It was Alfred North Whitehead, a mathematician and philosopher, who made the following observation:

“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

Whitehead’s view is now strongly supported by cognitive science. It is widely accepted that working memory — roughly, the thoughts we are conscious we are having — is extremely limited, capable of processing about only four items at any one time. However, these constraints fall away entirely when dealing with organised knowledge held in long-term memory. Educational psychologists call these webs of knowledge that are related to each other by meaning, ‘schemas’. We can effectively activate an entire schema and solve problems without much conscious effort, provided we have a schema in long-term memory to activate.

In my PhD thesis, I used the example of 3*x*=18, find *x*. This consists of four elements but those elements are not discrete, they are in a relationship with each other. To solve the problem, we need to know what the ‘=’ represents and what it implies — that we must do the same to both sides of the equation. Processing all these elements in working memory is therefore hard for a novice to grasp. And yet, I bet that most of you reading this will have simply ‘known’ *x* was 6 a second or two after reading the problem because you solved it effortlessly using a schema in long-term memory.

You were not born with the ability to do that. Something happened to you.

The process of maths education is therefore about constructing these schemas so that we can automatically solve more components of a problem without conscious effort, freeing our limited working memory resources to attend to novel elements which, hopefully, will also eventually become incorporated into schemas.

And we know something of the most effective ways of doing this. If we overload working memory, giving it too many items to attend to, little learning takes place. This is exemplified by an early experiment John Sweller was involved with prior to his formulation of cognitive load theory. He demonstrated it was quite possible for students to solve a challenging numerical problem without noticing a repeating pattern in the solutions — a pattern that had they noticed it, would have made the problems simpler and quicker to solve. What was going on? Problem solving is so taxing on working memory resources, there was nothing left over to do the noticing.

It is cautionary to remind ourselves that students can be active, successfully solving the problems we give them and yet miss out on learning fundamental concepts we intended them to learn.

Which brings me to Peter Liljedahl’s book, *Building Thinking Classrooms in Mathematics*. From the outset, Liljedahl treats it as self-evident that the purpose of a maths class is to ‘get students to think’. Moreover, he is clear what this is not. Liljedahl visited 40 Canadian maths classrooms and found that most students were not, as far as he was concerned, ‘thinking’, when asked to solve problems. There is no suggestion that these students were comatose and so they must have been engaged in some kind of brain activity, so we have to infer what Liljedahl means by ‘thinking’ from the examples he gives.

Although some students in these classes were ‘slacking’, ‘faking’ and ‘stalling’ — these teachers probably needed some classroom management strategies — the majority were ‘mimicking’, which Liljedahl does not consider to be thinking:

“The question is not whether mimicking is good or bad. The question is, what is mimicking good or bad for?… Mimicking is bad because it displaces thinking. Mimicking happens not alongside, but instead of, thinking.”

Given that their teachers had just demonstrated how to solve a particular problem type, the fact students were mimicking this in their independent work seems rational and hardly trivial. I suspect they were experiencing considerable cognitive load, especially since the teachers were in the habit of demonstrating a few different problem types before asking the students to try one — a practice I used to use before I became more familiar with the research on cognitive load and explicit teaching.

Liljedahl only considers the students to be ‘thinking’ if they try to figure out how to solve the problem themselves. So, to Liljedahl, thinking is synonymous with some form of discovery learning.

This clearly implies that teachers need to move to a discovery learning teaching style and this is what Liljedahl then pursues in his subsequent research and throughout the rest of the book in his own idiosyncratic way. It is unclear whether Liljedahl understands the history of these ideas. For instance, he writes about simply doing the opposite of whatever was standard practice, like a naïve revolutionary. Sometimes, this leads to unintentionally amusing consequences:

“To illustrate an extreme example of how far we were willing to go early on in the research I worked with eight teachers who taught for two weeks in classrooms without furniture. Furniture is an enduring institutional norm, and we wanted to see what would happen if we upended it. I learned three things from this experiment, First, student thinking increased—radically so. We had more students thinking and thinking for longer. Despite this positive result, however, I also learned that teachers don’t like to teach in classrooms without furniture.”

Exactly what the teachers thought of this maths education professor coming into their classrooms, dismissing their craft knowledge and taking the furniture away, remains a matter of speculation for fertile imaginations.

It is hard to find positives in Liljedahl’s approach. His methods and objectives exist in a self-referential bubble. Teaching approaches are judged on how well they deliver ‘thinking’ and so on. If I were going to attempt group work, his ideas on randomly allocating students to groups could be useful, although his odd insistence on using playing cards — and his fondness of ‘card tricks’ — make the text a little surreal at times.

Classrooms should not be too orderly because reasons. Disrupt. Do the opposite of standard practice etc.

Where Liljedahl does an objectively better job than many of his counterparts is in recognising that there is a conflict between messing about with fun but ultimately pointless maths games and reaching specific curriculum objectives.

Like other progressivist maths educators, Liljedahl loves a cool activity. You know the sort of thing: If it takes five giant ducks 20 hours to kill 100 tiny lions, how many tiny lions will fifteen ducks kill in ten hours? However, he realises that students will never learn how to factorise quadratic equations that way. Instead, he suggests starting with three lessons on ‘non-curricular tasks’ before easing students into asking similar questions relevant to curriculum content which he confusingly calls ‘scripted curricular tasks’. For instance, he suggests teaching the factorising of quadratics by reviewing the expansion of brackets and then saying, “if my answer were x^2 + 7x + 6, what would the question be?”

There are no area models here or manipulatives. I can imagine other maths education professors silently screaming with rage at the arbitrariness of the mere procedure Liljedahl wants the students to learn.

I can see his approach working well for some students — those who are already doing well in maths and have schemas to draw on — but it’s quite convoluted. I can imagine plenty getting off the bus.

Liljedahl claims his sequence is superior to the business-as-usual alternatives used in the schools he worked with. Maybe. But we have to take account of placebo-like expectation effects — these are the classrooms the fancy researchers are working with — and the fact that by Liljedahl’s own description, the business-as-usual classes were not optimal examples of explicit teaching. Some mini-whiteboards or a little cold-calling would probably help. So would presenting pairs of examples and problems for students to complete. As I mentioned above, I used to teach in a similar way and was angry when I learned there was research — such as that summarised in Rosenshine’s principles — on how to do this better. Why had nobody told me?

I am extremely skeptical that any benefit would be obtained by maths teachers picking up Liljedahl’s book and applying it to their own classrooms. Every chapter has a set of questions to consider at the end of it and with breathless self-regard, the first one is always, “What are some of the things in this chapter that immediately feel correct?” My answer: Not much.

Liljedahl’s is a recipe for confused and overloaded students. His peculiarly specific and arbitrary prescriptions may entertain some of the more able students in the short term. I cannot imagine them being the basis of a sustainable and equitable teaching programme.

We would quickly run out of fresh horses.

## Peter Liljedahl wants to make kids think about mathematics

I’ve been waiting for your take. I was recommended this book then given it to read as assigned PD. While I am only three chapters in, and have anecdotally tried a few ideas I can feel a lack of explicit teaching and not enough scaffolding for the learners in my classroom to feel confident and successful.

BTC has been an unmitigated disaster in our District....https://bhcw.substack.com/p/berkeley-heights-public-schools-btc-parent-survey-final-results