Thursday, April 10, 2014

A Math Major Makes the Difference

This semester I have had the unusual but distinct pleasure of supervising two math majors in their elementary school student teaching placements. Using the Bridges in Mathematics program with their first/second multi-grade classrooms they have had a profound impact not only on their students' math skills but also on the way the students feel about math. They have brought math alive for their students through their deep and comprehensive knowledge of mathematics as they have implemented lessons on a variety of math topics. They have been able to go beyond "transmitting" knowledge about math to helping children construct meaning and make sense of the concepts and ideas presented in the program. This making sense is difficult to achieve if you don't have a deep understanding of what you are teaching.

But their achievements go beyond helping children make sense. Their self confidence with math has rubbed off on the students, especially the girls they have been working with. The student teachers are comfortable with math, they are relaxed and smile a lot, they show no anxiety, and they use relevant examples to illustrate concepts that associate math with real life and pleasant things. I read about this phenomenon, or at least the inverse, in an interesting piece of research at the University of Chicago several years ago. Young children can pick up on the most subtle clues as to how a teacher feels about what she is teaching. Here's another more recent study reported in the Journal of Curriculum and Instruction that goes further to show the relationship between teacher anxiety and student performance in mathematics in the elementary school.

It's interesting to note that one of my two math major student teachers has already secured a teaching position while the other has an interview.  

Thursday, March 13, 2014

Pi Day or No Pi Day? that is the Question.

The annual Pi Day, March 14, is a time for mathematicians everywhere to celebrate that wonderful ratio between the circumference and diameter of a circle. Well not quite everywhere. The form of the date 3/14/14 that gives rise to Pi
day (pi is 3.14 to the nearest hundredth) is not universal. It appears that the US is one of only a few countries that uses the month/day/year format. However, as with most things, the origin of the difference is not easily determined as this discussion shows. This Wiki shows where in the world the different formats are used. And yet we typically say the 4th of July which is a nod to the British way of doing things which seems  a little odd since it was the British from whom we gained independence on July 4th, or was it the 4th of July.

Pi is much more interesting. It is a constant, irrational number that is a ratio between the length of the circumference of any circle and its diameter. The circumference of any circle is just over 4 times its diameter. In itsfraction form 22/7, 22 is the circumference and 7 is the diameter. There are so many lists of facts about Pi, from 5 to 50 on-line, and it's probably the most blogged about mathematical symbol in the history of maths. It's also probably one of the most poorly taught concepts in school as it is usually presented as a long, long number, often graphically represented on the walls of school hallways and yet seldom explored for what it really is. Here's a neat Pi Facts blog site that lists just 5 facts but they are really good especially the fact about the mirror image of 3,14 being PIE.Here is a list of 50 facts; probably TMI.

In a lecture I gave last October I raised the issue of there being no Pi Day in the UK. Later, someone in the audience emailed me to say there was, in fact, 12 of them this year, the third day of each month could be a Pi Day (1/3/14, 2/3/14, 3/3/14, 4/3/14, 5/3/14, 6/3/14, 7/3/14, 8/3/14, 9/3/14, 10/3/14, 11/3/14 and 12/3/14.). 

Being both a British and a US citizen I have always found it difficult to remember which way to write it. Luckily I was born on Dec. 19th so 12/1946 always works in the US.

Wednesday, March 12, 2014

It's A Small World

Last Saturday, in the McCarthy Performing Arts Center at St. Mike's, we held the ninth annual Concert for S. Patrick; and what an evening it was. Designed to shift the focus away from the traditional binge drinking associated with this particular US celebration, the Concert for St. Patrick brings together the best in internationally recognized Irish musicians with the best SMC Irish performers in the form of the Celtic Knights Dance group. The local Irish scene is also represented in each concert by the McFadden Irish Dance studio.

The headliners this year were the Alan Kelly Gang who brought us a selection of Celtic songs and  tunes from a variety Celtic  places up and down the west coast of Europe from the Shetlands in the north all the way down to Normandy in north-west France. Their music blends the wonderful traditional familiarity of the old Irish tunes with a more contemporary interpretation of the sounds of music.

On the personal level, the most remarkable event of the evening was my conversation with Alasdair White, the fiddler in the band, after the concert was over. During the concert he had mentioned how he had performed in the Shetland Isles but was more familiar with a part of Scotland further to the west. Knowing the area well this could only mean one thing, the island of  Lewis and Harris in the Outer Hebrides.

I asked Alasdair if he had played at the Stornoway Folk festival to which he replied that he had many times. I then asked him if he knew my brother Alastair Whiteford. "Oh, you mean Ali? Yes, of course, I grew up in Stornoway and went to the Nicholson Institute where he used to teach. Now, he fixes my fiddle when it needs fixing. I know his three daughters really well too".

It doesn't happen very often but when you meet someone who has intimate knowledge of the people and placed one grew up in many thousands of miles away it gives you a great sense of connection and grounding and makes the world seem so much smaller and more intimate. Ever since emigrating in 1977 I have missed most the people and places I knew rather than the lifestyle or anything else associated with life in general in the UK. I especially don't miss the rain but I do miss the wonderful Spring which is happening there right now. It's what I call the Robert Browning syndrome.

Tuesday, March 11, 2014

Attributes and Assessment

All four of the student teachers I am currently supervising are in the same K - 2 school which makes for some interesting connections between the lessons I observe. The school, Allen Brook in Williston, Vermont, has just started using the Second Edition of the Bridges math program published by the Math Learning Center.  Last Friday, for example, I watched a super lesson on attributes presented by Erika in her second grade class. The focus of the lesson was to develop  students' understanding of the way we use attributes of objects to sort or classify them. This is one way the Bridges program introduces young children to the idea that maths has structure as identified in the seventh Math Practice Standard.  The way Erika did this was brilliant. She had the students describe a shape on a piece of card in terms of  three attributes, size,  shape and color. In other words, each card could be identified according to three different attributes. As the children described and identified the cards she held up Erika  had the students arrange them by attribute on the board. After several minutes there were two columns of cards but with  with cleverly designed gaps of missing cards. Erika then asked the students, for example, what should go here, pointing to one of the gaps. The students would look at the structure of the pattern of shapes and work out which card needed to go in the gap. The look on their faces when she held up the exact card they thought it should be was priceless. The amount of reasoning and thinking it took to identify say, a large, red rhombus, was amazing.

Later that day, I observed Caitlin in a Kindergarten class also presenting a lesson on attributes but this time there were only two attributes, shape and color. The other neat difference was that the children had to move around and form groups based on the attribute card they were holding. First they grouped by color and then they grouped by shape. Then, Caitlin had them group themselves by shape and color so that they really had to think about how who they should stand with. Developmentally, it was much more important for the kindergartners to be physically involved in forming the groups.

Last Friday I attended  a presentation about the new SBAC "assessments " designed to measure student achievement according to the Common Core Math Standards. Having assured the audience that the materials included in the SBACs were indeed assessments and not tests it was interesting to note how many times the speaker referred to them  as tests during the presentation; five times on one slide alone. The word 'assessment', as more and more academics are reminding us, is from the latin assidere or ad sedere, meaning 'to sit down beside'. There will be no-one sitting beside the students with the SBACS when hey are scored 3000 miles away.  Doesn't that make them tests?           

Monday, March 10, 2014

Arrow Cards Are The Best

This morning I watched one of my student teachers teach a wonderful lesson on place value to a group of first grade students using arrow cards. According to the new Common Core Standards for Maths first graders are required to be able to count, say and read numbers up to 120. The Arrow Cards are a perfect way of teaching both the procedural and conceptual knowledge associated with this skill. The really neat thing about this lesson was that Amy had made a set of Arrow Cards for each student in the class (sometimes, college room-mates can come in handy especially as there is a large amount of cutting out required). Amy used the Arrow Card model I developed which reduces significantly the amount of cutting required. (To make the "arrows" cut diagonally across the black square on each card).  There's also an interactive on-line arrow card activity but it doesn't seem as good as actually having the number cards in your hand to work with. Here's a paper on Arrow Cards I wrote several years ago

It was really neat to watch the students get their own set of Arrow Cards. The first thing many of them did was to count through them, first by ones, then by tens and then by 100s. The full set also include 1000s but 100s is enough for first grade.

There are two interesting discussion point about using arrow cards. One involves color coding the cards and the other has to do with including a 00 card that some people like to include. As far as color coding goes I would rather keep all the cards in one set the same color. I think this focuses the students' attention on the numbers and not on the colors. If all the ones were one color and all the tens were another color the students would focus on the color and not the numerals. I do like to make the sets in different colors so that children sitting next to each other will not get their sets mixed together.
As for the 00 card I don;t think it is necessary as it is not a real number and students seem to manage fine without it.

The moist important thing to remember when teaching place value is to treat 0 as a number and not use that terrible term "place holder". Every number is really a place holder!      

Tuesday, February 25, 2014

New Education Minor at St. Mike's

For the very first time the Education Department at St. Mike's is offering an Education Minor. Designed for those who are interested in learning about education but not wishing to go all the way to get a major and teacher licensure the minor gives students several options of different paths of study. With the help of an adviser students can put together three elective courses to go with the two required courses. There are even choices of 2 out of 3 in the required courses.

There are so many careers where an Education Minor can make all the difference during the job interview. The minor is designed to fit easily with any other major so that a student can virtually double the job opportunities and prospects they mght otherwise have had.

There is also the possibility of  completing your education major and obtaining teacher licensure at a later date through graduate study if the decision is made at a later date to go into teaching. Many graduate students in the Graduate Education Program at St. Mike's have decided later in life that they want to pursue a career in teaching.

And finally, the Education Minor provides students with a safety net if they decide, as a junior or senior that the Education major they have chosen to pursue is not the one they want to complete. Since they are also completing a liberal arts major they can still graduate with their liberal arts major and a minor in education. All in all, it seems like a great addition to the Education Department at St. Mike's offerings.

Monday, February 24, 2014

Grit in the Math Class?

I was recently asked by a colleague associated with a professional organization related to math education what I thought were the best ways to develop "grit" in children. I must admit that I was a bit taken aback as the only context in which I have ever encountered that word before is in the movie True Grit. Intrigued, I decided to investigate why some people thought we should be developing  children's grit.

The first online definition I found was "courage and resolve, strength of character". Merrian Webster said "mental toughness and courage" while the OED also says :courage and resolve, strength of character". Some of the synonyms are interesting too, bravery, mettle, pluck, backbone. All the time I'm reading this I'm trying to visualize the first/second grade students in the classrooms I visit.

But before I dismissed the idea as just too non-academic or unrealistic in terms of an expectation in the school classroom I discovered "How Children Succeed" by Paul Tough. Little bells also started ringing in my head because the more I read the more it seemed like there was a direct connection between grit and the growth mindset described by Carol Dweck in Mindset, a book I am currently reading. Further exploration confirmed this connection when I came across this TED talk by Angela Lee Duckworth in which she also makes the connection between grit and growth mindset. Interestingly, she gave no indication of how grit should be developed,  and I really do like the idea of the growth mindset.

The original question was posed to me in the context of the word grit appearing in the Common Core Math Standards math practice standards. I've reread the practice standards several times and cannot find any reference to grit. There is certainly a reference to perseverance but is this the same as grit? Does perseverance really involve courage and bravery or is it the disposition of stick-to-it-ness that we all admire in people who have it? Do we really want children to be brave and courageous when they are engaged in math problem solving? For now I prefer 'perseverance' but I will reserve final judgement until after I have read Paul Tough's book.