Wednesday, November 12, 2014

Traditional Math Just Doesn't Work Anymore

I recently fell into an illuminating conversation with an old friend about math and math education. He was telling me how his daughter is homeschooling his grand-daughter and how much his grand daughter loves math and science. As the conversation continued he told me how his daughter was teaching her child traditional math and "how well" it was working out. He went on to say how he didn't think these new methods of math instruction helped at all and wondered why schools were not using the same methods to teach math his daughter was using and that he had learned 50 years ago..

I then shared with my old friend how I had had an unusual experience in my undergraduate math education course yesterday when one of the students had asked me why young children need to know why there are 180 degrees in a straight line. "Can't you just tell them?" the undergraduate student had asked. "I know there are 180 degrees in a straight line and that's been fine for me", she continued.

A little taken aback I then asked the student how she would explain to a young child how there were 180 degrees in a straight line when there was no angle to measure. She didn't say anything.

So, with my old friend watching I then tore the three corners off a paper triangle and lined them up so that they made a straight line by putting the three points together. There, I said, 180 degrees in the three corners of a triangle and a straight line. Seeing  a somewhat disbelieving glint in his eye I then took a 360 degree rotating protractor and rotated it through 180 degrees saying that degrees are a measure of rotation about a point; in this case the point where the three corners of the triangle come together. Here's a nice demonstration of the concept that angles are measures of rotation.

After a few seconds of thought he said "Hmmmm, why wasn't I taught math this way?" I wonder if he will say anything to his daughter.

The persistent  perception of math as a process of telling and memorization seems such an enigma in this day and age.

Sunday, November 2, 2014

Why a Rigorous Math Curriculum?

Why oh why did the maths education community have to choose the word "rigorous"? The greatest problem about choosing a word like this is that "rigorous" has so many different meanings. In an incredibly informative NCTM Summing Up statement, Linda Gojak, NCTM president describes how many different ways there are of defining the word rigorous. The problem with having such a loose word is that it can be interpreted in so many ways depending on a particular individual's agenda. For example, there are myopic people like Angela Lee Duckworth who have taken the word to mean "grit" and are advocating that all children develop the type of rigor that is all consuming and totally at odds with theories of child development. Clearly some people will do anything to make a name for themselves and get to deliver a TED talk, even if it is the worst one I have ever watched.

How much better it would have been to use a word like "interesting' or "relevant" or "meaningful" or even "inspiring" to describe a math curriculum. Why does maths have to be rigorous for kindergarten children? It's like requiring them to engage in rigorous play, or rigorous reading. It's like asking fifth graders to engage in rigorous creative writing, or rigorous music or P.E. lessons.

Why can learning not be a natural process through which we hold children to achievable standards through activities they find rich in interest, relevance and motivation.  Why can maths not be inspiring, wonderful, relevant, meaningful, and simply interesting?

How many adults would classify the pursuit of their adult lives as rigorous? 

Thursday, October 30, 2014

Homework Hotline's dismal math.

  I wasn't feeling at all well today and so I stayed home. When I do this I invariably watch Homework Hotline on Mountain Lake Public Television. This is never a good idea because every time I have watched it the math teachers solving the math problems called in by students have tended to solve math problems using rote learning strategies devoid of any conceptual knowledge and completely separated from the real worldin which math problems occur.

Today, in the Halloween edition with the teachers dressed in costume, one of the math problems called in by one of the students was this. If a table is 1.75 meters long and 1 meter wide and a chair is 39 cms wide, how many chairs will fit around the table.

So, the math teacher, Sir Lancelot I think he was dressed as, solved the problem by  adding all the sides of the table together, then divided this total by the width of the chair. The total of the 4 sides was 5.5 meters. When he had divided 550cm by 39cm he got 14.1. Correctly ignoring the .1 he said 14 chairs will fit around the table

Although this answer is mathemtically correct it makes absolutely no sense in reality. To fit 14 chairs around the table you would have to fit 2 1/2 chairs at each end and 4 1/2 chairs along each side. The correct, realistic answer should be 12; 2 at each end and 4 along each side; a solution arrived at by dividing the width of a chair into the length of each side of the table. This would leave space between each chair and not require 2 of the chairs to be cut in half.

This is an example of why we, as a nation, struggle to teach math effectively. So often the math of the classroom, especially in middle and high schools, bears no relation to the math required in the real world.   

Wednesday, October 22, 2014

Odds and Evens and Maths

One of the most difficult things for young children to explain is the idea of odds and evens. What makes an even number even and what makes an odd number odd? Their difficulty is compounded by the usually inept definition most of us adults give which is an even number is divisible by two while an odd number isn't. This does little to help young children who have yet to encounter the maths word "divisible" and it does much to add to their confusion in later life when they encounter division and realize that you can divide an odd number by two.

Several years ago, a wonderful kindergartner teacher whose face I can recall but whose name escapes me told me of this method she uses to help children conceptualize odd and even numbers. She said the odd numbers are like old school buses, the ones with a hood out front, and even numbers are like the newer ones  that are straight down with no hood. She then demonstrated it using small colored blocks.

There's a lot more to odd and even numbers than meets the eye. Dr. Maths has an interesting history of them and for the real math nuts here's a way more complicated explanation at Wolfram Maths World. And here's a wonderful article in Wired, a British journal about how we are hard wired to prefer even numbers; ain't that the truth!

Meanwhile I continue my mission to turn math into the much more grammatically correct and sensible maths.

Friday, October 10, 2014

Maths is More Than the Search for Answers

I often wonder why it is that so many people are convinced that maths is nothing more than the search for answers. Sometimes it is even seen as the search for right answers.  Usually, I don't wonder for long as I soon realize it is because we teach math, especially at the elementary school, as a problem solving or problem based activity.

For some reason we seem, as a profession, or perhaps even as a culture, incapable of seeing the study of math as anything more than a closed endeavor in which the one correct answer is the ultimate, prized goal. Sadly, I fear things are not going to get much better as we rush headlong back into the "high-stakes" testing frenzy that seems to afflict our Education system every decade or so. Tests, especially written ones, and especially those administered through computers and scored thousands of miles away thrive on the selection and use of right answer, problem-based questions.

There is so much we can learn in math that we can do for the sheer pleasure and joy of discovering patterns and relationships in the world. We can find all sorts of interesting and intriguing mathematical relationships such as the Fibonacci sequence, in entities in the real world that are just a joy to behold. Even things as mundane as multiplication facts can be learned through the number patterns that the multiples make. Just think about the 9s for example. 9, 18, 27, 36, 45, 56, 63, 72, 81, 90, 99, 108, (if you must go to 12). The digits in each number add up to 9, the digits in the ones place increase by 1 each time, the digits in the tens place decrease by 1 each time. Isn't that amazing. You can always tell if a number is divisible by 9 if the digits add up to 9 ( 1,672,947 is but 1,284, 982 isn't). And don't you think it's neat when you see five red cars in a row of traffic or 3 people in a crowd rearing identical clothing?

So much of maths, or quantitative literacy, is simply appreciating numerical relationships and has nothing to do with finding answers or solving problems. 

Tuesday, October 7, 2014

Sustainability Academy; Shaping SMC Students.

 The Sustainability Academy at Lawrence Barnes school in Burlington is, perhaps the most remarkable school I have ever been associated with. Back in 2004/5, the year before I started at S. Mike's,  I was the math coach at the school back when it was just Lawrence Barnes elementary school. That experience changed my life when I realized that math was not the same the world over. Working with children and their parents from all over the world I soon realized that the math they were bringing with them form Asia Africa, and easter Europe was quite different from the math the we were expecting them to learn here.

As a cognitivist educator, I believe that what we already know affects what we are learning and so it became incredibly important for me to know and understand the nature of the math the newcomers were bringing with them. They clearly brought different languages with them but what was the nature of the math they were bringing? Answering this question has been my primary professional interest for the past ten years.

The Sustainability Academy is now an even more remarkable school than it was back then due to the dedicated principal and faculty who inspire children from all over Burlington, and the world,  to learn through the application of  the ideas of sustainability. As you walk through the school you get a sense of purpose, the students seem to have a commitment to learning and to being part of the community that is not always apparent in the typical elementary school.

Earlier this year the school won an award  when the  National Wildlife Federation presented the school with the Eco-Schools USA Green Flag Honor, the first school to receive such an award in Vermont. The school was also featured in the local press this past weekend, and again in The Atlantic.

The most wonderful thing for me,  however,  is that I get to place most of the students in my Teaching Elementary Math and Science course at the school for their 20 hour (2 hours per week) public school classroom experience each semester. This practicum experience has such a profound effect upon my students as they learn how to listen to and observe students from all over the world as the learn and celebrate the mathematical aspects of the elementary school curriculum. 

Friday, October 3, 2014

St. Mike's grads Teaching at Berkshire School

Towards the end of every Spring semester for the past couple of years I get a call from Lynn Cota, the wonderful principal of the Berkshire Elementary School asking if we have any good grads looking for a teaching job in Vermont. For the past couple of years I have given her the names of several grads and, after the usual lengthy, exhaustive interview that characterizes job applications these days, Lynn has hired them. In fact, Lynn says she is always so  impressed with the quality of preparation pre-service teachers receive through their participation in the Saint Michael's College Teacher Education Program. This is so good to hear.

Lynn was an undergraduate student of mine when I was a professor at Trinity College in the
early 1990s and it is so wonderfully rewarding to see one's students graduate and do so well as professionals in the public schools system. As you can see from the Berkshire school website Lynn has spent the last year guiding the school and community through a lengthy, complex and comprehensive school building rebuilding process in which just about everything about the original building was changed. Pride of place is a large state of the art gym to which all members of the local community have access throughout
                                                                                    the year.

Above is a picture of Jess Neill's kindergarten classroom. Jess graduated from SMC several years ago and is really enjoying working in the Berkshire school in this community just 5 miles south of the Canadian border.

 This is a picture showing Erika Gravelin with her second grade class just a few weeks after starting to teach at Berkshire this semester. Erika was a math and elementary education double major at SMC and is a math whizz.  

This is Natalie Cowden teaching her first grade class. I supervised Natalie in her student teaching experience at Williston Central School two years ago. Natalie is an unbelievably creative teacher who, for example, has the children sing songs while they transition from one activity to the next. It helps them focus and keep on track, she says. 
And this is Andi Nelson who is also in her second year at Berkshire having graduated from St. Mike's last year. Andi was also a  graduate student in my Teaching Science in he K - 8 Classroom course this past summer and was as good a graduate student as she was an undergraduate student. Andi is always quick to smile and, from this pic, it looks like her students have picked up this wonderful habit too. As a friend of mine says, a smile is happiness right under your nose.

I will always remember and cherish my visit to Berkshire Elementary School last Monday, September 29, 2104.