Thinking about Zero

Just think, without the numeral 0 there would be no place value system, 10 would be 1, and we wouldn't have a clue how to show a million. Traditionally, in our base 10 place value system of numeration, 0 has been called a "place holder" which is pretty meaningless when you stop to think about it. In the number 307, the zero means there are no tens in just the same way that the 3 means we have three hundreds and the 7 signifies seven ones. To call it a "place holder" reduces the real meaning of zero to a valueless grapheme that in no way contributes to young children's understanding of our place value system. Zero, or nought,  means nothing, none of, the starting place, neither positive nor negative,

Whether we should teach children to start counting at 1 or 0 has been a point of discussion among academics for some time and has recently been complicated by the addition of subitizing, the instant  identification of numerousness without counting. It has recently seemed to me that we need to start at 0; here's why. All children know what it's like to have no ice-cream or no M&Ms or no Hotwheels so why not begin counting with this idea that all children are familiar with. We still need to teach number naming, the sequence of the number names similar to learning the alphabet, and we can still teach subitizing skills. If we start with zero, however, the introduction of fractions and negative numbers later will make so much more sense.

If children see 0 as the point of reference they will better be able to see how fractions fit between 0 and 1. Fractions are initially taught as being parts of wholes, which on a number line, for example, is the space between 0 and 1. All the models used to teach fractions rely on students being able to take one apart into its fractional parts.

It's a similar issue with developing the idea of negative numbers. Without 0 as a reference point many children develop the misconception that anything less than one, such as a fraction, is a negative number.

It's as if 0 is a state of equilibrium, a fulcrum, an origin, a place at which all life begins.


  

Students Learn in Different Ways

Sometimes you read something from a reputable source, such as the BBC website, and you have to stop and say "Did I just read what I thought I read?" In an interesting article on the BBC Website entitled Teaching Maths the author describes how we need to do a better job of helping  parents understand how we teach maths in the twenty-first century. We have known this for years and yet we seem to somehow completely ignore the fact. There's nothing worse for a parent of a ten year old than not to understand how she is doing her maths homework. Many years ago I taught workshops for parents entitled Math for Moms and Dads in which we explored mathematical concepts and ideas. I notice that in the article Nick Eastaway has just written a book title Maths for Mums and Dads.

But what is so staggeringly and utterly senseless are the reported words of the Schools Minister for England and Wales, Nick Gibb, "who has stated publicly that he favours the older methods that he himself was taught by, and that he believes it is not an issue if children do not understand why they work". The fact that anyone connected with education can say that we should teach children things by rote, without understanding, seems completely without foundation. Imagine teaching young children vocabulary words for which they did not know the meanings; imagine teaching young children names of cities but not where they were located; imagine memorizing 2x3=6 but having no idea what it means. It would be like learning "spig x toog = dasc".  

This is not the first time I have heard the cry, "If it worked for me it should work for everyone else". It is not difficult to understand how certain individuals who have been successful in life should think that what worked for them should work for everyone. In my experience, these individuals are usually middle-aged men who are usually, like Mr Gibb, of a conservative persuasion. The problem with this point of view is that not all people learn in the same way. There are vast amounts of research supporting the ideas of different learning styles and multiple intelligences, to name but just two theories, that show there are many ways of learning information and developing an understanding.

'Maths', by the way, is the British abbreviation for mathematics, as opposed to 'math' as used in the US. 'Maths' always seems more grammatically correct!

Cool New SMC Website

I've just spent a few hours exploring the new St. Mike's website; it's very, very cool. I almost feel as though I work at a different college now: I had no idea so much happened within a quarter mile of my office. Well, I probably did but now that it's all in vivid HD color it seems so much more real.

I also want to take the Horses and Healing First Year seminar. I never though a great deal about horses until I read the book, then saw the movie, War Horse. It seems like an incredible way of adding an important dimension to one's life.

I've also always been a great advocate for undergraduate study abroad. The See The World page is absolutely amazing with the 2011 Global Eyes page as well as the Service Learning Opportunities resource. The more good Americans travel the world the more the world will see good Americans and the more good we do abroad through Service learning the more good will come of it.

The neatest thing about the new website is that it really speaks about St.Mike's, about what a wonderful place it is; about the opportunities students can take advantage of; about the true value of a liberal arts education. The world needs more worldly knowledgeable people who understand what it really means to be a global citizen. What better place to do this than a liberal arts college like St. Mike's.

Math Leadership Conference

A week today I will be hosting a Math Leadership conference at St. Mike's. I have invited twenty K-6 school-based math leaders from around Northwest Vermont for a day of exploring what it means to be a math leader. My good friend T.J. Jemison, an independent math consultant, will be co-hosting the day in which we have two specific goals.

The first goal is to develop a sense of what it means to be a math coach/leader. The math leaders will be invited to share the types of things they do in schools on a daily basis; what sorts of things they have found successful and those they have found challenging. By doing this we hope to initiate a support system that will provide the math leaders with a "learning community" through which they can share their knowledge ad experiences with others, as well as develop their own professional skills. During the seven years I spent between my two experiences in H.E. I served as a school-based professional development specialist in three school districts; first as a science education specialist and then as a math education specialist. I enjoyed the positions immensely as, after 17 years in higher education, I really needed to refresh my understanding of what it is like to work in schools and classrooms on a regular basis. I have always felt the theory connected with my college teaching needs to be well-grounded in practical application.

The other goal for the Math Leadership day is to generate ideas and information for the development of a grant to support the ongoing work of school-based math leaders especially in the context of the impending Common Core Curriculum. We will spend part of the day thinking about those things that would help make a smooth transition to the Common Core as well as those things that will improve student learning that, perhaps, local funding cannot support.

  

Commencement - The Pearls of Wisdom

Commencement is but a precious moment in time. Yes, I know that it is longer than a moment, in fact it's usually about 3 hours, but in the big picture of life, graduation is but a fleeting moment. One second you are a HS graduate, the next a college graduate. One second you have a B.A., the next an M.Ed.. One second the tassle hangs on the right, then it is on the left. One second you are an undergrad, the next an alum.

Apart from the joys of graduation, made even more so when it is one of one's own who is doing the graduating, Commencement is a time for collecting pearls of wisdom and there were quite a few this year. "Those who have been helped have a duty to pay it forward" advised Kuanapawa Nangula, a native of Namibia graduating with her Masters degree in TESOL. Kuana gave the graduate student address so eloquently with a wonderful dash of humor when she threatened  to give the full five minute presentation in her "native tongue".

Michael McKinney, the senior class speaker, created a powerful image when he described a dream of "burning stars" -  the light of St. Michael's College. He went on to describe many experiences and lessons learned at St. Mike's that have provided the fuel for those stars to burn well into the future.

And then there was the Commencement speaker, Thomas Freston (67), the creator of MTV who suggested that students should "tread in the footsteps of the future". He also advised students that there is "no rush to be normal", something that I could really identify with. Perhaps his greatest pearl of wisdom was that "passport ink is better than tat ink". I've never heard anyone make a plea for worldliness and global understanding  quite like this before but he is absolutely right. The way forward really is in developing a global perspective through travel and the use of one's passport regardless of the field one chooses.

As for me, my pearl of wisdom which my daughter Marie has heard  so many times is "it just goes to show" as in "it just goes to show what you can achieve when you really try". Congrats on your M.Ed, Swish.  

Thinking about Dan Meyer's TED

One of my colleagues sent me a link to Dan Meyer's TED this morning and if there's one thing a TED is good for it's to get you thinking about something. In this case it is High School Math and I must admit Meyer seems quite young for one with so much wisdom; but wisdom he has in abundance.  He has the ability to cut through the mathematical jargon as well as the tricks text book publishers use to get students to follow the prescribed steps to getting the right answer without really ever understanding what they are doing. His Two-and-a half men analogy is great.

One of the most accurate, critical points he makes is that we provide students with all the information they need as well as the steps required to solve a problem as if that is the way problems come to us in real life. Mathematical problems, he suggests, and exemplifies, should come to us through genuine inquiry where students need to identify what they need to know to solve a problem and then work out how to solve it. The art of the math teaher is to provide them with the mathematics that will help them do this. Mathematics, he says, is the "language of intuition".

What is so neat about what he is saying is that we need to help students think about what they are doing as opposed to following prescribed steps, mechanically and dispassionately. This is the same reason why I think rubrics have stopped student thinking in terms of how they are going to construct a paper. Half of the work in writing a term paper is learning how to construct it, how to put it together so that it presents one's learning in the best light. This is like reducing learning to painting by numbers; (paint all the 2s red and all the 3s green; make sure the paper has an opening, a body and a conclusion).

Meyer concludes by saying that we need "more patient problem solvers". I would add to that that we also need students to see math as relevant, interesting, motivating, and aesthetic. To to this we need to teach it this way.
  

Knowing, Missouri and the Internet

If you've spent any length of time around a person from Missouri, or somewhere close like western Illinois, you'll know that it's quite unlikely she/he will take your word for it. "Show me" will be the cry which until recently, say the last 10 or so years, was frequently next to impossible. So one lived one's life with a sense of uncertainty about what one knew, or thought one knew.

The other day in the science part of my undergraduate class we had been doing a neat GEMS activity to explore our fingerprints when someone asked if identical twins had the same finger prints. Since no-one knew we looked it up on the Internet and discovered that indeed they did not. Someone then asked  if they had the same DNA to which Kristen, one of the students replied confidently, "Yes they do".

Unbelievably, my first reaction was to say "OK, lets check on the 'net to see if Kristen is right". Immediately, I stopped myself remembering how I had felt the previous week when no-one believed me when I was talking about cow magnets. I should have realized that Kristen's confident response meant that she was sure of what she knew.

Is this, I wonder, an example of the changing source of knowledgeable authority that researchers are seeing in high school students? Are we becoming a nation, or even a world, in which personal knowledge is suspect because the information, like God, is just a mouse-click away?  Is the accumulation of knowledsge and wisdom by the older generation being replaced or usurped by instant access to everything?

As long as we can access it instantly we don't need to remember or know it, perhaps? On the other hand, is everything we know subject to scrutiny because of the instant verification afforded by the internet? Are we becoming a nation of doubters?

I sincerely hope not; personal knowledge and understanding is what makes us unique and  uniquely human.
  

Elementary my dear Watson

In the introduction to the PBS series, Sherlock last night the host described how Sir Arthur Conan Doyle was greatly influenced by Dr Bell while working as his clerk at the Edinburgh Royal Infirmary in Scotland. Dr Joseph Bell's greatest asset, according to Doyle, was his ability to critically observe his patients and it was this singular trait that led Doyle to create in Sherlock Holmes one of the greatest  close observers of human kind who ever graced the pages of a good 'who-dunnit?'.

In a completely unrelated 'walk of life' the idea of observation has greatly affected the way I see teaching and learning ever since I read this wonderful book by Michael Armstrong. In it he describes how important it is to observe children as they go about their daily learning activities in the classroom. It is, he implies, the only way you can truly begin to understand whether children really understand and appreciate what they are learning or whether they are just going through the motions.

It's not enough, however to simply watch and listen. What makes being a teacher different is that we observe, or listen and watch, through a deeply developed lens comprising our understanding of the content we teach and the characteristics of child development. Observation in this context means making sense of everything we see and hear children do.

Observation is then the essence of assessment and reduces things like rubrics, tests and worksheets to mere tools that can be used to supplement the authentic and real assessments resulting from careful and systematic observation. The thing that makes for such a compelling connection between the observation skills ascribed to Sherlock Holmes and the observation skills described by Armstrong is that in one of the assessment strategies I use with my students I tell them to act like detectives, raising and testing hypotheses about what they think the student knows and understands based on their observations. I think Sir Arthur would approve. 

Habits of Mind, Body and Soul

One of the benefits of getting on in years is that one has accumulated a considerable amount of experience upon which to dwell and reflect. Now while I am no Marilu Henner there are certain things that I remember quite well. Some are images and some are events like the first time I fell off my bicycle. But many of them are facts and ideas that I learned or became aware of that have stuck with me throughout the years. Facts such as Arun, Ouse, Rother, Stour, Medway, Darnet, Mole and Wey have been of little use to me once I got through my Geography exam at the end of my third year in College. There are other ideas, however, that have stuck and gone on to serve as a foundation for what I believe teaching, or even life, is all about. These ideas are often called the 'residue' of educational experiences we have had both in and outside the classroom.


The idea of dispositions, or habits of mind, body and soul, was first presented to me and my fellow grad students in a graduate class I took at the University of Illinois. The class was team-taught by Jim Raths and Lilian Katz and was always known as the "Rats and Cats" class. I probably took the class in 1980 as part of my doctoral program.

Katz's brief article (the first in this Google list) has always been, for me, the definitive explanation of why it is so important to make the teaching of dispositions such a vital, if not difficult, part of the educational process. Katz defines a disposition as "a tendency to exhibit frequently, consciously, and voluntarily a pattern of behavior that is directed to a broad goal." The classic example is that we can teach children how to read, a skill set, but will they regularly pick up a pick to read for enjoyment, a disposition. 


Over the years, the inclusion of dispositions in the school curriculum has faded primarily because of the difficulty of both teaching them and assessing them in an education world dominated by the setting and evaluation of specific performance standards. Perhaps it's time to take up the gauntlet again and look at ways we can help students, and ultimately, adults, develop positive dispositions to guide their actions and thinking. 


The list of facts I learned for my third year Geography exam? They are the rivers that drain the southeastern part of England below London!! I never know when I might need that desperately important piece of information! On the other hand it's fun to be able to recite them whenever I feel the need.       

  

Amazing Graduate Class

Every teacher I know can recall that one class of students that just seemed to get it, the one class of first graders or sixth grade students that gelled together as a learning community; the tenth grade AP English class that celebrated Shakespeare as never before or the 9th grade Math class that could not stop viewing Vi Hart videos.

My never-to-forget class was the last fourth grade class I taught in the UK before emigrating; that was, until I met the students in my graduate class this semester. There were 13 of them in the class and were, to a person, dedicated to making sense of the K - 8 math curriculum. They actively sought out understanding by holding each other accountable in their groups during class activities. They reveled in each other's successes and aha moments which frequently only occurred as a result of deep and frequently painful introspection.

I think the whole semester is summed up in the comment of one student who said "Now I know what I know instead of what I don't know". "To understand" is my mantra and to hear a student say that they finally understand something after 20 years of not knowing why is the best thing I could ever hear from one of my students. Even one of the math majors in the class celebrated finally understanding something that had frustrated her for years.

The photo? I have to admit it's one of my favorites; taken on the Durham railway station in northern England. I like to wonder if the person or people who painted the sign worried about how to write the fact that the arrow points the way to platforms 1, 2, and 3. I wonder if they saw the double entendre in the use of the word "to". Did they request a decision from someone in the higher echelons of British Rail or did they not even notice it since that's the way they wrote all signs like this? Noticing things like this is why I love maths.