[Please note that I in the past I have been guilty of most of the nonsense I describe here]

First, lets stop the nonsense

- a ‘celebration wall’ of students’ mistakes
- letting students work things out for themselves
- students teaching other students
- letting students discover things for themselves
- activities that have no teaching input
- judging a lesson by how enthusiastic the students were
- judging students’ learning by ‘thumbs up/ sideways/ down’
- teaching maths in the context of a game or a real life situation
- individual lesson plans
- ‘teaching’ critical thinking/ problem solving/ communication skills
- letting the students decide for themselves whether they have learned something

Now, lets teach some maths

- planned lesson sequences
- testing the students to see if they have learnt what has been taught
- teaching that responds when students haven’t learned what has been taught
- teaching maths in the context of the discipline of maths
- teaching mathematical knowledge
- judging a lesson based on assessments of how much the students learned

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Teach them some maths. Make sure they understand it. Teach them some more maths. Yep, sounds about right to me!

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I would disagree about teaching maths in real life situations, particularly given the contextual questions we are seeing in exams. Also teaching problem solving is one of the three core elements of the new curriculum so can’t be avoided. The rest I pretty much agree with.

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I agree with the lot.

I don’t disagree with Mike’s points here but I’ll say that the pressure/drive to “contextualize” mathematics has done much harm. There will be natural contexts and to fill out math that HAS been learned contexts can be much fun and helpful as a SUPPLEMENT. Further, contexts can occasionally (not as a hard rule!) be used to motivate a new topic. But the best rule is to let math stand as a subject on its own and not immediately go to contexts as if math was somehow deficient.

As for problem solving, I am a 100% problem solving person. That is the heart of mathematics. I am the director of a major provincial math contest, which is all about problem solving. I train mathletes for competition, and I keep volumes of original and collected problems for bringing into my own classes and for enrichment when I go to talk to students. With that in mind, be clear on this: I think treating problem solving as a “core element of the curriculum” as NCMT and CCSS have done is extremely misguided. Problem solving isn’t “content” — it is a skill that one develops by doing it. Problems are at the heart of ANY effective and comprehensive mathematical course of education, and the skill is not so much taught as developed through comprehensive experiences in which problem solving arises by necessity. Although I am a fan of Polya I cannot say that I have seen any system of “instruction” of problem solving that I consider effective for “teaching” these skills. In a certain sense “problem solving” isn’t teachable. But what is probably more appropriate to say is that problem solving — as cognitive scientists believe they have shown in numerous studies — is highly context-dependent. That is, you can be proficient in solving problems in context X without gaining any discernable improvement in solving problems in context Y. Polya is not wrong in his delineation of the elements of problem solving, but simply teaching these steps as a mantra appears to confer no proficiency — it merely injects the topic of problem solving as an “artifact to be studied”. It does not “teach problem solving”. What is effective for conferring ability to solve problems in a given context is to give students problems to solve in that context, provide scaffolding where possible and expose them to excellent templates and worked examples to help them find their own feet in new problems they encounter in that context. Developing their knowledge of content and known approaches, and giving them problems by which to consolidate this knowledge is the ONLY way to help them improve this domain.

And none of that is accomplished by pushing aside mathematical content and “teaching problem solving”.

Teach the math.

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Thank you, this is a great elucidation of the issue with ‘teaching’ problem solving.

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The bottom line is Be excited. Everything else will follow.

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Thoughts on each point…

a ‘celebration wall’ of students’ mistakes

-While this could be useful, I always wonder about structures. Sometimes we hear something NEAT and want to try it. Maybe it has been successful for a group of students who didn’t realize that their mistakes were actually quite close to understanding… then transformed these mistakes as part of the learning process. Others then see the wall and try to copy without using the same process. It doesn’t work for others. Should this be a focus in a school or a district… NO

letting students work things out for themselves and letting students discover things for themselves

-I think I have mixed thoughts here. While I worry about those who just throw things at students and hope they will be able to learn the material… I also worry about those who believe teaching mathematics is transmitted from teacher to student… and students’ job is to copy the teachers’ thinking. When we do this, we often miss an important piece of mathematics – student reasoning!!! For this, I think we need to analyze our goals and our roles a bit more clearly!!! Take a look: https://buildingmathematicians.wordpress.com/2016/06/09/teaching-approaches-what-does-day-1-look-like/

students teaching other students

-Depending on what you mean here, I would either agree or disagree. In classrooms where students learning through problem solving, having students share their strategies both promotes the development of deepening understanding of the mathematics, and the development of mathematical reasoning. However, this DOES NOT mean show-and-tell where students present their thoughts TO others… it needs to be about US orchestrating a conversation that leads to the deep understanding of the mathematics at hand. On the other hand, if you mean that students should become experts then tutor each other, I don’t see this as being a good model. This might promote those that can and those that can’t do math.

activities that have no teaching input

I’ve never heard anyone doing this

judging a lesson by how enthusiastic the students were

-While I agree this shouldn’t be a way to judge our effectiveness. I do think if we have a group of students who are disengaged in the learning… then struggle with the math, it might be worth finding things that help these students participate actively in the learning process. Listening to others might not be the thing these students need!

judging students’ learning by ‘thumbs up/ sideways/ down’

-While we do want our students to be good self-assessors, this doesn’t mean that this should be our only way to see if things are working… nor is a thumb a good enough way for us to allow our students to self-assess anyways.

teaching maths in the context of a game or a real life situation

-The focus needs to be on the math itself. If a game or context is the most helpful thing, then great. If we are attempting to make things gamish or real-life because we want our students more interested, then we need to be a bit careful here. There are many games that would easily help our students make sense of things, or get the practice they need, however, these need to be very purposeful! Sometimes when WE are not comfortable with the math, or WE don’t find the math interesting we want to try to make the math more fun… and actually lose much of the mathematics itself. So this one, I think need much more consideration than you are giving it.

individual lesson plans

-I agree completely here. We need to allow all of our students access to rigorous material… not giving each student something different.

‘teaching’ critical thinking/ problem solving/ communication skills

-If you mean teaching these things in isolation, I completely agree… If you mean including these things as part of your daily expectations then I completely disagree.

letting the students decide for themselves whether they have learned something

-Again, I want all students to be able to self-assess. However, this obviously isn’t good enough.

——————

planned lesson sequences

-Of course, this is one of the most important aspects to good teaching. We shouldn’t be assessing a lesson’s success, the connections between lessons and the construction of understanding over the course of time is what is most important

testing the students to see if they have learnt what has been taught

-I wonder if a paper-and-pencil test is the only way you “test” students? Does this translate into students thinking mathematically or just being able to answer questions IN school?

teaching that responds when students haven’t learned what has been taught

-Again, this is a really important part of teaching, but this requires us to have a good sense of what to do when things haven’t been working. Please read: https://buildingmathematicians.wordpress.com/2016/06/24/challenging-our-understanding/

teaching maths in the context of the discipline of maths

-YES!

teaching mathematical knowledge

-Knowledge is important… but don’t forget understanding… and thinking… and reasoning… these things can’t be “taught”… they need to be developed!

judging a lesson based on assessments of how much the students learned

-Again, I wonder what our assessments look like? If it is a replica of what WE just did, then I do wonder if this is good enough?

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Mark, first of all I want to say thank you for such an excellent, comprehensive response. Let me try to address your points.

1. ‘celebration wall’: excellent point, and we agree that it shouldn’t be a focus. I would say we must, must, must direct teaching discourse towards more important issues.

2. ‘students working things out for themselves/ discovery’: First of all let me draw your attention to a conversation where I propose an activity to teach Congruence, and David Didau argues against me: http://www.learningspy.co.uk/learning/conversation-best-way-teach-new-concept/. In that conversation it could be argued that I make the point FOR students working things out for themselves; I too have mixed thoughts about this issue. In short, I think that student activity MUST be tightly controlled by the teacher, student reasoning can be integral to this, but it is distinct from giving students a mathematical situation and letting them come up with mathematical knowledge themselves. I take your point here.

3. ‘students teaching other students’: I wrote this point in response to a conversation with a colleague. I had pointed out that a particular activity in a lesson wasn’t good because lots of the students didn’t have the mathematical knowledge to properly tackle the task. He said that the ones who could do it were teaching the students who couldn’t in the course of the activity. I reject this. I am not going to absent myself of my teaching responsibility and delegate this professional, expert job to a child (however, I have to admit, I’ve done this plenty of times in the past.)

4. ‘activities that have no teaching input’: an activity given to students where the idea is that they learn simply by doing the activity, it happens a lot.

5. Yes, there has to be a strategy to get students to participate in the lesson. My point here is that students can be engaged in all kinds of things they enjoy that don’t necessarily have to be about maths. If they are enjoying it and ‘engaging’, it doesn’t determine that they are learning anything. Also, see this post for the problem with talking about things like ‘engagement’: https://clioetcetera.com/2016/06/18/genericism-and-the-crisis-of-curriculum/

6. Students assessing their own learning/ thumbs: Assessment is such an important part of teaching. Daisy Cristodoulou is a genius at getting to the heart of it. She writes about assessment including review of important recent work such as that by Dylan Wiliam here: https://thewingtoheaven.wordpress.com/about

One (of the many) good ideas that I have gleaned from her is the use of Multiple Choice Questions to assess. For example the following website can be used to set and then respond to questions for whole classes at a time: https://www.diagnosticquestions.com

There is a lot to read and say about assessment, but leaving it up to the kids is not helpful.

7. ‘teaching’ problem solving: I think we probably agree. Again, see Michael Fordham’s blog (clioetcetera above) and also this nice one from Jemma Sherwood in which she invokes the idea of cognitive science as made popular by Willingham: https://jemmaths.wordpress.com/2016/06/28/cognitive-load-and-problem-solving/

That’s enough for now. This has been fun, thanks for the comment

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