Why Do Americans Stink at Math?

An article by in the NYT; it is about America, but is very timely in the context of the National Curriculum reform in England. A quote:

It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.

The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them. [Emphasis is mine -- AB.]

Read the whole article.

 

 

Can you pass the maths test for 11-year-olds?

A sample KS2 test based on the official publication from Standards and Testing Agency,
2016 key stage 2 mathematics test: sample questions, mark scheme and commentary,
was published in The Telegraph. One question attracts attention. In The Telegraph version, it is

A question as published in The Telegraph.

The answer given is £12,396.

And this is the original question from 2016 key stage 2 mathematics test: sample questions, mark scheme and commentary

The official version of the same question

In my opinion, both versions contain serious didactic errors. Would the readers agree with me?

And here are official marking guidelines:

Official marking guidelines

And the official commentary:

In year 6 pupils are expected to interpret and solve problems using pie charts. In this question pupils can use a number of strategies including using angle facts or using fractions to complete the proportional reasoning required.
Pupils are expected to use known facts and procedures to solve this more complex problem. There are a small number of numeric steps but there is a demand associated with interpretation of data (or using spatial knowledge). The response strategy requires pupils to organise their method.

The Math Myth

D. Edwards,   The Math Myth, The De Morgan Gazette 5 no. 3 (2014), 19-21.

Abstract

I’ve been concerned with what skills those who are working as scientists and engineers actually use. I find that the vast majority of scientists, engineers and actuaries only use Excel and eighth grade level mathematics. This suggests that most jobs that currently require advanced technical degrees are using that requirement simply as a fi lter.

[A version of this text appeared in the August, 2010 issue of The Notices of The American Mathematical Society.]

Nick Gibb: “Join our battle against progressive educationalists”

in The Guardian:

As an ex-schools minister I see value in the unions. But they are wrong not to join our battle against progressive educationalists. [...] This might seem like an odd thing for a Conservative MP and former schools minister to say, but teaching unions are not the problem with our schools. [...]

[...] who is to blame for our education system slipping down the international rankings? The answer is the academics in the education faculties of universities. This is where opposition to the use of phonics in the teaching of young children to read lies, despite vast evidence from this country and other English-speaking countries that systematic synthetic phonics is the most effective and successful method.

Within these education departments lie the proponents of so-called progressive education, which advocates that education should be child-led rather than teacher-led; many advocate a play-based classroom until children are seven years old. It is an approach that espouses learning by discovery rather than having teachers directly teaching children. For decades many education academics downplayed the importance of spelling, punctuation and grammar. Textbooks are regarded by many in the education departments as appalling teaching tools, and in the 1970s they virtually disappeared from primary schools. Progressive educationalists oppose testing and believe that a knowledge-rich education is pointless in the Google age.

It is challenging the hegemony of the education departments of the universities that must be the focus of any serious education reformer and anyone who believes, as Gove does, that the attainment gap between those from poorer and wealthier backgrounds needs to be closed. There are many in the teaching profession who share this view. There are many on the left who hanker for the type of education provided in the independent sector – largely untainted by the progressive ideology of the education faculties – but who want their children educated by the state. They, too, should be railing against these educationalists.

 

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.

An email to Elizabeth Truss MP (waiting for a reply)

Dear Ms Truss,

I am a Secondary Mathematics Specialist Leader of Education and was lucky enough to be amongst the group of teachers who travelled to Shanghai this January. I am very pleased to hear that you made it to Shanghai this week to see for your self how teachers and pupils work.

I am disappointed that several crucial facts seem to have been overlooked in the reports I have read about your visit so far.

Firstly, I agree that there are lessons to be learnt from the Shanghai model of education. I was thoroughly impressed by the professionalism and commitment of both pupils and teachers when I visited China (although I saw no teaching in Shanghai itself).

Teachers collaborate to produce lessons and worksheets of an extremely high quality. They carefully chose the best questions that contain a new idea or adaptation to a demonstrated problem. The worksheets quickly move students through a series of challenges and this “imitation” was mentioned a number of times as a reason behind pupils success. I also saw examples of multiple choice homework sheets where every question was a hinge question (as defined by Dylan Wiliam). This climate of not asking questions for the sake of it, to fill time or to simple practice things again and again was refreshing. I noticed that lessons were always pitched at the highest level. Hence ‘extension’ activities were very rarely (if ever) needed. Instead pupils who didn’t understand had to seek help outside of normal lesson time (for which they had the self-motivation).

This teacher collaboration and teaching to the top, rather than the middle, is something I am developing following the visit.

However, I hope whilst observing these kinds of ideas that you have also taken careful note of the lesson commitments of maths teachers in Shanghai. It’s all very well saying that we need to adopt ideas from Shanghai but I very strongly believe that the fundamental reason behind their success is the huge amount of time they have to plan, prepare and reflect. Every teacher we spoke to taught no more than two lessons a day (many had those two lessons with the same class). Teachers plan lessons together, reflect on their pupils learning together and are able to give same day feedback to pupils. Every single maths teacher was a subject specialist from primary through. ‘Weak’ teachers don’t seem to exist due to this careful joint planning, reflection and support. As a previous AST, head of department and assistant head, current SLE and as someone who runs workshops (KS2 – KS5) for teachers around the country the biggest barrier to teachers working in a similar way is there are simply not enough hours in the day and not enough teachers to teach the classes (even if we made classes larger).

There are cultural differences that mean many Chinese students have different attitudes towards maths and family support that many students I teach do not. However, I strongly feel that if we collaborated more, developed suitable resources (not necessarily along the Shanghai designs) to suit our students, understanding and results would improve. Through this collaboration we would be able to support those teachers not comfortable with their mathematics and meet the needs of our pupils.

I realise that this doesn’t fit a nice easy (and cheap) way to solve the issue of problems in maths education that you are searching for but it would be the right thing to improve results and mathematical understanding. It’s also not a short term commitment.

I have worked with poor teachers and teachers who do not have commitment to the pupils they teach. However the very large majority of teachers I have ever worked with have wanted the very best for their pupils, have tried to teach to the best of their ability and tried to produce stimulating and challenging resources. Sadly a large number of these have also suffered with stress, depression and anxiety. Many have also left the profession as it was simply too much. I myself have had moments where I have doubted my ability, considered a different career but I cannot imagine ever not being a teacher.

shanghai night

I would be keen to discuss this with you further, if you have an interest.

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 $latex y = x^2$  aren’t $latex x$ and $latex y$  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.

Rebecca Hanson: 2014 Primary Mathematics Curriculum is Not Fit for Purpose

Re-posted from Authentic Maths.

Rebecca Hanson:

Following the writing of my first report on the 2014 Primary National Curriculum in December I have been corresponding with the key people involved in its development.

As a results of their comments I have written a second report which calls for the immediate suspension of the implementation of the 2014 Primary National Curriculum for Mathematics. This new report dated 6 Jan 2014 can be downloaded here: Fundamental Problems with the 2014 Primary National Curriculum for Mathematics.

The press release which accompanies this report can be downloaded here:
Call for Suspension of New Primary Mathematics Curriculum.

6 JAN FAULT: If you experience problems downloading the report an alternative version (without hyperlinks) can be downloaded from the Times Educational Supplement site.