# Mathematics teaching in China: reflections from an Ofsted HMI

By Sean Harford HMI, National Director, Initial Teacher Education, Ofsted

Reposted from TES Connect.

In late February I was a member of a delegation representing HM Government that visited the three Chinese provinces of Shanghai, Beijing and Hubei with a specific focus on mathematics education.

I have waited until now to reflect on my visit to China because I wanted to go back into some English schools to test out the thinking I developed while there. The differences in maths outcomes for our young people between the two countries are stark and worrying for us, unless we act now to catch up – and I do not mean just in terms of PISA test scores. I am coming at this not only from an inspector’s point of view, but also from my background of being a physics teacher and so frequent user of maths, reliant on pupils being able to handle and manipulate numbers confidently. In this respect, Chinese children are streets ahead of ours, so the benefits of their high standards in mathematics go way beyond just this core subject.

As everyone knows, Her Majesty’s Inspectors are not concerned about the ‘how’ but ‘how effective’ with teaching. This approach requires a clear focus on the outcomes for the pupils and their response to the teaching, including crucially the evidence of learning and progress over time in their work books and folders. These were impressive in the classes we observed in China, and told a story of a consistency of approach and expectations that has led to the pupils being confident mathematicians, willing to have a go and able to tackle problems in different contexts.

For example, given this problem…:

X = 2√ (7/14 x 28/7 x 3/9 x 24/8 x 18/9)

… none of the 12-year-old pupils reached for the calculator; they couldn’t because they have been banned from their classrooms. They calmly looked for the potential to cancel and reduce the fractions, and spotted that this expression is really just the square root of 4. Not a job for the calculator; not for them at least. This was clearly not about them learning ‘tricks’ either. This problem was one of just 4 or 5 set by the teacher in a 5 minute burst of practice, to help the pupils master the concepts covered by her in the latest part of the lesson before they moved on confidently together to the next stage of increasingly challenging maths. The key was not the teacher’s ‘performance’ in this lesson, but the demonstration of the depth of the pupils’ mathematical learning over time and the impressive armoury of knowledge and skills they had built up to deploy as and when needed. Evidence of solidly knowing their times tables was absolutely apparent across the pupils, as was the ability to use efficient methods of calculation without having to really think. Their mathematical toolkit was there to be used as surely as a mechanic’s spanners, or a surgeon’s scalpel

Read the rest at TES Connect.

# “5-Year-Olds Can Do Calculus” by Maria Droujkova -collecting comments

What if we figured out ways for young kids to play with ideas from calculus, algebra, and other mathematical subjects beyond arithmetic? Last week, The Atlantic published Dr. Maria Droujkova’s interview, “5-Year-Olds Can Do Calculus” by Luba Vangelova. It started a broad international discussion, with follow-up interviews by Canada’s “Globe and Mail” and UK’s “The Times,” and translations into Japanese and Russian by news agencies. Droujkova and her colleagues at Natural Math are aggregating major themes from the comments:
• How can we create and sustain environments where kids are free to learn, and adults are free to help them?
• Can young children understand abstractions? Can they deal with the formal language of mathematics? If they can, will it hurt their development in some way?
• Many grown-ups believe that young math will finally give them a second chance at making sense of algebra and calculus.
• But what about calculating and memorizing? We need more research on balancing concepts and technical skills.
• What can young kids actually do with algebra or calculus? How can they play with these ideas, or apply them to their daily lives?
• Many people recognized our activities as similar to what they are doing with their kids – or what their parents did with them. What difference does this casual, everyday early math make for kids whose parents understand and love mathematics?

# Howdy!

Alexandre Borovik invited me to join the writers here at The DeMorgan Forum. (Thank you, Alexandre. I am honored.) He got me started by reposting my Top Ten Issues in Math Education post, which I have now edited. (There were links to dead blogs and disappeared posts, mentions of ‘last week’ four years ago, and opinions I’ve changed.)

I blog most often at Math Mama Writes, and will bring some of my favorite posts from there over here. If you want to hear more from me, please follow me there.

# My Top Ten Issues in Math Education

[Originally posted at Math Mama Writes. Revised for The DeMorgan Forum.]

10. Textbooks are trouble. Corollary: The one doing the work is the one doing the learning. (Is it the text and the teacher, or is it the student?)
Hmm, this shouldn’t be last, but as I look over the list they all seem important. I guess this isn’t a well-ordered domain. A few years back I read Textbook Free: Kicking the Habit, an article by Chris Shore on getting away from using a textbook (unfortunately no longer available online). I was inspired to take charge of my teaching in a way I really hadn’t before. Now I decide how to organize the course. I still use the textbook for its homework repositories, but I decide on my units and use the text as a resource. See dy/dan on being less helpful (so the students will learn more), and Bob Kaplan on becoming invisible.

9. Earlier is not better.
The schools are pushing academics earlier and earlier. That’s not a good idea. If young people learn to read when they’re ready for it, they enjoy reading. They read more and more; they get better and better at it; reading serves them well. (See Peter Gray‘s post on this.) The same can happen with math. Daniel Greenberg, working at a Sudbury school (democratic schools, where kids do not have enforced lessons) taught  a group of 9 to 12 year olds all of arithmetic in 20 hours. They were ready and eager, and that’s all it took.

In 1929, L.P. Benezet, superintendent of schools in Manchester, New Hampshire, believed that waiting until later would help children learn math more effectively. The experiment he conducted, waiting until 5th or 6th grade to offer formal arithmetic lessons, was very successful. (His report was published in the Journal of the NEA.)

8. Real mathematicians ask why and what if…
If you’re trying to memorize it, you’re probably being pushed to learn something that hasn’t built up meaning for you. See Julie Brennan’s article on Memorizing Math Facts. Yes, eventually you want to have the times tables memorized, just like you want to know words by sight. But the path there can be full of delicious entertainment. Learn your multiplications as a meditation, as part of the games you play, …

Just like little kids, who ask why a thousand times a day, mathematicians ask why. Why are there only 5 Platonic (regular) solids? Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (A question asked and answered by James Tanton in this video.) Why does the anti-derivative give you area? Why does dividing by a fraction make something bigger? Why is the parallel postulate so much more complicated than the 4 postulates before it? Then came “What if we change that postulate?” And from that, many non-Euclidean geometries were born.

7. Math itself is the authority – not the curriculum, not the teacher, not the standards committee.
You can’t want students to do it the way you do. You have to be fearless, and you need to see the connections. (Read this from Math Mojo.)

6. Math is not arithmetic, although arithmetic is a part of it. (And even arithmetic has its deep side.)
Little kids can learn about infinity, geometry, probability, patterns, symmetry, tiling, map colorings, tangrams, … And they can do arithmetic in another base to play games with the meaning of place value. (I wrote about base eight here, and base three here.)

5. Math is not facts (times tables) and procedures (long division), although those are a part of it; more deeply, math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.
My favorite math ed quote of all time comes from Marilyn Burns: “The secret key to mathematics is pattern.

U.S. classrooms are way too focused on procedure in math. It’s hard for any one teacher to break away from that, because the students come to expect it, and are likely to rebel if asked to really think. (See The Teaching Gap, by James Stigler.)

See George Hart for the artform. The language of math is the language of logic. Check out any Raymond Smullyan book for loads of silly logic puzzles, and go to islands full of vegetarian truthtellers and cannibal liars. Or check out some of Tanya Khovanova’s posts.

4. Students are willing to do the deep work necessary to learn math if and only if they’re enjoying it.
Which means that grades and coercion are really destructive. Maybe more so than in any other subject. People need to feel safe to take the risks that really learning math requires. Read Joe at For the Love of Learning. I’m not sure if this is true in other cultures. Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons. (Perhaps they feel stressed but safe. Are they enjoying it?)

3. Games are to math as picture books are to reading – a delightful starting point.
Let the kids play games (or make up their own games) instead of “doing math”, and they might learn more math. Denise’s game that’s worth 1000 worksheets (addition war and its variations) is one place to start. And Pam Sorooshian has this to say about dice.  Learn to play games: Set, Blink, Quarto, Blokus, Chess, Nim, Connect Four… Change the rules. Decide which rules make the most interesting play.

Besides games, consider puzzles, cooking, building, science, programming, art, math stories, and math history for ways to bring meaningful math into your lives. (Here’s a list of good games, puzzles, and toys.)  If you play around with all those, you can have a pretty math-rich life without ever having a formal math lesson.

2. If you’re going to teach math, you need to know it deeply, and you need to keep learning.
Read Liping Ma. Arithmetic is deeper than you knew (see #6). Every mathematical subject you might teach is connected to many, many others. Heck, I’m still learning about multiplication myself. In a blog conversation (at a wonderful blog that is, sadly, gone now), I once said, “You don’t want the product to be ‘the same kind of thing’.  …   5 students per row times 8 rows is 40 students. So I have students/row * rows = students.” Owen disagreed with me, and Burt’s comment on my multiplication post got me re-reading that discussion. I think Owen and I may both be right, but I have no idea how to do what he suggests and use a compass and straightedge to multiply. I’m looking forward to playing with that some day. I think it will give me new insight.

1. If you’re going to teach math, you need to enjoy it.
The best way to help kids learn to read is to read to them, lots of wonderful stories, so you can hook them on it. The best way to help kids learn math is to make it a game (see #3), or to make dozens of games out of it. Accessible mysteries. Number stories. Hook them on thinking. Get them so intrigued, they’ll be willing to really sweat.

That’s my list. What’s yours?
What do you see as the biggest issues or problems in math education?

[You may also enjoy reading the discussion my original post prompted back in 2010.]

# Play this book: “Moebius Noodles”

Moebius Noodles. Adventurous Math for the Playground Crowd

Text: Yelena McManaman and Maria Droujkova
Illustrations and design: Ever Salazar
Copyedits: Carol Cross

This brilliant book is published under Creative Commons Attribution-NonCommercial-ShareAlike license, and this allows me to reproduce the entire Introduction:

Why Play This Book

Children dream big. They crave exciting and beautiful adventures to pretend-play. Just ask them who they want to be when they grow up. The answers will run a gamut from astronauts to zoologists and from ballerinas to Jedi masters. So how come children don’t dream of becoming mathematicians?
Kids don’t dream of becoming mathematicians because they already are mathematicians. Children have more imagination than it takes to do differential calculus. They are frequently all too literate like logicians and precise like set theorists. They are persistent, fascinated with strange outcomes, and are out to explore the “what-if” scenarios. These are the qualities of good mathematicians!

As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary, and dangerous than sailing on high seas!

But most of the time math is not presented this way. Instead, children are required to develop their mathematical skills rather than being encouraged to work on something more nebulous, like the mathematical state of mind. Along the way the struggle and danger are de-emphasized, not celebrated – with good intentions, such as safety and security. In order to achieve this, children are introduced to the tame, accessible scraps of math, starting with counting, shapes, and simple patterns. In the process, everything else mathematical gets left behind “for when the kids are ready.” For the vast majority of kids, that readiness never comes. Their math stays simplified, impoverished, and limited. That’s because you can’t get there from here. If you don’t start walking the path of those exotic and dangerous math adventures, you never arrive.

It is as tragic as if parents were to read nothing but the alphabet to children, until they are “ready” for something more complex. Or if kids had to learn “The Itsy-Bitsy Spider” by heart before being allowed to listen to any more involved music. Or if they were not allowed on any slide until, well, learning to slide down in completely safe manner. This would be sad and frustrating, wouldn’t it? Yet that’s exactly what happens with early math. Instead of math adventures – observations, meaningful play, and discovery of complex systems – children get primitive, simplistic math. This is boring not only to children, but to adults as well. And boredom leads to frustration. The excitement of an adventure is replaced by the gnawing anxiety of busy work.

We want to create rich, multi-sensory, deeply mathematical experiences for young children. The activities in this book will help you see that with a bit of know-how every parent and teacher can stage exciting, meaningful and beautiful early math experiences. It takes no fancy equipment or software beyond everyday household or outdoor items, and a bit of imagination – which can be borrowed from other parents in our online community. You will learn how to make rich mathematical properties of everyday objects accessible to young children. Everything around you becomes a learning tool, a prompt full of possibilities for math improvisation, a conversation starter. The everyday world of children turns into a mathematical playground.
Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry, and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive. Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.

Can math be interesting? A lot of it already is! Can your children be strong at advanced math? They are natural geniuses at some aspects of it! Your mission, should you accept it: to join thrilling young math adventures! Ready? Then let’s play!

# Correlation for schoolchildren

A few comments on MEI‘s draft “Critical Maths” Curriculum. They list

Glossary of terms which students are expected to know and be able to use [...]

Association: A tendency for two events to occur together.

Correlation: An association between two variables which is approximately linear.

This definition of correlation seems rather odd.  If $$y = x^2$$  aren’t $$x$$ and $$y$$  correlated?   What does “an association” mean here?  The suggested definition of association given above is for events, not “variables”.   Presumably the authors have in mind random variables.
There is a serious problem here in the use of language.  It needs to be made clear whether the notion being described is an intuitive one or a mathematical definition. I am not a statistician, but it seems to me that there are (at least) three common distinct types of usage of the word “correlation”,  none of which is captured by the “definition” proposed:
(1)  The vernacular usage. The  Merriam-Webster dictionary gives
”a relation existing between phenomena or things or between mathematical or statistical variables which tend to vary, be associated, or occur together in a way not expected on the basis of chance alone”
which seems to me a reasonable description of the vernacular or intuitive non-mathematical meaning of the term.    This is clearly much broader than the meaning suggested above.
(2)  The intended meaning proposed seems to correspond closest to the use of the  (Pearson) correlation coefficient  in statistics, although even then it is not  accurate, since  the correlation coefficient is not always a  reliable indicator of the existence of a linear relationship.   This meaning is that which tends to be used by a large class of people who have had some minimal exposure to statistics.
(3)  More generally correlation can be used to indicate a variety of mathematical measures of probabilistic interdependence  (e.g. mutual information).
On a separate point the very heavy concentration on statistical reasoning to the exclusion of other mathematics (including perhaps more elementary logical reasoning such as manipulation of quantifiers and logical connectives) rather worries me, since it may encourage the idea that  almost the only practical applications of mathematics are statistical.
Another  serious danger in my opinion is that statistics at this level tends to be more  like cookery than mathematics and it would have to be extremely well taught by a gifted and highly educated teacher if  conceptual precision is not going to be completely lost.  The danger is partially raised by Gowers in Objection 5 listed in his blog (though he doesn’t mention cookery), but I think his own answer is rather optimistic.
Somewhat in this connection there is an interesting passage in Noam Chomsky on Where Artificial Intelligence Went Wrong where Noam Chomsky is interviewed on various topics concerning science, in particular AI and  cognitive science, and what he clearly regards as a modern deviation from the classical scientific method, which has been indirectly caused by the power of modern computers .  The article is quite long, but I found his example of “how to justify the abolition of physics departments” very nice;  it could  equally well used to justify closing down everything in mathematics departments except statistics.

# Nesin Mathematics Village

The Nesin Mathematics Village is a small village of  about 13,5 acres, approximately 7,5 of which consist of olive groves. It is owned by the Nesin Foundation and is located 1 km away from the village of Şirince (tied to the Selçuk district of Izmir). Perched on a hillside and overflowing with greenery, it is a place where young and old learn, teach, and think about mathematics in peaceful remoteness. Unpretentious and unostentatious, the houses made out of rock, straw and clay give off a simple welcoming air.

Apart from the crickets, any factors which could prevent concentration and deep thought are kept away, there are no televisions, no music is publicly broadcasted. But traces of civilization such as electricity, warm water and wireless internet are nonetheless present. There is no shortage of insect life!

Most activities take place in the summer months; however in spring and autumn it is also an ideal environment for various types work groups, meetings and rest. It could for example be used as a place for an alumni reunion, a honeymoon in the “wild” or a mathematics workshop.

From teaching at the primary school level to the most advanced research, mathematical activities of any level can take place simultaneously at the village.

We now have the capacity to lodge 150 people, but there is the possibility of pitching tents if more capacity is required.

# Why I an teaching a math circle

At some point, I have compiled a short list of reasons why I get a lot of satisfaction from teaching a math circle.  I love:

• -the equality and feeling of mutual respect and attention that develops between me and math circle participants
• the democracy/lack of authority that shows us the “right answer”
• seeing the  value alignment and  deep intellectual friendship that develops among the participants
• sharing children’s excitement when they realize their own powers
• the feeling of freedom they develop when they get rid of their own mental blocks
• the intellectual stimulation of choosing the problems and personalizing and  teaching them to a particular audience
• when children  realize that they feel happy from doing a challenging job
• observing their self-discovery
• observing as children come up with amazing solutions and counter-intuitive discoveries
• getting a fresh view of the beauty and awesomeness of the world we observe and create  - thus multiplying my own happiness

# Tony Gardiner: National curriculum – Comments and suggested necessary changes

Published today:

From the Introduction:

The Education Order 2013 was “made” on 5 September 2013. The relevant details were “laid before parliament” on 11 September 2013, and will come into effect on 1 September 2014. Some of the details for GCSE were published on 1 November 2013. Further elaboration of GCSE assessment structure, and curriculum guidance for Key Stage 4 (Years 10–11, ages 14–16) are awaited.

It is generally agreed that the curriculum review process adopted over the last 3–4 years has been seriously flawed. Those involved worked hard, often under very difficult conditions. But the overall approach (of relying on civil servants and drafters whose responsibilities and constraints remained inscrutable) has merely demonstrated that drafting and maintaining curricula is a specialist task, requiring dedicated professionals with specialist experience.

Whatever flaws there may have been in the process, we will all have to live with the new curriculum for some years. So it is important to have an open discussion of the likely difficulties. This article is an attempt to indicate aspects of the National curriculum in England: mathematics programmes of study that will need to be handled with considerable care, and revised in the light of experience.

After three years of widespread unease about the process of the curriculum review and its apparent direction, it is remarkable that there has been almost no media coverage, and no clear professional response to the final mathematics programmes of study for ages 5–14. There is therefore a real danger that insights that emerged along the way will simply be forgotten, and that the same mistakes may then be made next time. [...]

The details laid before parliament are statutory’; but they incorporate basic flaws, and significant contradictions between the statutory list of content (which could all-too-easily be imposed uncritically) and the declared over-arching “aims” (which could get forgotten, or ignored). Given these flaws, the fate of the new programmes of study will depend on how sensitively their implementation is handled—whether slavishly, or intelligently. Teachers—and Ofsted, senior management, etc.—need to be alert to those aspects of the stated programmes of study that incorporate predictable pitfalls.

We summarise here what seem to be the two most important flaws.

Some material in Key Stage 1 and 2 is very poorly specified (especially from Year 4 onwards).

Some items are listed unnecessarily and unrealistically early, and so may be introduced at a stage:

• where they are not yet needed,
• where they will not be understood,
• where they will be badly taught, and
• where – if the relevant requirements were relaxed – the premature material could easily be delayed without causing any subsequent problems.

The listing of content for Key Stage 3 is in some ways reasonable, but too many things are left implicit. The programme of study is less structured than, and contains less detail than, that for Key Stages 1 and 2. Hence the details of the Key Stage 3 programme need interpretation. At present:

• the words of each bullet point are rarely elaborated;
• the connections between themes are mostly suppressed; and
• there is no mention of essential preliminaries.