A Maths Test

One of the wonders of the world is why math reform in the US has never seriously happened. It's as if traditional maths is part of the original US Constitution that cannot be changed. Recent commentaries such as this one in the NYTimes  and this one also in the NYTimes  begin to shed light on just what it is that causes this state of affairs. In spite of the overwhelming research evidence (e.g. Jo Boaler) that helping children understand math and not just memorize it is by far the best approach there are constant cries for "back to basics" and the traditional approach of memorizing facts and formulas.
So, for everyone who thinks this return to basics is the way to go here's a test to see how well you know the math that children  are expected to know in 2015.

1. What is counting?
2. What is addition?
3. What is multiplication?
4. What is division?
5. What is Pi?
6. What does the 0 mean in 308?
7. What is 1/2 divided by 1/4?
8. Make up a word problem for # 7 above.
9. What do  you get when you reduce 4/8?
10. Why do you  put a comma after every three digits in a large number?
11. What is area?
12. What is a degree measure in geometry?

Answers next week.     

I Really Love Maths

 Tomorrow night I'll be giving the key-note presentation at the VMLC (Vermont Math Leadership Council)  annual meeting in Randolph, Vermont. The title of the presentation is I Really Love Maths. I plan to present a sequence of examples of things that have caused me to love the subject I teach to prospective and practicing teachers in three main areas; the theories that have influenced my thinking, the people who have influenced the way I teach, and the life experiences that have caused me to develop a lifelong passion for teaching maths.
  

Perhaps the two most influential theories that I was lucky enough to encounter early in my career were Skemp's idea of instrumental (fragile) versus relational (robust) understanding, and Shulman's conceptual and procedural knowledge. Being able to look at preK-6 maths through these incredible lenses has allowed me to clearly see what matters more and what matters less, as well as the flaws of some of the traditional instructional practices we have had to endure in years past, and sadly still do in some places today.

Two of the people that have most influenced my thinking are smiling at you right now. At least Sir Ken Robinson is. I looked long and hard but could not find a picture of John Dewey smiling. Sir Ken gave me permission to be creative while Mr. Dewey impressed upon me the value of experience in the educational process. To these names I would add Jo Boaler of Standford  whose work in teacher education in math has been completely illuminating in so far as it has shone the light squarely on the need for teachers to help  children  understand the math they are learning. I would also include Vi Hart who's videos on irreverence in math class are so inspiring. And finally I would add Carol Dweck who gave us Mindset theory with the unbelievable idea that everyone can learn math if they have a Growth Mindset.

And finally my experiences working with English Learners has taught me the humility that comes with standing back  and listening to the way people from other countries do math and think mathematically. The diversity  in the ways we count and communicate mathematically are one of the hidden riches of global thinking. I also think of all the students with disabilities I have worked with some of whom think very differently in terms of the maths in their lives. Some function, and very well too, with an almost exclusively  understanding of nominal number as opposed to cardinal number while others can seemingly compute in milliseconds..

I have been lucky indeed to embrace such diversity of thought, experience and practice during the past 50 or so years.   

Math Fact Fuency

Nicky Morgan, the British Minister or Education, recently announced that "all pupils must know, by heart, their times tables up to 12 x 12". Apart from the use of the archaic term "times tables", which one would expect from a Tory, the whole issue of fact fluency, to give it its current term, is a really interesting one.

I have always believed that fact recall, but up to 10 x 10, is an important part of the mathematization process children go through. It's the math equivalent of being able to spell words, but the facts are by no means the "basics" of mathematics. The basics are everything that is included in the field of numeracy; being able to count, to recognize number patterns, to subitize, to see numerical relationships and so on. Remembering the math facts makes math easier and more efficient.

Memory, remembering things, is a crucial part of education, but it is pretty useless when we memorize things with absolutely no understanding of what we are memorizing. Memorize this list of words; Arun, Ouse, Rother, Stour, Medway, Darnet, Mole and Wey. Now use any of these words during a conversation you have with someone  over the next few days.

If we are going to require students to remember their math facts they must understand what they mean. The multiplication facts for example can mean 'groups of' as in 5 groups of 4 people are 20 people. They can mean area as in a carpet 5 yards by 4 yards has an area of 20 square yards. They can also mean the muliplicative comparison as in "I have 20 Hotwheel cars which is 4 times as many as you have if you have 5".  Each of these concepts of multiplication is different but each can be solved with recalling the fact 4 x 5. Or is  it 5 x 4?

Talking of which, when you see the fact written 4 x 5 do you read it as 4 groups of 5, or four 5 times? I asked my grad class this the other day and half saw it one way and half the other way.

For a great read on the topic of fluency read Jo Boalers incredible article Fluency Without Fear which includes a very relevant criticism of EngageNY's approach to fluency.

The names above, by the way, are the Rivers that flow out of Southeast England.

Jo Boaler Almost Has it Right

I really like what Jo Boaler has to say about maths education. I was one of the 40,000 who followed her Stanford course this summer on teaching maths. Her research over the years has shed much light on many different aspects of mathematics from the best way to teach to how to get people to open their minds about what maths and maths education are all about.

Sometimes, however, I fear she is guilty of making  over-generalizations in a way similar to those who believe that math should be memorized and learned the way it was 50 years ago.

In a recent article in The Atlantic she  avers that  "Speed doesn't matter, and there's no such thing as a "math person."". While I would agree that the two ideas of a) the need for speed, and b)  only people good at maths can do it, are two misconceptions that plague the real and joyful  study of maths there is difficulty in stating the situation categorically as Boaler does.

Clearly, the argument she gives against the need for speed in problem solving and so on is exactly right but I believe there are times when speed is a good thing such as in recalling facts. Having quick access to certain pieces of information is empowering and makes life in general easier and better. Generally, waiting for your mom to cut your food or for your dad to tie you shoe laces is OK when you are 3 or 4  but not when you are 9 or 10.

Again, to say there is no such thing as a "math person" seems to overstate what we really believe. There are clearly some people who will always be better at maths than others in just the same way some people will excel in sports or art or as musicians. I guess I still have to  agree with Howard Gardner that some people have a propensity for being able to think in certain ways while others may not. What I think is the heart of Boaler's assertion is Carol Dweck's idea of fixed versus growth mindset. In other words, so many, many people end up believing they are not good at math because they couldn't solve problems quickly, couldn't remember facts and were made to feel foolish by uncaring teachers resulting in a lifelong belief  that they were not good at math.  No-one should ever say again "I'm no good at math".

When we teach young children maths we should be sensitive to the way we respond to their efforts and accomplishments in maths class. We should encourage them to take time to think through what they are doing and to use what they know and understand.  We should encourage them to do the best they can but recognize a situation where, for one reason or another, a student might need extra help in the form of a different example, method or strategy to understand a particular idea or develop a particular skill, Everyone should have the opportunity to become the best "maths person" they can be. 

  

How To Learn Math

For the past week I've been auditing the on-line course by Jo Boaler of Stanford University. The course, How to Learn Math raises so many wonderful  issues about how we need to develop a more user friendly and conceptually-based  way of teaching math; how math needs to be seen as a creative activity and how we need to get away from the idea  that math is a closed set of procedures to be memorized. Now while these are things I have been advocating for for almost my entire professional life there is something in the course that is having a radical influence  on the way I see my role as a teacher of teachers of elementary school math.

This is the idea of fixed versus growth mindset. developed by Dr. Carol Dweck as described in her book Mindset.   This is clearly an  incredibly important issue for helping children become better at math especially those students who, because of our current system of  testing, have pretty much given up on ever achieving anything mathematically. To be told you are a mathematical failure at age 6 and have no hope of changing that is the world of the fixed mindset indeed.

But there's another application of this incredible idea which is to apply it to parents in terms of their attitudes toward math education. Over the years I have engaged in may discussions, arguments and  even confrontations, some even in public forums, with parents who truly believe that the math they learned by rote in school 30 years ago "was good enough for them so it should be good enough for their children". Sadly, the proponent of the fixed mindset are frequently successful businessmen who point to their success as the reason for maintaining the"no pain, no gain" approach to learning mathematics.

If we are to bring about the evolution of math education to a truly conceptual approach we have to do much to bring about a cultural shift in thinking about mathematics education, a task made more difficult by the disaster of the "new math' activities of the early 1970s in the last century. Perhaps the only way to succeed is to show that the methods advocated by Jo Boaler and the rest of us really do improve scores on tests.