# Teaching mathematical grammar

Consider these questions:

1. 3361 × 100 = 336100
2. 3.361 × 100 = 336.1
3. 33.61 × 100 = 3361
4. 0.3361 × 100 = 33.61
5. 336.1 × 100 = 33610

To a maths teacher, or a pupil who achieved old-money KS2 level 5 in maths, these are utterly trivial questions that require barely any conscious thought.

Yet to any pupil who finds maths difficult, these questions can be bewildering.

On the surface they are all the same process – ×100 – yet the process performed on each number looks radically different. Take question 1: 3361 × 100. Simple, right? Put an extra two zeros at the end of the number, and it’s right. But apply that process to questions 2, 3, and 4, and you’ll get it totally wrong. Question 2: 3.361 × 100. Similarly simple: all you have to do is move the decimal place two places to the right . But in question 3, the decimal point has disappeared altogether in the answer – where’s it gone?! Then over to question 4 and 5, and two weird things happen: whilst the decimal points have moved as expected, a zero has disappeared from the left of the number, whereas in question 5, a zero has appeared on the right of the number.

To illustrate: have a look at these mistakes, and see if you can spot the inferences made:

1. If 3361 × 100 = 336100, then: 3.361 × 100 = 3.36100
2. If 33.61 × 100 = 3361, then 0.3361 × 100 = 03361
3. If 3.361 × 100 = 336.1, then 336.1 × 100 = 3361 .

Of course pupils find it bewildering.

I’ve started to think that this topic is a great example of mathematical grammar, and as such, requires a specific form of teaching for pupils to learn it well. What do I mean by this? And how can we help pupils get it?

# Starting at Michaela

In September I started teaching at Michaela.

What drew me there? The first thing was Michaela’s mathagogy. Over the summer, I reflected on my experiences as a maths teacher in the series ‘Is this the best we can do? Doing better’, looking at issues with UK maths education such as work ethic, textbooks, pedagogy, subject knowledge, expectations, behaviour. Much of the inspiration from this series has come from Michaela teachers. (I didn’t quite write all the posts I was intending to write – but I’m hoping to finish off the series, especially in light of my experiences at Michaela, trying to put some of these into practise.) I’ve been eagerly following Michaela blogs since the school started, and found myself nodding along to(/exuberantly absorbing) many of their thoughts – for example, on recall, or the foundations of teaching problem solving; or teaching efficiency in strategies; or on making textbooks. So it felt like a natural fit.

And then I visited. Watching one of the maths team teach an hour of maths was one of the highlights of my year, and that one hour hugely transformed my own teaching practice when I returned to my previous school. (To list a few things, I started speaking much faster in lesson, expected 100% attention, encouraged pupils to take more responsibility for their results, heightened my own expectations, and showed my own personality more. All from watching her for an hour!) I knew I wanted to work there. More than that, I knew that I couldn’t bring about my thoughts on maths education all by myself – such as making textbooks, or setting daily homework, or regular testing – but Michaela seemed to be doing it all, and more, already.

Those things drew me to work there. I’ve been there for a month. How has it been? Here are some reflections.

# Is this the best we can do? Part 5: hard work & homework

“No food without blood and sweat.”

“Farmers are busy; farmers are busy; if farmers weren’t busy, where would grain to get through the winter come from?”

“In winter, the lazy man freezes to death.”

“Don’t depend on heaven for food, but on your own two hands carrying the load.”

“Useless to ask about the crops, it all depends on hard work and fertilizer.”

“If a man works hard, the land will not be lazy.”

“No one who can rise before dawn three hundred sixty days a year fails to make his family rich.”

This is part of a series offering my views on some problems with UK maths education. The first part looked at the state of affairs with regards to GCSE and PISA results, the second part looked at my attempt at a diagnosis, the third part looked at pre-existing maths education success stories, and the fourth part looked at how textbooks offer the largest potential to improve pedagogy across the nation. This post will look at the role of hard work.

Malcolm Gladwell’s Outliers is a great read. Chapter 8, on ‘Rice Paddies and Math Tests’ should be compulsory reading for maths teachers: Gladwell sets out to answer that perennial question: why are East Asians so good at maths? Even when they have grown up abroad? The following chart, from a UK Education Committee report is absolutely staggering (though it looks at educational attainment more generally, rather than just maths):

Take a look at the topmost teal line: that represents the average grade of Chinese students in the UK according to their wealth levels. As you’d expect, it has a similar trend to every other line – Chinese students who come from richer backgrounds tend to do better than those from poorer backgrounds. But now compare the lines: The most deprived 10% of Chinese students in the UK outperform the richest 10% of every single other ethnic group, save for Indians. That is staggering.

Gladwell quickly and rightly disputes the claim that Asians have higher IQs – instead, he even cites James Flynn’s research claiming that ‘Asians’ IQ… have historically been slightly lower than whites’ IQs, meaning that their dominance in math has been in spite of their IQ, not because of it‘. Yet the fact that East Asians transplant their educational success even outside of their native countries means that the answer isn’t totally located in specific education systems, either: since it isn’t genetic, it must be something cultural.

So what else can explain this dominance?

# Is this the best we can do? Part 4: better pedagogy through better textbooks

The demand for quality texts has been a cornerstone of the Escalante Math Program. In the seventies I realized that my students would be held back forever unless they had superior textbooks, so I searched for the best and tested many different texts. When I found what I needed, I demanded these texts for my students.

– Jaime Escalante (‘the most famous teacher in America’)

This is part of a series offering my views on some problems with UK maths education. The first part looked at the state of affairs with regards to GCSE and PISA results, the second part looked at my attempt at a diagnosis, the third part looked at pre-existing maths education success storiesThis post looks at the first common theme in those success stories: textbooks.

In 1982, two years before Singapore’s 16th-place ranking in TIMMS. the Ministry of Education published and rolled-out a new, intensely researched & resourced maths program.What did it involve? Research-driven textbooks, a prescriptive national curriculum, and mandated methods of instruction across primary schools in Singapore. 13 years later, Singapore jumps 15 places in the TIMMS rankings. The curriculum and the textbooks continue to be revamped, and Singapore’s scores keep on improving.

The dominance of ‘Singapore Maths’ should be well known to all UK teachers: in this last year, it has reared its head on our shores and even onto national news. So, there is a massive change in educational outcomes following a different curriculum and high-quality prescribed textbooks.

It’s a very similar story in Shanghai. Teachers across the city use the same expertly-created textbooks:

Shanghai mathematics teaching is based upon high-quality teacher resources. All schools follow the same textbook, which is published by the Shanghai education commission and refined and revised on an annual basis. Compare this with English schools, where, according to the TIMMS international survey, only 10% of mathematics teachers used textbooks as a basis for their teaching.

Interestingly, their textbooks are what you might call crowd-sourced; as Tim Oates writes in his brilliant paper on textbooks, they ‘are based on accumulated theory in maths education, are written and edited by expert authors, and constantly are supplemented by ‘adjustments’ from teacher-research groups. These teacher-research groups exist across the school system. Competitions are held, whereby ‘top’ adjustments are routinely fed through into the texts.’ What a great idea, and one for us to think about as well.

It’s a very similar story with Jaime Escalante, as his own words quoted above show; he spent a huge time comparing textbooks (which he describes as the ‘cornerstone‘ of his program), demanding the money for them. Moreover, where he felt they weren’t good enough, he supplemented them and even started making his own.

From these successes, I believe textbooks are the best answer to the crippling problem of when students ‘don’t get it’, the second signature problem I identified in my last post. As I see it, this problem has two interlinked causes: students have problems understanding because of missing mathematical foundations, and students have problems understanding because we little coherent nor specific ideas of the best methods for teaching each topic in maths. And the root of these causes is the fact that we maths teachers in the UK don’t have the best maths pedagogy at our fingertips, in a a readily accessible and classroom friendly way.

So, how can mere textbooks make such a crucial difference?

# Is this the best we can do? Part 3: success stories

There isn’t a problem in teaching or learning that someone somewhere hasn’t solved. We just need to find them and take some field notes.

– Doug Lemov

This is part of a series offering my opinion on some problems with UK maths education. The first part looked at the state of affairs with regards to GCSE and PISA results, and the second part looked at my attempt at a diagnosis. This post addresses where to find potential solutions.

Adam Creen responded to my first two posts in this series with this correct observation:

This is true. After all, many agreed with my diagnosis and could identify with my experiences, but no-one offered any solutions. If the problem is widely recognized but there’s no readily acknowledged solution, then any solution must not be straightforward.

But: it doesn’t mean that no solutions exist. In fact, I’m optimistic they are out there. Why?

When I heard Doug Lemov of Teach Like a Champion speak briefly on his work, my abiding memory is of his deeply insightful focus: there isn’t a problem in teaching or learning that someone somewhere hasn’t solved. We just need to find them and take some field notes.’

So, in Lemov’s spirit, this post will begin investigate: what education regions/systems/schools seem to have solved these problems with maths education? Following posts will then address: how do they do it? Do they address the UK-specific issues I identified in part 2? And how can we transfer these lessons to the UK, to our schools, to our classrooms?

# Is this the best we can do? Part 2: Diagnosis

‘A correct diagnosis is three-fourths the remedy.’

– Gandhi

This is the second post in a series. In Part 1, I reflected on the vast numbers of 16 year olds who finish 11 years of maths education unable to answer the straightforward questions needed for a grade C. I argued that this is and continues to be a problem, despite rising C-grade pass rates over the past 20 years. This post reflects more deeply on the causes behind why so many 16 year olds are in this state.

Matthew Syed’s ‘Black Box Thinking’ is a great read. He examines extraordinary successes in various fields like medicine, healthcare, aviation safety – like Team Sky, or Google – to find what they have in common. The answer? Such successes ‘harness the power of failure’. They don’t shy away from mistakes, but rather expect them, learn from them, and milk them for all the information they are worth. The aviation industry does this extremely well:

Pilots are generally open and honest about their own mistakes (crash-landings, near misses). The industry has powerful, independent bodies designed to investigate crashes. Failure is not regarded as an indictment of the specific pilot who messes up, but a precious learning opportunity for all pilots, all airlines and all regulators.

Some examples from Syed that bear this culture out: flight accidents are automatically transferred to independent investigators to look into, and those involved in the flight are protected to full disclosure, as whatever they say is inadmissible in court. The report is then circulated and freely available to any pilot in the entire world, ensuring that the entire industry can learn from the accident. (This is a striking comparison to the education sector in this country– but that would require another blogpost in itself…)

As Syed goes on to explain, this culture is particularly useful since the dissemination and circulation of information on failures can quickly reveal ‘signatures’ – particular patterns that keep recurring in various mistakes, problems, and accidents. For example, in one week in 2005, a whole host of reports of near-misses came from pilots landing in Lexington Airport. The investigators quickly cottoned onto the problem from the info from the pilots’ reports: lights had just been installed on an adjacent plot of land, which was then being mistaken for the runway. Within days (apparently, quite slow for the industry – the adjacent land didn’t belong to the airport) the confusing lights were taken down.

In the spirit of this ‘black box thinking’, I’ve thought about what problems and mistakes I frequently observe in my own classroom, together with the mistakes and honest frustrations I hear from colleagues across various schools. I’ve focused particularly on my experience with bottom sets, since it is these groups that usually fail to make grade C (though my conclusions may be more widely true too). From the mistakes with these classes, what are the ‘signatures’ – the common recurring patterns? What are the issues I keep experiencing and hearing about from other teachers? And, in the following post – what way forward can be discerned from these signatures, from these diagnoses?

# Is this the best we can do? Part 1: the problem

In 2014, 62.4% of teenagers sitting GCSE maths achieved at least a grade C – the ‘gold standard’ grade needed to progress onto a wide range of further study and employment. For the last 5 years, the pass rate has hovered around 60%.

The flipside of this statistic: every year, at least 40% – that’s a quarter of a million 16-year olds – complete their secondary education without having attained a C in GCSE maths.

Is this the best we can do?

I looked at the 2014 GCSE Higher maths papers. To have gotten a grade C in it, you needed 57 marks out of 200 (a statistic that shocked me when I first discovered it, and continues to shock). Below are 63 marks’ worth of questions from 2014’s two papers (click to enlarge) – if you got all of these questions right, give or take 2-3 of them, you would’ve done enough to get that coveted grade C.

To repeat: in 2014, as in other years, at least a quarter of a million 16-year olds finished 11 years of compulsory education unable answer all these questions.

Is this the best we can do? I hope not. Here’s some preliminary reflections and thoughts on this state of affairs.