## Thursday, December 29, 2016

### Evenly Odd or Oddly Even

If there's one thing that drives me crazy in maths it's the sayings and phrases we've come up with over the years to supposedly help students learn maths more efficiently. Two, in particular drive me nuts.

"You can't take 5 from 2" is a particularly useless thing to say when you are doing the subtraction algorithm such as 62 - 25. Children typically learn this when they are in second grade but then have to unlearn it when they get to 5th or 6 th grade and encounter negative numbers. It should be immediately apparent 5 is more than 2 so you regroup 62 into 50 + 12 so you can subtract 5 from 12 and 2 tens from 5 tend to get 37. Actually it's easier to use negative numbers. Just say 2 minus 5 is -3; 60 - 20 is 40, then 'subtract' the -3 and you get 37.

The other one that really gets to me is "an even number is a number that is divisible by 2".  This pearl of wisdom often crops up in crossword puzzles and should include "to give another whole number" to be accurate, Every number is divisible by 2 in some way, except 0. We can, of course say that even numbers end in 0,2,4,6,or 8 while odd numbers end in 1,3,5,7 or 9. Conceptually, even numbers comprise  pairs of things in them while odd numbers always have one thing by itself.

One way to think about this is to see the school bus on the left as if it was made of cubes. One at the front for the hood and then pairs stacked on top of each other for the rest of the bus. No matter how long the bus is, or how  many pairs you add, the number of cubes will always be odd.

Now look at the second school bus without the hood, or the odd block at the front. No matter how long the bus is, or how many pairs of blocks you add the number of blocks will always be even.

Another interesting thing about odd and even numbers is the way they impact our culture. For example, in Japan odd numbers are preferred while in the the US we prefer even numbers.

Odd and even numbers even elicit emotions as this Guardian report illustrates. Odds and evens even have gender assignments as this Kellogg study reports

## Sunday, December 18, 2016

### It's Never Too Early to Count

It's never too early for young children to interact with the world of maths. Children learn the fundamentals of what we call quantitative literacy in the same way they learn to speak and read at a young age: through experience and practice with someone who knows what they are doing.

As I mentioned a few posts back my daughter Marie and her husband Erik are doing an amazing job helping their son Lachlan learn the intricacies of counting and other aspects of maths They don't force it on him, make him complete math activities, or even call it maths. They just make him aware of the quantitative and geometrical aspects of his life  as he interacts with the world around him. Currently, he is coming to terms with the oral number name sequence up to twenty. He nearly has it except some of the teen numbers are a bit random.

Quite remarkably, at two and a half, he also is beginning to develop a sense of cardinality. This is when you put quantities to the number names, Right now when hes says, "one, two, three, four" etc he is just saying a sequence of words, a little like reciting the alphabet. He has cardinality with two; he can identify two objects that are the same. This is an important idea because you cannot count rationally unless you know what you are counting. He can identify two fingers or two tractors or two people. The fingers, tractors and people are the referents of the counting words, the things to which "two" refers. In early rational counting the identification of the referent is important because we can develop the idea of counting as the process of  "one more"; three is two and one more altogether. The word "three" now refers to the objects which were two and one before they were joined together to make three.

This is not as easy as it sounds because there is also the ordinal and nominal use of number. The ordinal use of number, first, second, third  really doesn't come into play at this point in the learning to count process. But the nominal use of number, using numbers to name things, does. In the picture above the numeral 1 appears above a single tomato, as do 2,3 and 4. It's easy for a young child to name each tomato as 1,2,3 or 4. This would be a good activity for teaching the numerals once they had been learned orally. But to teach  rational counting, or the cardinal use of number, you would need a picture with one tomato next to the 1, two tomatoes next to the 2 and so on so that the numeral becomes a number associated with that many.

Making maths a part of everyday life for young children is easy if you know what you are doing. Bedtime Math is a wonderful resource I have mentioned before.

## Thursday, December 15, 2016

### The 3Rs - Ridiculous Regulated Rubbish

Just about everyone thinks of Reuse, Reduce and Recycle when "the 3 Rs" are mentioned these days. Everyone, that is, except the British Government who still, inexplicably,  refer to primary education as "the three Rs" even though only one of them now begins with R (reading  writing and maths).

In addition to introducing the school "league tables", as if education was a sport, the Conservative Government has now increased the level of difficulty of the standardized tests they give to primary school (elementary school)  children which means, somewhat obviously, that the scores of students are going to be lower than they were with the old test last year. Just last week I blogged about how tests only measure other tests, an assertion that seems well supported by the British Government's latest advice to parents to "ignore the latest test results".

Unlike Canada, where each minister is an expert in her or his field, the British (and US) governments appear to revel in the idea of putting people in charge of things about which they know nothing. Nick Gibbs, the current Minister of State for Schools at the DfE in the UK was an accountant before he became a politician. It seems remarkable that schools and children are compared using tests that measure a narrowly defined set of skills and knowledge at a particular moment in time using a specific medium and think that anything useful can be obtained.

To then say that only 53% of children "meet the standard" is absolute Ridiculous, Regulated Rubbish.

## Wednesday, December 14, 2016

### Counting Grandchildren

One of the wonderful combinations in one's life, as I am discovering, is to be retired and have a grandchild. It took me quite a while to make the adjustment to being retired but no time at all adjusting to having a wonderful grandchild.

Since I was a professor of maths education for most of my working life my brain is pretty much dominated by the idea of number so when I retired all sorts of numerical thoughts began to spring into my mind. Most of them were things that had never occurred to me before or things that I may have taken for granted. For example, I no longer meet upwards  of 80 or so new people, students,  in my life each year. There were usually around this many students in the courses I taught each year. And then when I supervised student teachers in public school classrooms I would frequently meet another 100 or so  new children each year. Only now that I am retired have I become acutely aware of just how much I learned from all these new people I met each year; the interactions, the term papers I read, the classroom events and experiences all conspired to enrich my life each year, I miss these interactions so much and frequently wonder if I did take them for granted. I don't think I did but it all looks so different now.

So I now have time to watch and interact with my two and a half year old grandson as he comes to terms with the wonderful world of maths. He has already learned to say "Sierpinski triangle" and can pick one out in a whole bunch of different triangles. There is nothing quite like hearing him say those two words and pointing at an example of one, Numerically, he is going through the process of learning the number names and, dare I say it, has already the beginnings of a sense of cardinality at least with two and maybe three objects. His mom, my daughter Marie, took my maths ed. grad course several years ago and so is really in tune with the growth and development of a child's counting skills. She demonstrates so wonderfully the two most importance things in teaching maths. First the importance of observing the student, her son, and second, just how much a full understanding of the most basic mathematical ideas is to the teaching process. She doesn't push maths on him at all but just makes him mathematically aware of the world in which he lives.

Even something as seemingly simple as helping a student count requires a deep understanding of the ordinal, nominal, and cardinal use of number. Lachlan, my grandson, is currently learning the sequence of the number names. He can number name more or less up to twelve but hasn't yet quite got the teen sequence. I say number naming because he really is not counting yet in the true sense of the word  apart from, perhaps, "twoness" and "threeness". When he number names he is just learning the order in which the number names occur. He has, yet, no sense of "fiveness", for example. More next time on the nitty gritty of learning to count.

## Tuesday, December 6, 2016

### Tests Only Measure Other Tests!

The PISA test results were released this week and once again the US is slipping in math test scores when compared with test results from 60 or so other countries. Of the 35 industrialized countries included the US ranks 31st.

When I was a fourth grade teacher many, many years ago in England we changed from one reading test to another one year without changing anything else; the reading program and the way I taught children to read remained exactly the same. The most incredible thing happened. All my students' reading abilities improved by more than a year. In other words, my fourth graders looked like they were suddenly reading as well as fifth graders. What a remarkably effective teacher I must have been that year. Not really since I did nothing differently. It was because the reading test was clearly easier or standardized differently from the previous one. This was my first experience of tests measuring other tests.

In fact this very same phenomenon is mentioned at the end of the Hechinger report of the PISA results in these words:

"However, one test released last week —  the 2015 Trends in International Mathematics and Science Study (TIMSS) — showed a surprising gain for the U.S. in 8th grade math.  Historically, PISA and TIMSS tests have shown contradictory results for Eastern European countries and Russia, as they perform much better on TIMSS than on the PISA test. Scholars will need to explain the divergence for the U.S. this past year".

The PISA and TIMMS tests are clearly different and measure different things in maths. It is so easy to point fingers when you see bare test results such as these reported in the media but there are so many other things to consider.

Take the maths, for example. Maths is not the same the world over. There are vast differences in the math itself before you even start to teach it. In most Asian countries, for example, the numbering system is so much easier for young children to learn than it is in the US. The teen numbers follow  the pattern ten-one, ten-two, ten-three and so on rather than the difficult eleven, twelve, thirteen, etc system we use. Look at which countries are at the top of the PISA table! In some countries such as Singapore children are given vast amounts of homework to do each night and not all children have the opportunity to attend public schools. There are differences in the sizes and cultural diversity of the countries being compared which can all affect score averages on a simple test.

Like pretty much everything in our lives we can probably do better in maths education but we need to use caution when comparing test scores obtained from students in vastly differing cultures.

## Saturday, December 3, 2016

### Maths Is Easier When You Understand It.

I sometimes wonder just what we have to do in maths education to demonstrate that things are more easily remembered, used and developed if the learner understands what she/he is learning. This interesting piece of recently published research supports, for at least the millionth time,  that when students learn conceptual knowledge, along with procedural knowledge of maths, it is much, much, much, more effective. The neat thing about this piece of research is that it is for high school maths which has tended to lag behind the research on this topic at the elementary school level.

Probably the best way to distinguish between the two types of knowledge is the idea of pi. If I were to ask you what pi is you would probably say "3.14 but that's all I can remember". If that's all you know about pi then you have a small piece of procedural knowledge that you can plug into equations to find the area or circumference of a circle. Now, imagine that you have some conceptual knowledge to go with this. For example, knowing that pi is a ratio between the diameter and circumference of a circle would be immensely helpful to learning all sorts of more advanced maths. It would also be helpful for doing things in one's daily life. The fact that pi is a ratio means it is just over three times further around a circle than it is across the middle of that circle. If you think of pi as a fraction, 22/7, then the circumference could be 22 inches, or feet, or miles, and the diameter would be 7 inches, feet, or miles.

If I asked you to count by fives you would probably say, "5, 10, 15, 20, 25" etc assuming that I meant you to start at 0. So try counting by fives again starting at 3. The first few will be tough but you'll soon see the repeating pattern of 3s and 8s. Seeing the pattern is a piece of conceptual knowledge because you are applying your conception of counting by 5s and 10s rather than just parroting the numbers.

We have long known that conceptual knowledge of maths is as crucial to learning maths as procedural knowledge. Jo Boaler, the amazing Stanford professor and founder of YouCubed  was one of the first to demonstrate the importance of conceptual knowledge in a wonderful piece of research when she was at Liverpool University in the UK. She discovered that elementary aged students who were taught conceptual knowledge along with procedural knowledge did much better in high school maths that those who were only taught procedural knowledge at the elementary school, One of the interesting findings of her study, if I remember correctly, was that the conceptual/procedural knowledge taught students didn't. do as well on the end of elementary school maths tests as the procedural knowledge only taught students. The reason: the tests only tested the students' procedural knowledge of maths; a problem we still grapple with today.

## Thursday, October 13, 2016

### Cursive and Art History sadly axed.

This week we seem to have lost two aspects of education that contribute significantly to our civilized society. In the US it has been determined, by those who wrote the Common Core Standards, that cursive writing will no longer be a required part of the elementary school curriculum. The reason, apparently, is because of a) the time it takes to teach it and b) the prevalence of technology as a form of communication.

I find this second reason particularly irksome and bewildering. I remember when calculators first entered the elementary school classroom and all the students said they would no longer have to learn their math facts.Luckily, we did not fall for this strategy since we knew that fact recall, with understanding of course,  is an important part of being mathematically literate.

Te remarkable thing about the technology argument for removing cursive is that you could use the same argument for removing all instruction in writing from the elementary school curriculum since more and more electronic devices can convert speech into the written form. Even now, texting has auto-correct meaning that spelling is already a dying art.

On the other side of the Atlantic in the UK the History of Art  has been discontinued as an A-level subject of study. Remarkably, this was the reason given for the decision "In a letter to teachers, the board said it was struggling to recruit "sufficient experienced examiners" to mark and award specialist topics".

Sometimes, the decisions made by those who are, presumably, paid to know best are quite baffling..

## Sunday, September 18, 2016

### Retirement and Contributing

It's bad enough that Michelle van Etten  was billed at the NRC 2016 as employing 100,00 workers when she actually employs none. But when she said in her speech that her husband home-schools her children to "protect them from Common Core" I found myself once again temporarily  lost for words. If only a different term other than "Common Core" had been used to describe the essential elements of what children can explore and learn about as they grow up. The individual words "common" and "core" by themselves have such negative connotations; when you put them together they become a recipe for disaster.

But, I'm now retired and so feel no desperate need to raise such issues as part of my life mission to help everyone fall in love with the study of maths. Like many of us in the field of math education I have lost faith, a little, in the ability of our culture to embrace a conceptual and meaningful approach to the teaching and study of maths. So I will do what I can in whatever small way my failing health will allow me to do.

It's been quite a ride since I made the decision to retire two years ago. For some strange reason I did not want to still be working past my 70th birthday which happens on a few months time. Lat October I had a Pacemaker installed then had a major heart attack at the end of December. Things did not look good and I was given as little as 2 months to live. I went through a lot of soul searching as I recovered at home unable to work. I resolved the effects of the ultimate loss by confirming that I will not be here to deal with what happens after my death. I am at incredible peace with this realization.

Last May, I qualified, just, for a new aortic valve through the TAVR process. It was an incredible experience and has potentially increased my life span by several  years.That is, if I now manage to successfully negotiate a carotid artery stent insertion next Friday.. I knew I would need this procedure after I lost the sight in my right eye last Sunday morning after a mini stroke caused by something letting loose from the blockage that needs stenting.

So, a terribly self indulgent blog post but if it helps someone else deal with the same things I am going through than it is worth it. I am truly thankful for every new day in my life. I can continue to contribute.  .

## Wednesday, May 18, 2016

### The Power of Undergraduate Education

It's been a difficult week cleaning out my office and consigning the tangible elements of my career to the recycle bin. The task of deciding which books to keep, which to pass along to colleagues and which to throw away is not an easy one. I probably have close to 500 books on my shelves dating back to my undergraduate days in the late 60s and early 70s.

Deciding what to do with each book was absolutely impossible until I came up with an effective plan. Each book would be judged as follows. The books I would save and take home would be the ones that had impacted my professional life, my thinking and educational values the most. The second category would be those books that my colleagues or replacement might want. The third group comprised those destined for the recycle bin while the fourth and last group were the books my daughter wanted.

After three days of sorting, one spent with my daughter Marie, I have pretty well consigned every book to one of the groups described above. The pile of books that have touched my life the most are almost all books that I read as an undergraduate student. Books by Bruner, Dewey, Peters, DeBono, Whitehead to name a few. Looking back it seems remarkable that so few have had the same deep influence on educational values and core ideas as these great masters. Carol Dweck and Jo Boaler have been great influences along the way but it was these great writers who introduced me to the wonders of teaching in my undergraduate education who had he most profound affect on the way I think about teaching and teachers education.

The second group of books contain books all about teaching math, science and engineering; books that contain great ideas and activities for helping children at the K - 6 level learn to enjoy and succeed mathematically and scientifically.

By far the largest selection of books have filled a very large recycling bin to overflowing and contain books of several types. There are the what I would call the "fleeting fad" type of book while others are earlier editions of current books. Many were sent t me by publishers hoping  I would adopt the for my courses.

Undergraduate education today is promoted aggressively as a route to a better job, career preparation
and a pathway to fame and fortune in some instances. But the true value of undergraduate education is teaching students to be constructive and free thinkers; to develop cognitive and social values and a sense of who one is as an individual. Regardless of where I have worked in different parts of the world, the things I learned as an undergraduate have remained as the fundamental core of my beliefs.

## Thursday, May 12, 2016

### When You can't Hear the Song for the Singer

There's a line in my current favorite song, Black Muddy River,  that goes "When you can't hear the song for the singer". The song, written by Robert Hunter,  is sung by Bruce Hornsby, a quite unassuming "pop star" by current standards. His piano playing is quite remarkable and, although he didn't write the song, I find myself listening to the words and instrumentation intently. I hear every nuance in his voice, every note and cadence in the instrumental accompaniment. This is not always true. When I listen to Prince, or watch Celtic Women video I am preoccupied by the performance and the performer and not the song  With Prince it is his entire aura of who he is and what he stands for. His facial expressions and gyrations. I just can't hear the song for the singer. With the Celtic Women videos the sheer size and over production of the performance overtakes the actual words and their meaning. It's as if the song becomes a vehicle for everything else going on so you can;t really listen to it. The beauty of the Celtic women performers, the striking makeup on the drummers and the outside locations at castles and other notable venues.

I find  so much of educational "innovation" and "advancement" to be like this. You cant grasp or come to terms with what is being said because of who is presenting it. Much of current research is presented through emotional contexts based on  individual's unique experiences. It becomes difficult to grasp a theory of some sort that can become food for thought and reflection. Angela Lee Duckworth's Ted talk on the concept of grit in the classroom is a classic example. There are many others who shout about bullying, and dress codes, and all sorts of peripheral things that have little bearing on what education is all about.

The same is true inn the current political arena where the participants themselves become the object of the vote and not the idea and theories for which they stand. For example it's virtually impossible to determine exactly what Donald Trump stands for because his persona and delivery style far outweigh
what he really means.We seem to be becoming a nation of performers who know how to perform but but who's performances completely obscure the songs they are trying to sing..

## Monday, May 9, 2016

### Testing is not the way to Improve Education

It's absolutely remarkable watching the British Government's current attempts to destroy on of the best education systems in the world. Having already backed down on their lunatic plans to turn every school in the country into an "Academy", they now have come out with a standardized reading tests that appears to have the potential to destroy every student's self confidence according to this TES article  The  incredible pressure put on young children through this relentless regime of testing by a government that has minimal educational expertise  outlined in this BBC article is  absolutely staggering.  There was even a protest by parents who kept their children home for the day because of the testing. And all the time the British Government is clamoring for more and more rigor Finland lets children play more and still outscores the UK in global tests.  In fact there are so many articles on the dissatisfaction of the over-testing of children and other government mismanagement of education issues in the UK that there is almost an entire BBC webpage devoted to articles about it.

Thankfully in many States such as New York the "Opt-out" movement  is growing as people come to realize there is little benefit to children from constantly testing them. Assess, yes, test, no.

So why do Governments feel the need to test so much. In he case of the UK there is a terrible lack of trust of teachers on the part of the national government. They see education as "economic investment" and global competition first as opposed to helping children become the best they can be by whatever yardstick success as a human is measured. This is an incredibly complex issue, of course, but you simply cannot raise standards by raising the difficulty level of tests.

## Friday, April 29, 2016

### Schools in Finland

"In Finland the teachers are the entrepreneurs" is one of many interesting comments in this fascinating article from the Wall Street Journal.  The article is really well written because it describes in pretty good detail why the Finnish system produces such bright, well educated students. It describes the nature of the curriculum, the requirements to be a teacher in Finland as well as the government support for education, especially reading.

It also points out the many reasons why such a system would probably not work in the US which is rather sad and depressing. For example the Finnish school population, they say,  is much more homogeneous in terms of languages spoken. I presume this means that it is easier to teach students who are all the same!  Parents also take a much more hands-off approach when it comes to  raising their children from tying their shoe laces to college selection.  Things such as this are the reason why making international comparisons based on a single set of test scores are so difficult to analyze and interpret.Life in Finland is very  different from life in the US in every respect imaginable.  Tests such as TIMMS have long praised the math performance of students in Singapore but make no mention of the deplorable situation in that country where all students with disabilities such as Down Syndrome are required to attend special schools.

Nonetheless there is much to be learned from the schools in Finland, something Bernie Sanders has been advocating for for some time. Perhaps it's time to abandon the standards-based movement that, I think, has been responsible for much of the decline of student performance and happiness and give power back to well trained teachers so that they can, once again, teach students and not teach standards or text books.

Here's a wonderful short video by Michael Moore that gives us a glimpse of schools in Finland. Notice the words of the math teacher in particular.

## Wednesday, April 27, 2016

### I'm Counting on You.

OK, before you read any further start counting by 5s. You can stop when you get to 50. Done that? Did you go 5, 10, 15, 20 etc? Yes you probably did.

Now, before you read any further count by 5s starting at 3.

OK, done that? What you probably did, if you are representative of 95 percent of the population is that you started off really slowly and hesitantly
3 + 5 is 8 + 5 is 13 + 5 is 18 and then, you probably went 23,28, 33,38,43,48 gathering speed as you went.

To check try counting by 5s starting at 7. Chances re you stumbled to begin with until you saw the pattern. Math is the Science of Pattern and once you see the pattern you are a long way to having a meaningful understanding of what you are doing.

Try the same thing with fractions. Count by halves starting at 0. Try it before reading any further.   Now start counting by halves beginning at a quarter. It's just the same, once you see the pattern everything gets easier.

So why not try it with decimals. Count by .3 starting at 0. Now count by .3 starting at .5. A little bit more tricky but it works. As soon a you see the pattern you can reel the numbers off as fast as you can speak.

This is the sort of thing we should be doing with students and adults to combat the paralyzing fear or hatred of maths that is so pervasive in the US. We are hampered, held back and disabled  by the debilitating phrase of "I'm no good at math" spoken aloud every day by so many people in so many places with large and small audiences by people of all ages and genders. Even the phrase "do the math" sounds like a  threat sometimes. No-one ever says "do the reading" and no one ever says, out loud, "I'm no good at reading" as described in this very eloquent Washington Post piece
sent to me recently by Erik, my wonderful engineer son-in-law.

Everyone can do maths if they focus on understand it and don't just memorize isolated facts to get scores on tests. Maths is made to be understood and played with; just like language.

## Friday, April 22, 2016

### The Future is in Everything?

It's funny but it's not until  one is diagnosed with a terminal illness that one suddenly becomes acutely aware of how different perspectives come into play in one's life.

One such difference I've noticed recently is just how much the activities and thoughts that comprise our daily lives are focused on the future: I need to go shopping so I can eat later today and for the rest of the week; I need to publish an article to remain tenured; I need to brush my teeth so they will remain healthy; I need to not smoke so that I will live for ever (if only I had heeded this one 50 years ago). The list goes on. We paint our houses to protect them from the ravages of rain, we change the oil in our cars to keep them running. Of course, painting a house makes it look nice and choosing a car you like gives one a sense of pleasure. But when you stop and think and give everything a percentage of now/present or future it always seems the future part comes out way ahead, almost 90% it seems sometimes.

Everything except listening to music. Music is for the present. I listen  mostly to  Bruce Hornsby or Scars on 45. If you really, really listen to the music; to the voices and the instrumentation; the feelings and emotions being conveyed you are 100% in the present. There is nothing in the music you are listening to that is touched by, or touches, the future. Pure hedonism perhaps, but in the best possible way.

So I started thinking about this related to elementary school math. So often we call elementary school "preparation for the future" or :preparation for middle school". This is all wrong. Elementary school is for a child in the present, in the now, for them as children. For example, when we teach a set of multiplication facts we should help the student focus on the wonderful  patterns made by the facts, the relationship between the numbers, what multiplication means and how cool it is to have a tool that can help you work  out what 12 fifteens are.if we do this the student will remember the facts. The 'future' component of the activity is the result of being in the "present" while being involved in it.

Elementary school should be a celebration of what it means to be young, to be curious about the world, to make mistakes and have conceptions of things that are outrageous and wild until the reality of what they really are sets in. The British Government currently fails to see this where they see education as "economic investment". Their push for turning all public schools into private academies is as misguided as constantly testing children.  We re thankfully beginning to see that this focus on testing in US schools is seriously misguided.

I remember teaching a graduate course to teachers in a school in Monterrey in Mexico 30 years ago where children were treated as children. They were expected to behave and think like children  and not as mini-adults. Childhood was celebrated for what it was, a time of growing, learning,discovering and just being a kid.

## Tuesday, April 19, 2016

### Hot Moments or Difficult Discussions?

Sometimes the emails I receive at work make me wonder what is happening to the world of education. It's probably always been this way  but the use of sensationalism to grab people's attention seems much more in one's face these days; probably because of the use of social and mass media.

Today I was invited to register for an on-line seminar on the topic of "hot moments in the classroom"  presented by Dr. Tasha Souza of Boise State University. For only \$247 I can join in and there is a "100% satisfaction is guaranteed".  Just for fun, I google-imaged "hot moments"  and then I google-imaged "difficult discussions" the other part of the seminar title. When you compare the two sets of results it becomes fairly clear why the term 'hot moments' was chosen; obviously an effort to boost the number of people shelling out \$247.

I'm probably getting old and cynical, or maybe I already am, but this type of sensationalism is not what is going to help improve the educational experience for our students. For me, it is denigrating to use the term to describe what the author herself says are clearly important times during the educative process. Being able to handle different points of view between 20 or so adults in a classroom can be a tricky thing but a hot moment it is not. When difficult topics are discussed with open minds then we all can learn. When we all adopt and develop growth mindset skills and dispositions we are all able to develop the way we think, feel, sense and learn.

Theories such as Carol Dweck's Mindset theory are well research, developed and improved upon over time unlike the sensationalist ideas such as Souza's "hot moments" and Angela Lee Duckworth's idea of grit..

## Tuesday, April 5, 2016

### Classroom Stories Four

One of the best days of the year to teach a class of fourth graders is April 1 or April fool's day. The students are old enough to appreciate a good joke as well as to initiate a good prank on their teacher.

During the five years I taught fourth graders before moving on to college teaching I played some great jokes on my students and they on me. One of the most memorable was the April 1 of, probably 1975. It was a large class I had that year, some 35 students and for some odd reason it included several sons and daughters of fellow teachers. On this particular day I walked in to classroom after morning recess and all the students were sitting and working  quietly at their desks. Recognizing this was a somewhat unusual situation as I usually had to quieten them down or redirect a few. I sat down at my desk and  a line of students quickly formed with questions about what they were working on. Sometimes I sat at my desk and they came to me and sometimes they sat at their desks and I came to them.

As the students asked me questions each one had one hand at their chin gently stroking it as they asked a question. After I had responded to the third student in the row I looked up and realized that all 30 or so students still sitting at their desks as well as the few still in line were gently rubbing their chins; thumb up one side and fingers up the other in a pensive gesture. I stood up grinned and was about to say something when I realized that I, too, was gently rubbing my chin in exactly the same way all the students were. We all at once burst into unrestrained laughter and I never, ever rubbed my bearded chin pensively again.

Probably, one of the best April fool jokes I played on my students was to do a forty minute lesson on how spaghetti grows on trees in a certain part of Italy, I would draw a picture of a tree on the blackboard with bunches of long strands hanging from the branches. I would then draw the spaghetti harvesting tool that was a long rake-like implement with a  board on the shaft into which the bunches of spaghetti would fit when they were a meter long. This was the only sign that the spaghetti was ripe. The drawings would be accompanied by considerable writing explaining all the minor details such as farming licenses, climate conditions and so on. The class would end at 11:55 because in England April fools jokes are only good before noon. I would then tell the students that it was an April fools joke and that spaghetti really didn't grow on trees. I sometimes worry that the students, now in their early 50s still think that spaghetti grows on trees because "sir", as students called male teachers in those days, said it  did.

### Classroom Stories Three

Many years ago when we were advocating the use of genuine inquiry science there were what we called the "Alphabet" science programs. Things like SCIS, ESS and several more that I don't recall. They were all based on the idea that children needed to be actively involved in their learning, or hands-on-minds-on science as we used to call it. Activities involved interaction and engagement in every sense of the words. These programs were just like the current NGSS only much easier to understand and operationalize. At least that's how it seems to me.

One really neat activity in the SCIS program, I think it was, was called Clay Boats and involved the students in molding a piece of clay into a boat shape to see how many pennies it could hold. The main science concepts were sinking, floating, and density. The idea was to make the boat as large as possible to displace as much water as possible thus allowing it to hold more pennies, but still float of course.

So I selected this activity as the focus for a science ed. professional development session  for a group of teachers at a school in Plymouth Vermont I had been working with. Having run out of my regular clay I bought a packet of blue playdo that I thought would be a good substitute for the clay. After fifteen or so minutes introduction we filled large plastic containers with a couple of inches of water and started making our clay boats. To my horror, within  a minute of testing the boats in the water the 15 or so teachers' hands had all turned bright blue, as had the water. The playdo was water soluble; a fact that had never occurred to me. In panic, I searched the classroom for a substitute and for some reason, I'll never know how I thought of it, my eyes landed on a box of aluminum  foil. Quick as a flash I tore off 12 inch squares of the foil and showed the teachers how they could make their boats using the aluminum foil. The funny thing is that I have used foil every time I have done this activity since as it seems to work much better than the clay.

Another, brief classroom story happened just a few years ago when I was teaching math ed to a group of undergraduates. As I was teaching using the SMARTboard at the front of the class I noticed something fall from the ceiling to my right about four feet from the door. I looked up and saw that a couple of ceiling tiles had been removed for a maintenance project. I then looked at the floor directly beneath and saw a dazed mouse looking around as much as to say "where am I?". The mouse was about six inches behind one of the u/grad's chairs which was as close it had come to landing on her head. As I approached, the mouse found its bearings and disappeared under the classroom door. I followed it out just in time to see it disappear under the door to a fellow faculty member's office. By all accounts it spent the weekend dining on a packet of cookies left on the faculty member's desk.

There was not one "eek" from the students and no-one stood on a chair; how times have changed!

## Monday, April 4, 2016

### Classroom Stories Two

A couple of animal stories this time. When I was a fourth grade teacher in England I kept a pair of gerbils in my class. They were in a large aquarium with a chicken-wire cover on top. The students called them Brownie and Whitey because one was browner and the other was, you've got it,  whiter. Periodically the gerbils would produce offspring which always amazed the students especially when it happened during class time. The students would always get alarmed when the gerbils carried their babies around like the one in this picture. Each time there was a new litter students would sign up to take a pair home, with parental permission of course. I would build a cage out of half an old wooden desk and send it home with a water bottle, two gerbils and a week's worth of food. I always assured the parents that the gerbils were the same sex.

One afternoon I noticed one of the students walking slowly  across the back of the classroom and stopping every few steps and stamping his foot. I asked hi what he was doing and he said that Brownie had escaped from the cage and he was trying to catch him. When I asked how stamping on him could be an effective form of capture he stopped for a moment and replied "I was only trying to stamp on his tail". After that we kept a large net in the classroom for catching escaped gerbils.

The other story was told frequently by one of my doctoral advisers at the University of Illinois where I completed my doctoral program.As the story goes, a class of fifth graders had invited a member of the local raptor organization to bring a bird of prey to class for part of a thematic unit. It was a small hawk as far as I remember and the plan was to let it fly around the classroom a couple of times. After a fifteen minute talk about hawks the handler duly released the hawk which flew  straight to the classroom aquarium in which was coiled an unsuspecting, and very contented, small garden snake; the class pet. Instantly, the hawk snatched up the snake, flew up into the ceiling rafters and proceeded to devour the hapless snake much to the horror of all the students who were swiftly led from the classroom to avoid being splattered by the blood falling from the hawk's unexpected meal.

## Sunday, April 3, 2016

### Classroom Stories One

As my career draws to a close in just over a month it seems natural to recall some of the events and people that have shaped my life along the way as I have learned how to teach, a journey that is never finished or complete.

I started my professional career as a fourth grade teacher at Sefton Infant and Junior Schools in Bristol, England. As you can see it's still there. Hopefully it will remain under the watchful eye of the Local Education Authority and not become one of the Conservative government's dreaded Academies . It's hard to believe that they would even consider such a drastic, mandatory change in the way education is managed.

The event occurred on the same field trip when another school sharing the camp site took their students, also fourth graders, to the Isle of Wight for the day; a five or so  mile ferry trip off the coast in the English Channel. At the end of the day the students, tired and exhausted, boarded the ferry for the short trip back to the mainland. Five minutes into the trip one of the students told one of the teachers that Tommy was missing. After a quick roll call it was confirmed that Tommy was, indeed, nowhere on the ferry. There were, of course, no cell phones, or quick ways of communicating with Tommy wherever he was. As the teachers huddled to form a plan, which included asking the ferry captain to return to the Isle, one of the students interrupted to say that Tommy was standing on the bridge of a Royal Navy motor torpedo boat that was racing past the ferry at around 30 knots. A loud cheer went up from all on the ferry. Tommy was at the quay with his Royal Navy escort waiting when the ferry docked.

We shared several drinks with the teachers from Tommy's school that evening.

## Thursday, March 10, 2016

### Teaching Involves Engagement, not Entertainment

Someone said something at a meeting I was at yesterday that was so succinct, and was something I have always believed in, but have never thought to put it this way; "Teaching is engagement not entertainment". Sometimes when a sentence comprises only the important words it has much more impact.

This idea is particularly true in math education where we have, perhaps, been more guilty of entertaining children with manipulative materials and math games at the expense of engagement. The concept of engagement in learning means that the learner is actively, cognitively involved with the material to be learned. Something is happening to their thinking that is changing the way they think for ever; the permanent development of a concept, skill or disposition.

The difference between engagement and entertainment is not as clear or as easy to define as one might think. Here's a list of the perceived differences by Robyn  Jackson, president of ASCDedge .
There is clearly an interplay between the things listed in each column. And here's a list of myths by the same author.

Probably the place where the distinction becomes the most critical in the classroom is with the use of technology especially when games, activities or the internet are being used. Here's an attempt to differentiate between the two in relation to the use of technology in the classroom.

Finally, it looks like we won't have to worry about whether our students are engaged or not in the classrooms of the future. Students will all be wearing headsets like the students in this research project at Washington State University . This device will be developed to include a red light attached to the top of the headset. When the student is engaged with the topic under consideration the red light will come on and stay lit until the student becomes disengaged. The task of the teacher will be to make sure all the red lights are on, and remain on, for all 30 students in the class. Each head set will contain a device for calculating the percentage of time the red light has remained on for each student. The data collected each month  will be used to evaluate teachers. Yes, it's time to retire.

Here's something interesting I came across recently; Bloom's Digital Taxonomy !

## Sunday, March 6, 2016

### Maths Gets a Bad Rap

I've spent my entire professional career trying to bring math to life for my students; to make it meaningful, personally significant, interesting and even inspiring. I like to think that those students and teachers I have worked with have seen the light to some extent or another and have gone on to share with others the wonderful sense of accomplishment one gets from understanding and remembering anything mathematical.

Sadly, it's been an uphill battle primarily because of how our culture in general sees maths. I am an avid crossword puzzler and have lost count of the number of clues than denigrate maths. Clues like "division word" (into) are fairly harmless but then there are some that are completely wrong like the clue "Number indivisible by two". The answer is "odd" but it is so mathematically incorrect as to create the false impression that you cannot divide any odd number by 2. To make it accurate it should be "Number indivisible by two giving a whole number" or something similar. Every number is divisible by 2.

But probably more troubling is the clue in a crossword puzzle I completed a few days ago: "Member of  the math club probably". The troubling answer was "nerd" which is indicative of how people in general  tend to see people who are interested in math.THIS IS SO COUNTERPRODUCTIVE.
As long as we hold on to this view of math as being calculations, arithmetic and tedium nothing is ever going to change. Only when people start to embrace the beauty of fractal relationships, number patterns such as Fibonacci and Lucas numbers as well as the joys of geometric thinking will there be any change in the way we approach maths. I fear this will not happen in my lifetime, if ever,

I've been at meetings where. when the topic of math was raised, there was, sadly, an audible groan.

## Thursday, March 3, 2016

### Common Core Math Constantly Misinterpreted

The Common Core Maths Standards must be one of the least read and most criticized documents in the history of mankind. There are myriad websites devoted to its downfall and repeal; politicians calling for it's removal and parents posting ill-informed emotional sabotage videos such as this one on the People's Voice FB. The person in the video even uses the term "new math" to connect the Common Core Math Standards with the "New Math" debacle  of the 1970s, something that has been done with great success during the past 50 years to stop progress in math education.

The saddest thing of all is that the person who made this video almost certainly has not read the Common Core Math Standards. If she had done she would have read in the fourth grade Number and Operations in Base Ten standard the following;

4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

5. Fluently multiply multi-digit whole numbers using the standard algorithm.

2. Fluently divide multi-digit numbers using the standard algorithm.

The strategy that she so eloquently illustrates is just a step on the ladder to helping students

understand
what the algorithm is all about.The algorithms are so much easier to remember and apply

in appropriate places when students understand what they are doing.

It's all here in black and white in the Common Core Standards document. All you have to do is open

the link, read it and try to understand what most of us learned by rote, without any understanding

when we were in elementary school. The example shown in the video is an exercise a student might

do to develop understanding of the standard algorithm.

ps The picture above is a classic example of the misrepresentation of the Common Core Math

Standards.

### Cheating Watches and Calculators

Just when you thought you'd seen it all something new comes along. This is almost beyond my comprehension because they are actually marketed on Amazon as "cheating watches".

Anyone who gives in-class tests will now need to monitor student's wrists to see if they are wearing one of these cunningly designed cheating devices. They even have "panic buttons" that can be pressed to reveal a regular watch face that shows the time.

The deeper significance of this whole scheme is the way the popular culture  now views education primarily as achievement, success and failure. It seems that success, as measured by tests is the goal of education regardless of whether students are learning anything. The ability to succeed on tests is more a function of  the ability to memorize or retain specific information than it is about meaningful learning.  Here's what you can get on Amazon so that you can cheat your way to an A grade. There are even cheating calculators for sale as well as watches.

There's a neat story in the Atlantic, When Success Leads to Failure  that highlights a teacher's dilemma when she realizes a friends daughter has lost the love of learning because she is so focused on test and exam results, and grades.

The bottom line is that if you cheat on an exam you might get a good grade but you will certainly not have learned the information well enough to use it to your advantage later in life. Your temporary success will lead, in the long run, to failure

## Wednesday, February 24, 2016

### The American Dream

In the midst of this strange, and at times, somewhat scary election the idea of the American Dream is frequently raised by a politician keen to remind people that things could be better if you vote for him, or for  her.

An old college friend from England asked the question on Facebook this morning; "So  what is the American Dream?". I didn't have to think long about what it is for me, and probably for many thousands of other immigrants to this country. The American dream is opportunity; the opportunity to fulfill one's life ambitions through hard work and perseverance. And with opportunity comes hope and the belief that things can be better.

Growing up in England in the 50s, 60s and 70s one's life was pretty much mapped out by the time you left high school. If you passed what were called 'O' level exams at 16 you got to stay on in high school and take 'A' level exams at 18. If you got really good grades in these 'A' level exams you got the opportunity to go to University. I think it was less than 8% of the population got this opportunity at the time.

My 'A' level grades weren't good enough to get to University so I became an apprentice surveyor. At the same time I was playing field hockey for a mixed team of men and women at the weekends and it turned out most of them were teachers. They seemed like a really fun bunch and so I decided to enroll at a teacher training college, where my 'A'level results were good enough, to become a teacher. Four years later I graduated with a bachelors degree in education and started teaching a fourth grade class at a primary school in Bristol, England. A couple of years later in 1974 I was also  invited back to the college as a Resident Adviser which is where I met the exchange students from the University of Illinois.

At the time I had progressed as far as I could through the teaching profession; the next step being a headmaster (school principal) which I knew I did not want to do. The other career path was to be a college professor and teach other people all about the wonders of teaching. Unfortunately, my 'A' level results were still not good enough for me to pursue this avenue even though I now had a BEd degree. "So why not come to America and get your PhD at the University of Illinois" said one of the American professors who accompanied the US exchange students, "It's the land of opportunity" he continued. So I did. I took advantage of the opportunity to get my PhD in teacher education which then enabled me to become a professor in a teacher education program helping future teachers learn all about the wonders of teaching young children. The American Dream is opportunity.

## Monday, February 22, 2016

### Curiosity; the Acorn of Inquiry

Many years ago when I was a graduate student at the University if Illinois one of my professors, the renowned Dr. Lilian Katz, gave an in-class assignment in which we had to rank 26 characteristics of elementary school students. The task was to put them in order from most typical to least typical based on our experiences working with children in the K - 6 age range.

I was reminded of this experience yesterday as I watched my grandson, Lachlan, playing with different things. One of them was a set of colored, wooden stacking rings of different sizes. First he tipped them off the post and then put them back on in a random order. He then held the stack of rings close to his face and pulled them off one by one running them over his face and through his hair. He did it gently and carefully savoring the feel of each ring as it bumped over his nose and through his hair. He then dropped each ring behind his head and watched it roll across the floor. His curiosity about the things he plays with is inspiring. He wants to know, to find out, to learn and to discover the world around him. Later, he was eating a circular piece of provolone cheese by tearing it into strips. As he did so he was mesmerized by the way it came apart in  long, even  strips,

And, quite amazingly, he does this all without standards, performance criteria, blended learning, grades or any of the other verbal paraphernalia we have designed in the name of learning and education.

## Thursday, February 18, 2016

### A Son, an Uncle and a College Student

Every Monday and Wednesday afternoon I sit in my car on the south side of the campus quad at St. Michael's College and watch my son run across the grass on  his way to class on the north side of the quad. It doesn't matter if there is green grass showing or an inch or two of snow he always takes the same. most direct route always stopping at the same spot to retie a shoe lace. I sit and watch because I just want to share in his unbridled joy of attending a college class.

He's a student, a "sophomore" now, in Professor Doyle's class that meets in the Robotics lab. He absolutely loves the class and his fellow students and, of course, Professor Doyle. Taking classes at St. Mike's has added a major dimension to how he sees himself, to what he knows about the world as well as who he knows and socializes with.

Andrew is 23 and has Down Syndrome and is now a college student as he frequently tells me in his daily report to me of who and what he is. It always begins "I did hug Keith and Neil (two members of his favorite group, Celtic Thunder, whom he really did hug after their performance last year at the Flynn Theater in Burlington) and ends with "and I'm a college sophomore".

He takes from the class what is important to him. It's not a grade or a collection of assignments, papers or complex ideas. It's factual information about things that interest him related to the course content; facts he can recall at will in conversation and will always remember because he has a remarkable memory.  It's the interactions with his fellow class mates; listening to their stories, helping solve a robotics problem, feeling the joy of success when a project  finally works for his group.

But it's what he brings to the class that is, by all accounts, so wonderful; a boundless sense of optimism, an eternally positive outlook on life where there are no mountains too high or seas too wide. An individual challenge to each student in the class to look inside themselves and see who they really are, and then to grow in the knowledge that they have made a new friend, understood someone who thinks a little differently and, perhaps, overcome some assumptions.

## Tuesday, February 16, 2016

### Cost of College

I've often wondered why the cost of college is such a major issue when the cost of everything else seems to increase far quicker than the increase in what we earn to pay for things. A loaf of bread, when not on sale, now costs just under \$5. An oil change for my car is nearly \$80. Both are dramatically more expensive than they were just a few years ago. Yet, by themselves, both are fairly insignificant in actual dollars when compared with the vast amounts of money that need to be invested to become an educated person. And here is the essence of why, to my mind, it is such an issue. It is because all the things that we need in order to live are wrapped up in one phrase, "the cost of college" which includes all the increases in all the things one needs in one's daily life, which, when added together, become a dramatically large increase.

Anyone who supplies a product or service to a college that increases in price is responsible for the rising cost of college. This includes insurers, builders, lawyers, medical workers, caterers, and the infamous text book suppliers. For the past ten years I have used subsequent editions of an excellent math education text that has increased in price from around \$40 to \$215. True, students can rent a copy of get an ebook version for less money but I've always encouraged my students to develop their own personal professional libraries. I've always allowed my students to purchase the 7th edition instead of the current 8th edition for a fraction of the cost and there is very little difference in the content.

Anyone who, while searching for a college, has looked to see the extent of the amenities at a college under consideration is also responsible for the rising costs of becoming an educated person. In order to stay competitive, colleges have to make sure that they have the latest in technological advances and support services, the most complete recreational facilities and the most up to date living accommodations.

Anyone who has considered attending one of the notorious on-line, for profit, "colleges" is also responsible for the rising cost of becoming an educated person because they have reduced the number of people available to attend authentic colleges which increases the competition for students causing the situation described above.

Just about the only people who do not contribute to the rising "cost of college" are the people who work at colleges. Salaries for professors, for example, have increased woefully over the last ten years, in some cases, less than 15 percent. Lecturers (professors)  in Scotland have been engaged in strike action because of the state of their pay structure.

## Friday, February 12, 2016

### Bedtime Maths at Bed Time

When I first heard about Bedtime Math on the local public radio station a year ago I thought to myself "duh, why have we never thought of that before?". Now it seems like a no-brainer and there is evidence, although still tentative, that this program is working according to recent research at the University of Chicago.

We've known since the Stone Age that reading to children at bed time will help them become better readers as well as more curious and motivated individuals. So why not maths?

I think the answer to the success of this program lies in the way the author, Laura Bilodeau Overdeck, has managed to shed the nationally pervasive view that maths, especially at the preK - 6  levels  is nothing more than computation.  She sees math through a wider lens through which she weaves numerical relationships and patterns into everyday events and stories. The books contain questions at different levels that can be used by parents to match the developmental/ability levels of their children. One interesting finding from the study is that the books really help parents who are anxious about math or who did not have good math experiences in school.

Earlier today my daughter called to say she had just purchased her first Bedtime Math book for her 18 month old son Lachlan. She also added that he is seeing twos of things everywhere, and says "two"; even when he is breastfeeding :)

## Friday, February 5, 2016

### Teaching

You can probably remember a teacher who touched your life in a positive way at some point. Perhaps it was a third grade teacher who patiently helped you understand what the commutative property of multiplication was. Or maybe it was a fifth grade teacher who was passionate about literature and always found a way to help you make every story come to life. Or it might even have been a high school history teacher who dressed up in period costume for classes about the civil war.

Have you gone back to your school to thank that teacher?

There is nothing quite like the feeling that comes from being a good teacher; whether it's guiding children to a better understanding, sharing one's passion for a particular topic, compensating for a child's tough home life by caring a little bit more, or helping a student develop a growth mindset when they say they'll never understand fractions. Being part of a classroom full  of children for a year is an amazing experience.

It's not easy though, if you want to do it right. It' not something you can do by reading a book, taking a single course, spending a year in a classroom with a practicing teacher or just doing what comes naturally. Being a good teacher and getting the most fulfillment out of it involves a rigorous, challenging but incredibly rewarding intellectual process of thinking, reflecting, acting and developing the professional dispositions of a teacher. It involves having a good understanding of what you will teach, a practical grasp of how children develop and learn and the accumulation of a toolbox of teaching strategies you can call upon when needed.

A great place to do this is in the undergraduate or graduate programs at St. Michael' College where theory and practice are integrated in a way that enables you to be the best you can be.

## Thursday, February 4, 2016

### What is Maths?

For the past several year I have defined maths as the science of pattern and the art of making sense. This definition has probably developed as a response to the still somewhat prevalent view that maths, at least at the elementary school level, is the study of numbers.

I have tried to operationalize this idea by having student look at maths from many different angles especially from the aesthetic and relational points of view. I have included the study of fractals in my courses because I feel they epitomize the idea of numerical patterns and relationships which are so important as we try to get children to remember concepts, ideas and factual information.

I have read much by Jo Boaler and other illuminaries who believe that maths should be taught the way I believe it should be but I have never read anything quite as clear and eloquent as this piece by John Seibert of the Math learning Center, publisher of the wonderful Bridges Math Program.

The analogy of  defining maths as the study of numbers and comparing it with literature as the study of the alphabet is absolutely brilliant in its simplicity and absurdity. Numbers, like letter, are simply some of the things we use to convey meaning and communicate with.  And this sentence, "Perhaps it is through the lens of patterns that math can transcend the procedures that have come to define it" is exactly what we should be helping young children do.  Seibert then concludes with this masterstroke, "After all, when we teach our students their ABCs we expect them to one day write their own papers and poems. In teaching our students their 123s, shouldn’t we support the same creativity?".

## Wednesday, January 20, 2016

### The End Is Nigh

So, I was planning to retire at the end of this academic year but a bit of a serious heart attack got in the way just before the end of last year and I'm forced to take medical leave this semester. I had planned to really enjoy the last semester of my working life but sometimes things just don't work out the way we want them to; just  like some maths problems I have encountered over the years. No matter what you do, they seem to defy resolution. And so it is with my last semester. I was to have eased into retirement tidying up all the loose ends, making  sure my courses were complete and ready for my replacement to take over and make them hers or his.

The remarkable thing about teaching, something maybe only teachers know and understand, is that the courses one develops and teaches become incredibly, unbelievably, and almost inexplicably personal. The three maths education  courses I have taught in the Education Department at St. Mike's for the last several years are each unique and exclusively my own creations. Yes, of course there are curriculum standards and teaching standards that have to be addressed but my courses are my personal interpretations of those standards. I use the philosophical theories that I have experienced and developed over the past forty something years to operationalize those standards for my students. I use the experiences I have had teaching maths to children to illustrate and exemplify those theories and ideas with inspiring stories.

Each course I teach is characterized by an emotional set regarding the content I teach. I choose to focus more on those things that I truly believe work and cause my students to think. I minimize the time I spend discussing and exploring things that I know create negative feelings or are just "tricks of the trade" so to speak. I believe that to teach maths one must intellectualize the process, base ones practices on well reasoned theory or, as John Dewey said, we must use our "executive means" to get as close as we can to our "inspired vision". One must listen to children speaking mathematically and get to know what they think mathematically. One must know, understand, and love the maths intimately. My students were always actively involved in their learning by doing, talking, thinking and reflecting.

I cannot believe how difficult this is;quitting cold turkey, the sense of being shut out, stopping teaching, not meeting my students each week, not reading papers, not standing in front of the class, not telling stories, not conferencing  with students about their dreams and aspirations, not talking with colleagues, not helping students over their maths anxiety, not lighting up students' eyes about the joys and wonders of learning and teaching mathematics.

I hope there's a way I can continue to be a part of the maths education community and make a positive contribution in some way.