Tuesday, January 28, 2014

Culture and Education

At the beginning of this month I gave the keynote address at the ATMIM conference at Boston College. ATMIM, the Association of Teachers of Mathematics in Massachusetts, is one of the 50 state professional organizations for math teachers affiliated with NCTM, the National Council for Teachers of Mathematics. Since the conference is for all teachers, K - 12, I wanted to talk about something that would be relevant to everyone and not just those in my field of expertise, elementary education. The theme of the conference was New Methods, New Ideas for a New Year so I chose the topic of exploring how we select and implement new methods and ideas but included the ideas of new dispositions, new points of view and new perspectives all within the context of professional development. I based by address on two influential voices from the past, Matthew Arnold and TS Eliot, the former a really good guy and the latter, well, it took me a long time to find something he said with which I could agree. I use these two wonderful quotations that seem to sum up the educational beliefs of each man;


    " For there is no doubt that in our headlong rush to educate everybody, we are lowering our
       standards, and more and more abandoning the study of those  subjects by which the essentials of
       our culture—of that part of it which is  transmissible by education—are transmitted; destroying
       our ancient edifices to make ready the ground upon which the barbarian nomads of the future
       will encamp in their mechanized caravans". (Notes Toward a Definition of Culture)


     "The whole scope of the essay is to recommend culture as the great help out of our present
       difficulties; culture being a pursuit of our total perfection by means of getting to know, on all the
       matters which most concern us, the best which has been thought and said in the world, and,
       through this knowledge, turning a stream of fresh and free thought upon our stock notions and
       habits, which we now follow staunchly but mechanically, vainly imagining that there is a virtue
       in following them staunchly which makes up for the mischief of following them mechanically".
       (Culture and Anarchy - on the list of the 100 most influential books ever written)

Eliot's words are so outrageous but lurking somewhere in there is a grain of truth. For me, this is the idea that we must guard against the "Barbarian nomads encamping in their mechanized caravans". After recent events I might even cast Justin Bieber or the Syrian president Bashar Assad as Barbarian Nomads. But seriously we have to preserve the standards of human culture; honesty, truth, perfection and so on if we are to maintain all the good that we have achieved. 

Arnold's words, however, have an almost magical dynamism about them; the idea that culture is defined as the best that everyone can achieve as individuals while at the same time having a sense of growth and change as the world itself changes. Being the best plumber, the best engineer, the best librarian, the best truck driver or the best teacher one can be. Thinking the best thoughts one can think; now there's something to which we can all aspire.


Math and a Student's Mindset

I can't think of any better application that Carol Dweck's mindset theory than to the study of mathematics. For years so many students learning math have developed, and subsequently suffered from, the fixed mindset embodied in the statements "I can't do math" or "I'm no good at math". Remarkably, it's a phrase that is repeated so often that it is now quite an acceptable excuse for not being able to do a simple math exercise computation of problem solve. It's even reached the point where it's almost worn as a badge of h9onor by some people because it distinguishes them from the "less cool" people who actually like math and are good at it.

The reason for this is  most likely the way we have taught maths in the past as a dry, sterile subject with little relevance to life, other than balancing a check book, and with little aesthetic value. A situation that has to change if mathematics is to captivate students' interest and imagination.

For several years I have been advocating for the study of fractals in the elementary school math curriculum and it seems I have an ally in the Common Core Math Standards, at least in the Math Practice Standards section. Math practice standards 7 and 8 both suggest that students need to search for pattern in order to make sense of the world of maths. Talking of fractals I took my son Adnrew to see the movie Frozen last Sunday and there in one of the songs about the ice castle wwas the word "fractal" as she described the patterns the ice made. The study of pattern in fractals could do a lot to help students develop a growth mindset in math class.

And here is a neat website worth watching that focuses on the idea of Growth Mindset Maths. 



Tuesday, January 21, 2014

The Language of Mathematical Meaning

My wonderful mother-in-law knows that I like quirky math things so when I received this image and joke from her recently I knew she knew how I see maths. The image appeared at the end of these lines.

A husband seeing his wife is busy cooking a meal asks if he can help. His wife asks him to peel half the potatoes and put them in a large pot to cook. The picture shows what he did, much to his wife's amusement.

The wonderful thing about this mathematically is that it is a mathematical play on the word "half". When we teach fractions in school we seldom give much time to the identification of the one or the whole to which the fraction refers. When I start a class on fractions I always ask the students if they would like half the money in my right hand or a quarter of the  money in my left hand. The value of a fraction is a mystery until you know the size of the one to which it refers.

This is why operations with fractions can be so difficult. Think about 1/2 ÷ 1/4. The answer is clearly 2, but 2 what? If you learned this procedurally and instrumentally then it's difficult to know. If, on the other hand, you learned it relationally then you can see that it refers to the 1/4. There are two quarters in a half. You can put it in context by thinking about the number of quarters in the first half of the Superbowl.

The reason why the husband did what he did was that he used the potatoes as the ones and not the whole group. He should really have peeled 4 potatoes and not 8 halves. The result is the same in term of the amount of peeled potato but pretty useless in terms of meal preparation. 

Monday, January 20, 2014

Additional Common Core Math Practice Standards

Tonight, in the second class of my graduate level teaching Mathematics in the K - 8 Classroom course, I will be introducing the students to the Common Core math Content Standards and the Math Practice Standards.

We will explore the content standards using what I call the  Vertical Articulation of the standards. In other words, instead of grouping them by grade level they are grouped by content standard to that students can see the progression of the content; a critical element when trying to differentiate instruction.

For the practice standards I have added an additional two practices which I believe to be central to the study of mathematics and which have been totally neglected by the authors of the Common Core. Here they are:
9. Enjoy and celebrate mathematics

Mathematically proficient students enjoy and appreciate the aesthetics of quantitative and spatial relationships. They are captivated by the challenges of resolving mathematical problems and are able to use their mathematical understanding in creative and novel ways. They will demonstrate genuine curiosity when faced with novel mathematical situations. Younger students will share their excitement about finding several different ways of making 6, of understanding why a square number is so called and that pi is a ratio between the circumference and diameter of a circle and not just a number that goes on forever.  Older students will recognize and celebrate the artistic elements associated with fractals and the aesthetic characteristics of algebraic relationships. 

10. Recognize linguistic and cultural diversity in mathematics

Mathematically proficient students will recognize that math is not the same the world over. Living in diverse communities students will recognize that there are differences in mathematics and the ways we learn mathematics based on local and global cultural differences. As they work with students from different cultures they will be aware of the ways language development, as well as the language used in  mathematics, are major factors in learning math for all students as well as those who are English Learners.  

Thursday, January 16, 2014

Lincoln Portrait Penny Mosaic

Happy New Year everyone; time to start putting pen to paper again, metaphorically speaking, sine I haven't for some time.

So I have decided to complete the Lincoln penny mosaic that one of my students, Lydia, began the past summer. She is currently student teaching so has little time for things like this. The mosaic in 51 pennies wide and 61 pennies deep so it's just a cse of using coordinates to get the righ shade of penny in the right place; at least that is the plan. Each penny can be located by using a grid coordinate system, the fist number will be the column and the second will be the row. The key to success will be keeping things vertical and horizontal.

There will be 3111 pennies altogether and it looks like I will need an addition 2500 to finish what Lydia has started. Any penny donations will be gratefully accepted.

I was observing one of my student teachers in a kindergarten class this morning when one of the kindergartners said "I subitized" when asked ow she knew how many dots there were. This is the first time I've ever heard a child use this intriguing term.