For the Love of Maths

Reflections from Teaching: It’s More Than Just Doing

“Students learn more when their teachers know the content, and when they can anticipate student misconceptions.” -Daniel Willingham

Today in class as my student were presenting the solutions to the surface area and volume questions they had been assigned and an interesting opportunity came up. So I decided to do a little experiment.

In order to understand what I would like to share, we have to quickly refresh on some basic arithmetic when dealing with surface area:
(1) To calculate the lateral surface area of a right pyramid, one can relate this to the perimeter: L.S.A. = 1/2 x Perimeter of base shape x Slant height of pyramid. Note that this formula is equivalent to summing the various triangles that compose the lateral faces.
(2) To calculate the lateral surface area of a right prism, one can also relate this to the perimeter (in a much easier to see way): L.S.A. = Perimeter of base shape x Height. Note that this formula is equivalent to summing the various rectangles that compose the lateral faces.

Now that we have that under our belts, here is what happened. One student was calculating the volume and surface area of a right square-based pyramid (no diagram was given – but we had side length of 5ft and height of 12ft). The following was given as a solution:

L.S.A. = (4s) x h = 4 x 5ft x 12ft = 240 square ft.

After  the question was complete, I asked the class to analyse the response. I mentioned that often as a teacher of mathematics it is useful to be a detective (I call this Maths CSI). Where is the error? What went wrong? How was the student thinking and what should we say to help with the misconception? After several minutes, they were unable to find the error – mind you, they were looking at four calculations instead of one, so maybe the scope was too large. In our case it turns out that the student calculated the lateral surface area of a square-based prism instead of a square-based pyramid.

To me, this brought back the discussion of factual/procedural knowledge versus pedagogical content knowledge. My teachers could very well utilize the formulas given, make substitutions and explain exactly how they arrived at their solutions. That is, their factual/procedural knowledge was quite strong. But, when I asked them to reflect on what went wrong in this particular example, this was pushing them a bit too far out of their comfort zone.

Often I hear the argument that if our future teachers only took calculus (or some other high-level mathematics course) then they would be good teachers. I find this argument rather empty – I’m not convinced that more factual/procedural knowledge necessarily leads to a more effective teacher. I am certain we have all had instructors who were quite knowledgeable – yet lacked the aspects that would make them an excellent teacher. Now, I am certainly in favour of teachers knowing the progression of mathematics – that is I would like my high school teachers to have seen some university/college level calculus and algebra courses. But do our elementary level teachers have to take calculus to be good teachers? I feel this is out of the scope of beneficial progression for them to see. Would it not be more beneficial for a teacher teaching K-4 to understand deeply the progression of mathematics to Grade 8 instead? And those teaching middle school to know deeply the progression up to precalculus?

Rather than taking calculus, would it not be more beneficial for our future teachers to take a mathematics course in pedagogical content knowledge where they study aspects such as the progression of mathematical concepts, common misconceptions students have, questions to ask to understand how students are thinking, and how to know when students are ready for deeper learning? I would argue yes. Prominent researchers such as Dan Willingham, Deborah Ball and John Hattie remind us of the important role pedagogical content knowledge of teachers plays in the success of students. This is not to say that factual/procedural content knowledge has no place – in fact, I would argue that this type of knowledge is necessary before teachers can develop worthwhile pedagogical content knowledge. However, it may also be important to ask the question “How much factual/procedural content knowledge is sufficient to help teachers acquire pedagogical content knowledge required for success in the classroom?”

Reflections from Teaching: Fraction Talks

“We need to increase the explicit instruction on unit fractions and we need to explore fractions continually rather than as a single unit of study.” -Cathy Bruce

In this post, I want to describe my experience with the Fraction Talks Activity I originally borrowed from Nat Banting’s site. He has since become a curator to, a website devoted to various templates teachers may use – if you haven’t seen it yet, it is very worthwhile to check out!

After pondering about the original blog post for a few minutes, I was struck with just how versatile these templates are at connecting certain mathematical aspects together, and providing a visual representation of certain operations/ideas. The fraction talk template that I used when I worked with two classes of Grade7/8 was the following:


Concept of Fraction

Perhaps one of the most basic uses of the template is to connect it to the general concept of a fraction: a shaded portion of a whole, in which the whole has been divided into equal parts. To do this, we can shade in a certain area of the overall square (taking the big square to be the whole) and ask what fraction of the whole we have. For example, we may shade in the rectangle below and ask students to determine what fraction of the whole is shaded in.


Probing Question: “What fraction of the whole is shaded in?” (Answer: 1/8)

In this case, you can reinforce that the regions under consideration must be of equal size. Be on the lookout for answers of 1/16 since this means the student has partial understanding of what a fraction means in terms of shaded regions.

Extension questions (try on your own):
“What fraction of the whole is not shaded in?”
“Shade in a region of the whole representing 1/4.”
Shade in multiple different-sized regions and ask “What fraction of the whole is shaded in?”

Of importance is which region we use as our whole. For example, let’s use the same shaded region, but restrict our whole to a smaller square.


Probing Question: “What fraction of the whole is shaded in?”  (Answer: 1/2)

Here, you can check to see if students recognize that the shaded region depends on what region we agree to be the whole unit.

Extension questions (try on your own):
“Shade in a region representing 1 and 1/2.”

Fraction Multiplication

Once students are comfortable with the big ideas up top, you can start to play a little bit! When I used this activity for the Grade 7/8 classes, I had two end-goals in mind: I wanted them to utilize multiplication and addition of fractions to help them determine certain shaded regions. Let’s first look at how we can connect multiplication into this.

First, identify a region you want to work with. Using the big square as the whole, I have shaded in the purple region below as the area of interest to start:


See this region as 1/4 of the original whole.

Next, we want to identify a sub-region with our original 1/4:


See this region as 1/2 of the original 1/4.

Finally, we can view this sub-region in terms of fraction multiplication. The sub-region is 1/2 of the original 1/4. But we already know that this region is 1/8 of the whole. Hmmm…


See this region as 1/2 of 1/4 which is 1/2 x 1/4 =  1/8.

From here, once students can see the visual argument, you can shade in various smaller regions and have them write a multiplication statement in order to determine the fraction of the whole.

Extension question (try on your own):
“Shade in a region representing 3/4 of 1/4.”

Fraction Addition

Fraction addition can be visualized in a similar way, and may potentially be easier than multiplication. Here we identify two disjoint regions we want to work with. Using the big square as the whole, I have shaded in the purple region below as the area of interest to start:


Most students were happy dividing the overall shape into small squares to get a fraction of 3/16.


Sub-division of whole into 16 equal parts. The shaded region is 3/16.

Great! However, my end-goal was connection to fraction addition, so I went back to the shaded region and had the students think about what the fractions were as separate entities. We agreed:


1/8 of the whole


1/16 of the whole


1/8 + 1/16 of the whole

I asked the students to add the two fractions, and then two observations were noted. The first was that 1/8 + 1/16 = 2/16 + 1/16 = 3/16, which is what they had noted before. The second was that when the grid of red lines was added originally, it shows the connection to the equivalent fractions 1/8 = 2/16 (can you see it?). The teachers helping me facilitate noticed that the grid the students formed gave a visual of the LCM of 8 and 16, as well.

Extension questions (try on your own):
“Shade in a different region that also represents 1/8 + 1/16.”
“Shade in a region representing 1/32 + 3/64. What fraction of the whole is this?”
Shade in multiple different-sized regions and ask “What fraction of the whole is shaded in? Use an addition statement to help you determine your answer.”


I see many other applications of the fraction talks templates, such as perimeter or area, and I am sure they will come as you explore them a bit more. You will notice that the template I chose was slightly challenging. I actually gauged my audience with a simpler template (to decrease extrinsic cognitive load since the extra lines would add to the overall cognitive load) and moved on to the more challenging one since they were comfortable talking about the simpler template. The students I was working with had already studied fractions and fraction operations quite extensively, so this was essentially an extension activity for them to help solidify some of the bigger ideas they had been working with. This said, I still think some of the templates are very accessible and could be used as an introductory activity or as an intermediate activity to gather information on how your students are perceiving fractions. The versatility of the activity is certainly one of the major selling features for me. In closing, check out and try it our yourself – it makes for a pretty interesting and informative activity.

Here is the BLM from my fraction talk: Fraction Talks BLM
Feel free to use it for your own class.

Cognitive Load Theory is Wrong?

“[Rethinking the boundaries of CLT] may allow reconciling seemingly contradictory results from studies of the effectiveness of worked examples … and studies within the frameworks of productive failure and invention learning that have reportedly demonstrated that minimally guided tasks provided prior to explicit instruction might benefit novice learners.” -Kalyuga & Singh


An Introduction

For those of you who know me and follow my blog, you know that I am heavily influenced by Cognitive Load Theory because I believe it is an excellent framework to help me think about teaching and learning. However, you may have noticed that there are certain aspects of Cognitive Load Theory that have been bothering me as of late. For example, in my last post, I reflected a bit on the expertise-reversal effect and how Cognitive Load Theory tends to view most learners as novices. To be frank, I find this rather uninspiring. Similar to the ending of The Force Awakens – it leaves me wanting more.

Over the past couple weeks I have had the opportunity to discuss some really interesting articles with a couple of awesome educators. One article that fell into my lap was “Rethinking the Boundaries of Cognitive Load Theory in Complex Learning” by Kalyuga & Singh. As an aside, it may be important to note here that Kalyuga was one of the authors contributing to this important text I commonly refer to as The Bible – so I utterly respect and value this fellow’s opinion. Let me tell you, I was not disappointed.

Potential Conflicts

The reconceptualization of cognitive load theory proposed in this paper was triggered by the attempts to reconcile some empirical evidence that seemingly contradicts established findings of this theory.

So what empirical evidence is Kalyuga & Singh speaking of? And how does it contradict Cognitive Load Theory? Perhaps a good place to start are with the assumptions of Cognitive Load Theory.

The typical way Cognitive Load Theory is framed is based on the assumption that acquisition of schemas in the long-term memory is the end-goal of an instructional task. As such, the varying types of cognitive load are defined with this assumption in mind, which leads to particular instructional techniques being selected to decrease working memory load in certain situations. For example, it is suggested for novice learners that explicit instruction should be favoured to minimally guided techniques, since minimally guided techniques may involve using search methods (such as means-end analysis) that overload working memory space, impeding schema acquisition. The situation is different for expert learners, since experts already have well-developed domain-specific knowledge. Explicit instruction tends to be ineffective for experts, since too much attention is spent trying to integrate the pre-existing knowledge with the externally provided support, decreasing working memory space.

A growing body of knowledge in the areas of productive failure and preparation for future learning seem to contradict Cognitive Load Theory. That is, certain well-designed minimally guided tasks given to novice learners before explicit instruction tend to be beneficial when compared to beginning with explicit instruction. (For a taste, one of Kapur’s articles is here.)

It Gets More Complex

In lieu of the above empirical evidence, Kalyuga & Singh suggest that we need to rethink the boundaries of Cognitive Load Theory in the context of complex learning. In this article, complex learning refers to tasks involving “multiple learner activities that may have different goals.”

Some of such goals may indeed differ from the acquisition of domain-specific schemas and therefore require corresponding learner activities and instructional methods for their achievement that are different from the activities and methods that are best suitable for learning domain-specific schemas.

Now this is interesting! If we relax the assumption that our end-goal is acquisition of domain-specific schemas, then we can allow the empirical evidence that Kapur and Schwartz (and others) have given us into the realm of teaching and learning. In fact, Kalyuga & Singh give a possible simplified version of instructional goals in complex learning. First, one might consider low-level goals, or goals related to creating necessary prerequisites for schema acquisition. Items that may belong to this level include motivation to learn, activation of prior knowledge, engagement with the task, searching for deep patterns (as opposed to surface characteristics), or making students aware of gaps in knowledge. From this level, we move to mid-level goals, which include any goals that may be related to acquisition of domain-specific procedures and concepts. Here, we can turn to Cognitive Load Theory as a well-developed and tested framework. Finally, Kalyuga & Singh suggest a third tier of high-level goals, which may include generalization of procedures or flexibility in performance. [NOTE: this is simply a suggested model. For a more tangible model, see this article by Kirschner & van Merriënboer.]

Borrowing and Reorganizing

As Cognitive Load Theory began to emerge, Sweller suggested connections between natural information processing systems and the architecture of human cognition. One premise that came out of this evolutionary perspective was the borrowing and reorganizing principle. This principle states that natural information systems will borrow information from other stores (in human cognition, this information is borrowed from other people) rather than create new information. For example, during sexual reproduction the information held in your genome contained borrowed information from both our mother and father that was reorganized into a unique genome.

The borrowing and reorganizing principle underpins the worked example effect (that is we borrow the solution procedure from the long-term memory of a knowledgeable individual). Much research around the worked example effect has shown us that if acquisition of domain-specific schemas are the end-goal of an instructional activity, then worked examples are an efficient way to reach this goal. However, might it be that learning is not as simple as natural information processing systems? This is exactly the argument of Kalyuga & Singh:

The processes of learning biologically secondary information patterns (schemas) in human cognition may require additional learning activities that are not present or even required in other natural information processing systems.

This means the borrowing and reorganizing principle, and its associated consequences, may not extend to these particular activities.

In closing, I want to make note that this post is not meant to downplay the significance of Cognitive Load Theory. I like to think of it more as an attempt to nest Cognitive Load Theory into a larger schema of learning. Currently, this means we may need to be considerate of borrowing aspects from other well-developed learning theories and reorganizing them into a bigger picture of best-practices for our students.

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?

Reflections from Teaching: Multiple Strategies Converge into One

“Then all the colors will bleed into one.” -U2

Yesterday was just one of those days when you realize that you are in the right profession. I had an amazing and unscripted teaching moment while leading my pre-service teachers through a guided-discovery lesson in permutations/combinations.

The Task

They had split themselves up into small groups and I gave each group 5 unique playing cards. Their task was to determine the number of ways to select 3 cards and arrange them in a row. (Note: before this task had started, we had already solved the problem of selecting 3 cards from 5 and not ordering them. This task was upping the ante and asking how the problem changes when order becomes important.) It was unscripted in the sense that I had left them to their own devices – they were certainly expert enough that this task fell within their ‘zone of proximal development’ and I was confident enough in my abilities with permutations/combinations to connect the dots.

The Most Specific

I walked around to chat with the various groups to determine how they were thinking about the problem. The first group said they thought it was 5x4x3=60 ways by explaining that for the first card, they had 5 choices, then for the second card they had 4 choices, and the last card there were 3 choices. Awesome.

The Most General

The next group was playing around with 6s. So I asked them why they were thinking about 6s. They said they knew they had to order 3 of the 5 cards, and to order 3 cards this is 3!=3x2x1=6, so they were trying to figure out how many groups of 6 they had. I thought this was really interesting and offered to them that they already knew how many groups of 6 there were (ie. they knew how to determine how many way to choose 3 cards out of 5). It wasn’t too much of a push to get them to see they had 10 groups of 6, or 60 ways to order 3 of 5 cards.

In all honesty, this one caught me off-guard – but turned out to be the most interesting conversation! I had no idea where the 6s were coming from and I was glad I asked them to share their thoughts, otherwise I may not have connected the dots to see they were actually thinking about 3!

The Intermediate

For the final group, they had yet a different strategy. When selecting 3 of 5 cards, we can use the formula 5C3 = 5!/(2!3!), and they knew that the 5! in the numerator orders all cards; while the 2! and 3! in the denominator remove the ordering for the group of 3 cards, and the 2 left-over cards. This group had originally come up with 5!/3! ways; however, a slight chat about which set they wanted to order (did they want to order the group of three cards or the left-over 2 cards?) had them change their answer to 5!/2!=60 ways.

Converging to One Idea

Now that I had had an opportunity to discuss with each group and see they ways that they were thinking about the problem, I led a class discussion allowing each group to share their strategy for thinking about the problem. I filled in any gaps and wrote the three strategies on the board. It was now time to show how each strategy was actually the same idea!

I began with the ideas of the first and last groups. I asked the class to take 5!/2! and simplify to get a numerical value.

5!/2! = (5x4x3x2x1)/(2×1) = 5x4x3 = 60

Perhaps you can see that the first group’s answer to the problem is hidden within the third group’s solution! The next phase was to try to connect the second groups’d idea.

(#of ways to select 3 of 5 cards) x (# of ways to order 3 cards)
= 5C3 x 3!
= 5!/(2!3!) x 3!
= 5!/2!
= 5x4x3
= 60 ways

Now we can see all three strategies connected into one larger picture. From here we made note that to select AND order a subset of items, we can use nPr = nCr x r! and the connection between permutations and combinations was made.

It was a lot of fun to see that each group had a different way to solve the question. And for me it was a nice way to see how educational theories come into play – each group was trying to answer this new problem by starting with something they already knew. Group 1 likely saw this problem as a tree (for each of the first 5 choices, I have 4 choices, …). The second group knew how to order 3 cards so they likely altered the original question to ask “How many groups of 6 can I make?” The final group was comfortable knowing that 5!/(2!3!) removed the order from the group of 3 and the left-over 2 and likely asked “Is there a way I can alter this formula to solve the current problem?”