Maths Isn't Difficult

There's absolutely no reason why learning maths should be any more difficult that learning anything else. The main reason why it is seen, almost universally, to be difficult is that for generations it has been so poorly taught. Ever since the first math classes were held the focus in math education, especially at the elementary school level,  has been on learning and memorizing procedures that have no conceptual basis and personally significant meaning.

The analogy in the English language arts would be to teach children to decode words so that they could read them without giving them any sense of what they mean. For example, what does "Splad the yuricles into the bundecloupus ando tpzig" mean? We can decode words in any language so that we can pronounce them but without knowing what those words mean it is pretty much a fruitless exercise. So in math, when we say "carry the one" or "invert and multiply" or six fours are twenty-four" without any sense of what these phrases actually mean we might just as well be using a foreign language or gobbledegook.  By the way, David Pimm called this "teacher patter". The result of this is to develop dis-empowered learners. If we know something but don;t understand it there's not a whole lot we can do with it. If you know that pi is 3.14 and goes on for ever without recurring but have no idea what pi is then you can do little more than reply 3.14 when asked what pi is on a test.

If we teach math with understanding then learners are empowered to use the knowledge they are developing. If you understand that pi is the ratio between the circumference and diameter of a circle then you can apply this understanding to all kinds of situations involving round things. The advent of technology has significantly improved out ability to teach math with understanding as these Math Gifs my daughter Marie recently sent me show. Once you understand the visual/spatial relationships of a particular mathematical phenomenon it then becomes much easier to abstract or "mathematize" it through the use of a symbolic relationship such as an equation or set of symbols.     

Standards, Standards Everywhere

As the new semester begins it's easy to get a sense of being completely overwhelmed by the onslaught of "standards" that prospective teachers are expected to have mastered by the time they graduate and start looking for their first job.

As I prepare for my Elementary Math, Science and Engineering course there are three sets of standards that I have to address; the recently adopted InTASC standards for teachers in the State of Vermont, the Common Core standards for math content and practices, and the newly released Next Generation standards for science and engineering.

The InTASC Model Core Teaching Standards also include the Learning Progressions which further break the Core Teaching Standards down to form a sort of rubric that can be used to determine a student teacher's level of competence. There are 10 Standards each of which addresses a separate aspect of becoming a teacher. Each Standard is then divided into Performance, Essential Knowledge and Critical Disposition Standards that are then further divided into specific pieces of human endeavor giving roughly 200 discrete bits of activity. Each of the Core Teaching Standards is also divided into around 60 separate items in the Progressions section giving about 600 more descriptions of bits of activity.

The Common Core math Standards for Math content and practices includes, on average, 30 content standards for each grade level K - 6. There are also 8 math practice standards which are developmentally appropriately identified in each grade level giving a grand total of around 375 discrete standards.

The Next Generation Science Standards contain between 3 and 5 Standards for each grade level K - 5. Each Standard is then further divided into Science and Engineering Practices, Disciplinary Core Ideas and Crosscutting Concepts for a rough total of  around 200 discrete, but relatively broad  standards. 


 

Maths Through Art

It's been an exciting couple of months as I have met up with two other people who also think that maths should have an aesthetic component when we teach it at the elementary school level.  There's nothing quite as exciting as talking with someone about maths when the conversation doesn't center on the Common Core, SBACS or "grit". Instead, it includes words like Fibonacci, fractal, multiplicative reasoning, primes, Sierpinski, pattern, children loving math and so on.

Both Laura Sommariva and Nancy Benerofe see maths in the elementary school as significantly more than arithmetic and the memorization of facts. These things are, of course still important but can be so much more easily and effectively learned if they are associated with interesting, relevant and meaningful activities such as those Laura and Nancy advocate for. There's another person I should add to the group too, Karyn Vogel, the BSD math coach who also speaks our language.

Come to think of it, I think we should add Vi Hart to our group as she was probably the one who started it all.  Interestingly the Khan Academy lists Vi Hart's work under the heading "Recreational Math". One day, I am certain, the educational community will wake up to find that fractals and numerically and spatially defined patterns will form the basics of what we consider to be math education at the elementary school level. 

Bury NCLB

When you receive public applause from Diane Ravitch you know you must be doing something right. Such is the case with Rebecca Holcomb's letter to parents and caregivers concerning the damage to our education system that continues to be reeked by the No Child Left Behind act. Holcombe, the recently appointed Vermont Education Secretary eloquently and passionately describes how unrealistic it is to expect every student in every school throughout Vermont to score as proficient.

She also highlights how absurd it is that 97 percent of schools failed to make "adequate yearly progress". This would have been 100% if not for the fact that some schools were exempt because they are field testing the SBAC tests, the next travesty to be inflicted upon out schools.

The single-minded, myopic vision of NCLB has created a situation in which so much of what was good about US schools has been sacrificed for the sake of testing a narrow band of material much of which was learned simply to be recalled on a test, just like we did in the 1950s and before.

To label schools as failures has always seemed to me to be one of the nastiest, meanest things you could possibly do to an organization. We would never think of publicly labeling individual children, or even adults for that matter, as failures so why is it acceptable to do this to a whole community.

There are so many other things that schools, especially elementary schools should be free to teach that are not testable with paper and pencil or a keyboard and screen. The science of attention in the field of cognitive neuroscience illustrated by the work of Adele Diamond, for example, is something that many schools are already beginning to embrace  according to a recent PBS program.

Math and Diversity

So what does "carry" really mean? What does "carry the 1" really mean. Several years ago David Pimm, a British linguist, calls this "teacher patter", a word or phrase used by teachers to get children to learn to do something by rote, The "guzintas" as in "2 guzinta 10 five times" is another example of teacher patter.

The word carry" is of course a metaphor used to shortcut the process of regrouping the 14 ones into 1 ten and four ones. It was done for generations, and sadly still is, so that children could learn a procedure quickly without any sense of what was really happening in the quantitative relationship between the two quantities, 15 and 29. Without any sense of the concepts behind what was happening children were never able to develop the skill any further, were never able to check their accuracy, other from memory, and develop absolutely no sense of what the number system was all about.

This is bad enough for neurotypical children but presents greater problems for children who have  special needs or math anxiety. Teaching such meaningless rote bits of knowledge to children who will most likely never be able to apply this skill to anything meaningful is a terrible waste of everyone's time, especially the children.  Instead, it is so much better to teach children with special needs what addition and subtraction mean and when these operations should be used, and then use calculators to do the arithmetic.

For English language learners there are similar problems with the use of metaphorical language. In many other languages the linguistic equivalent of "carry" is used but in some languages different words such a "put" are used where math instruction is based on rote learning. Other metaphors such as "reduce" when changing 4/8 to 1/2 also cause learners to develop misconceptions between. Most children, and many adults, who have been taught fractions with the use of this word will think 1/2 is smaller than 4/8 because a) "reduce" means to make smaller and the numbers 1 and 2 are smaller than 4 and 8.

In my graduate course GED612 Math and Diversity this fall we will discuss many issues related to teaching math to children who have diverse needs. Here are soe resources for Math and Students with Special Needs and Math and English Learners.

Theory and Practice and Practice and Theory.

Learning to teach is one of those things where you need both theory and practice. Yes, some people may be born teachers but in order to teach in an institution such as a school there is so much more to learn than the simple act of teaching; say teaching a child to tie his shoes.

There are theories of learning, epistemologies of disciplines, management techniques and  and something like 2000 pedagogy and content standards that have to be mastered at the elementary school level.

In the undergraduate Education programs at St. Michael's College where stduents can ultimately obtain a license to teach the role of theory and practice and the interplay between them are taken very seriously. From the very first Education course, students get to experience what it is like in a public school classroom. They get opportunities over and over again to experience what it  is like to work with young children based on a  solid grounding in theory in their coursework.

Over the years I have experienced all kinds of variations to this theme both directly and through the stories and experiences of colleagues. I once had a prospective student ask me which 3 credit course she needed to take to become an elementary school teacher. I've had students ask me if they can do their Education courses as independent studies and other students who said they didn't have time to fit in the field placement attached to a a course. I remember a story from an old  friend from graduate school I met recently who was teaching in a small college in the Mid-west who said she had a student who was so poor in her student teaching experience that she was going to fail her until her program director discovered that she was a member of a family with which the director had a  lifetime of close ties. The student teacher was just unable to put theory into practice, she said.

This last example is what I mean by "Mind the gap", a famous sign on every underground station in  London. There is, of course a gap between theory and practice; just like between the train and the platform, that the student must negotiate with the help of her/his supervisor. I remember another student I was supervising in Illinois many years ago who, in the middle of reading a sotry to a class of first grade students, put the book down, burst into tears and said "I just can;t do this". She was a straight A student in her coursework but just could not negotiate the gap.

So when I'm working with anyone who wants to become a teacher I'm always aware of the gap that exists between theory and practice but also very aware that you can't have one without the other.

Arm-Folding Luddites

Last week I attended a workshop given by an Apple exec. in which he extolled the virtues of issuing  iPads to undergraduates when they enter college so that virtually everything they do during their college education can be done virtually on their iPads. I like the iPad as both a teaching tool and for recreation and it would be neat to see how this could work through a pilot study at St. Mike's.

What I found particularly interesting and something that caused me to stop and think seriously was the Apple exec's use of the term "Luddite". He seemed like a really bright chap, the sort that has probably done a TED talk and, for the life of me, I cannot remember or locate his name. What he said that caused me to stop and think was that he really enjoys the challenge of the "arms-folded-Luddites" he comes across as he spreads the word of the use of  all-encompassing personal computing technology.

As I quickly unfolded my arms and looked around to see who else was engaged in the same activity I immediately felt challenged; am I an arm-folding Luddite?  As my years advance I become more conscious of  the increasing difficulty of keeping up with the changes. I've been learning how to use Canvas, the electronic communication program that has replaced eCollege this week, and it seems to be going OK. So, in terms of learning new things I still seem to be cognitively able to do it given the right learning environment and support.

Then I started thinking about my professional life as a teacher. I recalled my first years of teaching as a 4th grade teacher in England in a school where there were 3 other teachers in their middle to late 60s where I am now. Two of them, both male teachers, had taught 5th grade the same way each year for 30 years. even the books they used were published in the late 1940s. The third, a female first grade teacher, and I wish i could recall her name, had modified and adapted her teaching each year to keep up with changes in pedagogy and the prevalent local culture. She had embraced multiculturalism, new teaching methods and had, along with us all, tried and rejected New Math in  the early 70s.

As I think about the theories and practices upon which I base my teaching I think I am like the female teacher I admired all those years ago. At least I hope I am.