For the Love of Maths

Cog-nitive Overload: “Counting Cogs” from

“This problem requires children to think about factors and multiples and, in particular, common factors, but it is not necessary for them to have met this term prior to having a go at the task. It offers opportunities for pupils to ask their own questions, find examples, make conjectures and begin to generalise.” -excerpt from Youcubed website

Today I want to review one of Jo Boaler’s lessons from I recently came across a grade five class where the teacher used the “Counting Cogs” activity, which can be found here.


The task can be summarized as follows: take two of the cogs, rotate them about to see if one tooth from the first cog will land in each gap of the second cog. There are nine cogs in total, having 4, 5, 6, 7, 8, 9, 10, 11, and 12 teeth, respectively.


While the instructions are fairly straightforward for an adult mind, no consideration is given to students who are novices to following instructions. The working memories of the grade five students quickly got overloaded and I had to intervene by telling them one direction at a time. Almost every group came to me or the teacher to ask us what to do since the instructions for the students were not clearly expressed on the handout. It was not that the students were not listening at the beginning of the lesson, it was because they would read the lesson, be overwhelmed by the number of tasks they had to do, and lose their indication of where to begin.

To counter this, I began by telling the pairs of students to cut out the nine different cogs. This was a bit of an annoyance, since it took more than half the class for the grade five students (even while they were working in pairs) to cut the cogs out. So now the students were learning how to cut along the lines properly, rather than focusing on the mathematics that the lesson was supposed to be engaging the students in. I felt more like a babysitter for the first half of this class than a mathematics teacher.

After a pair of students had successfully cut out the cogs that they were going to use, they again got overwhelmed by the instructions. Think about it: students were expected to colour in one tooth, move the cogs around, follow where the coloured tooth would land, keep track of all the places where the coloured tooth landed, know what a cog was, know what a “tooth” (in reference to a cog) was, and then start to find patterns all while working with a partner. Talk about working memory overload! For every group, me or the teacher had to model how to colour in one tooth and how to keep track of the gaps where the coloured tooth landed. Again more wasted time.

Key Questions:

The lesson focuses around the following key questions:

1) Which cogs have you found that work so far?

2) Which pairs didn’t work and can you explain why?

3) How could you predict if a pair will work before you try them?

Before we even get too far into this section, what is meant by the word “work”? It is such a vague word in this context. How will a grade five mind interpret this question, especially if he/she has no schema for cogs? Does “work” mean the coloured tooth enters the same gap each time? Does it mean that the coloured tooth enters only a few gaps? Maybe all the cogs “work” in the sense that they can rotate around each other! So there is a lot of potential confusion around this vague term. The teacher and myself took it to mean that the coloured tooth would land in each of the gaps of a different cog. Yet another aspect of the lesson that had to be explained to the students.

So now we get to the meat and potatoes, so to speak. I want you to note that the students had no prior knowledge of LCM, GCF, factors, prime numbers, etc. I know the big ideas being promoted here are the ideas of the GCF and factors. Two cogs will “work” when the GCF of the number of teeth is 1. For example, I know that GCF(5,7) = 1 so the cogs with 5 teeth and 7 teeth will “work” in the sense that all gaps will be filled by the full rotation of one coloured tooth. I also know that GCF(4,8) = 4, so the cogs with 4 teeth and 8 teeth will not “work.” This knowledge comes from a deep understanding of prime numbers and their relationships with other numbers – knowledge that a novice student does not have.

Next, go back and look at the choices we have for number of teeth. Do you notice a major flaw? The only odd number that is not prime on our list is 9, and we do not have a cog with 3 teeth, so GCF(5,7,9,11) = 1. Thus, all of the odd-toothed cogs are going to “work.” So a common misconception of the students was to state: “If two cogs have an odd number of teeth, then they work.” This is certainly true of the activity (because it has a major design flaw), but we know this is not true in practice: GCF(3,9) = 3, so a 3-toothed and a 9-toothed cog would not “work.”

Similarly, since the only pair of cogs sharing a factor of 3 are the 6-toothed and 9-toothed cog, most students didn’t even get to this pair (there are C(9,2) = 9!/7!2! = 36 different combinations to choose from, or a 1/36 = 2.78% chance of picking this pair). So they, again, often concluded incorrectly that a pair of cogs, one with an even number of teeth and one with an odd number of teeth, would always “work.”

Concluding Remarks:

The first major problem of the activity is that it doesn’t meet students at their cognitive level. The instructions overloaded the working memories of the grade five students, even after the teacher broke the lesson down into more manageable chunks. At this age level, students likely have no idea what a cog is, or what a tooth is (in this context) and will eat up a lot of working memory space just processing that. Then they have to add in the colouring of the teeth and the movement of the cogs – all of this definitely becomes very overwhelming for novice learners!

The second major flaw is the poor design. For a lesson that is centered around GCF and factors, one would hope that the misconceptions of GCF(2m+1,2n+1) = 1 and GCF(2m,2n+1) = 1 could be avoided. Realistically, addition of a 3-toothed cog, and removal of the 11-toothed cog would do the trick.

While the design of the activity is to have students using cogs to discuss the ideas of factors and the GCF of two numbers, the lesson falls utterly short of what its intended goal is. When introducing such an important idea as factors and the GCF of two numbers, does it not make more sense to use explicit instruction followed by discussion of some worked examples? This would not be a difficult thing to do using pattern-recognition (skip-counting and making lists for example). Also, explicit instruction has the advantage of not creating the misconceptions that this poorly-designed activity does.

I do want to remark that I actually enjoy the premise of the activity. I believe that it would be interesting to do after the students have an elementary notion of what factors and the GCF are. If students have some preliminary knowledge, then the discussion may reinforce their prior knowledge. Also, the lesson extends nicely to the ideas of prime numbers and the LCM. We can keep track of how many teeth hit the first cog before the coloured tooth comes back to the start to find the LCM of the two cogs, for example.

Altogether, as a teaching tool, the cog activity is not worth it. Maybe if it was redesigned a bit to account for some of the cognitive overload it places on the students, and to remove some of the possible misconceptions that students may create. Even then, I probably wouldn’t use it as an introductory activity for factors or the GCF (although it does have nice potential to reinforce prior knowledge in my opinion). Introduction of these key ideas is better done through explicit instruction and working with patterns of whole numbers. If I did use this activity, I would definitely decrease the number of cogs (remove some of the prime toothed ones), and cut them out myself so I am not wasting copious amounts of precious class time!

To leave you with some food for thought: Did you notice that my discussion of the GCF and LCM are not on the teacher handout? Go back and check for yourself – there is no connection to mathematics at all on this handout! Do you not find it odd that the most interesting aspects of this activity are not even mentioned to the teachers who would be using the activity? It makes me wonder if Jo Boaler has the best interests of the students/teachers in mind. How can students/teachers develop “rich and meaningful connections” to mathematics if the proper mathematical vocabulary isn’t even introduced? Very bizarre indeed.


Soft Skills Programs: Yea or Nay?

“Soft skills get little respect, but will make or break your career.” -Peggy Klaus

Quite a few teachers over the past few years seem to have bought into the idea of teaching “soft skills” in the classroom. From what I can tell, “soft skills” encompass (but are not limited to) the following: collaboration, citizenship, grit, leadership, conflict management, and a whole whack of others. This blog is a response to a photo that creeped across my twitter feed the other day by @danieldmccabe, who, to my knowledge is an assistant principle and co-moderator of a twitter chat #NYEDchat.

The real trouble I have with these so-called “soft skills” is that teachers really haven’t critically thought about what it means for their classrooms. Here is some food for thought on this issue.

Where did “Soft Skills” Originate?
While it may be possible to date “soft skills” programs back many centuries, there is one in particular that caught my attention as I was researching. In the 1960s, James Comer developed a program called the Comer School Developmental Program which had six cornerstones for child development in low socioeconomic areas:

1) Physical – helping children understand their development so that they may grow up as healthy individuals.

2) Cognitive – the ability to think critically and analyze information, but also to have mastery in required content areas.

3) Language – ability to understand, process and interpret written, verbal and non-verbal cues.

4) Social – helping students to maintain healthy relationships, even during times of stress.

5) Ethical – to help students with their capacity to judge situations with fairness and to maintain a sense of justice.

6) Psychological – ability to manage emotions, self-esteem and self-awareness, and to help foster a positive sense of self.

The original program had success in low socioeconomic areas.  However, let’s note a few key differences between this program and what is happening currently.

Teachers ARE NOT Psychologists
First, this program was designed by a PhD in child psychiatry, was implemented in a carefully designed and controlled setting for a specific group of individuals, and likely took years of precise and meticulous planning. Nowadays, it seems as if everyone and their grandma pretends they have a PhD in psychology and implements a curriculum that contains “soft skills” without critically thinking about what they are doing (take Growth Mindset as the current example). Teaching something like perseverance or self-awareness, which are complex psychological components, has become so over-simplified that it has serious potential to harm students. If you want to read more about the dangers of  “pop-psychology” and “soft psychotherapy” see this blog by @C_Hendrick. Also of interest is this blog by @polymathish that gives insight into Alberta’s Teacher Advisory program, where time and content has been cut so that teachers – who may or may not be trained in psychology – educate students in certain “skills” to give the illusion that “individual student needs” are being met. This is despite the research-based evidence that indicates there is no distinct advantage of having a Teacher Advisory program.

Removal of Content Knowledge
And this ties in to my second beef with “soft skills” teaching (and discussed in @polymathish’s blog as well): notice that Comer’s original program does not remove the requirement that students need to demonstrate mastery of content knowledge. This is a key point that I feel is certainly not addressed in today’s “soft skills” programs – likely due to the fact that current practices are neither carefully planned nor based on good research.

Many of the teachers I speak with barely have enough time to cover all of the outcomes in the curriculum over the course of the year without any extra activities. This implies that there must be a trade-off for the inclusion of “soft skills” activities into an already packed curriculum. So what gets removed from the curriculum then? Content knowledge. Students are being asked less and less to recall facts and apply basic skills, and are being asked more and more to problem solve open-ended and “context rich” questions. Fact recall and basic skills are fundamental to any program involving mathematics or science, and the problem with a “soft skills” program is that repeated practice with the fundamentals are tossed out the window in favor of a more problem-based approach. Many research papers I have recently read have shown that the basic mathematical skills of Canadian students entering first-year calculus is dropping at an alarming rate. In one study, more than 50% of the first-year students entering university-level calculus had computational skills equivalent to a student leaving elementary/middle school (Gr. 6-8).

Some might argue that students who master the basic skills do not understand the material. This trade-off is centered on the fallacy that students do not need to memorize facts to understand the content. This fallacy is thoroughly debunked in Daisy Christodoulou’s book 7 Myths About Education. Long story short – automatic fact recall allows us to remove the constraints of our working memory, and the facts we commit to our long-term memory help us to create understanding and become efficient problem solvers.  In addition to this, a ton of research within the last fifty years has shows us that use of problem-based learning and open-ended questions that introduce students to “content-rich” problems do not lead to student learning (some even report alearning loss!). See this article by Kirschner/Sweller/Clark, Project Follow Through and The Sutton Trust Report to name a few.

Some Closing Thoughts
So hopefully now you can see why I want you to thoroughly think about your implementation of a “soft skills” program, and why I get a little edgy when teachers promote “soft skills” in their classrooms.

Are your students still required to master content knowledge in your “soft skills” program, or has content knowledge been removed at the expense of teaching these skills? Has the program in your class or at your school been carefully planned, and founded off of rigorous scientific research in psychology that shows that implementing a program dedicated to “soft skills” improves student learning/achievement? Are there dedicated psychologists working with you to ensure minimal harm, or have you become the “psychologist” despite not having the credentials? Some food for thought for you to ponder over.

CLT Part I – Problems with Problem Solving

“Primary knowledge is learned easily, automatically and unconsciously and cannot be taught. Secondary knowledge is culturally acquired, … learned consciously with effort and should be explicitly taught.” -Sweller, Ayres & Kalyuga

I have started to read “Cognitive Load Theory” by Sweller, Ayres and Kalyuga. I intend to try to reflect on each section as it arises to help solidify my own knowledge and to invite those of you willing to read into my thoughts.

Just as we often do with a mathematical lesson, we first begin with some definitions.

Biologically Primary Knowledge
Geary describes biologically primary knowledge as skills that are readily learned because we have evolved to acquire them for biological survival. Primary knowledge is learnable, but not teachable. For instance, we come to acquire our first language, not by being taught, but by determining ourselves the varying motions of the breath, lips and tongue required to produce certain sounds. We readily acquire this skill because we have evolved to do so, irrespective of our culture. Individuals simply need to be members of a functioning society to attain this biologically primary knowledge. For now, we will assume that any skill area in which we can acquire information without first being explicitly taught is a biologically primary skill.

Biologically Secondary Knowledge
Biologically secondary knowledge, on the other hand, is knowledge that we have not encountered in our evolutionary journey as a species. We can assimilate this knowledge; however, it needs to be explicitly taught within a culture. Secondary knowledge is both learnable and teachable. However, acquiring secondary knowledge takes much effort and is done consciously. For example, while learning to speak is biologically primary, learning proper etiquette when speaking, such as saying “I am a boy.” rather than “I is a boy.” must be explicitly taught.

Problem Solving as Biologically Primary
Reflecting a bit on the phrase “problem solving” we might believe that this is a secondary skill. However, deeper reflection allows us to see that “problem solving” in its simplest sense (i.e. solving a problem) is biologically primary. Certainly our ancestors developed general strategies to solve basic problems such as how to find an efficient way to cross a river. A general problem-solving strategy called means-end analysis is well-known, and is argued to be biologically primary due to the fact that it is utilized in many problem domains, yet is not an explicitly taught skill.

Means-end analysis uses forward and backward reasoning over a problem space. There are two states: the current state and the goal state. Differences between these two states are used to determine actions that can lead to smaller differences between the two states. When an action is performed, we check to see if the current state is equal to the goal state; if not, we recursively continue until the goal state is reached.

For example, we might pose the problem “Who was the seventh president of the United States of America?”. Our current state (unless we know this fact) is that we don’t know his/her name, and our goal state is the name of the seventh president of the USA. We might reason that a Google search would be helpful in determining this information, so we type “seventh president of USA” into Google’s search bar (an action leading to a smaller difference in our two states). Now our current state has a page of relevant websites. Since the current and goal states are still different, we continue by clicking the first link, which is a Wikipedia article on Andrew Jackson (another action in hopes to simplify the difference of the two states). Finally, we reason that if we read the first sentence of this article, we might have our answer. Indeed, Wikipedia reveals that Andrew Jackson was the seventh president, so our current and goal states are equivalent. We are done our means-end analysis.

Instructional Consequences
As you can see from our previous example, means-end analysis, while biologically primary, is actually more complex cognitively than simply knowing the fact that Andrew Jackson was the seventh president. If we know this fact, we can by-pass the strain of using means-end analysis to determine the goal state. Of course, since you are likely an adult, you know how to use a search page relatively easily (you have already assimilated this secondary knowledge) so the strain on your cognitive processes might be minimal. For a more novice learner, the action of searching for this information using Google may be quite a strain on their cognitive processes. In either case, it is important to note that having domain-specific knowledge (i.e. knowing that Andrew Jackson was the seventh president of the United States) helps reduce the cognitive processes associated with general problem solving strategies such as means-end analysis.

To continue this discussion, let’s think about a mathematics problem such as simplifying 51 + 52 + 53 + 54 + … + 100. We could attempt to use means-end analysis by adding 51+52=103, then 103+53=156, then …, but this would be terribly taxing on our working memory (and we might be more likely to make an error due to this strain). If we notice, however, that 51+100=151, and that 52+99=151 and that 53+98=151, …, we may note that 51+52+…+100 = 25 x 151 = 3775. But the crux here is that we may not have thought of the second strategy unless we have either been explicitly shown this, or we have encountered a similar problem (and we reason through this problem thinking of the previous one). Again, we see that having domain-specific knowledge (i.e. having seen a similar problem) reduces the cognitive demand of this problem.

So should we teach problem solving to our students?

Well this depends on what your end-goal is as you teach problem solving. General problem solving skills are biologically primary, so we do not need to teach these skills. For instance, if your end-goal is to teach your students a general skill like “think of a similar problem”, it is probably not a beneficial problem solving session. Teaching domain specific-knowledge, such as giving worked examples, is likely to be very beneficial for students, as they can then use these worked examples to help them reason through future problems. In addition to this, the more domain-specific knowledge a student has, the easier problem solving becomes since domain-specific knowledge often lessens the cognitive processes of a more general problem solving strategy like means-end analysis (see the two examples given above).

1) There are two types of knowledge to think about: biologically primary knowledge, which is learned, but cannot be taught; and biologically secondary knowledge, which can be both learned and taught (but may take more effort than primary knowledge).

2) General problem solving strategies are a part of our biologically primary knowledge. We do not need to explicitly teach these skills. In fact, teaching general problem solving skills such as “think backwards” or “eliminate possibilities” is probably no more useful than a means-end analysis.

3) Having domain-specific knowledge is helpful when faced with a problem solving situation. In fact, the more domain-specific knowledge we have, the more likely we are to be able to find a strategy that will be useful to solve a particular problem.

Of course, I appreciate any thoughts.

Just a headache

Everyone is entitled to their own opinion, right? Maybe not.

ACT 1: The Initial Tweet

Before I proceed, let me preface this post stating my personal position in mathematics education. If one drew a line in the sand with constructivist theory with its implications on learning on one side, and cognitive psychology and its implications on learning on the other, I would definitely find myself on the latter side. Not to say that I do not find myself utilizing some ideas from constructivist theory; but I certainly consider myself more aligned with cognitive psychology.

Now, with this said, I wish to share a story from last week. I came across this site through a tweet on my feed. It is a cute page giving teachers some ideas for maths-related activities that tie to art. I, however, saw it the other way around: the activities were actually art projects that drew in mathematics.


Now why would I say such a thing about these activities? Well, Dan Willingham says it quite nicely on pages 79 and 80 in his book “Why Don’t Students Like School”:

“I once observed a high school social studies class work in groups of three on projects
about the Spanish Civil War. … The teacher took students to a computer laboratory to do research on the Internet. … The students in one group noticed that PowerPoint was loaded on the computers, and they were very enthusiastic about using it to teach their bit to the other groups. The teacher was impressed by their initiative and gave his permission. Soon all of the groups were using PowerPoint. … The problem was that the students changed the assignment from “learn about the Spanish Civil War” to “learn esoteric features of PowerPoint.” There was still a lot of enthusiasm in the room, but it was directed toward using animations, integrating videos, finding unusual fonts, and so on.”

Willingham is showing us that teachers need to be critical of what their students are going to think about during a learning activity. For nearly all of the activities given in the link above, the mathematics is the after-thought of the activity. For example, in “musical fractions” students are primarily going to be thinking about the music, or perhaps the colours, and little about the fractions.

Now, it’s not that I am suggesting that the activities have no value. Indeed they likely do have some value in art class or music class. In addition to this, they help students see connections to mathematics. Mathematics should be the subject you draw into other subjects, not the subject that we replace at the expense of other subjects. (Perhaps more on this at a later time.)

ACT 2: The Response

I found it rather odd to come home to these two tweets from Dan Meyer.
blog2 blog4

Perhaps Dan was simply having a bad day. Although, one could argue otherwise: we do not follow each other, and I did not hashtag my original tweet, so he really had to go out of his way to find this tweet to quote it. Either way, I find his arguments rather off-putting. Where did I oppose any ideas? I was simply stating my opinion on the matter – using my 21st century critical thinking skills to connect what I have learned to a real-life situation, to be exact.

Second, notice that he attacks the character of the mathematics community. What does this comment add to the discussion? To me, it shows me that he and his following are less interested in actually having a discussion about my position on the original tweet, and more interested in painting a false image of myself and the mathematics community. Again, maybe this was not his intention at all, but this is how I see the conversation. As such, I left him a cheeky response.


What is even more baffling to me, are some of the responses Dan got on his initial tweets.


As I mentioned earlier, I doubted that any of Dan’s following actually intended to have a conversation with me. I, however, still replied to some stating my viewpoint to no avail. In all honesty Tom, I doubt reading Dan’s blog will help me discover why he thinks an ad hominem attack is necessary in this situation.  blog6 Again, it seems these fellows (Kyle and Ian) have missed the original opinion. One must critically think about what the students will be thinking about in the lesson. My opinion was that most students would walk away from these activities answering the question “What did you learn in maths today?” with nothing related to mathematics. Kyle, this has nothing to do with enjoyment. Students may very well enjoy these activities, but this does not mean that they will walk away having learned any mathematics.

Ian, again I have made no mention of never connecting maths to anything else. In fact, I mention in my first tweet that these would be great activities to bring mathematics into an art class! My my, we have really explored the depths of confirmation bias now, haven’t we? blog5School is really important. It is the perfect age for society to help children learn biologically secondary knowledge, or the knowledge that our culture wishes to impart on our young learners. If students fail to acquire the necessary foundations for mathematics in elementary school (by consistently seeing art lessons dressed up as mathematics lessons, for example), they are constantly playing “catch-up” in the later years, and can miss out on opportunities to advance in a STEM field (this is actually quite an issue in North America). So no, I do not agree saying “Chill out, school isn’t soooo important.” will work here Michael.

ACT 3: The Reflection

Now that we have taken time to rile ourselves up, let’s jump out of the fire for a moment and debrief. The major question brought to mind for me was “Are there other teachers with similar beliefs as me that are treated equally as unfairly?” And more importantly, if the answer is yes, then why is this the case?

Perhaps, in order for these individuals to feel successful, they must have a cohort whom they can call “the other.” Utilizing another cohort as “the other” and painting this cohort in a certain way allows their positions to be justified. For example, note that I represented the “bad way” of thinking and was branded “the other” in Dan’s initial tweets. This allowed him to justify his underlying message of “my way of thinking is better.” And this underlying message is seen in several of the other comments from his followers that I have posted above. Are either of our views the “best” way to think? Probably not, but as soon as I am branded as “the other” then there is a distinct victor.

Also notice that the labelling of “the other” creates a power differential between the two cohorts – when Dan branded me as “the other”, members of his cohort felt completely comfortable verbally attacking my character rather than criticising my opinion. There is a distinct difference here: critiquing my opinion means coming at me as an equal who is willing to discuss; insulting my character/cohort means you are attempting to put yourself unrighteously and psychologically superior to me. This unfortunately seems to be a common trend in mathematics education currently. I believe there is some kind of middle-ground to be sought; however, we cannot get there if we consistently are searching for the psychological upper-hand in the discussion.

So it seems to me that it’s not only maths class that needs a makeover, but our way of communicating freely and respectfully with each other.