What is it with fake Einstein quotes? I recently read an amusing blog where the author set out a quintessentially progressive argument for abandoning traditional exams, setting kids project work and focusing on generic skills such as problem-solving over knowledge acquisition. At the same time, he argued that the debate between traditionalism and progressivism is a pointless false dichotomy because all good teachers adopt a range of ‘teaching strategies’. How did he open his piece? With a fake Einstein quote about fish climbing trees.
Similarly, in the TEDx talk that I am about to discuss, we are treated to a fake Einstein quote about play being the highest form of research. The talk is by Dan Finkel and the fake quote is not actually the worst thing about it. It is called, “Five Principles of Extraordinary Maths Teaching” – a title worthy of the Extraordinary Learning Foundation™. Have a view for yourself:
To me, it is eerily reminiscent of Dan Meyer’s TED talk and perhaps Meyer is due a credit. The difference is that Meyer’s talk has more substance whereas Finkel focuses on the deep and meaningful presentation. There is also a six year gap between the two talks, suggesting that these ideas are an enduring theme in commentary about U.S. maths education. This is worrying because neither speaker feels the need to list any supporting evidence for the teaching approach that they both broadly suggest and that can be summarised by Finkel’s five principles:
- Start from questions
- Students need time to struggle,
- You are not the answer key,
- Say yes to your students’ ideas,
We have known since the 1980s that asking students to struggle at solving problems with little teacher guidance is a terrible way of teaching maths and yet this is exactly what Finkel proposes. He even suggests ways to avoid answering students’ questions and commends his approach as one that works for teachers who don’t know the answers themselves.
An early experiment conducted by John Sweller helps explain the surprising ineffectiveness of problem solving as a way of learning maths. Students were asked to get from a starting number to a goal number in a series of steps where they could either multiply by three or add 29. Although many of the students could complete the task, they failed to spot the pattern; for each solution, it was necessary to simply alternate the two steps. Why? The process of problem solving was so demanding on working memory that there was little left over to spot patterns and learn anything new.
Finkel’s example problem is even worse. In it, the numbers 1-60 are colour coded. The students are supposed to figure out that the coding relates to the prime factors of each number. Yet, in order to do this, students would already need to know the factors of a good proportion of these numbers. If they know these factors then they might figure out the coding but what exactly have they learnt?
This illustrates why problem based learning is so inequitable. It selects against those with low prior knowledge and in favour of those with high prior knowledge. Students who already know their factors may gain some practice and perhaps consolidate knowledge of a few more factors from this exercise, although it will not be an optimal way to do this. Those who don’t know their factors will be confused by the problem and will be faced with a teacher who insists that he or she is ‘not the answer key’.
They most certainly will not be developing a generic skills of ‘problem solving’ because such a skill does not exist.
I am sure supporters of Finkel’s approach would suggest that I have missed the point here. It is not the actual maths that matters, they might claim, but the process of thinking mathematically. And Finkel’s methods are surely far more motivating than the traditional approach where children in linen gowns write in chalk on slates and have to vomit up rote, disconnected facts for fear of the strap.
I am not convinced by this. What is ‘thinking mathematically’ and how is it different from being plain good at maths? We know how to achieve the latter and it involves a lot of hard work. Even if we accept that thinking mathematically is more than the sum of its parts then students still need to know all of those parts – as in the factors example – and the evidence is clear that explicit forms of instruction are the best way to achieve this.
And the argument about motivation is simply the wrong way around. It places motivation before achievement rather than achievement before motivation. The evidence suggests that self-efficacy is key to motivation. Self-efficacy is a belief that you are able to tackle a given task. We build self-efficacy in maths by making students better at maths. There is no evidence that I am aware of that extended periods of struggle add to motivation. In fact, it is common sense that sustained struggle is demotivating. Instead, we need to give students the feeling of success. This will sustain them as, later in training, they tackle difficult and complex problems.
I understand that this is not how some people would like the world to be. It does not fit the dominant learning-by-doing ideology. But sometimes we need to sacrifice our outdated beliefs if we want to make progress rather than make a record of them in TED talks.