A New Course this Summer

Several years ago I developed a new graduate course to help teachers and graduate students teach math to students with diverse needs. The course had four strands of diversity and focused on the math areas of numeracy and operations. The four strands of diversity were teaching math to children with special needs, with math disabilities, were in poverty, or who were English language learners. Sometime, of course, there are children who fall  into more than one of these categories. The course, GED612 has become very popular and is now taught by a wonderful teacher, soon to be Principal, Wendy Cobb who is one of only a few people who have extensive experience and qualifications in both math education and special education.

Last summer it was decided that the Math for ELLs  part of the course would be a required part of the TESOL program offered through the Applied Linguistics Department at SMC. This was a really exciting development for me because this is my primary area of research. Although several TESOL students took the one-credit part of the course it was too difficult to separate one credit from the 3-credit course and so this summer I will be teaching GED611, a one-credit course designed specifically for students in the TESOL program. This is a really exciting learning opportunity for me because I will get to work with a different population of students skilled in the wider area of teaching students who are English language learners.

The course is particularly relevant in the Burlington area of Vermont as around 45 different first languages are spoken by students in the Burlington and Winooski schools. Isn't that remarkable?

Here are a few thoughts about leap day. 

Student Teachers, Dancing Rice and School Buses

I have an incredible group of student teachers this semester and sometimes they even outdo themselves in terms of sheer brilliance. For example, one of them has been doing a science activity with groups of students involving dancing rice (this activity uses different materials but the idea is the same). After doing the activity with several students groups ( first and second grade students) she was not happy with the speed at which the rice grains moved up and down in the liquid. Convinced the activity should work much better she decided to have the students experiment with different types of rice. After several trials the students discovered that the activity worked best with brown rice. They looked at the rice grains closely, observing carefully like all good scientists do, and discovered that the brown rice grains were lighter but larger than the white rice grains and so could  attach more carbonate bubbles to themselves than the white rice grains. This was as far as she needed to go with the scientific explanation with children of this age. The neat thing is that she got them to think like scientists by looking at variables.

Another great story from my grad math class last week. Helping students understand the difference between odd and even numbers is often quite difficult. It's relatively easy to teach the even numbers as 0, 2,4,6,8 and the odds as 1,3,5,7,9, but to teach why is more complex. I usually advocate using the pairs idea; in each even number every object has a partner and in odd numbers they don't. Well one of my grad students says she uses the "old school bus/new school bus trick". Intrigued I asked "What is that?".

She responded that odd numbers are like the old school buses with the hood sticking out in front and even numbers are like the new school buses which are straight down at the front so that it is easier for the drivers to see if students are crossing in front of the bus. If you use cubes, in pairs stacked one on the other with each pair next to each other,  to represent each type of bus you can see exactly what she means. The odd numbers have an extra, unpaired cube sticking out to represent the hood of the old type of school bus.

See what I mean;  isn't that cool

The Magical Moment When You Know

I can remember the moment that I knew I was a teacher as if it had happened yesterday. It was half way through my student teaching experience at Sefton Park primary school in Bristol, England in the Spring of 1972. I was teaching a science lesson to thirty-four 4th grade students and I had to write a lot of information on the blackboard for the students to write down before we did the science activity. The blackboard was actually painted on to the plaster wall of the classroom and I remember having to carefully write around small divots and craters in the wall where sharp objects such as board compasses,  had been jabbed into it during the previous 70 or so years. This was long before photocopying and
at a time when handwriting was practiced during many different subjects including science.

I directed the students to copy what I was writing into their science notebooks and proceeded to write on the blackboard for the next five minutes. I was so intent on missing the wall "divots" that I didn't turn around once. When I had finished writing I finally turned around to discover that all the students were still sitting in their places and had almost managed to keep up with my writing. Just a few stragglers took less than a minute to finish. This was my magic moment when I knew I could teach. Even with my back turned to them the students did what I asked them. I had motivated them to write and complete the activity. There have been many, many times since that day that my ability to teach has been reconfrmed by the achievements of my students and comments from others, not to say that I still don't have much to learn.

Last week I was lucky and privileged enough to be present when one of my student teachers experienced this magical moment. She was teaching a math lesson and the students were totally into what she was doing. Suddenly, she stopped speaking, gave a huge smile and told the students they were being so good. From observing her I could tell that she could not have stopped herself doing this; it was as if her inner soul was letting the world know that she could teach; so overcome was she that her dream of becoming a teacher was now a reality.

Every successful teacher has had one of these magical moments.    

Worried Problems - Whirred Problems!

Word problems! the one thing guaranteed to turn just about every student off mathematics. Remember the one about two trains approaching each other?  And then there are the remarkably insane strategies that are used to help children solve simple word problems like the one in the image to the left. There are some amazingly daft suggestions in this strategy such as find the "key words" as if they would really give you a magic way to solve the problem.

 "Do the math" is equally ridiculous. Why is "math" limited to the word "compute". Isn't the whole thing math?

Several years ago Thomas Carpenter came up with the idea that all simple addition/subtraction "problems" could be identified as involving part-part-whole, combining, separating or comparing. He

also identified another set of concepts related to multiplication and division; things such as the equal groups, area and the multiplicative comparison concepts. The main task in solving a quantitative problem is to work out what is happening in the problem in the context of one, or more, of these concepts and what is being asked. Once this is done, the solution is easily found by using one of the four operations using one's head, paper and pencil or a calculator.

To show the significance of understanding this, consider these non-problems;

  1. If it rains for 3 hours on Monday how much will it rain for the next 4 days?

  2. If it takes ½ an hour for 3 friends to walk home how long will it take 5 friends?

  3. If 2 students have 5 pet gerbils how many gerbils does each students have?

  4. If 2 girls have 3 brothers how many brothers do 4 girls have?

  5. If the temperature is 62F today what will it be for the next 3 days?

  6. If one train is traveling at 35 mph (clearly an AMTRAC train) 
      and another similar train is going at 45mph what time will they pass each other?

  7. If 2 squares have 8 sides how many sides do 3 triangles have?

  8. If it’s 60F in Vermont and 72F in Maine what is the temperature in
     New Hampshire?

  9. If you eat 2 slices of your birthday cake today and ½ of what’s left 

     tomorrow how much will you eat the next day?

10. I 140 students are going on a field trip and a school bus will hold 60 
     students how many school buses will you need? 

You could get some really interesting answers by using the "Attacking Word Problem" strategy. e.g. (Each student would have 2½  gerbils)

Maths and Down Syndrome

I have always thought that the math we teach to children with cognitive disabilities should be the same as the math we teach to students who are neuro-typical. This belief is based on the idea that good math instruction, the type that focuses on student learning and understanding and not just factual recall, is the 'best practice' for all students. Apart from the different instructional strategies we use such as adaptive technology and so on, the major difference in our instruction with students with cognitive disabilities is the way we modify the activities and our expectations of what the student can achieve given our knowledge of the individual student.

During the past 15 or so years I have watched my son Andrew, who has Down Syndrome and is going to be 20 this year, grapple with the all too often  abstract task of learning mathematics. Sometimes, it seems that no matter how many concrete examples we use he just cannot grasp a particular concept. The concept of comparison, for example, seemed to be completely beyond his ability to understand. At basketball games he cannot work out how much one team is winning or losing by even when the score is 4 - 2. It's not that he cannot do the calculation, he just cannot comprehend the idea of difference in a comparative sense.

I had pretty much given up with this particular concept when, purely by chance, a couple of months ago he looked at the underside of one of his Hotwheel cars and discovered it was first made the same year as he was born; 1993 was stamped into the underside. He was so excited to discover that it was the same age as he is that he now looks at each of his Hotwheels, a collection of some 300, to see if they are older or younger than he is. Success at last, his understanding of the concept of comparison is born but not because of anything I did but because he was motivated to understand. So I looked back on all the things he has learned; everything is related to his level of motivation. He bowled a perfect game on the Wii bowling because he was motivated, he can move a spinning basketball from finger to finger on one hand because he is motivated, he understands the idea of  sequence because he is motivated by wanting to know what comes next.

Is he motivated to learn how to do the algorithm 45 - 21? Not at all, so he probably will not learn it (we stopped trying around 7th grade). He is, however, motivated to use a calculator which will work fine as long as he understands which operation to use; something he will be motivated to do as he shops more for himself and manages his credit card and bank account. Here's a presentation I gave about math and Down Syndrome at the Massachusetts Down Syndrome Congress last November.

          

Kindergarten Math- Not Just Counting

I think I must have been a kindergarten teacher in another life because I always feel so at home working with younger children. This morning I spent time in two kindergarten classrooms (Ms Tenenbaum and Ms Nido) while they were doing math and what a wonderful experience it was. Three of my ED325 elementary education students are placed in the two classes for their 2-hour a week field placement which is a part of the course.

In one class the students were learning how to problem solve by drawing pictures of the problem and the associated numbers. It was so neat to watch the different strategies the children were using and it was not an easy problem; (If twelve children each have 2 cookies how many cookies are there all together).  The neat thing about this problem is that it all the children in the class can have a shot at solving the problem regardless of their mathematical development in the same way that all children can write a sentence of some sort regardless of their writing ability given a good writing prompt.

Some of the students drew 12 children, gave each child 2 cookies and then counted all the cookies one by one. Other children drew similar drawings but were able to count by 2s. I'm sure there was probably yet another student who used multiplication in some way. In the other classroom the students were playing bingo and learning to associate spoken numbers with their written form, part of what we call the triangle of meaning in math education. The other part of the triangle is the numerousness concept of the number (e.g OOO -- 3 -- "three")

The wonderful thing for my SMC teacher education students is that we were talking about these exact things during our math ed. class on Monday afternoon. So within 48 hours, the students can experience with children the exact things we were discussing theoretically in the college classroom. They even got to see some numeral reversals which are usually the misapplication of the counter clockwise rule that governs the formation of lower case letters in 6 of the first 8 letters of the alphabet. If you form your numerals going counterclockwise the 2, 3 and 7 will be reversed. These are the ones that nearly all students reverse when they learn to write numbers. Isn't that cool!

A Merry Band of Education Knights

The Education Department at St. Michael's College is a wonderful place to work, and study. There's a great spirit of community and cooperation in which each faculty member is committed to the delivery of state of the art courses and programs in the field of teacher education. There are over 100 courses offered through the undergraduate and graduate programs with an average course size of 12 students; no class is larger than 21.

There are 14 full-time faculty each with a teaching and research specialty in a particular area of teaching and education. Professor Mary Beth Doyle, the Department Chair is a leading authority in the field of Special Education. She has just returned from a sabbatical semester in Italy where she studied  special education practices in that country. Also on sabbatical in Italy during the Spring semester last year is professor Jonathan Silverman who is the resident art education specialist. Professor Valerie Bang-Jensen teaches reading and literature education courses and is the driving force behind the Teaching Gardens at SMC. Professor Amy Saks-Pavese is the newest member of the Department and teaches courses in Social Studies education while professor Gehsmann, a reading and language specialist is currently authoring a book on teaching reading. Professor Aostre Johnson is a leading authority on the role of spirituality in education and teaches courses in early childhood and professor Beth Peterson is an expert in the area of teaching children with special needs. I complete the elementary education faculty teaching courses in math and science education.

The secondary education program faculty comprise professor Claudine Bedell, an evaluation specialist, Professor James Nagle who co-directs the CREATE grant for working with students who are English Language Learners, professor Karen Donovan, a special education specialist and professor Becky Wigglesworth who has a special interest in teaching English. Professor Brian O'Regan teaches courses in school law and supervision in the graduate program and  professor Ann Judson, director of the Graduate Education programs, is an expert in the area of IT in education.

Together we have just about every base covered in the world of Education both through the programs offered at SMC as well as outreach programs in the public schools.  

Maths can be Useful, Sometimes!

"When am I going to need this?" in the context of math is one of those phrases that drives me nuts. No one ever says it in a science or social studies or language arts class. We never seem to question why we need to learn these things but in math class it's different. Perhaps it's because until recently math education has been so deathly boring and difficult that students feel there had better be a good reason for all the hard work of memorizing and practicing. Things are different today in math classes, at least at the elementary school level. We now know that math has all kinds of aesthetic qualities such as pattern, symmetry, color, relationships and so on that make it a joy to explore; and not just in geometry. We now know that if students find math interesting, relevant and creative they will learn it more effectively and be more motivated to see it in their lives as just interesting and enriching rather than useful.

Today, my daughter Marie was at home helping with the Andy's Dandys boxes and suddenly let out a cry of f delight,"I just used some geometry". Although she is a musician and she teaches piano every day and so uses math just about every waking moment it was still exciting for her to use math math; the type she learned in school and that you could give a name to, such as the Pythagorean theorem. The pictures show her math and the finished product.

Andy's Dandys, by the way, is my wife Lucie's gourmet dog treat business. The business has the

wonderful mission of providing students with special needs the opportunity to develop business and work skills in their factory store in the middle of Richmond. 

NCLB Finally Left Behind?

"Obama frees 10 states from No Child Left Behind" is one of the headlines in my local paper this morning. The idea that this myopic legacy of the Bush-era could be finally disappearing is something that many of us have been hoping for since its inception. Never before in the history of the field of Education has such a misguided and damaging piece of legislature been enacted. To imagine people thought that individual as well as institutional success could be measure by a single test is beyond belief to those of us involved in a system of education that should be educating all students in all aspects of the human endeavor.

Teachers and schools certainly need to be held accountable for the work they do but there are so many better ways of doing this than mass testing where everyone is treated as if they had the same beginnings in life. Although somewhat cumbersome, the Vermont portfolio process was a valiant attempt at identifying authentic growth and development of individual students as well as schools.

A common complaint is that the standards set by NCLB were completely unrealistic. It was a bit like saying that all 10th grade students should be able to run a mile in four minutes or that all third graders should be at least four feet tall. Now we can focus on all students and provide them with effective and and appropriate educational experiences in all subject areas and not just those that are tested.

Perhaps we can return the joy and creativity to the act of teaching. 

On Becoming a Teacher

I cannot think of many things in life that are more rewarding
than being actively involved in the creation of a teacher. As a supervisor of student teachers I am privileged to work with young people as they make the journey from being a student of teaching to being a qualified, licensed teacher. My student teachers are active participants in the journey, of course, but they need feedback, advice, and guidance as they navigate the many intricacies of the elementary school classroom.

The journey is often marked by remarkable self-analysis on the part of the student teachers as well as "aha" moments of life changing significance. Several weeks ago, one of my student teachers shared a wonderful realization with me which involved her work with a rather difficult student in her classroom. The student has a reputation for being a little bit difficult so when he acted up one day and said he would not do the assigned work she automatically assumed he was just being difficult. Within a minute of sitting down with him she realized that he did not understand the assignment and wanted help to get started. In her reflection the student teacher described how she will never again automatically assume that a student's difficulty is a result of an attitude, especially if the student has a track record of being negative. Once she had sat down with him and gone over the assignment he was fine. From now on, she said, she will assume the best of every student until proven otherwise.

Another of my student teachers has a beautiful clear, but rather loud natural voice; a voice like a bell, as I call it. This is a wonderful attribute for a teacher but one that needs to be tamed and used to best effect. When she spoke to the young students in her classroom  they would tend to get louder and then she would get louder and so on. So we developed a strategy that she could become more aware of her voice as others heard it. Her cooperating teacher videotaped her and used a signal to help her become aware when her voice was getting too loud. In her last set of journals the student shared her excitement at how the students were responding so well to her quiet voice. Her whole student teaching experience has now changed for the positive because she can see the results of something she is actively doing to improve her teaching skills.

The journey of becoming a teacher is not a spectator sport.   

Rekenrek: State of the Art Instruction at SMC

I was observing one of my student teachers in a kindergarten class today during math time and ended up having my first experience with Rekenrek math. This is one of the wonderful things about supervising student teachers; I get to work with incredible teachers in public schools using state of the art instructional materials. I have known about this incredible approach to teaching young children about number but this is the first time I have seen it in action in a Vermont classroom. Developed in the Netherlands it is now promoted by many math organizations as well as the Math Learning Center, publisher of the Bridges Math program used in many of the elementary school in Vermont.

Looking a bit like a simple abacus, the math behind the simple beaded counting frame is just incredibly good math.  There are many different activities that can be done with this simple device but the one I watched this morning went like this. Two students each  draw a card from a pile of cards with numbers 1 - 19 written on them. They then make that number using the beads; but in a very specific way. If they draw 8, they move 8 beads on the top line to the right with "one finger push". This simple little command makes the student subitize as opposed to counting one by one. This helps students develop the idea of numeracy, the essential element of quantitative literacy. If students are allowed to push each individual bead across they will develop an over-reliance on counting and not develop good numeracy skills, the idea, for example of "eightness". If 17 was the number drawn, the top 10 beads would be moved to the right  and then 7 from the second row for a total of 17.

Once the number of beads has been slid across the frame  students take turns to roll a die with the words 'more' or 'less' on each face. If "less" comes up the student with the smaller number takes both number cards. This simple activity develops the difficult comparative concept of more and less. They then continue until all the cards have been used and a winner is declared. Jen Canfield, whose class I was observing in this morning uses the game aspect of the activity as an opportunity to teach children how to be good winners and losers. At the end of each game the two students look each other in the eye, shake hands, and the loser congratulates the winner. Isn't that cool. 

Climate Change is for Real

Given the remarkable number of climate related events during the past couple of years one must be in absolute denial of reality to question the existence of the process of climate change. Sadly, there are people in positions of responsibility like Gov. Rick Perry who still utter the remarkable words "I don't believe in climate change" as if it was a faith and not a a science.

One of the ways we can begin to educate people about the exact science of climate change as well as the ethical and moral aspects of our own actions is through educational events designed to bring this branch of science into the mainstream. One such event will be happening on the St. Mike's campus this coming March 10th when SMC geography professor Richard Kujawa will be the keynote speaker at the Climate Change Workshop for Teachers sponsored by the Champlain Basin Education Initiative (CBEI). An agenda and registration materials for the daylong conference are available at the CBEI website.

The workshop/conference will provide participants with "information, dialog and classroom practices" that will help K-16 teachers and others involved in education raise levels of awareness, knowledge and understanding o that intentions, dispositions and attitudes can be turned into actions and practices.

60 Years Today

For most of us living in the US the fact that the Queen of The United Kingdom and the British Commonwealth  is celebrating 60 years as the Queen probably doesn't mean a great deal.

For those of us who grew up in England it can bring an almost mystical dimension to one's life. I have never been an avid Royalist but the Queen has been an integral part of my life for all but five years of it. In 1972 my Aunt Marian received an MBE from the Queen at Buckingham Palace for 50 years of devoted service to the country. My aunt, mother and another aunt donned their Sunday best and made the 100 mile journey by train to the Palace for the ceremony.  I still have the medal and certificate complete with Her Majesty's signature. The Queen even looks a bit like my mum in her later years but then I think everybody probably thinks that too.

This is probably her greatest gift to the world. Despite all the ups and downs of her family  life and the life of the country and "empire" she has remained, at least for me, a distant symbol of permanence in a world that seems to be bent on speed and change for its own sake. I remember Coronation Day on June 2 1953 like it was yesterday but I don't remember accession day and the death of the King at all. I was once told you can take the Englishman out of England but you cannot take England out of the Englishman.

And how fitting is it that she chose to visit a school  for very young children on the 60th anniversary of her accession to the throne as if to say this is what is the most important thing in our lives; our children.

Note: I can't believe I did this but I referred to the Queen as the Queen of England when I originally penned this thereby alienating everyone who lives in Scotland, Wales, Northern Ireland, Canada, Australia, New Zealand and a few other places too. I apologize; I should have known better. 

Apps..olutely amazing!

My daughter Marie just arrived fresh off the slopes at Sugarbush, one of the local ski Areas. She skis most Sundays with her boyfriend Erik and seems to have great fun just like I remember having when I last skied,  probably some 20 years ago.  

The snow is still white and cold and the sport still relies pretty heavily on gravity but what has changed is what you can do while you are skiing. Using an app on her cellphone Marie was able to keep a record of just about everything she did during the six runs she made this morning.  The image above is a screen shot from her iPhone which shows the details of her skiing experience this morning.

This image shows the times and locations of some the different runs she made at Sugarbush; she actually has a record of all the runs which include the times, speeds, locations, distances and so on, In addition this app also has a map feature for each of the ski resorts in Vermont (and the entire US), links to their websites, ticket purchasing facilities and information about the conditions at each area (number of lifts open, depth of snow etc.). The entire skiing experience has been mathematically defined; isn't that cool!

I wonder if there's an app for fixing sore knees and providing one with instant fitness. Now that's something that would get me excited. 

          

Math Has no Grey Areas?

Until today I always felt it was a breakthrough when students would say or write that "math is not all black and white, there are grey areas". Although this might be a step forward it is clearly, now I realise, almost as incorrect as saying math is black and white. For many, brought up in traditional math classrooms, math was black and white with only one right answer and one way of getting there. Not any more.

Now, I want to state unequivocally that "MATH IN NOT BLACK AND WHITE, IT IS EVERY CONCEIVABLE COLOR, HUE, AND SHADE OF COLOR IN THE RAINBOW, AND BEYOND.

For several years I have known of the fractal work of Mandelbrot, Sierpinski and others, and more recently, the use of fractal math in the fields of phyisical, biological and earth sciences. I have also know of the relationship between music and math, design and math, and art and math. All of this and more was  beautifully presented by Sheila Weaver of the Vermont State Mathematics Coalition in a presentation at St. Michael's last Friday. The presentation can be viewed through Sheila's Tegrity presentation.

There is so much more to math than its "usefulness". There are so many other questions to be asked by K-12 students than "whenever am I going to need this". Mathematics is the science of pattern; it addresses the quantitative literacy component of our lives; it has aesthetics and beauty both in and of itself and through other media.

It's remarkable how many times balancing a check book is a main reason given for learning math. "Maths" deserves more than this!

Time Flies When You..........

are having fun. Or, as you get older, it seems. When you stop to think about it, or even when you don't, time seems to go by much faster the older we get. Here we are in February 2012 and I can remnember not so long ago, or was it, wondering what life would be like in an Orwellian 1984.

As one interested in math, there is  an explanation as to why time seems to go by faster the older one gets. Imagine you are 10 years old again  (or maybe you are ten years old reading this). The year just passing is about a tenth of your life so far so it's a pretty big deal in terms of the big picture. It's about a tenth of your life, so far. For those of us in our sixties, the year through which we are speeding headling is but a fleeting sixtieth of our life, so far. This sixtieth year (actually sixty-sixth for me) is crowded into one's memory along with all the memories and activities from all the other years; it doesn't take up much space, or time, relatively sepaking.

So, there are days when no matter what you do, one's age seems to be a part of the action. I had to change one of my five thousand passwords this week and I'm counting the number of times it takes me to remember to use the new one when I log in. I haven't forgotten the new one, just the fact that I have to use it. It's probably the same thing they say about forgetting where you left your keys. The time to really worry is when you forget what your keys are for. Luckily, I have yet to forget what my password is for.

It looks like there's a wonderful choice of walking sticks these days if and when I should ever need one.