For the Love of Maths

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.