For the Love of Maths

Reflections from Teaching: When Are Learners Novices No More?

“Novice learners may benefit most from well-guided low-paced instructional procedures, while more knowledgeable learners may benefit more from minimally guided forms of instruction.” -Slava Kalyuga

The Example that Led to Reflection

I never cease to be amazed at the level of knowledge that my teachers keep bringing to the table in my class. Last week we were discussing probability trees, and one student was leading the activity with the following tree (probabilities of drawing a yellow, green or black ball without replacement):Untitled

After the student was finished answering a couple of questions we had about the tree, I posed the challenge “Create a question where the final answer is 2/5.” I asked this question because I wanted them to get more comfortable with conditional probability. For example, the probability that we will draw a black ball, given that the first ball is yellow is 2/5, so P(B|Y) = 2/5.

Much to my surprise, the first answer given was “Determine the probability of drawing a black or green ball, given that the first ball drawn was black.” I had to sit back and try to figure out where this answer was coming from since I had not anticipated it (this is both the joy and challenge of allowing students to lead the discussion)!

Since the events of drawing a black ball and drawing a green ball are mutually exclusive, we can calculate

P(B or G | B) = P(B|B) + P(G|B) = 1/5 + 1/5 = 2/5.

Can you determine the branches used to create this question? After doing some of what Michael Jacobs calls “Maths C.S.I.” I had successfully determined how the student was thinking.

Are All of Our Students Really Novices?

Over the weekend I began pondering about how there is a lot of talk that mathematics students need to be treated like novices, especially in elementary school. For example, in Anna Stokke’s C.D. Howe Report, she states

To be effective, instructional techniques must cater to the limitations of a person’s working memory, which can hold only a limited amount of new information. This is particularly important for novice learners who have difficulty focusing on new concepts when their working memory is overwhelmed.

I don’t necessarily disagree with the statement above – one which is taken from Kirschner, Sweller & Clark, and heavily founded in Cognitive Load Theory – it is important for us as teachers to understand when learners may have limitations, and how to effectively combat these limitations. I do, however, think it is important for us to also reflect on how often we treat our students as novice learners, and realize their potential as non-novice learners. Those who argue in favour of CLT often view their learners as novices, effectively by-passing the expert-reversal effect. Stated briefly, the expert-reversal effect states methods that typically work well to elicit learning in novice learners are not necessarily the best methods to elicit learning in non-novice learners. For example, as one progresses in their knowledge of mathematics, worked examples become less conducive to learning.

In lieu of this thought, I pose some questions:

1) Are all of our students actually novice learners? Is it possible that our students are sometimes non-novices?

2) If we agree that at least some of our students are non-novices, what methods should we utilize to elicit learning in these individuals? Must it still be direct instruction and worked examples?

3) If we believe that our students are novice learners, will we ever see them as non-novice learners? Does this belief we hold affect their learning?

Priming the Mathematical Mind

“The merit of painting lies in the exactness of reproduction. Painting is a science and all sciences are based on mathematics.” -Da Vinci

Take a moment to read the phrase: “The hungry caterpillar ate the juicy leaf.”

Now quickly complete the word by filling in a missing letter: SO_P.

Out of curiosity, did you complete the word using the letter U to make SOUP? According to Kahneman, author of Thinking Fast and Slow, after processing the words HUNGRY and ATE in a sentence, we are primed to select the letter U in the word above since SOUP is associated with the words HUNGRY and ATE. Let’s explore this a little bit, and see if and how we might think about using this idea in our math classrooms.

What is the Priming Effect?

An idea in our memory is associated with many other ideas. These associations may be categorical, such as connecting the words FRUIT and APPLE, or property-based, such as connecting ADDITION or MULTIPLICATION to COMMUTATIVITY. Ideas may also be associated through effects like how we may connect ALCOHOL to DRUNK, or CIGARETTE to CANCER. When primed with one of the links in an association, our mind has the ability to bring the other familiar and associated words into our working memory.

What Does Priming Look Like?

When priming occurs it is subconscious and Kahneman argues that we are likely not to believe it is occurring due to the way our brain functions (our brain allows us to believe that we are in full control). He mentions several studies in his book, but I will touch on only two to give you a sense of how priming is at work. In the first, participants were primed with images of money. The group that was primed with money images became more individualistic – less likely to help others and less likely to ask for help – on tasks that followed.

In the second group, it was shown that actions can also be primed. In this study, children read sentences involving words associated with the elderly such as  FORGETFUL, BALD, GRAY, and WRINKLE. None of the sentences explicitly mentioned mentioned the elderly. When the participants were asked to walk down a hallway, they did so at a much slower pace than normal. The reverse association was true as well: children who were asked to walk slowly for a period of time were more apt to recognize words associated with old age.

Can We Use Priming in Mathematics Class?

I wonder if mathematics teachers have been using this idea already? In most classes and assessments, we tend to be explicit with word choice when we are asking students to perform a task. For example, if I want my students to think in a linear way, I could use an associated word like SLOPE or a similar word like STRAIGHT to help them recall ideas around linear functions. Use of certain cues to aid in recall are most likely beneficial since we know that recall of facts helps with both storage and retrieval strength. I could also see the argument of priming allowing students to access previous knowledge, which may be an appropriate action during the set-up of a teaching task.

On the other hand, we do have to be aware that priming may occur without our knowledge at any given time. That is, if we utilize unnecessary pictures or words to aid in a mathematical task, our students may be thinking about what we don’t want them to think about!


What would zombies in a fraction activity prime students to think about? Does it help them think about fractions? The use of zombies doesn’t serve the mathematics (or vice versa).


What would zombies in an exponential functions activity prime students to think about? Does it help them think about exponential functions? Here there may be an argument that the use of zombies serves the mathematics (and vice versa).

In closing, the priming effect is an interesting process to be aware of in our classrooms. However, Kahneman notes that the effect doesn’t work with all individuals, so we do not have to worry about students becoming zombies to priming effects. In addition to this, it seems that the priming effect has been under scrutiny for robustness, including replicability of certain findings. Perhaps we will have to wait to see what color the first coat is before delving deeper into this theory in our classrooms.


Reflections on OAME #2

“Diversity if not about how we differ. Diversity is about embracing one another’s uniqueness.” -Ola Joseph

Friday – May 12th:

Any of you who have attended both days at OAME and are out tonight, I am not sure how you are doing it. Here I am, nuzzled in a blanket at 9:02PM with a glass of Moscato contemplating writing a reflection because my internal battery is at 5%, wondering how long it will take to fully charge if I know at 9:22 I will be at 19%.  Hint: the relationship is surprisingly non-linear (despite a correlation coefficient close to 1)!

Surprise is not a Surprise with Desmos

If you did not get the reference above, then I am not mad… just disappointed that I didn’t see you in Dan’s presentation this morning regarding the functionality of Desmos. I feel like I have grown so much over the past year through using the graphing calculator and the activity builder in my calculus classes, but am still learning more as the months progress. I have started to realize that with Desmos, I am consistently amazed, but never surprised (anymore). Two awesome functions of Desmos we got a peek at today were the geometry beta, and Desmos for the visually impaired. This was one of the few times in my life I got to hear what graph sounded like (the other being when I studied Fourier Analysis).

By the way, if you want the animation from the cell phone 3-Act, Dan tweeted it out to us. Play away.

Discussing Discussion

One of my afternoon sessions looked at the five practices for facilitating effective discussion in classrooms. This was a great connection to to Deborah Ball’s Domains of Mathematical Knowledge for Teaching (below). Regarding a lesson on perimeter of connected hexagons, we observed student work and strategies during two points in time – near the beginning of their pondering, and much later during the process. We had the opportunity to ask questions of the students to elicit how they were thinking about the problem, connecting them and us to Common Content (generalized math knowledge not specific to teaching) and Specialized Content Knowledge (knowledge specific to teaching) present in the problem.


One interesting aspect of this presentation that was not present in others was time near the beginning to discuss possible misconceptions and strategies of students as the problem progressed. Thinking about student misconceptions would fall under what Ball calls Knowledge of Content and Students, and thinking about different strategies to tackle the same problem falls under Knowledge of Content and Teaching. At the end of this questioning period, we had the opportunity to decide which three student solutions we would present to the class and in what order. We opted to choose a visual strategy to solve the problem first, followed by a tabular strategy, then finally a solution containing both a table and a picture. Interestingly, no students opted for a graphical strategy; although I argued that perhaps it was unnecessary to get the information required for this task. Understanding the progression from visual to table to graphical would be an instance of Horizon Content Knowledge, or knowing how one idea/topic connects to another. All in all, a very well-laid out execution of the pedagogy of classroom discussion!

Alternate Assessments

In the alternate assessment session, we saw non-typical ways to assess students. I thought these assignments were beneficial to have ready to go for when needed (use for students on vacation or sick for example). I was fond of the interview, mostly due to the fact that I have used it before with my elementary teachers. Giving onus to the student to develop and justify their mark is awesome. However, a good interview with prompts does take a lot of time to prepare for properly, especially if you want to mark it objectively.

One thing that caught my eye was the use of an alternate test format. Students could choose the alternate test, which was more open in the sense that a typical question involved elaboration (or explaining all he/she could on a particular topic.) which is a well-known strategy for learning. Of interest, was that the students didn’t necessarily see a decrease in stress/anxiety levels when comparing a typical test to a non-standard test. That is, students writing the more elaborative test had roughly equal nervousness as a student writing a standard test.

Igniting my Heart

Fermi Problem: How many objects can Matthew throw on the stage during his Ignite?



A bag of socks, a deck of cards, a pair of dice, a handful of pennies, a lotto ticket…

Jimmy and Jon reminded us that we need to be our own teacher. We are not, and should not be, 40% Dan, 30% Marian and 30% Cathy with a dash of basil and sauce. We need to be thoughtful about which strategies and philosophies work for us and our students. This was a refreshing thing to hear, especially after the Twitter conversations I had had earlier in the day. Bouncing off of this, Kyle brought up the idea that the debate is not about automaticity, but how we get students toward automaticity. Often those arguing on Twitter forget that automaticity is definitely and end-goal of many teachers attending OAME, and to state otherwise is rude and uncalled for. I was reminded of this while having dinner with the teachers who inspired me last year to develop my interleaved project at Okanagan College. And to me, that’s exactly what OAME is: a place to gather with, and learn from, math educators of varying walks. Not everything that you see will resonate with you, and it does not have to.

As I finalize this post the morning of Saturday the 13th, I look back and realize how lucky I am to work with and be friends with such amazing educators. Here is to an amazing 2016-17 school year, and to many many more together.

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.