Incredible Math Night

Last Thursday evening I attended the Math Night at Flynn Elementary School in Burlington and what an incredible experience it was. Over 300 parents and children took part in the event in which they experienced a variety of neat math activities put together from the Bridges Math Program as well as some culturally diverse math activities from different countries represented at the school.

The evening was planned by Karyn Vogel, the school math specialist, and a recent graduate from the SMC Graduate Education Program, and Rebekah Thomas, the school ELL teacher . There were over 25 tables set out with math activities including the Hungry Caterpillar, the Sierpinski triangle, and Tangrams to name a few. Each table was staffed by one of the Flynn school teachers, a student from Burlington High School or a Flynn school parent such as Kyendamina Cleophace Mukeba who shared a popular game from the Congo called Kisoro.

Judging by the enthusiasm with which the parents took part, the happy buzz that emanated from the cafeteria where the event took place, and the excitement in the children's voices as they taught their parents how to participate in the activities the event was a major success.

We now know, from definitive research, that adults' attitudes toward math have a direct effect on the way students learn math whether they be teachers, parents, or older siblings. Anything we can do to demystify math, to make it more user friendly, and to help people see the aesthetic side of math, the more effective our teaching will be and the more complete will our students' learning be. The Flynn school Math Night was a huge step in the right direction,.

   

Teaching Is All About Students

Whether we teach young kindergarten children, older graduate students or anyone in between, teaching is all about the students. My sophomore level Schools and Society students have been spending two hours a week in a public school classroom for the past six weeks. For most of them this has been their first formal experience on the other side of the desk, so to speak. Each week, they have been journaling about their experiences based on a journal prompt I have given them. This week, I asked them to bring to mind six of the students they have been working with (K - 8th grade), codify the students' names and then write a few comments next to each student that describes each student as they see them.

I haven't read their journals yet but my hunch is that close to 99% of their comments will be in what we call the affective/social area of learning. In other words the comments will identify the students attitudes, values, and social attributes. Very few will describe the students' cognitive (thinking/knowledge) attributes or their  psychomotor (physical skill) development. This is quite natural because when we first interact with other individuals the first thing we are aware of are these types of characteristics especially when individuals are observed in a social context.

But the primary goal of teaching is to help students learn in the cognitive domain of learning. Of course, learning in the social/affective and psychomotor domains are important but it is the development of a student's knowledge and understanding, for the most part, that is the primary goal of teaching. So on Monday when I meet with the students, we'll go over their student observations and talk about how they can change their perceptions of their students through changing the way they observe and interact with them. The more they can learn how to find out what their students know and understand the better teachers they'll be.

And the  more I'll know about how my students think  in the context of becoming a teacher too.      

Elementary Education is not so Elementary

Many, many years ago someone asked me which 3-credit course she needed to take at College to become an elementary school teacher. When I told her that she needed a Bachelor's degree with a double major in Elementary Education and a liberal arts major she was stunned. Such is the public perception sometimes of what it takes to become an elementary school teacher. No longer is the elementary school teacher the unmarried, virtuous maiden required of the local community to staff the one-room school house of two centuries ago.

Today, elementary school teachers are knowledgable about everything, skilled at dealing with every possible type of human interaction, and familiar with just about every research-based teaching strategy known to the profession. Well, maybe not quite that much, but the demands we place on those who teach our under-eleven offspring are pretty significant in a fast-changing and complex world such as the one we live in.
The epistemology or elementary education comprises three broad areas; knowledge of child development, pedagogical content knowledge, and the teaching strategy skills that act synergistically between these two areas of knowledge. Knowledge of child development means understanding how neuro-typical children develop as well as the unique characteristics of children with special needs. It means having a sense of what students from other cultures might need to adapt to US classrooms as well as the customs of their families that might impact their child's education.

Pedagogical content knowledge is knowing and understanding the content of the elementary school curriculum at a developmentally appropriate level. For example, the science that kindergartners learn is very different from that in fifth grade. The reading skills we teach in first grade are also quite different from  the reading skills we might expect a fourth grade student to learn.

The third piece of the puzzle is knowledge of, and the ability to operationalize, the actual teaching skills that help students learn in an orderly and motivating classroom environment. These skills range from the ability to write an effective lesson plan to the way one uses one's voice during interactions with students in the classroom.

The Elementary Education program at St. Mike's helps students progressively develop their knowledge and skills in each of these three cornerstones that make being an elementary school teacher one of the best, most rewarding occupations on the planet.

The picture, by the way, is of a group of students from the Lawrence Barnes Sustainability Academy in Burlington who are taking part in the Vermont Community Garden Project.

The Albert Hall....How Many Holes Fill It?

Many years ago The Beatles assumed we all knew how many holes it takes to fill the Albert Hall. To be absolutely sure we'd need to know both the size of the Albert Hall as well as the size of the holes they were filling it with. We'd also need to know whether the holes were pressed together with no gaps or overlaps.

More recently, an article on the MSNBC website described the extent of Ted Turner's ranch as "just shy of three Rhode Islands". Since most of us have no idea how big Rhode Island is, beyond the fact that it is quite small as States go, we can only assume Mr. Turner's ranch is rather large.

The Beatles' "holes" and Turner's "Rhode Islands" are examples of referent units, things we use to give us a sense of the size of something.  As useful referents these two example are not terribly useful but there are literally millions of referents that are. In fact, mathematics in the elementary school, as well as our lives in general, would be pretty bleak and meaningless if we were not familiar with a good number of these referents. Just about every noun you can think can be a referent. A referent can be an orange, a house, a universe, a plant, a person, a feeling, a hope or a penny.

There are also things made up specifically to act as referents. A "byte", for example, is a referent as is a foot, a pound, a dollar, a gallon or a  degree Fahrenheit. There are also referents that are culturally defined such as a gram, liter,  or centimeter. How wide is the span between your thumb and little finger in inches? How far in centimeters?

This ability, or inability, to think in terms of referents is what we can call quantitative literacy and is a key component of the elementary school math curriculum where we need to try to avoid the use of "naked numbers" or numbers without referents. The use of a referent changes a numeral, a grapheme, into a number and gives it magnitude. For young children this is an important part in learning how to use mathematics in everyday life as well as learning the joys and wonders of mathematics.

I once asked a group of kindergartners what was the biggest number they could think of.  After most had strung together a bunch of jumbled, familiar number names one little girl responded "One Universe".  

John Dewey; a Vermont Legend

There was a wonderful article on John Dewey in the Burlington Free Press this weekend. Dewey is a native son of Vermont and was born and raised in Burlington where he taught at the University of Vermont for some years. His greatest gifts to the field of Education were probably his steadfast beliefs that students should be taught to think for themselves and to learn through discovering the true meaning of knowledge. Labeled a progressive and a pragmatist, he wrote many books about education and was a strong advocate of education for democracy and the role of discovery learning.

He also developed theories about thinking focused on the ideas of deductive and inductive logic. In How We Think, he relates two stories about the ferries that ply Lake Champlain between Burlington and New York State 13 miles away. In one story he describes how, standing on the car deck of the ferry, he notices what looks like a crane derrick, a 20 foot pole,  jutting out from the wheelhouse at a 45 degree angle. He notices that is had no rope attached to it so it couldn't be a crane. He then notices that it has a ball on the end and is the same height as the pilot in the wheelhouse. Being a ferry, there is no pointed bow, just a flat place where the cars drive on and off. From all these observations he deduces that the pole is in fact a false bow that allows the pilot some accuracy when steering the ferry across the lake. The ferry captain can line up the ball with something on the shore. I remember standing in the identical spot on the same ferry when I first moved to Vermont and looking at the pole in the same way Dewey had done..

In the second story he describes a strange sight of sparrows flying down and landing in front of cars parked waiting to board the ferry. As soon as the birds land they disappear up inside the engines of the cars only to reappear moments later with something in their beaks. Knowing that cars have radiators at the front and that all kinds of bugs get caught in the radiators as the car is moving he deduces that the sparrows have somehow learned that there is a veritable cooked feast waiting for them in every car that shows up at the ferry dock.
I have seen sparrows doing this too at the ferry dock. I wonder what they would do with an old VW Beetle with the engine in the back and no radiator.

Dewey's ideas are alive and well in the Teacher Education Programs at St. Michael's. His ideas of "Inspired Vision" and "Executive Means" are illustrated and exemplified in senior student licensure portfolios where they share their philosophical beliefs about teaching, their inspired visions,  and the teaching skills they craft in classrooms in local area schools, their executive means.  

Math Is For Everyone

I've just finished putting together my presentation for the Massachusettes Down Syndrome Congress conference this coming November. I've always believed that the math we teach to children with special needs is the same as the math we teach to all students. The difference lies in the way we teach it and how we adjust our expectations. In order to make these adaptations special education professionals need to have a deep understanding of the math they are expecting their students to learn.

So my presentation will have 4 parts; some theory, some math, some Andrew, and some applications. The theory part will address the way we can think of math as composed of both procedural and conceptual knowledge. This distinction can be illustrated by the idea that math is composed of symbols, rules and methods (procedural knowledge) and ideas, concepts and schema (conceptual knowledge). 3 + 4 = 7 is a piece of procedural knowledge comprising 5 symbols and a syntactic rule. It could be used for solving a joining, separating, combining or part-part-whole problem, the conceptual knowledge. This example also included some of the math I will be presenting.

The Andrew part will include pictures of my son Andrew in a variety of activities in which he has the opportinuty to develop his math skills while the final part of the presentation will include a selection of instructional strategies and applications of math I have used with Andrew; things like the iPAD2, the Wii system, his Hotwheel car collection,  the daily calendar he uses, clocks and board games to name just a few.

The interesting thing I have found working with Andrew, who has Down Syndrome, in math is that he finds it difficult to make the cognitive connections between procedural and conceptual knowledge. For this reason we have focused almost exclusively on the development of his conceptual knowledge. It is almost certain that he will not pursue a career involving higher mathematics so the most important thing for him is that he develops his quantitative literacy skills so that he can use the math in his life that he needs to use and can appreciate the numerical and quantitative relationships he encounters.

That's Andrew at 4,800 feet on top of Mt Mansfield, Vermont's highest peak, earlier this summer.  



     

There's More to 2+2=4 Than....

I observed a great math lesson this morning with one of my student teachers. She was introducing her first grade students to the idea of the horizontal  number line such as the one in the picture on the left (3 + 4 = 7).
Traditionally we have used language like "three plus four makes seven" or "three and four are seven". We now know that both these forms of language actually develop in children a misconception about what is happening in this piece of procedural knowledge. Children tend to think that the equals sign makes things happen. This can cause untold problems when they come to do serious algebra in middle and high school.

The key to later success in math, and algebra in particular, is to develop the equals sign as a sign of equality by helping children develop the appropriate concept-based language as they are learning the meaning of this number sentence. This key to later success is helping the children associate the equals sign with the idea, "is the same as". Three plus four is really just another way of writing seven. So is 2 + 5, and  1 + 6. 

The student, assisted by the cooperating teacher and school math specialist, then went on to ask if it was OK to write 7 = 3 + 4. As I would have predicted, the children had a really difficult time with this. Almost all of them decided that you could not write the sentence this way; "this is just wrong" as one student said. The teachers then did several activities  using small cubes in which they showed the children that it really didn't matter which way round they wrote it; it still meant the same thing.

It's attention to small details like this when teaching young children that make a profound difference years later when students learn more complex math ideas.  

SMC Teacher Ed. Programs Shine

For the past four days the Teacher Education programs at St. Mike's have been undergoing intense scrutiny by a team of educators appointed by the VT State Department of Education Standards Board. The purpose of the assessment is to make sure that all the SMC teacher licensure programs comply with the VT State DOE regulations and standards for teacher licensure. At the exit interview this morning we learned that not only were all the programs approved, we also received four commendations which is highly unusual.

The commendations described how our students are encouraged to excel in all they do, they are well-versed in working with students from diverse populations, they are committed to inclusive education and they use the assessment of student work to drive instruction.

For the past several years we have been assembling a 67 page self study which formed the basis for their assessment. In addition, they spent time talking with graduates, visiting local schools to talk with teachers and administrators, visiting college classrooms and reviewing course materials

We also assembled a Presentation of our programs to introduce the team to the teacher licensure programs at St. Mike's.

Many thanks to everyone who took part in the assessment; the graduates, undergraduates, graduate students, faculty, staff, administration, local area teachers, administrators, public school students and to those who were part of the DOE team.

Is it the 1970s again?

When I was an undergraduate in the late 60s and early 70s there always seemed to be some cause to join. Hardly a month seemed to pass without a sit-in for this, or a march for that, or gathering for the other. I remember one sit-in in the "Junior Common Room" (student bar) that lasted several days because the college would not let the students install a cigarette machine (I kid you not). Usually, however, the cause was much more celebrated such as supporting a miner's strike or going on a "ban the bomb" march. They were always peaceful and always had some positive effect on public opinion, or so we thought.

This week on the St. Mike's campus little signs have been appearing on walls and doors declaring "We are the 99%" and other topical slogans. I'm sure they are connected with the current Wall Street "sit-in" and I think it is wonderful. I don't want to use my blog as a political soapbox but I think this passion for an issue that the students are showing is great, as long as it remains peaceful and is focused on the issues. I remember the wonderful sense of hope and involvement our support for various causes invoked in us back in the middle of the last century. There's nothing quite like it to take one beyond one's own little niche in the world.

By the way, that Peace Sign, or Ban The Bomb sign as we used to call it, in the picture is on the back of my VW van. It's called a Peacewheel. It's my way of staying connected with the causes of my youth.

More Thoughts About Grades

In my last blog entry I mentioned how I was going to withhold the letter grades when I returned the weekly journals to the students in my Schools and Society class. It turned out to be a really interesting and worthwhile activity because it raised all kinds of thoughts and questions in the students' minds regarding the purposes and roles of grading in the world or K - 16 education.

As I gave the papers back, the students looked immediately for the small grade slip stapled to the backs of the papers. After a minute or so I asked them to talk in their groups about how they felt and what they thought about not getting a letter grade.

The general consensus was that since they were used to getting a grade they really missed it and really wanted to know what their grade was. They also raised several really interesting points that promoted some great discussion. They all said that they read my copious comments much more closely to see if they could work out what their grade was. They tended to agree that since most of my comments are supportive or tend to ask questions to further their thinking they found it difficult to work out the grade. They also confirmed what they had read in several of the chapters on assessment that grades can be motivators, give you sense of how you're doing, and provide a sort of feedback.

We concluded by deciding that since the State Dept. of Education requires a B grade average in order for students to be licensed to teach it was really important to have a grading system. However, none of us were convinced that there is a clear link between a grade and what has been learned, all the time.

Tomorrow, were going to be exploring issues of assessment especially the role of what one knows about the topic in the assessment of a student's work. Why is the picture above so strange? Or is it? What do you need to know to make that decision?

Letter Grades

This past week, my Schools and Society students have been reading about different points of view regarding assessment. Each week, each student sends me two journal entries. One is a focused response to the chapter they read in the Educational Foundations text we are reading and the other is a response to a question about their two-hour weekly public school classroom experience. The journals are great and give me a good insight into how the students are thinking about the profession of teaching.

The readings about assessment are promoting some really interesting responses from my students. Many of them, even those who get As, don't like letter grades and wonder why everything has to be graded with a number and a letter grade. I must admit I often think bout grading as a bit of an anachronism but it's required by the college as well as by the State of Vermont DOE for students wishing to get their teacher licensure. In fact, the State Department of Education requires a B average which, in effect, renders a C grade a failing grade. Perhaps this is one reason for grade inflation. I certainly don't give many C grades.

Most of the readings in the book present quite a radical view of teaching and education because in the Education Department at St. Mike's we believe the teachers who graduate from our teacher education programs should be thinkers and intellectuals in addition to caring about children's learning and welfare and having a passion for teaching. Two of the readings on assessment this week are by Alfie Kohn and William Ayers; two of the great writers and thinkers about issues in education.

To make the discussion more interesting I am going to leave off the small grade sheet I attach to the students' journals when I return them at the beginning of class. They will only get to read the copious comments I write on their papers. How do you think they will react? I have a feeling they will want to know what grade they got before the end of class.

My grading system is a bit tougher than the one above; a few points more are required for each letter grade.