Symmetry Rules

Do you remember in middle school, or even in elementary school, having to draw lines of symmetry on different shapes. There was usually a square, a rectangle, a triangle and of course the circle with infinite number of lines of symmetry. Later there were 3D  shapes with their planes of symmetry and lines with their points of symmetry..

Why did, and do, teachers not use everyday example to demonstrate the concept of symmetry?  Just about everything we use or experience is symmetrical. Math geeks like me know this because we are constantly aware of the impact of symmetry on our lives. Just about every face on every animal is symmetrical; every door you walk through or window you look through is symmetrical. Most cars or trucks we drive are symmetrical as are most houses in which we live.

Fruits and vegetables are symmetrical as is most packaging that things come in. Fractals are symmetrical and so are road signs. Furniture is symmetrical as are appliances. Light bulbs and candles are symmetrical as well as plates, bowls and forks. Knives are symmetrical about a plane of symmetry. Drawers and drawer handles are symmetrical as are most tools like screwdrivers and hammers; again a plane of symmetry.

Some words are even symmetrical; words like madam, level and radar. And, of course there are phrases and sentences that are symmetrical like: "A man, a plan, a canal: Panama".

One of the sad things about our symmetrical world is our sometimes  inexplicable intolerance for things that are not symmetrical when they should be. This is perhaps most vividly illustrated by the difficulties faced by Auggie in the wonderful book Wonder. Sometimes we must make sure we don't let our inherent need for symmetry get in the way of being a good person.

I'm a Math Geek not a Math Nerd

I often joke with my students that I'm a math nerd. I love math; at least the math that we expect young children to learn. But nerd am I no more! After listening to a great program on Vermont Public Radio this morning I now declare myself a Math Geek.

I see the world through maths-tinted glasses. I see how the angles between the branches and twigs on the same trees are similar and how they are different on different trees. I get a kick out of seeing a bunch of cars the same color or a row of buses; in other words, unexpectedly large numbers of things the same. I revel in subitizing and finding number patterns where there really shouldn't be any. From my increasing  understanding of fractals I am beginning to see the mathematical perfection of just about everything and how there really is very little in the world that is random or that does not follow a mathematical pattern of some sort.I even appreciate the land of my fathers differently ever since I saw the coastline measured using fractal mathematics.

So being a Geek is no longer a negative thing. According to the VPR commentary it is "as cool as being a homecoming queen or captain of the football team". The program also introduced me to the grammatically correct act of "verbing" the word geek as in "geeking out the library".

The really cool think about being a math geek is that it gives legitimacy to the idea that math is really cool; something I have believed in for many years. No, I don't mean that adding 29 + 59 is cool (unless you just see it as 30 + 60 - 2) . I mean looking for the next number in the sequence 0,1,1,2,3,5,8 is really cool; thinking about odd and even numbers as partnered numbers and non-partnered numbers; seeing how all circles are just over 3 times their diameter around the circumference; all circles are.

Math is the science of pattern and the art of making sense.

Schools as Places of Learning and Safety.

An article in the Burlington Free Press recently started me thinking about something I just cannot seem to stop thinking about. The article described in detail  how the Shelburne community school, built in 1967, was built with an open plan which was closely related to the idea of Open Education.  The words of the  School Board Chairman,  "The open classrooms represent an idea whose time has come and gone," pretty much sum up what the community thinks of the original idea upon which the school was based. It's a somewhat curt dismissal of something that made Vermont quite famous in the 1960s and 70s as schools like Shelburne bought into the progressive education movement known as the Open Classroom. As a graduate student in the 70s I worked with teachers in an Open Classroom school in Arlington Heights, Illinois,  called the Olive School. It was a magical experience where the focus was on helping individual children learn through a variety of exciting and meaningful experiences. I miss it dearly.

Back then, whether one agreed with it or not, we designed school buildings based on how we thought students learned best; a philosophy of education, if you like.

Today, we design school buildings based on ideas of how to keep children  safe.

Interestingly. back in those halcyon days, the architecture of the building could have little effect upon the way some teachers taught. Even in those open classrooms with no walls there were teachers teaching the same way they had for 40 years with desks in rows, lecturing the 6 year olds, and giving them work sheet after work sheet.  

Sadly, as events in Newtown, Connecticut have shown, the architecture of a school cannot stop people intent on killing young children. More recently in Pakistan, even a school built like a fortress can be breached.

It is clearly not the architecture of schools that we need to address but the nature of our society.

Math and Science Education is Not Enough

I've spent my entire professional life teaching math and science education courses and have always been an advocate for the inclusion of those disciplines especially at the elementary school level. During the 42 years since I graduated with my freshly minted teaching license I have seen many changes in the field of education. The disaster that was New Math in the late 1960s and early '70s, the "alphabet" science programs such as SCIS and ESS of the 1970s and 80s. I have also experienced many different trends in education in general from the total integration of all disciplines in the '70s and '80s through the child centered movement of the late '90s and early noughties to the current standards/high stakes testing movement we are currently suffering through.

There have also been subtle changes in how our cultures view children and education in general. I will always remember the times I spent teaching in Mexico where it was so clear that parents and teachers expected children to be children and to make the most of that developmental stage that we all pass through. I was always so impressed that they didn't seem to view children as mini-adults, or childhood as preparation for adulthood.

Today, as this article on the BBC website illustrates not only do some view childhood as preparation for the future but they see it as an economic force for the future. In British Prime Minister David Cameron's words "There's no secret to success in the modern world. If countries are going to win in the global race and children compete and get the best jobs, you need mathematicians and scientists - pure and simple"......"It will take time but it's absolutely vital for the success of our country that we teach maths and science and computing in the modern way, because that will be one of the things that will determine whether we succeed or not," What happened to the idea that children need to be educated in all disciplines if they are to lead  fulfilled and happy lives? To focus on math and science to the exclusion of anything else would not be in the best interests of anyone. 

In the US we are currently going through an interesting time of paradox regarding the teaching of math and science; a time when the need for good math and science teachers is becoming increasingly important and yet a time when pre-service teachers' interest and skills in these subjects seems to be declining.     

The Common Core and the Deficit Model

Last night, in my Math and Diversity graduate course, we had the most wonderful discussion about the deficit model of education.  The topic was briefly mentioned in one of the  four articles assigned, The Myth of the Culture of Poverty by Paul Gorski.

I first encountered the deficit model of education many years ago while reading about the ways in which Native Americans teach their children. Apparently, there was never any sense that their children were "below standard", or were deficit in their learning in any way. The development of  their skills, understandings, attitudes and values was seen as a natural progression through which children passed in an orderly, logical manner. If a child couldn't do something it was because they had yet to learn it or develop the prerequisite skills first.

Several years later, I was deep in conversation with Rick Marcotte, a beloved principal in a South Burlington elementary school when he began talking all about the concept of "yet". His whole outlook on the way children learned was based on the idea that if a child couldn't do something or understand something it was because they hadn't learned it yet. In other words he saw only the positives in children, the jewels of their passions and what they knew and could do; he didn't see them as deficit based on some arbitrary grade level standard designed to make all students conform to what someone thinks a 10-year-old should be like.

Our current obsession with testing and with identifying standards at each grade level that all students must achieve is destroying our education system in a way that is turning student away from the joy of learning.As so-called standards are raised, with terms like "high stakes testing", "race to the top" and so on, more and more students will fall by the wayside because they will see themselves as deficit failures, not good enough, sub-par, below standard, incomplete human beings.

How wonderful, in contrast, to celebrate the jewels inside each student, to build upon what they know and understand, think and believe, to allow them to pursue their passions within the epistemological structure of the academic subjects studied in school. That structure exists in the form of the Common Core if we strip away the grade level standards and expectations. The learning sequences and structures  exist so that we know what students need to know or develop an understanding of next, based on what they already know and  understand.

Here are two interesting articles that give one a sense of what it might be like if we were to abandon the deficit model of learning. This article, Ditching the Deficit Model, describes building on students strengths and interests, while this one, Discarding the Deficit Model, focuses on the application of the term to teaching minorities and children with special needs, an area where the application of the deficit model has caused the greatest grief.        

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.

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? 

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.   

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.

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. 

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. 

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.

Cognitive Academic Language Humor

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.    

 

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.           

Maths: The Science of Pattern and the Art of Making Sense

Last year I was awarded the Balomenos award for math educator of the year by ATMNE  the six State New England region of the NCTM. Here is the Powerpoint presentation that accompanied the  lecture which is below; my longest blog ever.

Maths: the Science of Pattern and the Art of Making Sense

Including a visual presentation;

Maths: the Science of Pattern and the Art of Using
Cents

Balomenos  Lecture
ATMNE Conference
Killington, Vermont
October 24, 2014

Tim Whiteford PhD

 

(1) First, I would like to thank the VCTM nominating committee and the ATMNE board for presenting me with this wonderful award. The more I have learned about Richard Balomenos the more I am humbled by the task ahead of me but I also feel a kindred spirit through my boundless enthusiasm for teaching math to young children and my tendency to speak my mind which characterized Richard’s approach  to life.

My life as a teacher began in 1972 at Sefton Park primary school on the edge of inner city Bristol in the UK. To one side of the school, on the hill, was a white middle class neighborhood, to the other, down the hill, was St. Pauls, the part of the city settled by Indian, Pakistani and West Indian immigrants. My first class of 34 fourth graders was composed equally of students from these four ethnic backgrounds  so my awareness of diversity in the classroom began the moment I started to teach. If this wasn’t enough of a challenge, the headmaster walked into my classroom 15 minutes before class started on the first day of the first semester of my life as a teacher and, with a wry smile,  said to me “ Do you realize that if all 34 of those students you are about to face decide, as a group,  to do nothing you ask or tell them to do there is absolutely nothing you can do about it”. Knees shaking, I began my teaching career determined that the students in my class would be motivated by their interests in what they were doing and learning  and not by the fact that I was their teacher. (2) Making sense in maths is a function of your motivation.  

(3) The images of pennies on a wall you will see throughout this presentation are taken from the hallway wall outside my office. Last semester, one of my students completed her eMath notebook assignment on the Math of Pennies. It was so well done that I made a paper copy of the Powerpoint presentation and put it up on the wall outside my office. Each time I walked past it I kept seeing more and more ways that pennies could be used to stimulate interest in the patterns inherent in the study of math. I began with this Sierpinski triangle and it has just kept going. My goal was to convey to my students that elementary school math can have an aesthetic component and can be more than arithmetic. I call the display Maths; The Science of Pattern and the Art of Using Cents. The response has been quite amazing.

My first couple of years of teaching were characterized by coming to terms with the fact that New Math (4) was just not working. It had been around in the UK for several years through the School Mathematics Project (SMP) and had been something I had studied as an undergraduate but it was clear that set theory notation and the lack of computing and calculating skill development were too radical even for the most ardent supporters such as a newbie teacher like me. This was my first experience of a math program designed to help children make sense of the math they were learning but it was just too radical for the conservative world of the classroom; a revolution instead of an evolution.  I have always found it odd how New Math has been held up as a warning every time we have tried to institute change in math education while ITA (Initial Teaching Alphabet), a disastrous form of new spelling instruction, died quietly and was never heard of again (5). Making sense in maths is a function of making meaningful connections.

I loved the five years I taught fourth grade at Sefton Park Primary school, the scale 3 post I obtained for developing the math curriculum in the school, and the time one of the parents called me aside one parent night and asked if I knew what was growing amidst the grasses in the student science projects. I told him that we had planted packets of birdseed purchased from a local store to observe and record the growth of the grasses. This one, he said, cradling a complicated leaf pattern in his hand is a grass of a very different sort. “This is marijuana”, he said. I thanked him, he smiled quietly, and I carefully removed the thirty or so offending grass plants from the student “gardens” in the classroom (6). Making sense is a function of overcoming one’s ignorance.

In 1977 I took a one year leave of absence from teaching fourth grade to spend a year in the US in graduate study at the University of Illinois, a leave that turned into the rest of my life, so far. As a T.A. my job, at first, was primarily to teach undergraduates. During my first class I came face to face with what George Bernard Shaw referred to when he said the UK and US were two countries separated by a common language when I asked students if they had seen the rubber so I could clean the chalk board. There were no computers or SMARTboards  at that time. My focus again in my graduate studies was math education and I met up with a remarkable person, one of my doctoral advisors by the name of Harold Lerch, known to everyone as Fuzzy (and no, he was not the origin of the disparaging term “fuzzy math” used so frequently to describe “new math”). I spent many hours collecting data in schools in Kankakee in Illinois for the math education text he was in the process of writing for publication. It was here, I learned the value of observing children while they are engaged in maths activities. I watched second graders bounce in their chairs, or count ceiling tiles to count when they didn’t know their add facts. I also learned that it was not cool to carry your bat to first base in intramural softball games, a habit I had learned while playing cricket in the UK.  I even had Max Beberman’s son in a graduate math ed course I taught in 1981. Max Beberman was a University of Illinois professor and to many,  the “father“ of new maths (7). Making sense in maths  is a function of your personal experiences in  life.

And so to Trinity College in Vermont in 1982 with my newly minted PhD in Elementary Teacher Education. I continued to advocate for a math education based on Richard Skemp’s (8) concept of relational understanding  versus  instrumental. Skemp’s identification of these two types of understanding provides us with a theoretical model for making sense of mathematics. We can learn instrumentally through tricks such as “change the sign and flip the second fraction” or mnemonic devices such as FOIL; or we can learn through making connections, constructing schema and seeing patterns as he suggested in his definition of relational understanding. The former is fragile, the latter robust. (9) It has also seemed to me to be incredibly important that we make a clear distinction between knowledge of symbols and procedures; procedural knowledge, and knowledge of ideas and concepts; conceptual knowledge, as defined by James Hiebert. To confuse one for the other can inhibit our ability to make sense. For example, it is a mistake to think that the procedural knowledge involved in calculating 34 + 29 is teaching the conceptual problem solving concepts of addition; ideas such as part-part- whole, joining or comparing. The algorithm, 34 + 29 is a procedure based on the ideas of place value and base ten. (10) Making sense in maths is a function of understanding the epistemology of maths.

As a cognitivist I focused more and more on how children interacted with maths and how prospective teachers interacted with children interacting with math. It began to seem to me that the private universes children were developing as they developed their mathematical understandings could harbor all sorts of weird and wonderful ideas; ideas that we needed to access if we were going to help them develop strong and realistic mathematical ideas. I shall never forget working with a young student teacher who was completely puzzled by a confident kindergartner’s assertion that 6 – 6 = 5. She had been developing the idea of 0 with the students by giving them some small cubes and then asking them to give them all away so that they had none left, or 0. Remember, this was before the time when we developed the idea of 0 as the starting point for counting numbers and was probably a left-over strategy from the new math programs, the idea of the empty set. Anyway, the three of us sat down one morning in the kindergarten classroom to have a chat about 0. I gave the student 6 Unifix cubes and asked him to show me how 6 – 6 = 5. So, very quickly, he placed the Unifix cubes on six of his fingers, five on one hand and the sixth on the first finger of the other hand. He then counted the 6 cubes and said  “1,2,3,4,5,6” . He then took away  his right hand with the one Unifix cube on the first finger and said “See, 6 take away 6 is 5” showing us the 5 remaining cubes on his left hand. He was in that ‘no-man’s’ land so clearly and eloquently identified by Bob Wright in the Math Recovery materials where children are partly using nominal numbers and partly using cardinal numbers to count with (11). Making sense in maths is a function of where you are in the conceptual progression of an idea.   

But it wasn’t until I returned to working in public schools (Trinity College was clearly not surviving) as a math professional development specialist in 1999 that I began to see just how much we needed to acknowledge that there was something about math in the context of students from different cultures that we, as a profession, were not addressing.

As a math coach in South Burlington I worked with a third-grade student from the Congo who couldn’t count orally beyond 8 and was diagnosed as having difficulty in math because of her limited English.  Curious, I interviewed her in French and found she could not count past ‘huit’ and had no cardinal sense of number at all. Upon further research I discovered she had received no formal instruction in maths before coming to the US. On another occasion, in the same school, a parent from India asked me for advice with her fifth grade son who, she said, just couldn’t grasp the idea of division. When I asked her for an example she said “he just cannot understand how 12 goes into 4 three times”. Also confused, I asked her to demonstrate what she meant. So she quickly shared 12 pencils between four imaginary children 3 times. Her idiosyncratic use of English, probably a function of the process of translation, made perfect sense to her and her son but not to her son’s teacher (12).  Making sense in maths is a function of the language we use in mathematics.

 A year later, in the Burlington school district, I was excited to discover a student from Bosnia doing subtraction in exactly the same way I had learned it when I was a child growing up in the UK. (13)  She even used the same language, “borrow one and pay it back” putting a small 1 next to the top number in the ones place and another small 1 next to the number in the tens place in the bottom number (12). The teacher was completely bewildered by the fact the student used a completely different procedure and yet got the correct answer each time. We talked about how this equal addition method had been used in the US up until the mid 1940s and how it was still used in many parts of Europe. We decided that as long as the student knew what she was doing it was perfectly OK for her to continue using this method as opposed to the standard decomposition method now required in the Common Core Math standards (14). Making sense in maths is a function of the cultural math of the learning context.

This same teacher, Lillian was her name,  also made special 10 x 10 squares like a chutes and ladders board that started in the bottom left hand corner so that her ELL students could see that 53 really was “‘higher” than 21 (15). Making sense in maths is a function of the instructional materials we use. (16) Making sense in maths is also a function of how much those around you care about you making sense. 

In 2004, I had the opportunity  to work with a group of newly arrived Somali Bantu students in the Burlington School District helping to teach 26 K – 6^th grade students the task of developing number sense in English. We started with the numbers 1 – 20 then went on to the decade names and how to count to 100. First, we gave the students large numbers of link cubes to make different numbers. Within five minutes they had all made guns out of the link cubes. We told the students they could not make guns and so they quickly made cell phones and started talking to each other in Maay Maay.  After several days of continued practice they appeared to have mastered the teen numbers and the decade numbers so we asked them all to count by tens to 100 as a group activity. They began, “ten”, “twenty” and so on but when they got to “ninety” they all followed it with “twenty”.  They hadn’t heard the ‘n’ sound at the end of the teen numbers and so thought we had the same words for 15 and 50 and so on. They must have thought we were crazy having the same number name for such different numbers of things. (17) Making sense in maths is a function of what you hear.

For a year I worked with teachers with students from all over the world (over 45 languages are spoken in the Burlington School District) come to terms with the demands of learning and making sense of US mathematics. I watched students from Bosnia recognize the importance of the number 3 and children from Asian countries worry when dealing with the number 4, a number associated with death in many Asian cultures. I watched children from different countries have a difficult time understanding the significance of the numbers used in many of the sayings we have in English and started wondering if other cultures shared in the same mathematical cultural characteristics that we have; do they have special names for 12 like a dozen?. Do they identify odd and even numbers? do they write the date the same way we do? (there is no Pi day in the UK because it is 14/3/13). So I started collecting information about the different maths around the world and compiling it into a web resource to which anyone teaching a student from another culture would have access. (18) http://academics.smcvt.edu/twhiteford/Math/Cultural%20Math/Their%20Math.htm
 I also started to realize that some student were coming from countries where girls and boys were educated separately, where boys were educated but girls were not. I also learned that in some countries such as Singapore students with special needs are not expected to attend school at all and if they do they have to go to special schools, something we abolished in the US some  40 years ago as unethical. It still amazes me how Singapore is so revered for coming top in the TIMMS report in math scores every four years when the school system there appears to be so elitist and based on different ethical values from those that underpin education in the US. Also, children in Singapore are sorted by exams at age 10 into those who are successful in math and those who are not; a practice that was terminated in the UK in the 70s (19). Making sense in maths is a function of the opportunities or restrictions in diverse national education systems .

I returned to higher education at St. Mike’s in the Fall of 2005, refreshed and renewed by my public school experiences. And as I was watching student from other countries trying to make sense of our mathematics I was also watching my own son Andrew, who has Down Syndrome, struggle to complete math activities with any sense of meaning. For years he would come home with pages of addition and subtraction algorithms in which he had learned how to plug in the correct numbers so that he received check marks or a smiley face at the top of the paper. This went on through upper elementary and well into middle school until I finally suggested that this excessive use of hand calculating really wasn’t going anywhere.  I suggested many interactive websites where he could learn some fundamental math concepts and begin to recognize mathematical patterns that would help him navigate his way through the mathematical and quantitative aspects of life. It seemed to me so much more important that he learn how to use a calculator to complete the arithmetic while focusing on the concepts of joining, separating, part-part- whole and comparison as outlined in Thomas Carpenter’s definitive work (20). Making sense of maths is a function of the maths we select and expect children to learn.

Like Conrad Wolfram I even began to wonder why we still spend so much time teaching children hand calculations : some say it takes up 80% of the time children spend studying maths at the elementary school level.  How much better might it be to spend  that time in elementary school teaching number sense and numerical relationships through equations, estimating…………. and, perhaps, even fractals?

At the risk, for a moment, of being an agent provocateur, a fox in the hen house, a cat among the pigeons, or an Englishman at the Boston Tea Party let me ask a question that borders on heresy.

(21) Why do we teach 28 in this form when 28 + 16 = 44 fits right in with making sense of algebra?                       +16                  
                                         44                                                                                          
How much better it would be if students spent their time learning about the relationships between 28 and 16 rather than learning a hand calculation that can be done by a calculator in a nanosecond. There are so many, many more wonderful mathematical relationships that can be studied instead of the desperately dull and dismal task of hand calculations.

What we must not do is confuse the procedural and conceptual knowledge required of algorithmic thinking for the conceptual knowledge identified by Carpenter in the different forms of basic problem solving. Selecting an appropriate algorithm is a totally different skill from completing the algorithmic calculation (22). Making sense in maths is a function of knowing why we are teaching selected mathematics skills and concepts.

Think about 28 + 16 = 44. As a number sentence it is poised to introduce children to the whole world of algebra. In this form it introduces children to perhaps the most important idea in algebra; that the real meaning of the = sign is equality or “is the same as” (23). Making sense in maths is a function of understanding the mathematical symbols we use.  

To simplify things let’s explore 8 + 4 = 12 a little more.

·         (24) How about two groups of people, one of 8 and one of 4 being joined together into one group? Maybe combining two families for dinner and needing to know how many chairs to set out (P).

·         (25)How about a class of fourth grade students, in which you know there are 8 boys and  4 girls and  you want to know the total number students in the class. There are no separate groups; just two parts of one whole. (P)

·         (26) How about two bags of candies? One bag contains 8 candies and the other bag contains 4 more than the first bag? This time we start with two groups and one of the numbers is not a group at all but the difference. Or one bag contains 4 candies and the other contains 8 more? (P)

·         (27) How about wanting to know how many free tickets you started with when you have given 8 away and now only have 4 left. You can even use addition to find the answers to a problem that involves separation. (P)

How many ways can 8 and 4 be related to each other through the addition, subtraction, multiplication or division operations ? (28) Making sense in maths is a function of understanding these procedural/conceptual knowledge relationships.
Perhaps this is similar to what Max Beberman really had in mind with his ‘new math’!  I wonder what would have happened to Beberman’s  ideas if he’d had the same access to technology and computers  that we have today? (29)  Making sense in maths is a function of the technology we have at our disposal.

But wait, it’s too easy to get carried away by the seductive idea of letting technology complete the graft for us. In her illuminating argument for the inclusion of the “standard algorithm” in   the Common Core Math Standards Karen Fuson eloquently points out that completing algorithms by hand is an important way for children to learn to think algorithmically and to practice and develop their knowledge and understanding of place value and the Base 10 system. Again, this is what Bob Wright refers to as the mathematization of students’ thinking. Tools and “concrete” manipulatives are critical in the initial development of ideas and concepts but sooner or later thinking has to become abstract; students need the skill of mental gymnastics developed through involvement with computational exercises. This mental  number sense is part of what it means to be mathematically literate (30). Making sense in maths is a function of developing one’s abstract thinking skills, of mathematizing oneself.

At home, my son Andrew does his calendar every morning, he uses a credit card and a phone, he is constantly thinking mathematically about the various things that make up his life, comparing quantities and numbers on his DS, his Wii games or his iPAD2. I well remember his excitement when, about  three years ago, on a Sunday morning at 6:30 he woke Lucie and I with the “good news” that he had beaten his Wii bowling score of 299. The perfect game.  He learned about dates by looking at the date on the underside of his Hot Wheel cars to see if they were older or younger than him after he had found one, quite by accident, that had 1992 (the year he was born) stamped on the underside.  As a young man with D.S. he has to be careful about what he eats so he is constantly aware of the dietary numbers  such as the calorie count on food wrappers. He has developed his comparative sense of measurement through our frequent weigh-offs which he almost always wins and he has learned how to use nominal numbers in place of measures when he wants to assess a situation. Temperature readings, for example,  name a certain level of warmth or lack of it for him rather than giving him a sense of difference on a scale. He just knows 75 is warmer than 62 but is not aware than the numbers are related through a temperature scale. Each number names a level of warmth. Does he really need to learn how to do algorithms (31)?  Making sense in maths is a function of a student’s individual cognitive abilities.

For the past ten years I have tried to emerge from my math education silo, a place in which I lived and felt comfortable for many years. I have tried to learn about the intricacies of the WIDA Standards, the SIOP model and academic language proficiency as they relate to mathematics but I have remained a true disciple of the credo that math is the science of pattern and the art of making sense. I have made Math and ELL presentations at TESOL conferences in front of many participants and have made the same presentation at ATMNE conferences to fewer participants. I have also made presentations on math and Diversity at Special Education conferences such as the MDSC conference to many  participants and have made the same presentation at STEM conferences to fewer numbers of participants (32). Making sense in maths for all students is a function of our dispositions.

And so as we begin the implementation of another evolution in math education, the Common Core State Standards for Mathematics content and practices, we owe it to all students the best that is thought of and said regarding math education, to paraphrase Matthew Arnold.  The students in our classrooms are more diverse than they have ever been; diverse in their ethnic backgrounds and experiences and diverse in their abilities and dispositions. Regardless of this broad band of diversity all students have the right to make sense of what they are learning, to see the patterns inherent in mathematics that make the recall of factual information easier and more precise and provide us with effective problem solving strategies. Patterns help us make sense, reason, construct viable arguments, model, be precise, bring structure, regularity and repetition to the quantitative aspects of our lives, to paraphrase the CCSSM practices; the type of patterns we find in the study of fractals. 

Along with the CCMS comes the Smarter Balanced Assessment Consortium or SBAC for short. By Fall 2015 in 27 States, I think it is, most students (some 99%) in grades 3 and up will be expected to take these computer based assessments. Having just spent many hours reviewing the sample test questions my hope for the finished SBAC assessments is that they assess whether students are making sense of the math they are learning as well as recalling  the factual and procedural knowledge they are learning (33). Making sense in maths must be included in any assessment activities if they are to be meaningful and relevant.   

Maths must make sense to everybody who studies it. Maths must also captivate students if they are to truly make sense of it. Patterns help us see the coherence between all aspects of mathematics in the same way Hung-His Wu describes mathematics as a tapestry in Phoenix Rising, his commentary on the Common Core State Mathematics Standards. He writes, “Mathematics is coherent; it is a tapestry in which all the concepts and skills are logically interwoven to form a single  piece”. As math teachers we must help children weave the different threads of mathematics together to form a coherent whole, a tapestry, that forms a picture that makes sense to each and every one of them.

As we implement the CCSMs we need to make sure that our students, whether in kindergarten or twelfth grade, or even in college courses, see that math has an aesthetic component as one would see in any tapestry; that there are poetry and creative writing equivalents of the English language arts in maths so that students find it exciting, interesting, motivating and captivating. We must not make the mistake again, however, of proposing these things in lieu of the rigor of remembering factual information and the ability to estimate and persevere. We do not give up on spelling and grammar when children write creative stories or poetry.

And finally, this past summer, I audited an online Stanford math education course developed by Jo Boaler, one of the current shining lights in maths education. Boaler, also an expat Brit continues to use the British term maths (with an s) which I have always thought a more logical abbreviation of mathematics than the word ‘math’, which is why I have used it today. As I followed the course I was reminded of how importance it is for all students to  develop and use  growth mindsets related to learning mathematics as opposed to the ubiquitous  fixed minds sets so often found behind the statement “I’m no good at math”. Carol Dweck’s work on fixed mindsets probably has more relevance to math education than to anything else. We must present mathematics in a way that helps students of all ages make sense of what they are learning through the patterns they see and construct, so that instead of saying  “I’m no good at math”’ students say “ah, now I get it, that makes perfect sense to me”. (34) Making sense in maths requires a growth mindset, something we all should have whether we are kindergartners  or veteran teachers.

Thank you.

References

Boaler, J. (2013) Stanford On-Line Course EDUC115 How to Learn Math;       
                  https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_
                  Math/courseware/9da2cc3b8dc24b648b210ed445c35d3e/1b3ac347ca
                  064b3eaaddbc27d4200964/

Carpenter, T.P., Fennema, E., Loef Franke, M., Levi, L. and Empson, S. (1999) Children’s
                  Mathematics; Cognitively Guided Instruction, Portsmouth N.H. Heinemann

Conrad Wolfram (2010) TED talk; http://blog.wolfram.com/2010/11/23/conrad-wolframs-ted-
                  talk-stop-teaching-calculating-start-teaching-math/

Dweck, C. Mindset (2006) New York, N.Y. Ballantine books 

Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics   
               Mahwah, NJ: Lawrence Erlbaum Associates.

Hung His Wu (2011) Phoenix Rising in American Educator;                       
                  http://www.aft.org/pdfs/americaneducator/fall2011/Wu.pdf

Skemp, R. (1972) The Psychology of Learning Mathematics, London UK, Penguin Press

Wright, R. (2012) Developing Number Knowledge London U.K. Sage Press