For the Love of Maths

Reflections on OAME #1

“Developers who understand the whole stack are going to build better applications.” -Mike Loukides


What a whirldwind of a week so far. Monday – interview for a recurring role at the college in BC. Tuesday – an affirmative nod that I will be back again (and again and again… hopefully) for a few more years with the college. Wednesday – fly to Ontario and commute to Kingston, all while ceremonies for the faculty awards I was a finalist for are occurring.

Thursday – Day #1 OAME.

The Humanness and Non-Linearity of Teaching

Matthew consistently reminds me that learning is non-linear and messy and that the process of learning is interesting. As teachers we need to accept and be aware of the non-linearity of our learners. One thing to remember is that we have all used mathematics so many times that the act of going through the consolidation process is already complete in our minds, so we tend to forget (as humans do with information that is not being used semi-consistently) what this process feels like. Oddly enough, I was reminded of the joys of being a student at dinner tonight with a good friend of mine taking the MMT program through UWaterloo. What an interesting role to be in – one where you are once again the learner. What an excellent way to gain perspective, remind ourselves to be humble and to accept the human element of being a teacher.

How Many Fermi Problems Can One Find in a Calculus Class?

I also began wondering what Fermi problems would look like in a Calculus class? Can one realistically develop a Fermi problem to discuss estimation with derivatives – or is there a certain magnitude component to a Fermi problem that allows it to escape more complex mathematics? I feel like making a Fermi problem related to derivatives would be awkward, but I’m open to suggestions and thoughts.


So interesting and fitting that I attend my next session on transitioning through high school mathematics to college / university mathematics –  as I recently transitioned out of a fairly precarious work environment into a very accepting one. It has been very interesting to see the amount of freedom and flexibility (within certain constraints, of course) that I am able to bring to the table at the college. For example some things that I have tried, that definitely would have been a no-go in my previous position, are: open book assessments, collaborative assessments (groups and pairs), an Amazing Race (calculus-style), take-home assessments, and a final exam consisting of mathematical stations. One day I will get around to blogging about the latter two, but until then I will leave you to ponder about the possibilities.

One thing that caught my attention is how sessions like this often boil down to both sides (secondary teachers and post-secondary teachers) complaining about the lack of content knowledge or skill sets of students. However, recall that students (because they are humans) are naturally going to forget mathematical information because they are not like us and are not using this information on a semi-regular basis. So, to me, the real question that needs to be asked is what methods are we putting into place to help students decrease the amount of forgetting that is happening as they transition from high school to university, and how can both sides contribute?

An Interleaved Approach to Interleaving

What can I say. Jamie and I work probably too well together. But in all honesty, I was extremely happy with what we were able to bring together considering that we were several provinces away from each other (how cool is a cross-country collaboration? It’s pretty cool, not gonna lie.). Many thanks to Doug Rohrer for his insight into interleaving and mathematics, as well as Yana & Fabian from the Learning Scientists for all of their hard work in making interleaving sexy and making a very accessible spreadsheet resource. If you want to know more about my interleaved project at the college, you can read my blog posts here, here and here, view our slides from our presentation, or connect with me via email. I’d be happy to discuss interleaving anytime.

There’s Something about Meyer

What is it about Dan “full-stack” Meyer that hits you at your core? There is definitely something about the way he thinks about the process of teaching and learning that leaves you walking away thinking “Yeah, that makes a lot of sense.” In Kingston, he spoke of lessons that should involve more action words than simply recall and compute. However, it does leave me wanting. I wonder if it is realistic for all knowledge to be consolidated this way? If yes, there is no argument to be had. If no, I wonder if there exists an optimal strategy that involves both the process he explained mixed in with something else?

My intuition tells me that cognitive science plays an important bit here, and it feels connected to Lucy’s book, where we learn that one of the Asian educational systems she visited had an interesting strategy toward mathematics learning. First, the teacher would ensure that students had appropriate background knowledge, typically done through a direct instruction method. After this, the teacher would break students up into groups, each tackling a challenging problem that hey had never seen before, and might contain the topics learned at the beginning of class. Discussion of the problems followed at the end. This feels about right to me: (1) introduce students to the tools they may need, perhaps done in an interleaved fashion, (2) work through a more complicated non-routine problem involving some of the concepts we wish for our students to recall within this process that Dan describes. Anyone looking to co-create a study involving these aspects, hit me up.

Journey to Interleaved Practice #3

“I realized that I had to study some material less because I knew how to tackle these problems. It’s kinda fun.” -Student

In my first post in this series I shared my thoughts on my motivation for the design the observational study, noting that discrimination was a key idea I wanted to explore. In my second post in this series I shared some of the tools and my thought process in designing the structure of the interleaved homework assignments. In my final post on my journey (for the season, anyway), I will share some preliminary results, some student solutions that I found interesting, and my overall thoughts on what I learned.

Preliminary Results

First, here are the overall trends in the assessments from this term.


Each solid line represents one of the 14 students who were involved in the observational study. The dashed black line represents the average progress of the class. A few things should be immediately apparent:

  1. The black line shows a general decline over the semester of about 20% if one observes Quiz #1 first and the Final Exam last. However, if one were to remove the quizzes, one would see a decrease of 15% from Test #1 to Test #2, followed by a slight increase of about 5% from Test #2 to the Final Exam. More discussion on this below.
  2. What the heck happened to that poor blue student? It might be that H found the interleaved structure of the course and homework overwhelming and needed more time for comprehension compared to the other students. Is it possible that students with special considerations benefit more from the structure of a blocked approach? I haven’t read much on this, but please feel free to share some research if you know about it.
  3. Aside from a few students who remained close to the top for the assessments, many students saw a drastic decrease around Test #2. Why is this? Test #2 contained 86 points dedicated to all the various integration techniques (substitution, integration by parts, strategies for trigonometric integrals, trigonometric substitution, partial fractions) and I told my students to do whatever questions they wanted to in order to obtain 50 marks. Perhaps this choice was too much, and a more structured test would have been better-suited.

If there are other items that are particularly noticeable, let me know and I will reflect a bit more on why that might be the case.

I also compared the scores of the 14 students on Test #1, Test #2 and the final exam from differential calculus to integral calculus. Since Test #2 was so varied from the structure of differential calculus, I decided to exclude it here (although there was a 10% decrease). Test #1 saw a change in scores of about 10% and the final exam also showed a slight increase in score of about 2%.test1test2

Student Solutions


A estimates the area under the upper half-circle using a parabola.


B shares how we know the integral is bounded above by 1 using properties of odd functions.


M shows the correct (and challenging) solution to two difficult integrations on Test #1.


What I find odd about P’s solution is that she gets the substitution in part (a), yet tries to use the power of -1 in part (b). This was a common misconception in the course that persisted all semester.


Trigonometric integrals offer interesting flexibility in answers. Here, I would have tried a substitution, but A decided to make everything cosines and use cosine reduction.


Here, a u-substitution would work, but N decides to use trigonometric substitution.


First and foremost, while I did select an interleaved approach due to the hopes that it would make integral calculus a bit easier in the long run by allowing students to discriminate between integral techniques, I also noticed that students’ mindsets changed a bit this semester. In differential calculus, where they might not venture an answer, in integral calculus they would try substitution or integration by parts, even if it led them down a dangerous path. There was a difference in both effort and execution. They persisted and often came up with insightful solutions. It was also true that there was less cramming for tests and the exam. In fact, N came up to me and said “I realized that I had to study some material less because I knew how to tackle these problems. It’s kinda fun.” It would be interesting to follow-up with them over the summer months to see how much of this knowledge they retained.

From my perspective, I know that any fluctuations in grades are highly likely due to random chance factors, and not necessarily due to the interleaved practice. This said, it was an interesting first-go at something this big and I definitely want to try it again. The main difficulties I had were:

  1. Time. It took a lot of time to work through the homework solutions in class. Due to the time I lost, I had to teach differential equations in the lab portion of the course, and lost time discussing some aspects of power series. I’m not sure I would have necessarily changed this, as many students appreciated the extra time spent on solving questions and being able to ask specific questions.
  2.  How do I measure whether or not the interleaved practice actually helped? I’m not sure that I effectively can do this based on the way the study is designed, but here is a thought. When a student tackles a question, either they use the correct technique or they don’t. What if I looked at the proportion of times a correct technique was used on Test #2 and compare it to the proportion of times a correct technique was used on the final exam? Maybe this would be helpful.

Unmasking Logarithms

“By shortening the labours [he has] doubled the life of the astronomer.” -Pierre-Simon Laplace

I had such an interesting conversation in my pre-service math class the other day. We were solving the equation


My goal here was to get them thinking about how they could us the power laws to help them. We worked our way down to


And someone offered the suggestions that the 3^8 and the 3^4 x 3^4 were the same, so all we really needed to determine was

CodeCogsEqn (2).gif

and we eventually settled down on x = 2, since 4^2 = 16. Then I did something weird. I told them to pull out their calculator and evaluate

CodeCogsEqn (3).gif

to which they found the answer to be 2. Now I had them intrigued. How were logarithms connected to this question?!

They knew that logarithms were a pre-calculus operation, but hadn’t made that connection between logarithms and exponentiation. It is likely that logarithms had been taught as a series of rules to follow, without a real explicit connection to how they are the inverse operation of exponentiation – or, more importantly (in my opinion) how they solve one piece of the “triple puzzle.”  You see, the process of exponentiation involves three values: the base (a), the power (p), and the evaluation (b).

CodeCogsEqn (4).gif

We could cover any one of these numbers up, so we have three different but related problems. (1) We could cover up b; this problem can be solved by the process of exponentiation.


(2) We could cover up a; this problem can be solved by applying a radical.


(3) We could cover up p; this problem can be solved by applying a logarithm.


I am not convinced that students get enough time exploring  and developing their sense of logarithms, so I suggest utilising a structure that was brought up by David Butler today (the triangle typically used in science courses to remember arrangements of formulas). I do think in our case, the structure of the triangle works a bit better than it does for science formulas. Here, the triangle works nicely for a^p = b:


What if we fill in two values:




Is it possible that we can create meaning about logarithms by using these diagrams to introduce the three similar, but related, problems? I think so. We know 2^3 = 8, so how might we reason through this?


Well, we know ? must be close to 3 since if ? = 3, we would get 8. We also know ? must be larger than 2 since 2^2 = 4. Hmm… here we might begin to introduce the clunky notation of logarithms. Perhaps log(7)/log(2), or log 2(7). Aha! 2.807 seems reasonable based on what we have thought about. And we can now flush out the problem of non-integer powers.

Anyway, I don’t think we will ever be able to get rid of the unfortunate notational issues with logarithms, but I do think we can do better making the connections back to exponentiation. Maybe there is some space in the progression of learning about exponentiation for triforce notation? As always, I welcome your thoughts.

Journey to Interleaved Practice #2

“Acquire new knowledge whilst thinking over the old.” -Confucius

In my last post, I gave some background to the study that I am undergoing with my calculus students this term. In this post, I want to share some of the tools and methods I used to make the path towards interleaving clearer to me.

I have been a fan of interleaved practice for some time since it is well-known in the scientific community to be a successful strategy for learning (here I am thinking about learning as a flexible and long-term change in long-term memory that can be measured through test performance). However, when thinking about how to successfully implement interleaved practice it feels like a very daunting task and there are a lot more questions compared to answers:

How many questions do I assign at each step?
How do I best mix-up all the questions?
Should some topics be more weighted compared to others?
In what order to I teach the topics? Should I also interleave the way I teach the topics?

So what I did was draw some inspiration from a Slack work-group where Yana‘s husband Fabian (congrats on the recent wedding!) put together an Excel worksheet that gave a potential teaching and quizzing structure using an interleaved approach. If you open this link, you can see space to enter the topics, as well as the number of classes you have, and finally the number of questions you want per quiz. Hitting the “Do Quiz” button will create two lists: one that suggests topics to teach during any particular class, and one that suggests the topics for each quiz (which I assume happens at the beginning or end of each class).

I took this basic structure and decided to create a list of potential topics for my integral calculus course. I divided this list into six “strands” each with a certain number of “lessons” (note: I am not done finalizing this list yet – it is a work in progress). Basically I sat down, went through each chapter of the textbook and made a map of how the topics were interconnected. For example, the Sequences and Series section of the textbook discussed geometric series. Well, I could easily do this in Strand One so that students have an introduction to sigma notation before working with sigma notation with approximations. Then I could circle back to sigma notation later in Strand Six when working with Taylor Series, effectively spacing out our work with sigma notation throughout the semester. Each placement of a lesson within a strand was a calculated choice to try to space out the important ideas as best I could.

Now that I had a list of topics, I could input this information into Fabian’s worksheet and get an idea of how to interleave topics. I decided that I had already interleaved teaching topics as best I could, so I ignored the top output. I chose 4 questions per quiz and focused my attention on the bottom output. Using the output at the bottom as a model, I created a new page that listed the four questions I wanted to include on each quiz. See Sheet 3 of this workbook for that page (again a work in progress).

My final decision was not to use quizzes, but homework assignments instead. That is, at the end of each lesson, I give a PDF handout like this one to each student that is due at the beginning of the next class. This particular PDF came after the lesson on the Fundamental Theorem of Calculus Part II (FTC II). Notice that there are questions about the FTC II, but there are also questions on the topics of geometric series and the definition of the definite integral as well (the limit of the Riemann sum question).

To ensure that students complete each 4-question homework assignment to the best of their ability, I check for completion only at the beginning of class. We then take the questions up as a class – focusing on the “hard” questions that students are having trouble with. So far things have been going very well. The first test is coming up next week, and I will definitely try to blog about any interesting information I gather from looking at their responses.

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.