Here are the first few pages of a recent calculus midterm of one of my students who has been diagnosed with ADHD. I’ll let you take a peek at what you see before I give my reflection.

Now, I want you to go back and take a look at the first page, where question #1 ii required the knowledge of the derivative of *log_{3}(x)*. You can see that the student set up the equation* log_{3}(x) = b* in order to help him determine the derivative using the quotient rule. But the giant “*?*” beside *b’* caught my interest. (Of course, if there is anything else of interest to you please leave a comment!)

Now, he begins playing around at the top of the page, recalling rules for how to deal with logarithms. There is a *y = 3^x* and a *y = log_{3}(x)* indicating to me that he was thinking about potentially finding the derivative using inverse functions or implicit differentiation. However, not much happens here, so we will catch up in a few pages.

The next page is nothing special, in that he tackles the next couple derivative questions without making any more thoughts on the log base three problem he is having. But check out the top of the third page. Here, he correctly gets the relationship between exponentials and logarithms: *3^x = b* means *log_{3}(b) = x* (or vice versa). Then there is a little bit of play at the bottom of the page trying to re-write this relationship in various ways to potentially get a nice equation to differentiate. Aside from now having the inverse relationship solidified, not much headway is gained on the initial problem.

Finally, on the last page, we see one last attempt to think about *3^x = u*, perhaps a nod to the variables I use when doing the chain rule (*dy/dx = dy/du * du/dx*). This is the final attempt to determine the solution to the log base three problem, and the rest of the test continues in a normal fashion.

The most interesting thing from my perspective is embedding what I see in a cognitive load theory setting. We know that the working memory has limited capacity to hold and synthesize information. This information can come from either environmental stimuli, or as schema entering from long-term memory. I was always under the impression that trying to cut back environmental stimuli for students with ADHD was a must, as this allows the working memory to focus more on the task at hand. However, seeing this test had me thinking a bit deeper.

At the college level, we are typically good at minimizing outside distractions; doors are closed, rooms are quiet and I cross my fingers that maintenance has fixed any lights that are in strobe-mode. However, as I do not have ADHD myself, I cannot comment on what outside stimuli might still be entering the working memory. Perhaps a song that was heard earlier that morning? Whether or not he forgot his lunch at home? What plans are for after school? So let’s assume that some working memory space has been allocated to this.

Now it’s test time. Since this particular student is quite adept at mathematics, most schema enter the working memory quite effortlessly. We can see this demonstrated on page 2, where some complex derivatives are handled. From my perspective, it is actually the snag of not knowing the derivative of *log_{3}(x)* that pushes the working memory over its capacity. Look at how often he returns back to the problem – at the top of page 1, the top & bottom of page 3, and at the top of page 4.

Just how taxing is it on the working memory to be subconsciously processing this log base three problem over the course of four test questions? How debilitating would this be if there were not well-developed schema to draw from when writing this test? How much more success would there have been if he was able to dislocate this log base three problem from his working memory, instead of it continually returning back to occupy his focus? I find these questions super interesting, and I have thoughts, but no particular answers. If there are any readers who have studied cognitive load theory from the perspective of individuals with ADHD, I’d love to read a bit more on this topic.

]]>In my last two blog posts, I discussed the concepts of element interactivity, as well as intrinsic and extraneous cognitive load. We say information has high element interactivity if there are many elements of the information that must be processed together at the same time. High element interactivity generally implies high intrinsic cognitive load. Here, intrinsic cognitive load refers to a working memory load caused by the intrinsic nature of information that we are trying to process. Finally, extraneous cognitive load refers to a working memory load imposed by the pedagogical nature of the information being taught.

**Defining Understanding**

Now that we know about element interactivity, we can use this concept to define understanding. In a cognitive load setting, understanding is the ability to process all interacting elements in working memory at one time. Since the focus is on interacting elements, it does not make sense to define understanding to individual elements, such as learning one French vocabulary word (cat = chat).

Let’s analyze our previous examples. Consider the math fact *3 + 5 = 8*. According to our definition, if a learner is able to answer *3 + 5 = ?* correctly, without having process all of the interacting elements, we would say that she has demonstrated understanding of the question at hand. I would argue then, that using a strategy such as tallying up three and five on her fingers would display a lack of understanding. Even beginning with three fingers and counting up to eight, whilst being a more effective strategy, still displays a lack of understanding as she is processing some or all of the elements individually. Of course, I am not arguing that students shouldn’t be permitted to use these counting strategies. It is likely that these are crucial stepping stones in the learning trajectory, and the instructor needs to be mindful of when the student seems ready to move beyond these strategies.

**Understanding and Incorrectness**

One aspect of the definition that I am curious about is when the learner makes a mistake in the process. Consider solving for *x* in *3x – 10 = 5.* Is it possible for the student to understand, yet be incorrect? Are these mutually exclusive events? Let’s say the student solution is

*3x – 10 = 5
*

This is incorrect, but it still shows us that they understand the process of solving for *x*, and that they can process all of this information in working memory at once. Does understanding come down to a judgement call on the side of the instructor in these cases?

**Instructional Implications – A Case for Quick Math Fact Recall**

Let’s try to deconstruct our current pedagogy in light of this definition of understanding. Consider all of the multiplication facts that our students must recall. There is element interactivity amongst one individual fact (*3 x 4 = 12*), as well interaction amongst all of the multiplication facts for three, as well as interaction amongst all facts up to *9 x 9 = 81*! Working memory might get overwhelmed, as intrinsic load is high due to the number of facts that must be remembered.

Think also about what our current curriculum states: students should be comfortable with knowing other concepts, such as knowing *3 x 4 = 4 + 4 + 4 = 12*, building array models, or knowing about the commutative property. All of this increases extraneous cognitive load; thus requiring more time and effort for the students to move the facts to long-term memory. I would argue that this is why we have seen a shift to moving recall of the multiplication facts to later grade levels. In British Columbia, students aren’t expected to recall facts for 3s or 4s until Grade 5; and there is no mention of the harder facts like 7s, 8s or 9s.

To compare, I had my multiplication facts memorized by the end of Grade 3 in the 80s in Ontario. Some might argue that we were taught without *understanding* (this alternate definition is a bit fuzzy, but typically is interpreted as knowing how to complete a question utilizing a model). This is false, as I have many documents showing that we indeed used models. But the key difference here is that * the focus of instruction was on automatization of facts*, and that models were used to introduce concepts and as help when students weren’t understanding.

For such a large task, such as learning the multiplication facts, why not have students learn the individual facts first? Using techniques such as interleaved and spaced practice, and introducing new fact families after long exposure to previous ones, would be beneficial for learning. After students are comfortable with recall of the facts, then we can focus our teaching on developing *understanding* (the fuzzier definition) of how multiplication is connected to other concepts. Of course, once students can recall the multiplication facts, they have displayed understanding in the cognitive load sense, as they can process all of the elements together at once. So why would we want to learn our facts first, before connecting to other concepts? Once the facts are remembered well, then the can be retrieved quickly and efficiently, leading to lowered intrinsic load, and more working memory capacity to work on the current problem of connecting the fact to another concept.

In conclusion, I am not saying that we shouldn’t explain why certain facts are the way they are! This can certainly be done as motivation to the problem, and mixed throughout as needed; however, this should not be the focus of the learning because this increases extraneous load and not all students will successfully move the facts into long-term memory store this way.

]]>In my last blog post, I briefly summarized element interactivity. When elements must be processed in working memory simultaneously due to them being logically connected, we say the elements have high element interactivity. By supporting schemata development in our pedagogical practices, we can combat the strain on working memory that element interactivity causes. There are also two other ideas to keep in mind when reflecting on our pedagogy: intrinsic and extraneous cognitive load. I would like the topic of this post to be dedicated to summarizing and exploring these topics.

**Intrinsic Cognitive Load**

Working memory load that is imposed by the intrinsic nature of the information we are trying to process is known as intrinsic cognitive load. Perhaps this can be explained nicely through the use of an example.

First, let’s think about solving for *?* in the addition statement *3 + 5 = ?*. We have seen that, for novice learners, there are many elements to process here, leading to high element interactivity. Novice learners may have to process all of these elements separately, perhaps first counting to three, then counting up again to eight. In this instance, the high element interactivity causes intrinsic cognitive load. It would be a significant challenge to process anything else in working memory since all of the processing power is dedicated to making sense of the symbols and using the counting-up strategy.

For those who know the fact that *3 + 5 = 8*, this whole element can enter working memory, freeing up processing space. For expert learners with well-built schemata, this problem has low intrinsic cognitive load since they are able to interpret all the symbols in *3+5 = ?* as one unit, and come up with a solution to their interpretation quickly.

In summary, information can have either high or low element interactivity. High element interactivity necessarily leads to high intrinsic cognitive load due to the complex nature of the information. This is especially evident in novice learners. However, as schemata develop in these areas, learners are able to process the interactions of the elements more efficiently, decreasing intrinsic cognitive load.

**Extraneous Cognitive Load**

Working memory load imposed by instructional design is called extraneous cognitive load. For example, open-ended problem solving is a challenge for novice learners since they may be unsure of where to focus their attention. Too much working memory capacity is being used to understand the teaching pedagogy, that little to no information can be learned. Based on the *Borrowing & Reorganizing Principle*, as well as the *Narrow Limits of Change Principle*, direct instruction through studying worked examples provides one of the best practices for learning novel information. In general, studying worked examples with expert instruction has low extraneous load. In novel situations with new information, instruction with little to no structure leads to high extraneous cognitive load.

Of course, this comes with some caveats, as worked examples can be structured poorly. The way the instructor approaches examples can also lead to high extraneous load. For instance, when working on related rates problems in calculus, most instructors will read the entire question, then proceed to working through the problem. Due to the high element interactivity and intrinsic load present in these types of problems, solving the problem using the typical approach causes high extraneous load in novice learners. A better approach comes through understanding the *Split Attention Effect*: interweave solution steps with information from the problem to decrease extraneous load.

**Instructional Implications**

We have seen that when there are many interacting elements in a given problem, intrinsic load is necessarily high for novice learners. As instructors, our primary focus should be on schemata formation, as this leads to decreased intrinsic load. Well-built schemata also enter working memory as single elements, freeing up more processing space for other novel information.

When information is presented in a way that causes the learner to focus on aspects unrelated to the problem, this creates unnecessary extraneous cognitive load, leading to decreased working memory capacity. To combat this, we can present novel information through the use of direct instruction & studying worked examples. This will free up working memory by decreasing extraneous cognitive load. As our learners move from novice to expert learners, it becomes easier to vary our teaching pedagogy, as well-built schemata help to decrease intrinsic load in instances when extraneous load is high.

]]>**Elements & Schemata**

In one of my earlier posts, I discussed biologically primary and secondary knowledge. In short, primary knowledge is knowledge in which we are biologically programmed to learn, such as how to communicate to others within our culture. Secondary knowledge, however, we are not biologically programmed to learn.

To keep things simple within the framework I want to discuss understanding in, let’s assume that facts and procedures can be divided into two classes: elements and schemata. Elements are single pieces of information that can be processed within our working memory, such as knowing that the number 3 corresponds to the numerical amount three. Once known, elements can be placed together to begin forming schemata. For instance, a schemata for “3” may include knowing that 3 can be mapped to the word “three” or to three objects (cardinal), is the whole number after 2 and before 4 (ordinal), or that the number 3 may be used on your football jersey (nominal).

Schemata, once well-known, can be linked. For instance, a schemata about prime numbers may include knowing that 2, 3 and 5 are the first three prime numbers. In addition to this, elements can form sub-schemata. Our reference to the ordinal, cardinal and nominal interpretations for the number three might all be considered sub-schemata of the overall schemata we have for three. As we know, the beauty of schemata is that, once well-formed, they can enter working memory as a single element, freeing up working memory space for other information.

**Element Interactivity**

Element interactivity occurs when two or more elements must be processed simultaneously in working memory because they are logically related. Think about the multiplication fact *3 x 4 = 12*. There are actually five symbols that must all be interpreted at once due to them being logically connected. There are three numerals: 3, 4 and 12. There is the multiplication operation, which could be interpreted in a couple different ways (as an array, as repeated addition, as a multivariable function that returns the product). Finally, there is the equal sign, which is a symbol referring to the idea of 12 being equivalent in some way to the product of 3 and 4. As a novice learner, all five symbols must be processed individually in the working memory; whereas an expert learner has a well-built schemata that allows them to by-pass having to process all of the symbols every time they see a multiplication fact. In essence, an expert processes one element; whereas a novice may have to process all five elements.

**Instructional Implications**

As mathematics instructors, we need to be mindful of how the elements of our problems are interacting within the context of teaching our students. High element interactivity necessarily causes more working memory capacity to be used, increasing cognitive load. One potential way to combat curricular competencies involving high element interactivity is to re-visit pre-existing topics and ensuring our students have the well-formed schemata required to ease some of this cognitive load. Think about how challenging linear equations are for our students: they involve complex understanding of integers and fractions, as well as comprehension of how to manipulate all four of the main numerical operations. Before introducing equations, it would seem logical to review operations with integers and fractions so that students can consolidate their knowledge in these areas. By helping to create well-formed schemata in these topics, students can apply more working memory capacity to the new procedures that are intrinsic to linear equations, without applying too much working memory capacity to previous curricular topics. If consolidation does not happen, it is no surprise that the student struggles with linear equations, as the element interactivity is high and too much working memory is being allocated to topics that are not the focus of the lesson.

In my next blog post, I will explore two more interesting topics: intrinsic and extraneous cognitive load. We will see the interplay of element interactivity with these two topics and discuss instructional implications.

]]>I was asked by a colleague last week to prove an identity involving radicals. The two expressions arise when considering cosine of the angle pi/12. Normally, one would apply a sum or difference formula

and this would simplify to

(1)

However, when one of his students used a calculator, the calculator returned back an unusual expression:

(2)

He and the student were able to verify that these expressions evaluated to a similar decimal expansion, so must be equivalent. But then his student asked him how to prove the equivalence of expressions like this. He tried for a bit, unsuccessful – then he tormented me with this problem all Easter weekend. Eventually, I was able to show the equivalence using an old right triangle trick I saw a few years back.

Attach two right triangles together in such a way so that the right leg of the second, and the bottom leg of the first meet at a right angle. On the hypotenuses of the smaller triangles write root 6 and root 2, respectively. This is done so that the hypotenuse of the larger right triangle is root 6 + root 2 – matching up with the numerator of expression (1).

Our goal is to apply the Pythagorean Theorem on the large right triangle, so we need to determine the legs of the larger triangle. To do this, we will determine the legs of the smaller right triangles. For the root 2 triangle, we have the obvious choice of making the legs (1, 1). For the root 6 triangle, we could make the legs (root 2, root 4), (root 3, root 3) or (root 1, root 5). Notice that in expression (2), we have a root 3. This suggests we might want to try the (root 3, root 3) combination for the root 6 triangle. This shows us that the legs of the larger right triangle are both root 3 + 1.

Now we can apply the Pythagorean Theorem on the large right triangle.

Taking the square root of both sides gives

And finally, dividing both sides of the equation by 4 yields the desired result.

I suppose that the moral of the story here, besides seeing some really interesting mathematics, is that I never would have solved this problem unless I had seen the previous problem involving something similar. In general, I believe it is safe to state that in order to be successful solving problems, one should be exposed to many different types of problems (ever wonder how those Math Olympiad contestants get so “smart”?). From a cognitive science perspective this makes sense – it allows us to create problem archetypes (schemata) that we can draw upon to help solve future problems. And the more well-connected these schemata become, the easier it becomes to solve problems.

]]>#1)* Cumulative Review — Why Isn’t Everyone Doing It?*

I have recently read “*Accessible Mathematics*” by Steve Leinwand, in which he outlines 10 instructional shifts to help raise student achievement. One of those shifts is to shift toward giving ongoing cumulative practice at the beginning of your math lessons. It does not have to be terribly extensive – perhaps just four or five short recall-type questions to ensure that students are not forgetting past concepts. It seems obvious that we should be doing this – but many of us are not!

Why should we be doing it? Well, this was somewhat tied to the presentation that Yana and I gave at researchED. It seems that interleaved and spaced practice are highly effective strategies to increase long-term learning in our students. For instance, I saw a 10% increase in the discrimination of problem type when I used interleaved practice in my integral calculus class last year. However, there are some things that we don’t know about interleaving that warrant future studies – like how many problem-types should we include, or how interleaving affects attention in our students.

Why are we not doing it? Efrat discussed some of the practical limitations of using interleaved and spaced practice at researchED New York. Teachers typically list time investment, lack of support, or an incompatible system as reasons for not utilizing spaced practice. What might change their minds? It seems that teachers are interested in ongoing professional development in cognitive science, and time to work with colleagues in order to help ease them into implementation of such tasks. As this is an area of interest to me, please contact me if you or your school is interested in ongoing professional development in cognitive science – I would be happy to help!

#2) *Depressing — Why Aren’t We Collaborating?*

Continuing on the conversation, we could ask why *aren’t* we collaborating more as a community? Let’s take a look at an example from my life. I had a student come into my calculus class with a TI calculator stating that his teachers at high school said they would absolutely require a TI calculator for college calculus. *Literally, what?!* With tools like Desmos at our fingertips, why is there a need to drag around a $200 brick? In addition to this, my department doesn’t allow graphical display calculators on major tests anyway. So it looks like I will need to reach out to the local community and try to spread the Desmos love. Why? Let’s look at it form the alternate viewpoint: If I teach Desmos to my students this year, but when they move on, the next teacher doesn’t know how (or doesn’t want to know how) to use Desmos, these students are now potentially disadvantaged. In essence, a teaching tool is greater when we share it with others in the profession and we develop long-term learning goals using similar tools.

#3) *Planning — Using Space, Not Time*

In a presentation by Nat Banting and Ilona Vashchyshyn, we were asked to consider planning a lesson using quadrants labelled as “Teaching Actions”, “Teaching Spaces”, “Anticipation”, and “Improvisation.” In other words, when it comes to planning, we need to consider our space (the room, manipulatives, desk arrangement) and our actions (modelling, watching, telling). And Nat and Ilona see our actions and spaces situated on a continuum between anticipation and improvisation. In fact, there has to be some improvisation within our classrooms, since it would technically be impossible for us to plan all the possible divergence that may happen in any given lesson.

Of interest to me was their belief that false dichotomies arise when we believe an individual spends all their time within one of the half-planes. For instance, if we believe an educator continuously anticipates and does not improvise in the class, then they are defined as a traditional teacher. On the other hand, those who are thought to improvise all the time are branded as reform or progressive teachers.

This also works for the horizontal half-planes. If an educator is too focused on the teaching spaces, the lesson might be branded as a differentiated instruction type of lesson; and if an educator is too focused on the teaching actions, the lesson might be branded an inquiry type lesson. There is probably more to this conversation, but I am still trying to think more on these two particular diagrams.

#4) **Synthesis — Finding Your Balance**

In Saskatoon I tried to synthesize some reading that I have been doing as of late. The first bit of information was regarding non-routine cognitive tasks I originally heard of from Dan Meyer at OAME 2017. The main premise is that a mathematical task can either have a real-world context or not. In addition to this, a mathematical task can involve “real work” or “fake work.” There are certain verb choices that we make in a math class that lead to real work (question, predict, analyze, debate) or to fake work (evaluate, simplify). Finally, doing fake work in a real world context is overrated; that is, dressing up a routine task with the air of real worldness is overused in math education. However, pushing students to do real work not in a real world context is underrated; that is, we often fall short of allowing students to use meaningful verbs like question, predict or analyze outside of real world contexts. Think “Calculate when the phone will be charged given the model.” (routine, plug ‘n’ chug, dressed up in real world clothing) versus “Predict the y-value given the data.” (non-routine, analyzing data to predict, non-dressed up mathy question).

In addition to Dan’s thoughts on non-routine tasks, I embedded Steve Leinwands idea to lead lessons with data. My thoughts were that if we are interested in moving toward doing real work, data can help drive questioning, noticing and predicting. Provided things go well with the lesson, we can follow up with verbs that allow us to extend, such as generalize or debate. If you are interested in seeing a bit more, my slides from the conference can be found here.

Realistically, I think it would be quite the challenge to create every lesson as a non-routine cognitive task. To me, it feels unrealistic. Also, I firmly believe that the verbs recall, calculate and simplify have a place in mathematics classes and that they should be respected. For instance, John Mighton of JUMP Mathematics consistently reminds me that cognitive load is important – that is, our students require some skill in order to begin a rich-task such as data analysis. This skill comes with practice, which can easily be acquired via spaced practice involving recalling facts. However, on the other side, Bjork reminds me of desirable difficulties. Could non-routine cognitive tasks be shaped in such a way to support learning and long-term retention?

As I continue to navigate the large divide of what feels like a *fake world* of mathematics and a *real world* of mathematics education, I can’t help but wonder how we might all be able to help shift the collective from *fake work* to *real work*.

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!

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.

]]>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?

Hilarious.

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.

]]>Wow.

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.

Transitioning

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.

]]>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:

- 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.
- 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.
- 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%.

**Student Solutions**

**Discussion**

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:

- 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.
- 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.