Dan Meyer, one of the foremost proponents of fuzzy maths*, has written a couple of blog posts (here and here) where he argues against calling a mathematical mistake a ‘mistake’. He illustrates it with an example where a student makes an error filling in a tedious linear function table. The student has assumed that the interval in the first column is constant and has filled in the second column accordingly.

That’s a mistake, right? However, Meyer would prefer us to see it as the right answer but to a different question:

“If I

labelit a mistake, even if I attach a growth mindset message to that label, I damage the student, myself, mathematics, and the relationships between us.”

This doesn’t make any sense. The idea of encouraging students to adopt a ‘growth mindset’, an idea based upon the work of Carol Dweck, is not without challenge. Recent systematic reviews of growth mindset interventions have shown little positive effect, but the basic idea of encouraging students to view mistakes as a normal part of the learning process seems reasonable enough.

Yet the students of a teacher following Meyer’s advice will not be able to do this because their mistakes will be covered-up by the teacher. Meyer seems to recognise this later in the post but can’t quite extract himself from his own quicksand.

As a student, I would have hated to be patronisingly informed that my answer would be correct if the question were different, but I have a deeper objection than just this. I think we should be honest with students. There is something manipulative and sneaky about this kind of approach.

So what is going on? It makes more sense when you see fuzzy maths as part of the progressivist tradition. One broad element of this tradition is the tendency to see education as the process of drawing something out of a child rather than putting something in. It is as if the correct maths lies within the child in the manner of a set of tangled-up Christmas tree lights. The teachers’s role is to help the child unwind them.

Of course, this is absurd and that is why fuzzy maths has such a poor track record.

But what if those who go about promoting fuzzy maths do not themselves recognise it as a product of progressivism? Does that invalidate the argument? Not really. Curiously, given the preoccupation of progressivism on students figuring things out for themselves, the cultural transmission of ideas is powerful and ideas thus transmitted may persist long after everyone has forgotten their origin. Peter Ackroyd describes this effect in medieval England in his *The History of England Voume I*:

“Customs could be an inexplicable mystery. If the king passed over Shrivenham Bridge, then in Wiltshire, the owner of the land was supposed to bring to him two white domestic cocks with the words ‘Behold, my lord, these two white capons which you shall have another time but not now’.”

We are all unaware of the origin of many of the ideas and practices that we hold to and assume are just common sense.

**Fuzzy maths goes by many names such as Reform, Constructivist, Discovery, Problem-Based and Inquiry-Based Teaching. I use “fuzzy maths” as a catch-all*

I’m conflicted. I’m not bothered by this, because it is a good place to be. Let me explain: I read Dan Meyer, Li Ping Ma, Robert Kaplinsky, Andrew Stadel, and many other ‘progressivist’ writers. I have been a huge proponent in my district for constructivist math programs and open ended activities. I enjoy doing math this way and I love teaching math this way. It is dynamic, unpredictable, creative, and, well, fun. I like this because I failed math repeatedly as a kid and as a young adult. Directly teaching me steps to solve problems and then giving me endless practice never encouraged me to think.

I wonder a lot about agency and authority in my own learning and how this might be manifested in my classroom: I’m much more satisfied as I make connections to things I have learned previously, when I see how math concepts are intertwined. When a teacher or mentor can lead me to these realizations through the use of masterful questioning so that I arrive at the aha through my own seductive reasoning, I gleam. I sit up straighter. I want to learn more. I know I can.

I rarely experience this with middle school children. Oh, sometimes, but often they have such baggage about mathematics, so many gaps, that they are unwilling to risk an idea (or even admit that they have one). I see your point when I question a student repeatedly with little or no progress and waste everyone’s time when we could get there more easily through instruction. But there is so little magic there much of the time. They stop listening, or make excuses. I have put a bandaid on a bleeder when it needs stitches.

I want thinkers in my classroom. More than anything else. I do believe that many of the building blocks are already in there, and maybe if I ask the right question I can help them make a connection on their own. But I do also acknowledge that there is a time to get down to business and teach them directly how to do something. My experience (nearly 18 years of it) has told me, though, that listening and practicing is not nearly as impactful and enduring as discovery.

“My experience (nearly 18 years of it) has told me, though, that listening and practicing is not nearly as impactful and enduring as discovery.”

A lot of people would agree with you but there’s very little evidence to support this and alternative narratives are available. For instance, it is possible that many students find discovery frustrating and demotivating, yet when they are actually taught maths clearly and in bite-sized chunks, they gain and sense of accomplishment that they find motivating.

I totally agree with you there! I think that often students appreciate the breakdown. I also agree that the struggle of discovery and investigation can be totally unproductive if you are asking students to work outside their bandwidth. I guess I struggle with how to balance the two ideas. This is why I read your blog even though I often disagree with you. It pushes me to consider all sides. Thank you for this.

I think there is a false premise here. That is that one method can’t be done badly. Either can be done badly. Traditional approaches that are too boring or don’t go beyond repetitive procedures are a bad idea. Discovery approaches that require every step, concept and connection to be discovered can be awful.

John Mighton of Jump math looks like a DI proponent who makes it very clear any learning problem is due to a lack of explaining some intermediate step. yet the jump material doesn’t look like boring repetitive work with no interesting challenges.

As an example I am looking at JumpMath 8.1. On page 121 question 5 asks students to solve which 3 consecutive numbers sum to 48 in two ways. The first is an exhaustive trial starting with 1+2+3. The second spells out the steps to do it algebraically – assign the label x to the first, write formula for the other two and then combine these as an equation and solve it.

Page 122 question 10 is just a more complex variation on this including a multiplication as well as addition but no steps are suggested.

The premise here is it only takes one example of both ways to get which is better. Anyone telling people to do it both ways to drum this is insulting their students.

Nor is anyone one is expected to discover how to solve equations or the detailed steps of converting words to algebra but once shown examples and led step by step through a few problems they are expected to apply this to new problems with additional twists to them.

With math it is always true that no matter what you show people explicitly there is a challenging question you can ask. There is never a shortage of problems that a student can take on that lets them make that aha leap.

Those criticizing DI should look at what it actually is in detail with the best available examples not what happens when the worst impression of it is done.

Andrew,

“I’m much more satisfied as I make connections to things I have learned previously, when I see how math concepts are intertwined.”

This sounds like transference to me. Taking previously learned concepts and applying them to new and (possibly) novel situations. Is transference a type of discovery? I don’t know, maybe it could be.

Nevertheless, as a music teacher, I too struggle with this idea in my lessons. When are enough students comfortable with a concept that they should be able to use it independently when learning new material? When should I let the class try to learn the new (carefully chosen) song by themselves first, rather than teaching it explicitly?

These are indeed tough questions.

I feel you may be being a little harsh here. The way I have interpreted it, possibly wrongly, and although it is not mentioned and perhaps badly explained is the difference between a ‘mistake’ and a ‘misconception’. The student(s) haven’t made a mistake (knowing the correct method but applying it incorrectly) but have not understood how to answer it. The teacher has made the mistake by giving the students the task before they are ready.

How do you get that from the link provided? http://blog.mrmeyer.com/2018/that-isnt-a-mistake/

I think I would have been physically ill if I had a teacher like DM, celebrating various wrong answers because apparently the kids don’t care about getting it right either.

He can’t even agree with himself about whether there is a mistake or not. Seemingly he just doesn’t want to tell the student they messed up because they are too fragile to hear that and it will destroy their interest in math.

This is why you always see kids will stop playing a sport the first time they lose the game or mess up. Every time a goalie lets in a shot he stops playing that sport and fears it forever unless someone tells them they actually won even though they didn’t in which case the child will always believe the adult.

Perhaps I was being a bit generous in my interpretation and projecting my own views. The article isn’t very clear and I’m not familiar with DM’s previous work.

The answers presented are wrong and should be treated as such, the way to approach a correction would depend on whether it was by mistake or misconception.

Yup, Given we can’t figure out what he means you can imagine his students are going to prefer it if they can write down anything that comes to mind.

I agree with your solution for the students error and it would be a simple question to prod the student to see if they appreciate the left hand column doesn’t increase by the same amount between each row. It doesn’t have to be such an emotional roller coaster for everyone.

Another oddity in those blog posts. DM talks about how is favorite questions in math are open ended ones with no right answer.

Does anyone think those are everyone’s favorite math questions?

The archetype of a keen math student is someone who does math competitions where there answer is exactly right or wrong. So it doesn’t seem like the exemplary students prefer these questions or the people who organize math competitions.

There are lots of popular math books by enthusiastic adults. They have names like “How not to be wrong” – lovely book or Creative Mathematics which is full of proof style questions and answers – where you are showing exactly that something is right.

Does anyone think struggling math learners prefer the sort of questions DM likes?

As a student, I’m disgusted by the idea that I would be glorified in my mistakes. This is maths we’re talking about: mistakes come from either bad reasoning or mechanical application of concepts with no thinking (which is also bad reasoning). I’m leaving contextual mistakes such as typos aside for this comment.

Now, one has to say what it means for something to be “correct” or “incorrect”. If you have a set of rules and a question, the answer is correct when you get it without breaking any rules. That is the closest to the etymology of “correct”, which is roughly speaking “supported by” and in this case “supported by [rules]”.

A student must not be vilified for their mistakes – it will hinder their will to learn – but this way they’re going to stop looking for the right answer by reason.

Making mistakes is a natural part of the learning process (and this includes self-study) and should be only worked upon in a way that puts the right approach – not necessarily the right answer – under the spotlight.

Education is probably meant to bring something out of someone. If I may indulge again into etymology, it is precisely “bringing something out [of a person]”. But I see it as also putting something in, like the author pointed out. It’s a mixture of both; students will develop a maturity whose seed needs to be planted by the teacher, and eventually it will be harvested.

Fuzzy maths doesn’t care about whatever grows in the student, apparently, and is not going to do anything about it.

Reblogged this on The Echo Chamber.