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.