Consultation on Key Stage 4 mathematics

The government response to consultation for key stage 4 English and mathematics on December 2nd 2013 can be found here; pdf.

DfE are now consulting on the draft Order and Regulations that will give effect to the new programmes of study for English and mathematics at key stage 4 from September 2015 and to extend the disapplication of the key stage 4 science programme of study for a further school year (2015/16).

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.

33rd MATHEMATICS TEACHERS AND ADVISERS CONFERENCE/WORKSHOP

33rd MATHEMATICS TEACHERS AND ADVISERS CONFERENCE/WORKSHOP
Friday 27th June 2014 13.00-17.00 – No registration fee

The 33rd Mathematics Teachers and Advisers Conference/Workshop provides an interface between the School of Mathematics at the University of Leeds and teachers in schools and sixth forms.

Teachers and university staff alike are given a rare opportunity to exchange valuable experiences and re-invigorate their perspectives on the ever-changing world of mathematics education.

Please book the date of 27th of June 2014 in your diary and attend the event.

If you have not done already so, in order to register, simply JUST SEND an EMAIL to:

D. Lesnic >>at<< leeds.ac.uk

and give your name, name of the school and email.

Programme:

Julian Gilbey (University of Cambridge) “Cambridge Mathematics
Education Project”

Currently in the development phase, the project will provide innovative online resources to help support and inspire teachers and students of A-level  mathematics. The aim is to help to make sixth-form  mathematics a rich, coherent and stimulating experience for students and teachers. Join to get a preview of the web site, and to work together on some of the new A-level resources.

David Kaplan (Royal Statistical Society Centre for Statistical Education at Plymouth University) “SAS Curriculum Pathways”

Plymouth University has endorsed SAS Curriculum Pathways as a free-to-use online teaching and learning resource in order to promote the uptake of STEM subjects in further and higher education. The resource has been developed in the US over a number of years and has been successful for three main reasons:

(i) Commitment to Teachers. SAS Curriculum Pathways works in the classroom in large part because teachers have shaped every phase of the planning and production process.

(ii) Focus on Content. Teachers, developers, designers, and other specialists clarify content in the core disciplines. Content difficult to convey with conventional methods is tageted topics where doing and seeing provide information and encourage insights in ways that textbooks cannot.

(iii) Approach to Technology. SAS Curriculum Pathways makes learning more profound and efficient, not simply more engaging. Audio, visual, and interactive components all reinforce the learning objectives identified by teachers. It stands apart from other online resources becuase of its interactive nature students obtain immediate feedback. The resource promotes subject specific terminology and leads students through sometimes difficult methods in a structured way. http://www.sascurriculumpathways.com/portal

Sue Pope (Chair of the General Council of the Association of Teachers
of Mathematics) -“Post-16 Mathematics Opportunities and Challenges”

Despite increasing numbers of students studying level 3 Mathematics, England is remarkable in its low participation rates. The government is committed to increasing participation, yet will we have a curriculum and associated qualifications to do this? Will linear A levels, core maths, critical maths (MEI Gowers’-inspired) and other qualifications in development fit the bill? Have policy makers learnt from Curriculum 2000, or the Mathematics Pathways project? How do we ensure students have the mathematical skills to thrive whatever their future? And what are those skills?

“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?
Some discussion and follow-up links:

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.

Brain finds true beauty in maths

From BBC: Brain finds true beauty in maths. A quote:

Brain scans show a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers.

Mathematicians were shown “ugly” and “beautiful” equations while in a brain scanner at University College London.

The same emotional brain centres used to appreciate art were being activated by “beautiful” maths.

The researchers suggest there may be a neurobiological basis to beauty.

The study in the journal Frontiers in Human Neuroscience says,in partucular, that

The formula most consistently rated as beautiful (average rating of 0.8667), both before and during the scans, was Leonhard Euler’s identity

\(1+e^{i\pi}=0\)

which links 5 fundamental mathematical constants with three basic arithmetic operations, each occurring once; the one most consistently rated as ugly (average rating of −0.7333) was Srinivasa Ramanujan’s infinite series for 1/π,

\(\frac{1}{π}=\frac{2\sqrt{2}}{9801}\sum_{k=0}^\infty  \frac{(4k)!(1103+26390k)}{(k!)^4\cdot 396^{4k}}\)

which expresses the reciprocal of π as an infinite sum.

Other highly rated equations included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler’s formula for complex analysis, and the Cauchy-Riemann equations. Formulae commonly rated as neutral included Euler’s formula for polyhedral triangulation, the Gauss Bonnet theorem and a formulation of the Spectral theorem. Low rated equations included Riemann’s functional equation, the smallest number expressible as the sum of two cubes in two different ways, and an example of an exact sequence where the image of one morphism equals the kernel of the next .

- See more at: http://journal.frontiersin.org/Journal/10.3389/fnhum.2014.00068/full#sthash.7b7Pdf5a.dpuf

 

EMPG 2014: Call for papers

You are cordially invited to attend the 2014 European Mathematical Psychology Group Meeting (EMPG 2014), held at the University of Tübingen, Germany, from Wednesday, July 30, 2014 until Friday, August 01, 2014.

Presentations

Proposals for paper and poster presentations as well as proposals for symposia related to all aspects of mathematical psychology are welcome. Relevant topics include:

  • perception and psychophysics
  • models of cognition and learning
  • knowledge structures
  • measurement and scaling
  • psychometrics
  • computational methods
  • statistical methods
  • mathematical models

Important dates

  • Abstract submission opens:  soon
  • Abstract submission closes:  April 30, 2014
  • Notification of acceptance:  May 15, 2014
  • Early registration deadline:  June 06, 2014
  • Start of conference: July 30, 2014
  • End of conference: August 01, 2014

Invited Symposia

Symposium in honor of Jean-Claude Falmagne celebrating his 80th birthday (organized by Michel Regenwetter and Jean-Paul Doignon).

Invited speakers

Andrew Heathcote, University of Newcastle, Australia
Ehtibar Dzhafarov, Purdue University, USA

For further information please consult the website of the EMPG 2014 (www.uni-tuebingen.de/psychologie/empg2014).

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.]