A Magical Place; The Teaching Gardens.

The 15 mile journey from my home in Richmond to my office at St. Mike's is so different in the summer from what it is in the winter. Apart from the obvious differences in the weather, differences that can be just about as different as they could possibly be, there are also differences in the route I take. In winter, it is straight from the car into the building as quickly as possible hoping that I can park somewhere close. In summer, however, when the sun is shining and it's warm I always meander through the Teaching Gardens taking in the magical atmosphere, reading the plant labels and trying to remember which is which. At this time of year the growth and colors are quite remarkable especially after the recent rains. But it's not just the plants. I'm always drawn to the stone benches many of which look very much  like toadstools from Alice in Wonderland. Knowing the gardens' creators I am sure they are designed with this in mind. Is that where the caterpillar sat?

What is so remarkable about the Teaching Gardens is the number of  ways they can be appreciated. They bring to life childrens' books and stories:  I've just reread R.L Stevenson's Garden of Children's Verse; "The world is so full of a number of things. I'm sure we should all be as happy as kings". They help us appreciate flora of all kinds through raising our consciousness and awareness of their existence. I'm determined to plant flower beds this year as well as my annual vegetable garden. The latin names of the plants on the labels also remind us of the scientific interpretation of the natural world where species are classified into kingdoms and so on. And the stones, some as big as rocks, are a geological reminder, and perhaps, a metaphor, for what Vermont is all about.

Now, if I could just find a way of celebrating math  and bringing  it to life in a similar, naturalistic, creative  way.    

Dignify the Remainder

Long division is by far the coldest, most frightening and sinister of the four arithmetical operations. It's the one that traditionally is the last one that students have to master, the final and highest hurdle in the traditional elementary school arithmetic program, It looks completely different from the other three algorithms as the line is on top of the numbers when you start and and not below them. It also works from left to right which, again is completely the opposite to the other three.

It also has its own language like "guzinta' in that  '5 guzinta 10 two times" as well as "bring down". And then there's that mysterious r3 or R3 or 3/5 or .6. It always seemed to me that the remainder was a nuisance because math is 'supposed' to be exact and tidy. There are no remainders in addition, subtraction or multiplication so why does there have to be one in division, I remember thinking when I was a fifth grader.  I can even remember using the remainder idea to decide which operation to use to solve a problem. For example, you always knew that if a word problem contained the numbers 4 and 11 it couldn't be a divison problem because there would be a remainder which would be untidy. Sadly, I suspect these disabling strategies are still taught in some places.

What we need to do is to dignify the remainder by giving the numbers in a problem referents. By giving a number a referent we turn it into a magnitude which enables us to know what to do with the remainder. Let's think about problems involving 4 and 11 as in;

Problem 1.  How many cars do we need to take 11 people to the movies assuming each car holds 4 people and the drivers are part of the group of 11 going to the movies. It would not be 2 r3 unless three people were left behind. It could also not be 2 3/4 or 2.75, for obvious reasons.
     
The answer would have to be rounded up to 4.

Problem 2  How much orange juice would there be in each of 4 glasses if you poured 11 pints of OJ 
equally into the 4 glasses. Again, it couldn't be 2 r3 but it could be 2 3/4 or 2.75.

What would the 2.75 refer to; glasses or pints of OJ?  It would have to be pints of OJ .

Giving numbers referents allows students to dignify the remainder so that they know what to do with it. It also helps demystify the concepts of division. Yes, there is a different division concepts in each of the two problems. The first is an example of the repeated subtraction where you know the size of each group (a 4-seater car) but not how many groups. The second is an example of the fair sharing concept when you know the number of groups (4 glasses) but not how much in each 'group' (glass). The first problem also involves a discrete variable (people) which requires the use of the term "many" while the second involves a continuous variable (OJ) which requires the use of the term 'much'..

         

A Million Million!

This afternoon I was watching this YouTube clip as I do from time to time when I need a shot of nostalgia and couldn't believe my ears when, at 3.40, I heard these words "at a cost of ten thousand million pounds". To most people this probably means very little but to a math nerd like me it sounds so odd because that particular piece of mathematical vocabulary usage has never existed in the US. Only in the UK up until 1975 was a billion another name for a million million. Everywhere else in the world, as far as I know, a billion has always been a thousand million like the number above. Perhaps the British Commonwealth also used the same mathematical definition for a billion as used in the UK. I don't remember the change occurring when I was living and teaching in the UK in 1975 but I guess it must have done.

Another interesting thing connected to this issue is the fact that the Oxford Dictionaries defines the change as a linguistic change as opposed to a mathematical change referring to "British English" and "American English".

I always wonder how much confusion it must have caused when the same number names referred to such vastly different quantities. Apparently there were similar disparities in the words trillion, and probably all the other words for large numbers. George Bernard Shaw was only half right when he said the UK and the US were two countries separated by a common language; they were also separated by a common, or maybe not so common,  numerical system. It's also 'Maths' in the UK and 'Math' in the US. Which one is the correct abbreviation for Mathematics, I wonder?

The Brit's billion was a thousand times bigger than an American's billion which just goes to disprove the old saw that everything is bigger in the US of A!.   

Math Must Be Warmed Up

For the past several years I have been arguing, pleading, advocating, promoting, and sometimes even begging for math to be warmer, more user friendly, relevant and just generally nicer. I want the study of math to be endowed with aesthetics, beauty, relationships, meaning, humor, and joy. I want it to have all the positive attributes that other elementary school subjects have, the social studies field trips, the science inquiry activities, the reading instruction recreational fictional reading, the language arts poetry and creative writing, the creativity of art and music and the sheer accomplishment of PE.

I want young children to develop their mathematical thinking and their mathematical memory skills through the use of pattern, relationships, reasoning and puzzlement. I want them to study fractals and all kinds of wonderful geometric relationships that can all be used to enhance the remembering of math facts and the more mundane aspects of math that have to be learned. I want arithmetic to be made obsolete, or at least to be made relevant to the problems solving skills it is designed to accompany.

I've just read a wonderful article in the Harvard Education Letter that speaks far more eloquently than can I to what I am trying to say and do. Changing the Face of Math is a wonderful plea to make math less "cold-blooded", to give students the warm fuzzies when they learn it. It is a plea to make math personally-identifiable-with in the same way that students identify with their favorite reading genre and their creative writing interests.  Everyone who teaches math in any way whatsoever should read this article and be resolved to act upon what they discover.

The Way Life Should Be

I've just returned from a 48 hour trip to the way life should be State. It was a long overdue visit to my friend Lee who was best man at my wedding 33 years ago and my main conspirator while negotiating the ups and downs of graduate school at the University of Illinois in the late 70s. The life of a full-time graduate student back then was one of little money, enthusiastic partying, considerable studying and, above all, hours and hours of stimulating conversation during which we sorted all the problems of the world of public school teaching and education. Upon graduation with our newly minted PhDs we set out to put into practice all the ideas we had spent the previous four years, and more, generating, refining, defending and espousing.

So how successful have we been during the intervening 30-something years?

Judging by the 24 hours of conversation Lee and I have had during the 48 hours I was in Maine I would have to say fairly successful with occasional setbacks. The one thing that seems to stand out clearly is how formative those years of full-time graduate study were in terms of grounding our beliefs and practices on solid ground. We each recalled with remarkable clarity the great educators and philosophers whose ideas we read and those others with whom we came into contact in those hallowed halls. The likes of Tom Sergiovani, Lillian Katz, Jim Raths, Delores Durkin, Ted Manolakes, Harold Lerch and Bud Spodek; names that trip off the tongue as readily as do their ideas still inform our thinking.

I still use Frances Fuller's Concerns Based Adoption Model for supervising student teachers because I have never found anything better. Fuller was a supervision guru at the University of Texas at the time. I still apply Lillian Katz's ideas of dispositions because they are so important in determining how we work with the diversity we embrace in the students we work with on a daily basis. Lee's research into colleague consultation and my own investigations into the allometric growth of pre-service teachers' conceptions of teaching may not have become household names in the field of education but the process of reflection and deep self exploration we were challenged  to conduct as we completed out dissertations enabled us to develop the skills necessary to assess and evaluate all the ideas and points of view we have experienced since.

During the intervening years I have probably been involved directly, in some way,  with the education of countless teachers through undergraduate, graduate and on-line courses as well as through conference presentations and publications. I would like to think that there is a little bit of the intellectual rigor I went through at the U of I passed on to each of those teachers.

I still earn relatively little money, party far less enthusiastically and frequently, still study considerably to stay current and, sadly, tend to only have really stimulating conversations about education when I visit Lee in Maine, the way life should be.    

Rethinking Rethinking Elementary School Math

I recently took to task the NCTM president, Linda Gojak, for what I interpreted as a call for ability grouping students in elementary schools. After several emails with Ms Gojak and after rereading the editorial I have to admit that I completely misinterpreted her argument. I did what is frequently all too easy to do when one is worried about a trend, I went too far with my inferences.

The focus of the "Summing Up" editorial was the need    

to have young children taught by those who love math, understand the math pedagogical content knowledge, and so can differentiate it more effectively to include all students of diverse abilities in the same class. A teacher's ability to differentiate instruction to include the diverse range of learners in a typical elementary school classroom is dependent upon the extent to which she/he understands the math and the way the children she/he is teaching learn math. If specialist teachers moved from classroom to classroom this could probably be accomplished and, at the same time, avoid the attendant danger of children being grouped by ability. Good specialist teachers working together could still integrate the disciplines through integrated thematic units.

But, and this is a big but, are we ready for this dramatic institutional and cultural change? Is there research suggesting that elementary school children can work with four or five regular classroom teachers on a daily basis in addition to the art, music and PE teachers they already see on a weekly basis? 

The top school bus by the way, represents odd numbers while the lower one is a model for even numbers.

Old Students, Old Friends Stay in Touch

During the last couple of days I've heard from several past students, Sebastian pictured left and Lisa. Lisa was a student in several of my classes in a previous life when I was a professor at the now defunct Trinity College in Burlington, Vermont. She graduated in 1988 and tells me she now teaches 8th grade biology and has twin 17 year-old children, a boy and a girl. She reminded me how she used to come and watch me sing in an Irish band at the Last Chance saloon in the days when the drinking age was 18.

The other old student, old friend, is Sebastian who graduated from St. Michael's just two years ago. A tall, 6'9", 'skinny kid' from northern Vermont who played center on the SMC basketball team,  Sebastian is one of the major success stories of the education program at St. Michael's but not because he has gone on to become an outstanding elementary school teacher or school principal. Sebastian is one of those rare people who is able to successfully admit to the need to adjust his goals in life at a formative stage after already committing to a particular path. Towards the end of his student teaching experience he realised that teaching was not what he thought it was or what he really wanted to do. So he graduated, set out for Alaska, discovered he wanted to fly aiplanes and is now a qualified pilot flying people and things around Alaska. 

Here's a short piece from Sebastian's email that I will always remember;  The most important thing I realized, however, was how I could never have ended up where I am today without each and every member of my Saint Michael's family. You all did things for me that you did not have to do. You all took time and energy out of your lives in order to set me on this path. Your high standards for achievement, your caring, understanding, and love have helped me to flourish and achieve everything I have set out to do in my life since Saint Mike's, and I am so deeply thankful from the bottom of my heart.

This is what life is all about. 

Who, or What, is an Authority?

Several weeks ago I arrived early for an appointment with a school principal at a local elementary school. I was shown to the school library to wait and as I pondered the meeting my eyes fell on the books on the book shelf closest to me.  It was labeled the 'Geography' section and I was soon reading the titles on the book spines and stopped, quite naturally, at one titled 'England'. As I flipped through the book I saw pictures of large cargo ships hauling British products to all four corners of the Earth; pictures of coal mines turning out vast quantities of coal and men dressed in collars and ties, with caps, working in the fields. There were also pictures of busy High Streets with mainly British-made cars and bustling people on the sidewalks. Curious, I looked inside the front cover to see the publication date; it was 1968. Returning the book to the shelf I noticed other books in the same series labeled, for example,  Sweden, France, Germany, and Spain.

In the courses I teach I advise students about using Wikis and other socially constructed forms of media for their research. I often provide them with good and not so good examples of on-line "authorities" they should, or maybe should not, use when conducting research or writing papers. It is important, I think, to help students become connoisseurs of the resources they use so that they can be appropriately informed. I always suggest they know who or which organization is responsible for what they are reading and the process it has gone through to be published on the web.

For many, the internet has replaced books, and people, as sources of authority in the sense of the truths, facts or ideas upon which our culture relies. It's even difficult sometimes being a professor when internet resources are viewed with such unquestionning belief.

So what of the geography books in the elementary school library to which children have access when, perhaps, they are completing a project on a European country? My suggestion would be to simply move them into the 'History' section and advise students to use the internet to find out what life is really like in England, France, Sweden or Germany.
    

A 33rd Commencement

Counting my own 2 commencements I think the commencement I attended yesterday at St. Michael's College  was the 33rd I have sat through not counting several high school commencements and, dare I say, even a kindergarten commencement!

They tend to be always the same except for two major differences; the commencement speaker and, of course, the individual students who are graduating. This year saw some wonderful students graduate with their teaching degrees and licensure recommendations. Many of them have already secured teaching jobs which is a wonderful testimony to the quality and reputation of the St. Mike's teacher education programs. One in particular, Callie Lumbra, was one of two valedictorians at this year's commencement and has been just a wonderful student to work with during the past four years. What makes Callie's achievements more significant is that her mother, Joan, was one of the first students I worked with at Trinity College when I first came to Vermont in 1982: I think she graduated in 1985. For this to happen on mother's day was especially significant and really brings special meaning to the phrase 'like mother, like daughter'.

The other wonderful thing about yesterday's commencement was the graceful wit and generative wisdom of Mark Shields, the commencement speaker. Having seen him many times on PBS and other, commercial TV news programs I always had the feeling there was more to him than meets the eye. This was certainly shown to be true as he regaled the audience with humorous advice, serious rules and finally, a word of hope that as difficult as it might seem to accept sometimes, politics is still the best way of solving our differences. His one piece of advice I think I shall always remember is not to worry about what other people are thinking of you as they are too busy worried about what you are thinking of them!

One of his other pieces of timely advice to the students was to "call your mother". I certainly wish I had heeded this advice more often before my mother passed away eight years ago.     

Rethinking Elementary Schools!

I usually enjoy reading the articles and features on the NCTM website but the recent 'Summing Up' feature written by the current NCTM president is alarming to say the least.  Here is my response;

As a long time member of NCTM I find the NCTM Summing Up, May 8 by NCTM President Linda Gojak extremely depressing and hope sincerely that this is not a direction in which she plans to lead NCTM. The two most distressing parts of her “solution” to rethinking the elementary school are #3 and #4 .

In #3 Gojak argues for homogeneous grouping in the elementary school classroom effectively negating IDEA and everything good that comes from having students of diverse abilities working together. Ability grouping in the elementary school will effectively consign some children to the lowest ability classes from kindergarten onwards since it is a well known fact that it is difficult to move upwards through ability levels and remove the stigma of being in the “lowest” class wheremany of the ELL students and most of the children with disabilities  inevitably end up. I started teaching in 1972 in the UK, three years before PL94-142 and remember what it was like to see classes of young children with disabilities all herded together. Ms Gojak’s strategy of ability groupings would bring back the stigma attached to the students in these classes as well as remove the wonderful benefits arising from all children with all sorts of diverse needs working together and understanding each other. As the parent of a child with D.S. I find this to be cruel and unusual punishment.

In #4 she suggests that ”tradition and costs” have been the argument against subject area specialists. In reality and in my experience, this has never entered into the argument. Teacher specialization in the elementary school has been argued against on the basis of pedagogy. I might be accused of being a “child of the 70s” but we still teach integrated units and still help children learn to write by using science and social studies. We still help children to see the value of mathematics by applying it to other subjects through integrated projects. This is far less likely to happen with specialized teachers in the elementarys school.

The depth of pedagogical content knowledge that Ms Gojak seems to think is too much is something that all teachers need even those teaching in a homogenously grouped middle school classroom.  Teachers who understand what comes before and what follows, in a certain grade level, are more able to help students explore misconceptions and extend mathematical ideas for those who need the challenge. It sounds very much like Ms Gojak is suggesting that teachers need only know and understand a thin band of pedagogical content knowledge required of a specific grade level. Perhaps she believes that children have minds  like  “vessels to be filled with facts and figures” rather than minds to be grown, developed and nurtured.

A Different Kind of Final Exam

At the end of my Teaching Elementary School Math and Science course students are required to solve an engineering problem for the final exam. The problem comprises the construction of a Rubber Band Roller (RBR), a vehicle powered by a rubber band. In essence the RBR consists of a cylinder of some sort, a rubber band and a stick such as a pencil. But in reality it is so much more as the students always find out. The definition of a good problem in any academic field is a situation that cannot be immediately resolved without the application of knowledge and thought. For example if you know that 3 x 4 = 12 then you don't have a problem.

If we define science as the exploration of questions in the natural world then we can define engineering as the use of science to solve problems in the world in general. The RBR is such a great engineering project because it never, ever works first time. As Aris, one of my students wrote, "Overall, my emotions with this project went from excited, to frustrated, to more frustrated, to feeling hopeless, to then suddenly having the wonderful feeling of accomplishment".  The project requires the students to use all the engineering process skills we discuss in class. These are; defining a problem, researching relevant information, designing, constructing, testing, adapting and improving, and completing and presenting. 

The 'presenting' part is what constitutes the final exam where students can enter their RBR in 3 competitions; distance, speed and creativity. Congratulations to Leanna who broke the world distance record with four times up and down the length of the classroom (about 60 feet), and to Jenn who won the speed trial with .6 seconds over three feet, and to Emily who decorated her RBR with a Peppermint Pattie motif. The winners each received a copy of an  elementary school science activity resource book but all participants were winners because they all overcame what really is quite a difficult engineering problem.  

The Seasons, Day and Night

Ever since, many years ago, I observed a student teacher erroneously  demonstrating to a 4th grade class how the sun orbits around the Earth I have made it a point to include a class session on the reasons for the seasons and why days and nights vary in length.

Apart from this obvious error there are all kinds of misinformation students can develop in their own "private universes" that we must guard against. On the left is a classic picture found in many textbooks that attempts to show the tilt of the Earth in relation to the sun. Unfortunately, the picture also shows the earth significantly larger than the Sun as well as showing how the Earth appears to get closer or further away from the sun as it makes its annual orbit. These misconceptions are caused by the scale and perspective of the drawing. Both of these pieces of misinformation can lead to the development of major misconceptions that can live in the private universe for a lifetime.

I usually try to demonstrate the changing seasons by having the students sit in a circle with me in the center holding a flash light. They then pass the globe around the circle making sure to keep the Earth tilted toward the front wall of the classroom at all times. The other piece that goes along with the tilt of the Earth is the idea of insolation. This is basically the different intensity of the suns rays on the Earth's surface caused by the angle at which they meet the Earth's surface. This can be modeled by holding the flashlight virtically above the desk and then at a shallow angle when the pool of light (and heat in the case of the sun) becomes much less intense and more spread out.

As the globe moves around the circle I try to shine the flashlight on different parts of the Earth's surface making sure that it is always horizontal. As the globe goes around the circle you can then see how the overhead sun moves from the extremes of Tropic of Cancer (our summer in the north) to the Tropic of Capricorn in the south (our winter). It also demonstrates how the north pole can rotate through 360 degrees and never be in the dark in summer or always in the dark in the middle of winter.  It helps to have the students imagine they are standing on the Earth's surface and observing the sun at these different times of year.

The same strategy also works for exploring day and night. Imagine you are watching the sun behind the Adirondack Mountains in the west across Lake Champlain. Now reconceptualize the mountains "coming up" to cover the sun rather than the sun going down and you can develop a much better sense of the rotation of the earth on its axis and the reason for day and night.

Finally, there's this remarkable website that gives you a great sense of global issues such as the days as well as the different lengths of day and night between the summer and winter. The shaded part on the map is night time. Look how much shorter the nights are in summer in the northern hemisphere than they are in the southern hemisphere. Isn't that way cool? You can also see where it is tomorrow.