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,
- Play!

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 *de*motivating. 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.

Addressing your comment about the Einstein quote:

While the specific quote is not Einstein’s, Einstein did encourage exploration and creativity. According to Quote Investigator, the false attribution started with a misreading of Scarfe, but QI does not challenge this attribution of Scarfe: ““The desire to arrive finally at logically connected concepts is the emotional basis of a vague play with basic ideas. This combinatory or associative play seems to be the essential feature in productive thought”” http://quoteinvestigator.com/2014/08/21/play-research/

I can’t find that specific quote. It might be a munging of this, from “Ideas and Opinions” (1954), or it might be from a similar piece: “It is also clear that the desire to arrive finally at logically connected concepts is the emotional basis of this rather vague play with the above-mentioned elements. But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought–before there is any connection with logical construction in words or other kinds of signs which can be communicated to others.” (https://namnews.files.wordpress.com/2012/04/29289146-ideas-and-opinions-by-albert-einstein.pdf pp. 25-26)

He’s responding to a question about whether, as a mathematician, he thinks in words or in some other sort of object. It’s indeed highly misleading to equate Einstein’s idea of “play” in this passage with the notion of playing with real world objects. He appears to be speaking far more about not constraining oneself with formal notation and codification before allowing the mental abstractions to dance around. This is still what I would characterize as a form of mental play needed for mathematical thinking, but “Play is the highest form of research”, particularly in the context of this TED Talk, is watering it down to an obscene level.

At the same time, Einstein DID say, “Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.” (https://en.wikiquote.org/wiki/Albert_Einstein) — which is consistent with the spirit of the TED Talk here.

So I do think there’s evidence that Einstein encouraged advanced mathematicians to allow themselves to push at limits, be creative, and play with concepts. The question is whether Einstein would support the idea that children and teenagers should be put through such paces. I do think that a mathematics education program that actively inhibits exploration and creativity is a tragedy, but at the same time I don’t think a mathematics education program should involve nothing but “discovery learning”.

And hidden in all of these Einstein quotes (real and fabricated) is an Appeal to Authority. He wasn’t a secondary mathematics teacher, so why are we all quoting him as if his is the final word?

I also think that we need to acknowledge that ideas are formed in their own time. Would he,if he were alive today, with the knowledge of how discovery learning and constructivist ideas have played out in real life, have still supported them? We don’t know. I do agree that Einstein can’t have the last word on anything any more than any person in the past can. Neither does that make what they have said better or worse than what is said now but we do, as the living, have the responsibility of revising, improving on and reconsidering ideas and beliefs with the knowledge that we currently have. To this end, it is clear that Einstein was not willing to simply accept Newton’s ideas and beliefs as eternal truths.

Good question – exactly how much time did Einstein spend teaching high school or primary school?

Well, I’ve had to pause to let my blood stop boiling.

0:27 Maths changed his life — so it has to change everyone’s life. Please spare us! Maths is not going to be everyone’s cup of tea, and teachers will kill themselves trying to make everyone like it.

0:57 “Maths used to be repetition, memorisation and disjointed facts”. What crap! It was repetition, rightly so. It was only memorisation for things like times tables, or for those who aren’t very good at it — I memorised effectively nothing during my time doing Maths.

And the “disjointed facts” is so wrong it’s offensive.My teachers worked hard to link all the skills, just as I work hard to link all the skills. That some students can’t follow the links is because some students are bad at Maths — just as I couldn’t follow my Music teacher’s best efforts. Maths has never been about disjointed anything.

3:45 “Currently teachers don’t start with a question”. Could not be more wrong, at least at high school.

I’m about to start Area with my Year 9 students. We’ll do area of a rectangle, then after brief practice I will put up an L shape and leave them to work it out. We’ll do area of a circle, and after brief practice I will put up a semi-circle and leave them to work it out. Then we’ll move onto circles taken out of squares, and arches.

Small leaps like that are part of any good teaching, and the students get the pleasurable sensation of working them out from previous knowledge. And it they can’t they have something else to do in the meantime and I rescue them within 10 minutes or so.

What I don’t do are vague “questions” like his crappy coloured prime activity and leave them to flounder unpleasantly. Some good kid figures it out in 2 minutes, is bored, and then tells everyone else so we can move on.

5:15 “Kids graduate thinking every problem can be solved in 30 seconds”. He really needs to go into senior Maths classrooms if he thinks that. What Integration questions does he think takes less than a minute? How many circle geometry proofs take 30 seconds?

It will depend on your curriculum, but in NZ we start long questions — 15 minutes or more — at Year 10 (Grade 9 in US). I will be giving assessments to my Year 11 class (Grade 10) that take two hours — one question, two hours. That is a curriculum thing, not a progressive teaching thing. I still teach how to answer long questions in a step by step explicit manner.

6:45 “Teachers are not the answer key”. The best way to lose children’s faith is when you clearly don’t know what you are doing. If you cannot answer the material you give to them they will think you are a fraud. Rightly.

Sure, there will be times that students ask me questions outside the scope of the curriculum that I don’t know the answer to. I go away and look it up, if the student actually wants to know. What I don’t do is break the lesson to go on some wild tangent that even the person asking isn’t that interested in.

(I wonder sometimes if teachers who like to answer such questions in class realise what is generally going on. If a teacher is prone to sidetracks, then students will sidetrack them deliberately to reduce the amount of work that they have to do. They don’t care about the answer.)

I’m going to stop my rant here. It’s not calming me down at all.

you need this

A cure for ted based hypertension from Canada’s CBC.

Then you have to remember that TED is a form of B Ark.

(http://hitchhikers.wikia.com/wiki/Golgafrincham)

“My teachers worked hard to link all the skills”

I’ve seen teachers work hard to link skills, and I’ve seen teachers step through the lessons without showing any links at all. I’ve seen textbooks, same thing.

I could see an argument that, rather than have a Grand Revolution in Mathematics, what is needed is better, more coherent, and more consistent training of mathematics teachers. But teachers who don’t link anything together definitely exist and (in my experience, which is in impoverished schools) are the majority.

The people you describe are really bad ‘teachers’ (I use the word in its loosest sense, meaning “someone employed to be in a room with children all day) and should not have been hired.

I agree with you about the need for better training of teachers. This needs to be *mathematical* training, rather than *education* training.

I agree Paul. But plenty of people conflate bad teaching with bad ideas about what good teaching is. Somehow that they had a bad teacher at school will be saved by making everyone progressive, utterly ignoring the actual issue wouldn’t be affected.

Imagine how utterly shit that coloured number activity would be with a

badteacher!I agree that education reform is focused on the wrong thing. It’s easier to focus on content than to admit that we’re knee-deep in lousy educators and sinking fast. And the competent ones are getting beaten down by the system.

I’m currently a fourth year student taking Pure Mathematics. I love what I’m doing! I’ve had the opportunity to take courses with graduate students and learn math from some of the best people on the planet who clearly love what they do! Before going to university, I never realized how beautiful mathematics really is, and how little of an effort my High School teachers did to portray that. They never asked the real questions, like why things are the way they are in math. After reading your rant, it makes me realize that it’s people like you that made me dislike my high school math experience.

If you love something, then it’s only obvious that you’d want others to see why and how a person could love such a thing. If you have no interest in sharing your passion, then why should anyone care. What’s the point of becoming a math teacher? Literally anybody else could do your job and it would be just as interesting. You don’t have to be intrusive when being passionate; but when people can see why you love something, they can come to appreciate it. They may not end up liking math in the end, and that’s fine, at least now they could see why someone would.

You clearly missed the point of the “Teachers are not the answer key” argument. The point was to teach students to justify their thinking and to build their deductive reasoning skills; Rather than just being told that they’re right or wrong. It’s okay to be wrong. It’s okay to make mistakes. Even the best Mathematicians made mistakes. If the teacher did something wrong, or questionable, a student should feel confident to point it out; Rather than just accepting everything the teacher says as fact. If a student asks you something you don’t know, you should try to solve it before giving up and going to the web. If students live their lives thinking that the teacher is the answer key, they will never question anything they’re taught.

In conclusion, you clearly don’t show your passion for math, and you should! I think people would appreciate it if they knew why you love mathematics so much. Maybe then, they would be more interested in you, and what you have to say about the subject.

Yes, I recognise this mathematical thinking argument as opposed to knowing maths rules in some discussions I’ve had with a set of teachers. I really pity the poor maths student at school today.

It’s all about balance. Educators have a tendency to bandwagon without looking at the whole picture. Learning Maths can be encouraged by engaging investigations but the skills need to be taught as well. It’s like learning a language; you haven’t learnt French if all you have done is learn how to recite a French poem off by heart without understanding the words but equally an interesting translation can’t be attempted until fundamental vocabulary and grammar have been taught.

Learning Maths can be encouraged by engaging investigationsIf you mean investigations like the ones shown in that video, then I believe you are wrong.

First, because engaging activities doesn’t lead to encouragement in learning Maths. It leads to encouragement avoiding actually learning Maths in favour of engaging activities. And that’s just the teachers.

Worse, that “investigation” leads nowhere, except back to the definition of a prime number. Which takes 1 minute to write on the board and isn’t a difficult concept that needs a long time to get to grips with. And, as Greg shows so well, that activity actually risks adding confusion.

So, I don’t think it is about “balance” in the way you suggest. Discovery based activities, with little learning and taking a long time, do not “balance” out your lessons.

There should be some variation in the way you teach, sure, but that variation need have no truck with discovery, investigation or project based learning.

I think the exercise shown would be fun for people who already know what prime numbers are, to look at cool patterns that are generated by them. Also, my son, who is 7 and still not all that attuned to prime numbers, loves “You Can Count on Monsters”, which does the same thing with monsters instead of colors. But the book explains exactly what it’s doing at the beginning and then is just a fun art book of playing with numbers. Which isn’t the same thing at all as an adult playing dumb when a kid figures out that all the even columns have orange.

Which is to say: This is a fun art activity for people who know primes, but i agree with your analysis about those who don’t know primes.

That’s not to say that there are no discovery learning or number talk exercises that aren’t useful. But the one chosen for the video is appallingly bad.

I think there is a lot of misunderstanding here, and this shows quite well where it is.

As far as I remember (it has been a while since I watched the TEDx talk), he never actually acknowledges that this exercise is about primes. And I don’t think we should see it as being about primes.

First, no, the definition of primes is not that simple. Most would say, “a number only divisible by itself and by one”. But this is wrong already, as it includes one and a lot of problems arise (other definitions stop working then). So correct would be “An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.” (source: https://primes.utm.edu/notes/faq/one.html) and it already becomes an intriguing question why this is the case.

In the visual depiction, this immediately gives a very good question: “Why is 1 the only number in white”? “What happens, if we would give 1 a color?” etc. From trying to answer this question, students would learn a lot about numbers and primes.

But even more: The definition of primes we should use to understand this exercise is “An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D. ” (source: https://primes.utm.edu/notes/faq/one.html) You cannot give this to elementary student, but it is at the core of why the concept of a prime is important. The reason to be interested in primes is not its one-sentence definition, that is just nice to have. The true importance of primes is that they are a generating system for all numbers. You don’t learn explicitly about generating systems until high-level maths usually, but I believe using exercises like this, we can teach children what a generating system is at a low age. You probably should also have other examples of generating systems and then have children figure out how they are related (may even have the same color coding, if the group is isomorphic to Z). Then prime numbers don’t become important as a definition that children have to memorize (ok, it’s just one sentence so it isn’t hard), but something that is important in itself. Knowing why something is important makes the content much more easy to grasp and provides motivation.

So it is not the “engaging activity” instead of math. Math should be an “engaging activity” because it can be once you show the “why”. Prime numbers should not be taught as this or that fact, but immediately compared to lego pieces to directly show students what they really are.

Brilliant essay Greg, thank you again for the work you do.

Speaking of teachers who believe this nonsense here is one claiming (as we speak) that the East do better in maths because they problem solve using content they haven’t been taught??????????

http://www.essentialkids.com.au/forums/index.php?/topic/1185821-should-naplan-test-content-that-hasnt-been-taught-yet/page__st__50

I am a maths teacher and have taught in KS2-KS5 in England (age 7-18), and find the main issue of maths teaching is the lack of a genuine product. I am not advocating PBL, but in order to engage young people’s minds there needs to be, as Einstein quite rightly put, imagination. Sparking this imagination needs to be embedded into maths teaching by the development of a ‘what if…’ Scenario that students could then use their skills and understanding to develop an answer themselves, with support from the teacher with development of new skills. One of the previous repsondents was quite right, the beauty of a French poem is both the understanding of the meaning of the words in the mother tongue, but also the enjoyment of the reading of the poem.

Another example could be science, where the teacher sets out an experiment with a goal of deepening the understanding of the scientific processes involved, which makes the teaching of the knowledge much easier.

Maths teachers needs to create this ‘need’ for a new skill rather than the teaching of a set of rote, disparate processes.

Put Dan Finkel in a maths classroom with 7-year olds. Then put you in that same class. The students that walk out of his class have an affinity and natural curiosity about maths, and Finkel has fed those needs. Most of the students that exit your class leave hating maths or believing they are not good at maths. That is the difference, mate. Think about that for a minute.

Don’t know how you know this with such certainty. Your rather harsh criticism of Greg brings to mind something a fellow middle school math teacher, Vern Williams, has to say about the opposing factions of math reformers (progressivists) and traditionalists:

“I have always stated that if a reform minded teacher produces competent, intellectually passionate students, they will absolutely escape any criticism on my part. But the opposite seems never to occur. Regardless of stellar results, the traditional teacher will always be criticized for being a self centered sage on the stage, controlling student learning and running a draconian classroom. Their students may be the happiest most accomplished students of all time but the teacher will never be good and pure until they cross over to the reform side.”

Just a friendly correction.

“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.”

I couldn’t get this to make sense (seemed like the numbers would quickly get really large) so I read Sweller’s paper. Sure enough, it’s multiply by 3 or *subtract* 29.