believing “I am not like them.” For example, when I discuss students being fluent with math facts, I am often misunderstood to mean rote memorization. Note that ‘constructivists’ labeling me this way promote out-group homogeneity, and define themselves by not being one who believes in rote memorization. Continuous reinforcement within one’s in-group may lead to that moment when you are looking around the room and everyone is blindly nodding while the presenter admonishes the ‘traditional’ group for believing in rote memorization.

The honest truth is that I certainly do not believe that math facts require rote memorization (perhaps early coding of numbers, but this is a topic for another time). By definition, rote memorization means any comprehension of the meaning is removed. When I say fluency with math facts, I mean that comprehension and recall must work hand-in-hand. I do, however, believe that more recall of facts needs to be implemented in early grades – I want high storage and retrieval strength of these facts for later use! The act of labeling me as one who believes in rote memorization removes the possibility of bringing forth a more diverse conversation from a more diverse individual belonging to a more diverse group.

(3) Intergroup contact is prevented.

One possible way to dispel our own bias of out-group homogeneity is to spend time within the out-group (called intergroup contact). By speaking with individuals in the other group, we begin to understand their points of view, and they ours. That is, we begin to see them as a more diverse group than we had originally gave them credit for. Healthy conversation and movement forward is then possible. However, recall that individuals in a group tend to want to boost their self-esteem by looking more favorable within the group. Since these actions come at the expense of the out-group, the divide deepens and a cycle forms which may prevent intergroup contact.

I remember “scheduling issues” at my last place of employment. During my first year, my colleagues were very happy to allow me to volunteer my time in a classroom that we all would have labeled as more ‘traditional’. However, during the next school year when I was spending more time with an individual they deemed as ‘constructivist’ my timetable for the winter semester changed so that I was unable to volunteer any longer. The worst part was that I had developed a rapport with the local teachers, and we were all genuinely excited to be able to work together during the winter semester. I felt as though I had let them down – despite all of this being out of my control. In a similar vein, there were a few times within the past year in which members of the ‘traditional’ group nagged to determine my status within the group:

“I cannot tell which side you are on.”

“If you pursue a PhD, ensure you work on a question that will help our cause.”

These were probably some of the lowest lows I encountered. There is nothing quite like the feeling of your in-group consuming you whole, stripping you of your self-esteem, and leaving you to rot (yes, this is a fairly accurate description of how I felt at the time and sometimes still do). Psychologically I think this created a divide between me and those who identify as ‘traditional’, which caused me to look for social acceptance elsewhere (and also prompted a change in locale).

It has been challenging to build some of the rapport back with individuals whom I deem to be ‘traditional’ since psychologically I worry that all of this will happen again. However, it has been a worthwhile trip since I now feel the intergroup connectivity I have created. It turns out that the more time I spend with those who I had originally labeled ‘constructivist’ the more I see that this group is a largely diverse group. Sure, there are individuals that carry ideals that I disagree with – I don’t think this will change (and I don’t necessarily want it to). But the important lesson that I have learned is not to walk in with labels. Those individuals who would call themselves ‘constructivist’ are as diverse as those who would label themselves ‘traditional’. In fact, if we are able to look past our own out-group homogeneity biases, we may come to realize that our out-groups are more diverse than we think.