## 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.

Sometimes when you are teaching a class of college students there occurs a moment when humor completely takes over and you just have to let it play itself out. Such a moment happened in my graduate class, Math and Diversity, last night when we were discussing the difficulties posed by homophones and words with multiple meanings for English Language Learners. The words were sum or some and product in the context of addition or multiplication. The discussion of the homophones (sum and some) went fine and then we came to the word product when some one said you can put product in your hair.

This led to all kinds of visual comments of people having their hair multiplied or of cards with 2 x 3 = 6 wrapped around their heads or of multiplication facts sticking out of their hair. The English language has such a vast array of words and of words that have multiple meanings that it must be incredibly difficult to learn.

Even in math class, the way words are used academically is so different from their everyday or social meaning. Seemingly simple words like 'find', 'degree', and 'count' all have very specific meanings ini math.
The classic exam question asking the student to 'find x' is a wonderful example of how specific the mathematical usage of a simple word can be. In the example to the left 'find' mean calculate or compute and not just look for as the student thought.

The word "reduce" used in the context of reducing 4/8 to 1/2 is another word that creates problems not only for English learners but for everyone. Is it any wonder that so many young children develop the misconception that 1/2 is smaller than 4/8 when we tell them to reduce 4/8. Even the numbers get smaller.

It is so important to teach the mathematical meanings of words in context, through discussion, free of metaphor or idioms so that English learners can identify the appropiate use of certain terminology in math class.

## Wednesday, October 1, 2014

### Are You Five Ten or One Seventy-Five?

I was walking along the first floor hallway yesterday and was just about to get into the elevator when I heard one of our international students ask someone from the TESOL office how tall he was. "Five ten" was the reply to which the international students responded "Five, ten? What is this five, ten?" He was smiling and clearly was amused by the use of two numbers to convey someone's height. They had apparently been in conversation for several minutes as the TESOL employee began to elaborate by saying "five feet and ten inches" to which the international student replied "what is an inch and what is a foot?". I wanted to ask the international student how tall he was; anticipating a reply of  175 or 180 or so but the elevator arrived and I had to go and besides, I didn't want to intrude on what was a wonderful internationally defined conversation.

The use of the implicit referent is commonplace throughout our culture. Prices are usually given in the form of three ninety-nine or two twenty-five. We talk about the size of a motorcycle as a seven fifty or a twelve hundred and snowboarders and skateboarders routinely do one-eighties and three sixties. The time is always given without a referent and sportscasters  could not survive if they had to give the referent for every number they used. An ERA of point three five eight, hitting three forty, winning five to three and declaring at five hundred twenty six for six.

This is all well and fine when you are a native to the culture but when you are not it can cause considerable difficulties. For example, in the above examples what does "declaring for five hundred twenty six for six" mean?  And if you had to fill your car with gas in Montreal about how many litres would it take?

When we teach math to English Learners we must remember to a) make the referents explicit and b) give students a sense of the size of each referent so they can understand the significance of the measurements they are using.