A colleague is teaching Year 12 maths methods next year for the first time since the introduction of the new course. As part of the process, I sent her through the materials that we had created for the 2017 course. She spotted a potential flaw. “There needs to be more examples with literal terms in them. The sooner students see these, the better, because they find them hard.” She was right and we have added in examples of this kind.

In discussing this issue, my colleague and I both implicitly grasped an important point; early exposure to these examples would ultimately lead to a more positive emotional response to these kinds of questions. I don’t think anyone has ever articulated this reasoning to me and so I’ve probably picked it up through experience, both as a teacher and a student. It seems obvious to me that being left to struggle with a new kind of problem could lead to anxiety.

This matters because those who promote the use of inquiry-learning and problem-based learning in maths lessons, methods that leave students to struggle with new kinds of problems, have latched onto a concept known as ‘maths anxiety’; a form of stress that is so consuming that it can even harm maths performance.

Jo Boaler, advocate of problem-based maths teaching and the closest maths education has to a rock star, has suggested that maths anxiety is induced when teachers use timed tests. This certainly seems plausible and you may intuitively agree with the idea of eliminating time limits. However, timed tests also have some advantages. For instance, we really want students to just *know* many maths facts, rather than have to work them out, because this will then free working memory resources to focus on higher level aspects of a maths problem. By timing students’ retrieval of maths facts, we can ensure they have reached this level of automaticity. So this is a great question to test with research; where does the balance of cost and benefit lie?

In her book, Mathematical Mindsets, Boaler alludes to research that demonstrates the harm of timed tests. And yet, when reviewer Victoria Simms of Ulster University attempted to trace this claim to its source, she drew a blank.

Which brings us back to those examples. In a new Canadian study, a group of university students were surveyed on their levels of maths anxiety and their school maths experiences. They found that a greater perceived level of support from teachers was associated with lower maths anxiety and they also found that, “…there was a significant decrease in [maths anxiety] when participants reported that their teachers provided plenty of examples and practice items, and this remained after controlling for general and test anxiety.”

This is, of course, a correlation. However, in this area, correlations could be the best kind of evidence we are likely to get because, in order to do experiments, we would need to manipulate the anxiety levels of test subjects and that might be hard to get past an ethics panel.

Given that the finding is supported by common sense and a plausible mechanism – familiarity with example types reduces anxiety – then I think it perhaps provides yet more evidence for the superiority of explicit teaching.

For students who are good in Mathematics and want to be top students in their school I would suggest what my son did for the year 11 examination.

He had a site where any of his friends could pose question that could not answered by them.

My son would then try to do the problem and give them the answer. In the rare occasion that he could not he poses the question in Yahoo answers and someone from around the world will almost immediately answer the question.

He would then pass the answer to the person who requested the answer to the question.

As such my son got to answer all sorts of difficult question and did exceptionally well in his mathematics as well as in other subjects.

brilliant. Your son will do well in the world. 🙂

If you want a site with more of a math community see aops.com .

For example, here is an example of what you can in 2 minutes.

https://artofproblemsolving.com/community/c2052

or see

https://math.stackexchange.com/

It is a crime (or should be) that not all kids who enjoy math are taught about these sites.

thanks Stan. Will pass them along.

Maths is the ideal discipline for Sweller’s approach of modelling where the problem type is worked, then subsequent examples of similar problems are less and less ‘worked’.

If explicit teaching was such a remedy, we would have generations of successful math students and a general positive disposition towards math. Is that what you observe? Yes there are facts to recall, and procedures in which to be fluent, but is that what is most important for math learners?

Research that shows the benefit of a worked example is not directly support for explicit teaching. Much of traditional teaching is telling students what to do instead of demonstrating how we, expert problem solvers, approach a problem. A problem, meaning that we don’t already know how to solve it.

No, Maths is difficult. Previous students failed for that reason. Trying to pretend that everyone can get Maths — if only teachers used the right methods – is a pious myth.

What we DO see, is that as teachers move away from Explicit teaching our students do worse. PISA scores are falling. Maths anxiety hasn’t reduced all.

Chester,

What is your estimate of the percentage that can get math in a realistic ideal situation? By that I mean a normal class size, un-streamed, but an expert math teacher using the optimum method? This includes this starting in grade 1. So for grade 10 it includes these conditions in all prior grades.

I know you can’t give an exact percentage. This is just prompted by your reference to everyone and trying to understand what that means.

Wow. Well that’s where we part ways. Math is difficult, yet I believe all students can do significant mathematics. If your starting point is that this isn’t for some of your learners, it really lets us teachers off the hook. I will agree that math presented as mostly lecture is very difficult for some students to make sense of. I was successful in math despite, not because of, the way I was taught.

I hear this argument a lot. Unfortunately, there is little evidence to support it. The evidence we have is that explicit teaching is the best way to ensure that the maximum number of children learn maths or any other academic subject. Explicit teaching is not lecturing. https://gregashman.wordpress.com/2017/03/15/what-is-explicit-instruction/

goldenoj,

You are arguing against a straw man with “as mostly lecture”. Even in university where lectures are less interactive as a rule students are expected to spend more time working on problems on their own that in lectures. The perhaps poorly named explicit instruction Greg and others argue for is not mostly lecture. If you are not sure look on this page and you should see Greg’s link to What is explicit Instruction. (Greg gets an A+ for the 21st century skill of define your terms on your blog and put a link on every page to avoid pointless debate about straw man positions.)

Stan,

I’ve only taught in two schools really, so it’s not really an area I am comfortable with. Below is just my thoughts, and I certainly don’t think they are gospel.

It’s a Bell curve. Like most biological systems it has to be really, because there must be evolutionary pressure to raise intelligence, and a cost to do so.

At my current school we get about 95% to the point where they can read a question in geometry/trigonometry/linear algebra etc, convert it to Maths, solve, and answer in a context. Of those that do not reach this standard, poor attendance and a lack of desire are our main issues, and then there are a handful who are just too limited intellectually to reach that standard. To that extent I think “everyone can do Maths” is a useful approximation.

However, test them four weeks later and the number that could still do it would be closer to 75%. When people say “everyone can do Maths”, they seem to leave out the large numbers who can do it, but cannot retain it,

because it never really makes any sense to them. If you try to teach those students by insisting on them understanding before they start doing it, you are just wasting your time and theirs.I estimate about 50% of people, if reasonably taught their entire schooling, can handle serious Algebra and Calculus — quadratics, calculus, closing the square, maxima etc.

At my school about half of all students could do Year 13 Calculus, if they wanted to and about 30% actually do so. That’s because I am lucky to work in a department which is uniformly good, so students get multiple years of quality teaching, and the result is that large number of students like Maths, despite being near the median in terms of ability.

But a lot of people just don’t have the mental architecture to do this. I had a couple of boys come to after school tutoring before their crunch Year 11 algebra test this year. Keen to do well, putting the time in, and completely unable to pass. As soon as you taught them one skill, another would leach out. It wasn’t the teaching — they could understand it and do it — it was the retention of sufficient amounts of it. It wasn’t the motivation — because they were highly motivated.

And on the top side only about 5% of people really “get” Maths. For example when first shown a quadratic inequation and its visual representation as the area inside a parabola immediately make the connection and its implications.

I don’t blame my Music teachers for the fact that I cannot remember a tune. I can be taught one, only to forget it within days. I cannot even properly retain tunes I have sung dozens of times. Yet somehow we expect Maths teachers to be able to teach

everyoneso they remember everything?Chester,

Thanks for sharing your thoughts. I hope that the number is higher and that people such as John Mighton of JumpMath are right that most can do a lot more than currently expected. But the jumpmath work only deals with grades 1 to 8 so there is nothing there to say how his experience would apply to Calculus.

As your reply points out there is a massive range of what is meant by “getting math”. .

Goldenoj,

I think the other counter argument to what you have said is that if inquiry worked we would have generations of successful students come out of that tradition and we don’t. Whole schools and even systems have followed that philosophy over the last 150 years and they have consistently been shown to perform worse on every measure including general problem solving. For evidence look at Greg’s sources.

There has hardly been 150 years of inquiry learning, nor any broad implementation of it to this day in the US. Research done in TIMSS and PISA has found very positive results for where it has been implemented with supporting curricula.

“Research done in TIMSS and PISA has found very positive results for where it has been implemented with supporting curricula.”

Can you point me to this? The PISA data I have seen shows a negative association between inquiry learning and achievement e.g.

https://gregashman.wordpress.com/2017/12/05/what-have-we-learnt-from-pirls-2016/

Ah, Greg, you missed the “for where it has been implemented with supporting curricula.”.

It’s the No True Scotsman fallacy. If it works, it’s real Inquiry Learning. If it fails, it wasn’t implemented with “supporting curricula”.

Hey, it makes a change from the usual reason, which is that it works but that teachers need to be properly trained and motivated to make it work.

Maybe 150 years is a bit of a stretch but Montessori goes back 110 years and Steiner 100 years and I would say they were pretty broad.

Dewey was highly influential on systems a century ago (and today).

TED talks have been influential for a decade;)

Project Follow could only look at it in the 60s because it was so widespread.

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