Twelve, Twelve, Twelve

Today is the last day this century that the date can be written the same way in the US as it is written in the UK. There is no Pi-day, 3.14.12, in the UK, for example, since the date is written 14.3.12 on that fair isle. It took me ages to understand what US HS teachers were talking about when they said Pi-day was coming. I though it was connected with mud pies because of mud season!!!!! Every time I write the date I have to consciously remember by birth date, 12/19//46, so that I write it the right way round. In the US the day fits right before the year making 1946. This didn't happen in the UK where it was 19/12/46. I must admit day/month/year does seem more logical. Why do we hop around here in the US and do month/day/year? Whose idea was that? Mr Webster's perhaps?

12 really is an interesting number. It's far more interesting that 10, the number around which our base system is constructed.  There is, of course, the duodecimal system or Base 12 that's used in several African cultures and, apparently, by Tolkien's Elvish language. 12 even has another name, 'dozen', just like twenty can be a  'score'. 12 also figures prominently in our culture with many things being purchased by the dozen, or the baker's dozen (13) that was still in effect when I was a child; the insurance against the crime of short measure. There's dozen roses too, a bottled or canned drinks are sold by the half dozen or less 'literarily' appealing 'six-pack'.

12 follows eleven which together cause more problems for children learning to count than just about anything else in western culture. In most Asian cultures the far more logical  equivalent of 'ten and one' and 'ten and two' are used.

There's the 12 days of Christmas, the 12 apostles, 12 people on a jury,12 signs of the zodiac, 12 inches in a foot and there used to be  12 old pennies in a shilling in the UK. There's also the dreaded 12 times tables. Why was it all the way up to 12 and not 10?  Why the added agony of those extra double digit facts to learn?

Time likes 12s too. Seconds, minutes and hours are all multiples of 12 and there are 12 months in a year. 360 degrees in a circle is also a multiple of 12 and we know circles are related to time in many different ways.

I know exactly where I was at 12:12 12/12/12. I was passing the whale tails on I89.
    

Elementary Education majors practice Engineering

Up until recently it was called Design Technology, the art of using our scientific knowledge to solve problems and create solutions. Today, thanks in large part to the Boston Museum of Science it is now called Engineering. Yesterday, in my math and science education class I challenged my elementary education majors to support a small washer as high as they could using two copies of the Defender, the student newspaper. I made sure the students spent a few minutes first reading the paper so that we could genuinely recycle it through the engineering activity.

In science the questions and curiosity arise from the natural world; How do seeds grow? What is magnetism? What are genes? In engineering the questions and curiosity arise from the ways we can use our scientific knowledge to overcome problems or improve our lives. Building a better mousetrap is probably the epitome of an engineering project. Just about everything we use in our lives owes its existence to the application of some type of engineering.

There are other differences between science and engineering that we need to be aware of when we are teaching students. The skills we want children to learn in science class include observation, classifying, measuring, predicting and communicating. In engineering we want children to learn all about researching, designing, constructing, testing, adapting and improving and presenting. Engineering provides students with authentic situations of problem solving and creativity. And it is incredible fun too so students are highly motivated to succeed.  

The Math of Aging

The older I get the faster time seems to pass. I've always felt this and have heard many others make this observation. There is, of course, a mathematical explanation since each successive year in one's life is a smaller fraction of one's lived life so far. At age 6 a year is a whole sixth of one's life whereas at 60 it is a sixtieth.

The speed of life also tends to slow down with the passing of the years. Athletes lose a yard, reaction time increases and and tasks around the house that used to take half an hour now need two hours to complete.

There also seems to be an inverse correlation between the accumulation of wisdom and one's ability to recall facts, names and fundamental relationships. I've always thought that wisdom is the ability to use one's knowledge and understanding in a beneficial way within a social context. I think it  would be difficult to be wise in the forest, so to speak, in the same sense that if one were not there, the falling tree would make no sound.

The finances of aging also take on a different character and dimension. The things one buys change since one has accumulated most of life's essential material things. One learns to do with less, want less, and wish there was less sometimes.

Geometrically life seems to take on a downhill slope having passed over a crest of some sort. When you look back you start looking uphill instead of downhill all the way back to one's birth. I find myself sneaking a look around the crest or the hump to see things way back in the past. The geometric shape of the individual years also seems to have changed. It's flatter and has fewer peaks and valleys; summer tends to resemble a wide valley while winter looks like a plateau. Spring and fall don't seem to count in the shape of the year any more.

The frequency of doing mindless things seems to increase too. I poured a can of dry dog food into the garbage can the other day instead of putting it in the dog's food bowl. A week ago I mislaid the potato peeler and finally found it in the draw under the computer table.

The geometry of one's physical shape also seems to change but I don't think I'll go there. The dog did eventually get fed by the way.

Zero is an Even Number

There was a neat, short article in the NYT this past Monday (thanks VBJ) about the dilemma posed by whether 0 was an odd or even number. It was a high stakes decision because it determined when you could get gas during the gas rationing imposed in the wake of hurricane Sandy. After consulting a number of mathematicians Mayor Bloomberg correctly called it as an even number so that owners of cars with license plates ending in 0 knew exactly when to fill up. The article also went on to describe how the French police in Paris just ignored the issue during a smog alert in 1977. While odds and evens drove on alternate days 0s could drive whenever they wanted to.

I grew up calling 0 nought and played the game noughts and crosses which seems a much more logical name than the tic-tac-toe used in the US. The name has 3 parts yet there are only two options - very illogical. The name also gave rise to the early (2000 - 2004) and late (2005 - 2009) noughties in the UK. The word also features frequently in British history, literature and lore while Zero for me, growing up, was the name of a Japanese fighter plane.

The recent inclusion of 0 as the starting number in teaching counting in elementary school math is, however, a serious issue. When we teach children to count these days we teach them to begin with zero. The reason for this is that researchers have discovered that it makes fractions much easier to teach, and for student to learn,  if students realize that there is a space on, say, a number line, between 0 and 1. Fractions are introduced as parts of a whole using a variety of different models. One of these models is a linear model such as a ruler. This is really useful because rulers are usually marked with fractional parts beginning at the 0 or origin.

This idea of counting from 0 also helps young chidlren learn about measurement. Life begins at 0 as does time, distance, weight, volume, capacity and angles. Since all measurement is the repetition of the referent unit it is important for children to realize what that referent unit is; what is the one or the whole of the referents being counted.

The really interesting thing, though,  about the numbers on the license plates is that they weren't even being used as counting numbers but as naming numbers so it didn't matter at all if it was odd or even in a cardinal or counting sense. The fact that it was called an even number just made everything tidy and mathematically correct. 
   

Geography; the Most Important Science

If you want to be licensed as an elementary school teacher in Vermont, and in many other States too, you have to have a second academic major in addition to your education major. This requirement came into effect in the late 1980s and I remember well being part of the team of HE teacher education folks and  VTDOE staffers who met for several days to work out a rationale and how the requirements would fit. The primary idea behind the regulation was that an in-depth knowledge of something other than how to teach would bring scholarship to the teaching profession and help teachers become subject leaders in their schools. It would give them an academic passion to go along with their passion for teaching. I thought, and still do, that this was a great step forward, not least, because I had gone through this process some 10 - 15 years earlier in the UK.

My academic major was geography which complimented my teacher education studies so well. I remember learning all about geomorphology, a little geology, human geography, some social geography as well as physical geography. I learned about probablism v possiblism, Gondwanaland, drumlins, eskers, eratics and Monadnocks. So much of what I learned is still with me (there's a great eratic in the middle of the Bolton golf course) because I loved learning it so much and I used what I knew with passion when I was a fourth grade teacher. It wasn't that I didn't enjoy teaching math, language, reading and science but geography was just something else.

I remember teaching an eight week unit on the canals of England in which the students read and wrote about just about everything related to canal construction, canal life, influences on commerce and on and on. The student even brought in old pots and plates and cups which they then decorated in the traditional roses and castles artwork of those who lived on the canals. So when Richard Kujawa, the geographer extraordinaire at SMC, put this link on his FB page it brought a little sadness into my life to know that Geography was no longer a top ten favorite discipline in British high schools. I was glad to know, however, that I was in good company with my love of geography as the author of the article was none other than Michael Palin, president of the Royal Geographical Society.

British Government bans Calculators

Here's a remarkably backward step by the British Government concerning the use of calculators in primary school maths (elementary school math). They are banning calculators from the test taken by 11 year-old students.  It's almost inconceivable to believe that a group  of educated people could consign young children to the laborious task of nineteenth century pencil and paper computations. It's like not allowing people to use word processing technology to write.

This is such a bad idea. First, it is guaranteed to do nothing to help children develop more positive attitudes toward math. There are so many more wonderful things to learn about in math class than the unbelievable drudgery of doing algorithms by hand.

Secondly, the incredible amount of time wasted in schools teaching children how to complete laborious algorithms could so much better be spent teaching them how to solve problems. According to Thomas Carpenter, there are 22 different types of simple mathematical constructs involving joining, separating, part-part-whole, comparison, equal groups, area and so on that can be solved through the application of the addition subtraction, multiplication and division operations. We need to spend our precious time we have teaching children how to identify the nature of a particular math problem so they can then know which button to press on the calculator in order to solve the problem.

We cannot, of course, allow calculators to replace mental math. We now teach children from a young age all about the intricacies of our number system and how to use number mentally. Wonderful math programs such as Bridges focus extensively on numeracy or quantitative literacy as it is sometimes called.

It's interesting to note the use of the word "sums" in the BBC article. In the UK a sum is any one of the four operations, not just addition as an Harry Potter fan knows.

Vi Hart's Revolutionary Mathematics

For years now I've believed that Mathematics has an aesthetic, creative, and joyful component to it that is sadly ignored for the most part in the way we teach it; especially at the high school level. I have also believed that one of the main reasons why students at the K - 12 level, as well as the 13 - 16 level, dislike it so much is that it is seen, and taught for the most part, as a functional, objective discipline with little intrinsic value.

In math education there tends to be no equivalent of  poetry, creative writing, literature or anything that really helps children develop a sense of enjoyment or creativity. Imagine if reading and writing were taught from a totally functional perspective? One of the most remarkable things is the number of times students will ask of a particular math skill or concept "when am I ever going to need this" compared with the number of times a student will ask the same question of something related to creative writing.

One person who is doing something about this is Vi Hart, a self proclaimed mathmusician. She could also be called a mathartist or a mathologist as what she does flings wide the boundaries of  what we traditionally think of when we hear the word 'math'. You can also now see her videos on the Kahn Academy site which is a place many people go if they are having a difficulty understanding a particular mathematical idea.

Voting for the First Time

My son Andrew was able to vote today for the first time in his life. He was very excited and wanted to make sure he voted for the right people. It's a neat experience voting for the first time. It's almost like a rite of passage and it suddenly gives you a sense of responsibility and a sense that you really matter; at least that's how I felt both times I voted for the first time.

The first time I voted for the first time was in the UK in 1965. It was different there, as it still is, from how we vote in the US. In the UK you vote for your local candidate for one of the political parties. In each of the wards throughout the country individuals who represent the entire spectrum of political viewpoints, and usually a political party, "stand" for the House of Commons and  try to win your vote. The political party that has the most candidates elected to the House by a certain margin wins the election with the prime minister coming from that political party.

The second time I voted for the first time was in 1986 after becoming a US citizen in 1985. It was the same level of excitement all over again even though it was quite a different system. It has always seemed strange voting for a single person, a President, but in retrospect voting for a political party in England was probably the same thing because you always knew before you voted who the Prime Minister would be if a certain political party ended up being elected.

So, I'm off to vote.

Bonfire Night

Oh how I miss Bonfire Night and everything that went with it on November 5th. The incredible anticipation or waiting for my dad to get home from work with boxes of fireworks when I was but a lad. It was one of the annual rituals of growing up in England and, as a celebration, rivaled Christmas and birthdays for the sheer excitement of it all.

It is, of course, the annual celebration of the discovery of a plot to blow up the houses of Parliament in London by Guy Fawkes and his band of anarchists.

              "Remember, remember the fifth of November, the Gunpowder treason and plot
                    I know of no reason why the Gunpowder treason should ever be forgot"

The full verse is much longer but this s all we ever had to remember. For weeks leading up to Bonfire Night, every school boy and girl would be collecting combustible material to build the largest bonfire they could manage in their back yard. It was tended with loving care as the days grew shorter. Guy effigies would be made by stuffing old clothes with whatever was available to make them as life-like as possible. The Guy was always topped off with an evil looking mask. As the day approached praying in earnest would begin that it would not rain, for rain was the absolute scourge of everything related to bonfire night. The fire would not burn and the damp fireworks would not light, and this in a country renowned for its rain.   

The fireworks were the highlight of the night and only my dad was allowed to light them, at least until by brother and I were old enough, around 12 or so, to be given the lighting honor. Catherine wheels were always pinned to the same tree and jumping jacks always terrified me. The rockets were spectacular but I was always worried that the burning embers would start a fire somewhere. They never did, of course. Later, when we were teenagers we were only interested in "bangers" which were virtually small charges of gunpowder that made the most incredible bang.

Memory Pegs are Great

A light year ago in another life I was teaching a class of pre-service teachers all about problem solving in science. Today it's called engineering but back then it was called design technology. I was happily telling the students that they could get help completing the activity by asking their uncle, brother, father, son, or grandfather when Cathy, at the back of the room slammed her hand down on the desk and said "you can't say that". Somewhat stunned I asked "Why not?" and before she answered I realized just how gender biased I had been. It had never occurred to me before but from that moment on I resolved to rid myself of any gender biased language, disposition, intention or anything else that would discriminate on the basis of gender. What it took to raise the issue to my awareness was a shock that shook me out of my usual behavior; something that caused me to change. I am forever indebted to Cathy who I see at conferences from time to time.

So, mindful of the trauma of that moment I resolved to help pre-service teachers change their habits in less traumatic ways. Such a strategy is the 'memory tag' or "reminder" that a student can use to raise her consciousness concerning something about her teaching behavior she wants to change. For the past two weeks I have been working with Natalie, one of my student teachers, to slow her rate of speech when she is teaching. She's doing an outstanding job in the classroom but I felt she would get less tired and frazzled if she slowed down a little. To do this I suggested she put a Bandaid on one finger so that when she felt it it would remind her about her rate of speech.

Today when I observed her I couldn't believe the difference. She was speaking at exactly the right speed and everything about her  was calmness. The fourth grade students were also noticeably less frenetic and so after I had observed the lesson I complimented her on her new found skills and asked her if she had tried the Bandaid idea. She held up her hand to reveal the largest costume jewelry ring I have ever seen. She said that every time she felt the ring she remembered to slow down. Isn't that neat!

Tommy Sands is coming to St. Michael's

Tommy Sands is coming to Saint Michael's College in March for the next Concert for Saint Patrick. Probably best known for his incredible antiwar song Roses, Tommy Sands has been a voice for peace and justice for as long as I can remember. I have sung many of his songs with my band, the Highland Weavers, over the years so to meet him in March is something incredible to look forward to through the long, dark winter months.

Here's more about the legendary Tommy Sands. 

The concert will once again be a fund raiser for the Sustic Fund for Families of Children with Cancer.

Clocks Fall Back

I am determined not to miss the clocks "falling backwards" this year.  It's already happened in the UK so I have been warned. It's been so warm and un-fall-like recently that it's easy to miss the event. I completely missed the "Springing forward" because it was such a dark and dismal time of the year this year.

The changing of the clocks in Spring and Fall is one of the most difficult concepts to teach young children. It's such an abstract thing and yet it affects everyone. Time, in hours and minutes, is a man-made construct. It's really the daily rotation of the Earth on its axis that gives us day and night. The two things are related but not "glued" together, so to speak. We can "slide" the hours and minutes construct/scale back and forth on the Earth-related time scale to optimize the daylight time available to us. Here's a neat web-site that shows day and night as it's actually happening as well as the time anywhere in the world. Put a small piece of "post-it" on your monitor screen next to the line that shows night-time and leave the website up for half an hour or so. When you return to the screen you'll see how much it has changed.

Here's another web-site that shows you the time instantly anywhere in the world. It's very difficult for children to grasp the idea that it can actually be tomorrow or yesterday at this very moment somewhere else in the world. In fact, a fourth grade student once asked me if he went to Australia and watched a horse race and flew home could he bet on the winner he had seen win the race and win the bet because he had watched it yesterday which was really now!

The Moment you Know you can Teach

The mid-term of student teaching is a time of great anticipation in the lives of everyone involved in the student teaching experience. There are probably as many definitions of what constitutes a good teacher as there are teachers. In fact, there are probably as many definitions of what a good teacher is as there are teachers, parents and legislators combined. Just about everyone has experienced a teacher of some sort in their life so just about everyone "knows" what a "good" teacher is.

Regardless of what constitutes a good teacher there are signs to look for when you are learning how to teach that let you know that you can actually teach, in the institutional sense, that is. These are not things that other people tell you you are doing but things that come from within. There's something quite unique about teaching; it's the only job where you are face to face, within one room, and in control of up to 30 other people for around 180 days a year. This is what sets teaching, in the institutional sense, apart from all other human endeavors. So knowing, for yourself, that you can do it is a huge milestone on one's life. This "knowing" usually occurs in an instance, in a trice, momentarily; it's not something that creeps up on most people.

I knew I could teach when I was called from my class one day to deal with an urgent issue to find, when I returned, some 7 minutes later, all 34 fourth graders were still happily working on their individual assignments. Last semester I finally saw it happen with Teal, one of my student teachers when she stopped during a class and said quietly  "you're all being so good". Yes, I know that teaching is infinitely more than classroom management but that institutional sense of teaching comes primarily from that threshold of confidence across which all successful teachers step at some point.

Then begins the life  journey of learning how to teach in the pedagogical sense where the learning of each individual student in your class becomes the number one priority and your knowledge and understanding of the subject matter enables you to optimize the learning of each of these  individual students. OK, so this is how I define good teaching!

Outstanding Teacher Recognized

It's not every day that someone you know gets a $25,000 award and incredible recognition  for being the best there is. This is what happened to Matt Hadjun, a St. Mike's grad., current graduate student, cooperating teacher, fifth grade teacher, and all around great person. On the morning of Monday October 15 Matt was presented with a Milken Family Foundation National Educator award. Each year the award is presented to forty teachers nationwide in a surprise ceremony in which only a handful of people are privy to before the event. This creates one of the most exciting momentary situations one can imagine.

In a school assembly at Champlain school where Matt teaches fifth grade, all the students, teachers, a selection of local dignitaries, and past Milken award winners gathered under the pretense of listening to a presentation given by the Vermont Commissioner of Education. Commissioner Vilaseca started to speak but quickly turned the proceedings over to the Milken representative who then skilfully led the audience on a journey of what it means to be a great teacher until she finally identified Matt as the winner of the Milken Family Foundation award in Vermont. Sheer joy and pandemonium broke out as a battery of TV cameras rolled to capture the moment. Some of Matt's fifth grade students even had tears of joy rolling down their cheeks as he stepped forward to receive the giant check.

I have known Matt for several years and had the pleasure of working with him last Spring when he hosted one of our student teachers. His classroom is a remarkable  place of learning. With the subdued lights, intimate places for students to learn, and Matt's ever present gentle demeanor and sincere caring for his students and the subject matter he teaches, it is not difficult to imagine how wonderful it must be to be a student in his class at a time when the wonders of the world are just beginning to raise curiosity and a passion for learning.

Well done, Matt, you are an inspiration to all our students who want to be elementary school teachers.

Place Value and Reader's Theatre

In a previous life and many years ago I taught a language arts and social studies course as part of a teacher education program. My favorite part of the language arts part of the course was Reader's Theatre (and I intentionally put the r before the e). I think I enjoyed it so much because it was an opportunity for students to come to terms with the most incredible teaching tool we all have, our voices. I used to tell them that they had to imagine they were performing a play on the radio so no-one could see any of their actions or facial expressions; everything had to be communicated through the voice.

Yesterday evening I spent half an hour with Stephanie, a graduate education student to help her develop her project which was to teach place value through reader's theatre. Since I believe math should always have an aesthetic component this seemed like a wonderful way of developing the fundamental concepts of place value in an artistic and motivating context.

We talked primarily about place value, how it is groups of tens of tens and how we have ten numerals with which to make every conceivable number possible. We also talked about the misconceptions caused by phrases such as "0 is a place holder" and how 0 really means "none of". For example, in 103 the 0 means there are no tens. We also talked about the reason for putting a comma every three digits to help us read large numbers. Most people remember being taught this but few people ever remember being taught why. If you think about a large number such as 21,487,439 the first 3 digits from the right, 439 refer to ones, the next 3 digits, 487,  refer to thousands while the 21 refers to millions. Between each set of commas, from right to left are ones, tens and hundreds. 439 is ones, tens and hundreds of ones. 487 is ones, tens and hundreds of thousands and 21 is ones and tens of millions. This pattern of ones, tens and hundreds repeats itself between each comma for ever.

Teaching and Learning Abroad

It's advising week this week which means I get to share in my advisees plans and goals for their lives. I spend 15-20 minutes with each student as they share their excitement about their courses, professional goals and their work with children. Every so often a student is really excited about a program they have experienced and they want other students to know about it. So when Elizabeth Watts shared her excitement about her participation in Projects Abroad this past summer I had to add a link to the Education Department  webpage.

We frequently hear negative things in the news about study or work abroad programs so it is really good to hear about one that really works well. Elizabeth went to Ghana through Projects Abroad and was really impressed with both the educational component of the program as well as its administration. It's a volunteer program but everything is really well organized and coordinated. She says she always felt she was well looked after and knew exactly where she was supposed to be.

Another great overseas program is the Advanced Studies in England program which is a more formal educational experience but with a significant field placement in a public school classroom. One of my advisees, Sara Denton is currently studying with ASE in Bath, England. You can read all about her experiences in her SMC blog.  

Inspirational Teacher Voices

In one of the undergraduate courses I took when I was learning to be an elementary school teacher in the late '60s/early 70s the instructor would have us all periodically lie on the floor and project our voices up to the ceiling. Sophie Williams was her name and she taught Theatre courses as well as Education courses. The idea behind this was to help us develop our teacher voices so that we could be heard and understood without shouting. She taught us how to project our voices from our diaphragms as opposed to our chests. She taught us all about voice intonation, pitch and volume. These, she assured us, were the tools of our trade. They would help us use our voices in a survivable and effective way; something we would need for a lifetime of some 45 years of teaching.

Since that time, I have treasured Sophie's words and come to realize the wisdom in what she said as I try to help my student teachers master the infinite qualities of their voices. I have added to Sophie's advice by reassuring my students that they will get much more enjoyment from their teaching if they see it as a "theatrical art" form in which they can play the part by using their voices effectively and enjoyably.

For example, I tell them to sound curious and genuinely interested when they are asking a question; soften the voice or use falling or rising intonation so that is sounds like a genuine stimulus for thought or search for an answer. I tell them when they want student to follow specific directions to say what they mean and mean what they say. It doesn't really work very well if you want to let children know you are not terribly happy with their behavior if you say it with a big smile on your face.

So although I want my students to develop their teacher voices the idea is a long way from what is traditionally known as a "teacher voice". This is something left over from a Dickensian classroom. Effective teacher voices are inspiring, encouraging, enlightening, motivating, sensitive and theatrical. If they are this, they never need to be "teacher voices".

Presidential Debate Numbers

The last presidential debate was really interesting from the perspective of the numbers used by the two candidates. Both candidates started off with small numbers then went to big numbers and finally finished up with specific numbers.

By this I mean the numbers used by the candidates when they are trying to score points. At the beginning of the debate the numbers were mostly single digit with Obama using 4,5,2 and 46 while Romney used 4,3,5,1,2,3,4,5,4, 30 and 4. This went on for some 5 to 8  minutes or so and then they got into the big numbers such as millions, billions and trillions such as 2 million, 3.5 trillion and so on. Then came the specific numbers like 4,300, 3,600 and so on. The rest of the debate was a mixture of all sorts of numbers with the large numbers being the most frequently referred to.

The other interesting way of analyzing the debate numerically is the type of numbers they used; were they cardinal, nominal, or ordinal numbers? For the most part they were cardinal or counting numbers but Romney did use quite a few nominal numbers in referring to his 5-point plan. There is yet a third way of looking at the numbers which involves identifying whether the candidates use a referent or not when using a number. For the most part the referents are not included especially when the number is in the mill-, bill-, or trillion (e.g. "the national debt is at 3.5 trillion").

After you've heard these big numbers over and over again they begin to lose their intended meaning because they are not intentionally connected to referents. It's much easier to throw around numbers rather than use logical, reasoned arguments to make a point in a discussion. In the end, though, the numbers will just become hollow words without any real meaning.

Here are some even bigger numbers the candidates could use to further their arguments. They've used up to several trillion. They could go on to use quadrillion, sextillion, nonillion or, to really make a splash, they could use septendicillion or the very apt novemdecillion (this would be 1 followed by 60 zeros).

It's fun to note the numbers people use when you are in a meeting. Keep a note of the numbers  and develop a PNP, a personal numerical profile, for each person.
  

A Student Teacher's Words

Sometimes the words of a student are far more eloquent that those of the professor.

An Excerpt from a Student Teaching Journal

"I really enjoyed teaching the students the game “An Hour or Bust” this morning. I was a little nervous giving the directions since I was not sure if the students were understanding the concept of the game but once we started playing I could tell that they were getting the hang of it. In this “me vs. you” game as we like to call them I played against the entire class. Each team could spin up to five times on a spinner that had six different increments of time on it ranging from 5 to 20 minutes. The goal is to get as close as possible to 60 minutes in five spins without going over. The team that gets the closest is the winner, and both teams can choose to stop spinning at any point in time. The students followed along with me as I filled out the game sheet on the overhead, coloring in the increments of time that were being spun on the clocks and writing down each increment. The students were really engaged in the game and I could feel the effects of this engagement on their behavior. When I asked for students to show me they were ready to move to the next spin by putting their materials down they really followed through with that signal and we were able to have a good pace throughout the lesson. Students were not playing with their crayons or having side conversations and they were really excited about the game.

.

            I think students were so engaged in the lesson today because I really had a good understanding of the math concept that was being taught and I was excited to share my knowledge with them.  Students could tell that I was excited about playing this game and as a result they were really giving me their attention. The game was so much fun for me to teach because of how visual it was for the students. It was fantastic to show them how to count by 5’s around the clock and be able to shade in increments of time on the clock. At one point we had half of a clock shaded in and I asked the students how many minutes were shaded in. S___ raised her hand and shared with the class that we had half an hour shaded in on the clock, or thirty minutes. When I asked S____ how she knew this she was able to tell me that she could see half of the clock was shaded in which must mean 30 minutes since half of 60 is 30. Having the shaded in clock as a visual to refer to was so helpful for so many students today.  Near the end of the game the students had 50 minutes shaded in on their clock and I asked M____, one of our students who struggles with math how many minutes the students needed to reach 60. I pointed out to her the empty space that had not been shaded in and she was able to tell me that the students needed 10 more minutes to reach 60. I explained to the class that instead of counting all the way around the clock up to 50 minutes they could figure out that they have 10 minutes until 60 by thinking about 60 as a whole and knowing that they are missing a part that is equal to 10 minutes. I could have gone on all morning about subtraction, addition, fractions, and probability using this simple game.

 It was fantastic to see the students feeding off of my excitement about number corner this morning and having them so engaged and on task was a great feeling. Towards the end of the game when the class had to make a decision about whether or not to take their final spin because they were close to 60 minutes the students were on the edge of their seats with excitement. Our math enrichment students were at this point late for their enrichment class and I had to almost kick them out of the room with promises that we would fill them in later on the results of the game. It’s a credit to the Bridges program to have the students so engaged in an activity and utilizing visuals to make concepts concrete for students. I hope that I can carry over the good vibes from today into future lessons and keep in mind that enthusiasm is contagious."

This is why I love what I do. 



Ambition, Distraction, Uglification and Derision

In my math ed class this week I talked about ambition and distraction and uglification and derision but no-one knew what they were so I asked who had read Alice in Wonderland and could remember the Mock Turtle. Perhaps three students out of the eighteen raised their hands. I was initially somewhat alarmed by this but then I started thinking that what is termed a classic piece of literature  probably changes over time. But I'm not an expert in this area so I'll just wonder about it.

Something else we did pursue in class through a really interesting discussion is  whether the incredible amount of time we spend teaching algorithmic procedures  such as 54   is still
                                                                              +43 
worth it given the culture in which we live. Is the use of endless tedious practice exercises really worth it given the extent of the work we now do on numeracy skills as well as the technology-based tools available for computing. Shouldn't we be spending our valuable time teaching children to compute simple problems mentally and more complex problems using some sort of calculator? Shouldn't we be spending our time teaching children the meanings of the operations such as recognizing joining, separating, comparing, part-part-whole, groupings, Cartesian products and multiplicative comparisons?  The essence of solving a math problem is deciding which operation to use. Once this has been done the problem becomes simple arithmetic which can be solved either mentally of through the use of a calculator. In the discussion with the students I was playing devil's advocate; well maybe!

There are very few places in more advanced mathematics where the use of paper and pencil algorithms is required over mental or technology based computations.

STEM to STEAM

As I mentioned in a recent post I am involved in the development of a STEM Academy at one of the local elementary schools. At the first meeting someone raised the idea that it should be a STEAM Academy. We kind of joked about it for a few minutes, wound up the meeting and all went home. Yesterday, one of my colleagues, Professor Jonathan Silverman, an art ed. professor extraordinaire and instrumental part of the Integrated Arts Academy elementary school in Burlington sent me a link to a site advocating for STEAM Academies.

The idea behind STEAM is the integration of creativity and the arts into what has traditionally been called STEM. As I said, when we first heard this we treated it somewhat humorously but as this video, STEM to STEAM shows this is far from a cute acronym or humorous matter. My initial concern was that if we are going to integrate the arts then why not language, reading, social studies and so on which would basically lead us back to a school with an integrated curriculum; just like the great innovative schools of the 1970s and '80s.

But, I keep thinking about this and the more I think about it the more sense it seems to make sense because what it really does is to take us out of our traditional interpretations of what math, in particular, is all about especially in the elementary school where it tends to be somewhat destroyed by the dominance of arithmetic. Combining math and art is what Vi Hart is all about. It is what the Bridges math program  is hinting at; it is what I have believed for many years we should be doing in math; looking at the aesthetics of math, the pattern of number and the joys of probability and geometric shapes.

Science, engineering and technology have for many years been blessed with a strong creative/artistic component but never math. Perhaps this is the real opportunity for the salvation of math at the elementary school  level.



    

Classrooms Around the World

One of the significant elements of my research into how English Language Learners learn math in US classrooms is the nature of the schools and classrooms the students experience before ariving on US shores. For several years I taught a graduate course for a week at a school in Monterrey, Mexico and was alsways impressed by the way adults treated children as children and celebrated childhood. Children were not seen as mini or potential adults as we often tend to do in the US.

Valerie Bang Jensen, one of my colleagues, recently sent me a link to an interesting website that shows the diversity of classrooms around the world. The website, Brain Pickings is the work of Maria Popova, a self confessed  "interestingness hunter-gatherer and curious mind at large".

By looking carefully at each picture you can get a sense, to a small degree, of what education might be like in that particular school. One of the things that strikes me is the incredible diversity between each classroom and the almost total lack of diversity in some of the classrooms. There are classrooms where all the students are either male or female and there are classrooms where all the students are wearing identical clothes.

Perhaps the most remarkable lack of diversity is in the facial expreessions of all the students. Not a single student in any of the pictures is laughing, smiling broadly or even smiling at all. The pictures are clearly posed for the camera but I wonder why no-one said the usual "say cheese" or whatever it is in the appropriate language. The author describes how the students have to concentrate during the photoshoot but why do they have to all look so glum?

Is school really like that? Perhaps Sir Ken Robinson is right!   

This Teacher was a "Super Hero"

When a teacher is described as a "super hero" in the local media you know that he was something special. Sadly, George Cannon passed away suddenly last Monday. He was a chemistry teacher at the South Burlington High School and was clearly a very, very special teacher in the eyes of his students. At a time when the teaching profession is undergoing assaults from all sides it is incredibly encouraging to know that there are teachers like George who light up students' eyes to the joys of learning.

I never met George but I wish so much that I had. I can, however, listen to his words of wisdom in this Youtube clip in which he describes so clearly his approach to teaching and his beliefs about how high school students learn best. R.I.P George: we will all miss you. 

Mindstorms Storm the Mind

For many years I have been including design technology activities in my teaching Math and Science courses. I have used a variety of "found" materials to teach the design technology skill sets to my students. Design technology is basically the application of scientific knowledge and understanding to the improvement of the human condition through problem solving and invention.

Today, the term 'engineering' has replaced 'design technology' but everything else remains the same. The change is a good one because the discipline was constantly being confused with 'information technology' and other forms of  technology; a neat example of how word meanings change over time.

To upgrade the engineering aspects of my courses I want to introduce my students to the Lego Mindstorm materials similar to those used in Mike Thomas's Williston schools engineering program I blogged about several weeks ago. The Lego Mindstorms materials are a long way from the Rubber Band Rollers I usually have my students create but they offer infinitely more in terms of opportunities for developing genuine engineering skills in young children as well as in college students. Just imagine the sense of accomplishment when you build something such as a Lego robot or a vehicle and then program it to do things using your laptop computer. You can even program it to solve a problem such as climbing a step or getting out a a 'room' using light  and motion sensors. Here are some YouTube videos of many of the possibilities.

Just think of the language development, the social studies, math and science students can learn while they are engaged in authentic inquiry with projects like this not to mention the motivation!

  

And a Hush fell Over the Crowd

Many, many years ago when I was a fourth grade teacher I remember trying several things during my first year of teaching to get the students' attention when I wanted to say something to the whole class. I don't  have a particularly loud voice and I knew I didn't want to spend my life shouting and so, after trying the usual, "Everyone look at me" or a loud handclap I finally found a totally silent strategy that worked to perfection. I stood, with my arms folded, next to my desk and waited. It took about two weeks of repeated attempts and reminders to "condition" the students (34 of them one year) to stop talking and look at me. It became so effective that occasionally I would accidentally stand next to my desk and the room would suddenty go silent. The next minute I would be asking the studnets what was wrong and they would let me know that I was standing "on the spot" as they called it.

One of the neat things about working in so many different classrooms obcserving my students is that I get to experience a variety of ways that teachers get their students' attention. Many have little bells, chimes or gongs, others use a rhyme such as "1 2 3, eyes on me" to which the students respond "1 2 eyes on you", while others flash the lights or use a verbal command.

Every so often I come across something quite out of the ordinary such as the teacher at the Olive school in Arlington Heights, who I worked with many years ago, play the scale on a recorder when she wanted her students' attention. The effective part of this strategy was that she didn't resolve the octave scale until all the children were paying attention. This drove the more musical students in the class crazy and so they would make sure that their peers were soon giving the teacher their full attention so she could restore harmony through the resolution of the octave scale of 8 notes.

One of my current cooperating teachers has a wonderful way of getting her third graders' attention. She says "And a hush fell over the crowd" to which all the students reply "Hush". Isn't that neat?

The Emotions of Numbers

I started blogging 239 blog posts ago; or about two and a half years ago. It was great fun to watch the readership grow month after month and I started keeping track of the different countries where my blog was being read. It was really neat to see that it was being read in over 65 different counties by upwards of 75 people a day. Perhaps not earthshaking numbers in terms of the big picture but enough to keep me motivated to keep blogging. Occasionally people would respond with comments and I had 7 followers.

Then it all changed. Some time in the summer of 2012, the blog address changed as St Michael's College launched a new website and I was once again assigned to obscurity. My readership plummeted from 65 - 125 pages a day to 0 - 6 and I was devastated. I was told it was like "shouting in the dark" or "screaming in a storm" and that my readership would return as long as I kept writing. Well I have kept writing but very few people read what I write.

As of this moment, 3 people read my blog yesterday and no one has so far today. I've started pinning my blog posts on a bulletin board outside my office so that at least my students can read my pearls of wisdom but that seems to defeat the purpose of a blog. Perhaps this communication medium has run its course and our culture has tired of on-line titilation. I hope that the other 42 people who blog have better readership than I do otherwise there's an awful lot of wasted energy.

  

Student Teaching

I'm supervising four student teachers this semester, two undergraduate and two graduate students. Three of them are at Williston Central School, a 3 - 8 grade school while the other is at the K - 2 Allen Brook school, also in Williston. Their classroom experiences are each unique but they all experience the same developmental process that we all go through when we embark on a new life adventure.

Many years ago I came across Frances Fuller's Concerns Based Adoption Model (CBAM) of socialization into the teaching profession and have used it ever since as an assessment guide to the progress student teachers make during their student teaching semester. The CBAM provides me with a way of assessing where the student is in her development as a teacher. Through reading entries in her daily journal and weekly observations in her classroom I am able to tell which of the three stages of development the student is going through.

The three stages are basically this; concerns for personal survival, concerns for teaching and, finally,  concerns for student learning. Students usually pass through the first stage within a couple of days of being in the classroom although sometimes self doubt can creep in later when things don't go as planned or they experience a particularly tough day. The second stage of concerns, "am I teaching properly?", can last for several weeks or even a month or two. In this stage the student is preoccupied with her ability to plan lessons, manage the classroom, keep records and deal with the everyday demands of being an elementary school teacher. The final set of concerns are characterized by a focus on student learning and is quite a magical time. The student is usually confident in her teaching abilities at this point and knows her students well. She no longer accepts whatever quality or level of work they hand in but bases her expectations on what she knows each student is capable of  doing.

She has arrived at the point where she is concerned that each student is learning to her or his potential. This is both the art and science of teaching and is what it is all about..    

Nu, Numb, Numbers

I've found the perfect activity for getting through long, tedious meetings. Every time someone says out loud a number write it down. You then have some wonderful raw data to play with and do all sorts of things with. For example, if you have to attend lots of meetings on the same day like I did last Friday you can compare the numbers mentioned in each meeting. This will give you a Numerical Meeting Profile or NMP. Sometimes the meetings are dominated by small numbers while at other times they can be characterized by large numbers. Meetings can also be differentiated by the different types of numbers that are mentioned; ordinal (sequencing - third grade), cardinal (counting - sixpack) or nominal (naming - 2011) just to mention a few number types. The recorded numbers can only be those spoken and not any written on any form of presentation. This could get out of hand very quickly.

Another way of exploring the numbers is whether they are naked numbers; in other words does the speaker include the referent with the number. A naked number is one that has no referent attached such as "two-ninety nine" for a price or "six two" when referring to someone's height. This activity can be amped up considerably by pretending one is from another country, planet or occupation. For example a "two point five" would not cause any alarm if it came up in a discussion about engine size.  When discussing student GPAs, however, it can have the most dire consequences for someone.

It would be interesting to develop NMPs for different groups of people. For example, would an NMP for a group of mathematicians be different from an ensemble of historians at their monthly department meetings? What would the NMP be for a group of 20-somethings out for the night on the town?

Smoking in the Principal's Office!

President Reagan was in office when I moved to Vermont and supervised my first student teacher at Williston Central School. That was 30 years ago in 1982. The amazing Marion Stroud was principal at the time and I can remember sitting in her office chatting and smoking cigarettes; I kid you not.  I listened to her dreams of a new Williston school building with Kivas and "houses", ala Harry Potter's Hogwarts, and a real theatre and an Olympic size swimming pool. Such was her dynamic nature that within just a few years she had accomplished everything except the swimming pool.

Marion was a rare visionary in a world now dominated  by the bottom line and adherence to rigid standards. She had worked for some time at the Bankstreet school in NYC and brought with her to Williston three critical elements of education which were to form the foundation of a remarkable school; experiential learning, interdisciplinary learning and collaboration.She forged the school into "houses" of grade groups 1 - 4 and 5 - 8 where teachers worked together across grades and students learned in communities designed to promote real learning and not just give them an education.

I found Marion to be a kindred spirit for she too was British and a proponent in the US of the British Open Education movement upon which the Bankstreet school was based. I watched the school go from strength to strength as teachers bought in to her belief that a caring, collaborative and conscientious environment was the most conducive to the all around growth and development of every student and every teacher. The school became a leader in the application of technology in the field of education and was actively supported by Seymour Pappert, one of the great educators of the last century.

Thankfully, the school still retains much of Marion's vision through the dedicated work of many of the teachers who still inspire children through their enduring beliefs in experiential, interdisciplinary and collaborative learning.

The Museum of Science and Engineering in Boston is an amazing place; more so since Dr. Ioannis Miaoulis has become director. What he has done is to create an entirely new dimension of education and learning that brings science into the world of engineering (what we used to call design technology).

Instead of just learning about the natural environment as we do in science, Dr Maoulis has focused our attention on the idea that we use what we know about science to improve and protect the world we in which we live. In science education at the elementary school level the focus is on inquiry. Students are encouraged to use their natural curiosity to learn about the world in which they live. Which substances are magnetic? What do plants need to grow? How do rivers create valleys? In engineering, students are encouraged to use the science they have learned to solve problems. How can you make an object move using magnetism? How can you grow a plant without any soil?  There is a neat program at the Boston Museum of Science (EIE) designed to help students develop their engineering skills from a very young age.

One local school that has an incredible engineering program is Williston Central school where Mike Thomas has been guiding students in grades 3 - 8 in the art of design and construction using a variety of motivating activities and materials. On any given day the students can be engaged in designing and making a medieval  trebuchet or programming a Lego robot they have made using a computer program. 

Math is the Science of Pattern

Every semester when I start a new math ed. course with a new group of students I increasingly see the value of exploring and including the ideas associated with pattern in the course. When you stop to think about  it there is almost nothing random about math, in fact, trying to come up with complete randomness requires the sue of sophisticated computer models.

There are several different definitions of "pattern"; pattern as a design, pattern as a model or something to be copied, and pattern as a regular sequence or set of events. It is this last definition that is most relevant here. If we can help children see patterns between different mathematical entities it will help them remember, recall and make sense of what they are learning. We can do this from the earliest stages when children learn to count by 2s, 5s and 10s. What makes this type of activity even more worth while is to count by 5s starting at 3 or count by 10s starting at 7. Whenever i do this with my students I can see them initially stumbling and going slowly. Then, as they see the pattern emerge they speed up and end up rattling the number sequence off.

The Fibonacci number pattern identified by the squares above and dun flower to the left is a classic number pattern that occurs all over the place. It can also be demonstrated in the Sierpinski triangle, a classic fractal, as well as in Pascall's Triangle.

Here's another amazing number pattern. Add 1+2 (=3)+3 (=6)+4 (=10) +5 (=15) +6 (=21) +7 (=28) +8(=36). These are called triangular number because they make triangles (imagine 1 + 2 next to each other like steps etc). Now add consecutive triangular numbers together and what do you get? Yes, a square number. Imagine turning 3 upside down and fitting it together with 6 to make 9.

Math is, indeed, the science of pattern. 

Handwriting on the Wall

We recently received an email message implying that one of the goals of Higher Education was to eliminate paper forms of communication. By utilizing all the different forms of technology-based communication programs at our disposal we could actually run our college courses without our students ever having to handwrite a single thing. They could complete their papers using Word and submit them for assessment through eCollege. We would then comment on the paper and return it complete with a grade which would also be entered into the Gradebook feature in the same program.

Since I am a blogger I am clearly an advocate for technology in all it's various forms but something worries me, a lot, about wanting to achieve a paperless culture. Yes, we'll save lots of trees just as Kindle and other book forms are doing, and we would save hours and hours of painful handwriting practice sessions for young children in schools. But is this really what we want?

If our cultural goal is a paperless society then the art of handwriting will no longer be valued and if it's not valued it will not be taught. We will then have to make sure that we always have a keyboard of some sort with us complete with a form of printing off notes, letters, shopping lists and anything else that we might incidentally need to communicate. Perhaps all those hours, days, weeks  and months spent learning how to do joined-up writing, as we used to call it in England, could be spent teaching keyboarding skills instead. But is this what we really want?

Several years ago I read an interesting research paper in which parents were asked whether their children's teachers should spend writing instruction time teaching keyboarding skills or handwriting skills. The results of the research were overwhelmingly (80%) in favor of teaching handwriting skills.

There are many ways in which handwriting can be integrated with technology. The IPAD2 a well as the SMARTboard and many other forms of technology accommodate handwriting and some can even convert it into print. If one of our goals is to create a paperless culture then we must make sure that we don't also create a keyboarding only culture and bring about the demise of handwriting skills.

    

The Numbers of Our Lives

A new semester, new classes and another new beginning. I'm always so nervous for the first half of the first class but I usually settle down once the students have laughed or smiled a few times. It's been the same since I taught my first fourth grade class in August 1972 and I wouldn't have it any other way.

There are a couple of things I really want to focus on in my undergraduate Teaching Elementary School Math and Science course this semester. I want my students to be able to see the relevance of school mathematics. I want them to see that it isn't just a means of balancing your check book or working out how to solve mathematical problems. I want them to go beyond the "do the math" syndrome that our culture seems to have fallen into when it comes to any form of quantitative experience. I want them to see the poetry and creative writing equivalents in mathemtics.

I want my students to appreciate the aesthetics of math in the same way that Vi Hart does. I want them to see how wonderful it is when you add two successive traingular numbers such as 3 and 6 and get a square number. Imagine three small wooden blocks next to each other so that a stack of 2 is next to 1; like a step. Now add a third stack of 3 to make another step and a total of 6. If you take another 3-step and invert it, it fits with the 6-step to make 9; two successive triangular numbers make a square number. You can make triangular numbers just by adding an additional step. 6 plus a stack of 4 is 10. Just like this: invert the 15 and it will fit with the 21 to make 36, a square number.

Isn't that just way cool? 

Fourth Place in the Olympics

Go forth and multiply, come fourth and get nothing.

I would imagine it must be quite devastating to come fourth in an Olympic event especially if it is by the merest of a hundredth of a second or fraction of a centimeter. No medal, no standing on the podium, no name in the record books, nothing to take home and share with friends and coaches, and nothing to bite on.

On the other hand coming fourth means that you are the fourth best in the world at something. Of all the people on the earth only three can do that particular thing better than you can. This is the problem with ordinal numbers, they say so much and so little all at the same time. They also often tend to turn into nominal, or naming numbers, once they have been identified, such as in dates. Although July 4th is the fourth day in July it is more widely used to identify or name a particular day.

Of all the different uses of number, the ordinal use is the one we use when we want to compare things numerically. They are the type of number that seem to carry the greatest social meaning. When we are in elementary school it is way cooler to be a fourth grader than a first grader and it's probably better to live on fifth avenue or ninth street in New York rather than 12th Avenue or 42nd street.

Floors of buildings are identified first as ordinals but this can get really confusing. In most western cultures there is no 13th floor while in the east there is no 4th floor; each number being consider unlucky in their respective cultures. Then there is the confusion created in Britain where the first floor is above the ground floor making every high-rise building in the UK actually one floor higher than anywhere else in the world.

The Olympics really have been incredible to watch and a credit to London, my home town. I certainly wouldn't  mind coming fourth in any Olympic event.