Nu, Numb, Numbers

I've found the perfect activity for getting through long, tedious meetings. Every time someone says out loud a number write it down. You then have some wonderful raw data to play with and do all sorts of things with. For example, if you have to attend lots of meetings on the same day like I did last Friday you can compare the numbers mentioned in each meeting. This will give you a Numerical Meeting Profile or NMP. Sometimes the meetings are dominated by small numbers while at other times they can be characterized by large numbers. Meetings can also be differentiated by the different types of numbers that are mentioned; ordinal (sequencing - third grade), cardinal (counting - sixpack) or nominal (naming - 2011) just to mention a few number types. The recorded numbers can only be those spoken and not any written on any form of presentation. This could get out of hand very quickly.

Another way of exploring the numbers is whether they are naked numbers; in other words does the speaker include the referent with the number. A naked number is one that has no referent attached such as "two-ninety nine" for a price or "six two" when referring to someone's height. This activity can be amped up considerably by pretending one is from another country, planet or occupation. For example a "two point five" would not cause any alarm if it came up in a discussion about engine size.  When discussing student GPAs, however, it can have the most dire consequences for someone.

It would be interesting to develop NMPs for different groups of people. For example, would an NMP for a group of mathematicians be different from an ensemble of historians at their monthly department meetings? What would the NMP be for a group of 20-somethings out for the night on the town?

Smoking in the Principal's Office!

President Reagan was in office when I moved to Vermont and supervised my first student teacher at Williston Central School. That was 30 years ago in 1982. The amazing Marion Stroud was principal at the time and I can remember sitting in her office chatting and smoking cigarettes; I kid you not.  I listened to her dreams of a new Williston school building with Kivas and "houses", ala Harry Potter's Hogwarts, and a real theatre and an Olympic size swimming pool. Such was her dynamic nature that within just a few years she had accomplished everything except the swimming pool.

Marion was a rare visionary in a world now dominated  by the bottom line and adherence to rigid standards. She had worked for some time at the Bankstreet school in NYC and brought with her to Williston three critical elements of education which were to form the foundation of a remarkable school; experiential learning, interdisciplinary learning and collaboration.She forged the school into "houses" of grade groups 1 - 4 and 5 - 8 where teachers worked together across grades and students learned in communities designed to promote real learning and not just give them an education.

I found Marion to be a kindred spirit for she too was British and a proponent in the US of the British Open Education movement upon which the Bankstreet school was based. I watched the school go from strength to strength as teachers bought in to her belief that a caring, collaborative and conscientious environment was the most conducive to the all around growth and development of every student and every teacher. The school became a leader in the application of technology in the field of education and was actively supported by Seymour Pappert, one of the great educators of the last century.

Thankfully, the school still retains much of Marion's vision through the dedicated work of many of the teachers who still inspire children through their enduring beliefs in experiential, interdisciplinary and collaborative learning.

The Museum of Science and Engineering in Boston is an amazing place; more so since Dr. Ioannis Miaoulis has become director. What he has done is to create an entirely new dimension of education and learning that brings science into the world of engineering (what we used to call design technology).

Instead of just learning about the natural environment as we do in science, Dr Maoulis has focused our attention on the idea that we use what we know about science to improve and protect the world we in which we live. In science education at the elementary school level the focus is on inquiry. Students are encouraged to use their natural curiosity to learn about the world in which they live. Which substances are magnetic? What do plants need to grow? How do rivers create valleys? In engineering, students are encouraged to use the science they have learned to solve problems. How can you make an object move using magnetism? How can you grow a plant without any soil?  There is a neat program at the Boston Museum of Science (EIE) designed to help students develop their engineering skills from a very young age.

One local school that has an incredible engineering program is Williston Central school where Mike Thomas has been guiding students in grades 3 - 8 in the art of design and construction using a variety of motivating activities and materials. On any given day the students can be engaged in designing and making a medieval  trebuchet or programming a Lego robot they have made using a computer program. 

Math is the Science of Pattern

Every semester when I start a new math ed. course with a new group of students I increasingly see the value of exploring and including the ideas associated with pattern in the course. When you stop to think about  it there is almost nothing random about math, in fact, trying to come up with complete randomness requires the sue of sophisticated computer models.

There are several different definitions of "pattern"; pattern as a design, pattern as a model or something to be copied, and pattern as a regular sequence or set of events. It is this last definition that is most relevant here. If we can help children see patterns between different mathematical entities it will help them remember, recall and make sense of what they are learning. We can do this from the earliest stages when children learn to count by 2s, 5s and 10s. What makes this type of activity even more worth while is to count by 5s starting at 3 or count by 10s starting at 7. Whenever i do this with my students I can see them initially stumbling and going slowly. Then, as they see the pattern emerge they speed up and end up rattling the number sequence off.

The Fibonacci number pattern identified by the squares above and dun flower to the left is a classic number pattern that occurs all over the place. It can also be demonstrated in the Sierpinski triangle, a classic fractal, as well as in Pascall's Triangle.

Here's another amazing number pattern. Add 1+2 (=3)+3 (=6)+4 (=10) +5 (=15) +6 (=21) +7 (=28) +8(=36). These are called triangular number because they make triangles (imagine 1 + 2 next to each other like steps etc). Now add consecutive triangular numbers together and what do you get? Yes, a square number. Imagine turning 3 upside down and fitting it together with 6 to make 9.

Math is, indeed, the science of pattern. 

Handwriting on the Wall

We recently received an email message implying that one of the goals of Higher Education was to eliminate paper forms of communication. By utilizing all the different forms of technology-based communication programs at our disposal we could actually run our college courses without our students ever having to handwrite a single thing. They could complete their papers using Word and submit them for assessment through eCollege. We would then comment on the paper and return it complete with a grade which would also be entered into the Gradebook feature in the same program.

Since I am a blogger I am clearly an advocate for technology in all it's various forms but something worries me, a lot, about wanting to achieve a paperless culture. Yes, we'll save lots of trees just as Kindle and other book forms are doing, and we would save hours and hours of painful handwriting practice sessions for young children in schools. But is this really what we want?

If our cultural goal is a paperless society then the art of handwriting will no longer be valued and if it's not valued it will not be taught. We will then have to make sure that we always have a keyboard of some sort with us complete with a form of printing off notes, letters, shopping lists and anything else that we might incidentally need to communicate. Perhaps all those hours, days, weeks  and months spent learning how to do joined-up writing, as we used to call it in England, could be spent teaching keyboarding skills instead. But is this what we really want?

Several years ago I read an interesting research paper in which parents were asked whether their children's teachers should spend writing instruction time teaching keyboarding skills or handwriting skills. The results of the research were overwhelmingly (80%) in favor of teaching handwriting skills.

There are many ways in which handwriting can be integrated with technology. The IPAD2 a well as the SMARTboard and many other forms of technology accommodate handwriting and some can even convert it into print. If one of our goals is to create a paperless culture then we must make sure that we don't also create a keyboarding only culture and bring about the demise of handwriting skills.

    

The Numbers of Our Lives

A new semester, new classes and another new beginning. I'm always so nervous for the first half of the first class but I usually settle down once the students have laughed or smiled a few times. It's been the same since I taught my first fourth grade class in August 1972 and I wouldn't have it any other way.

There are a couple of things I really want to focus on in my undergraduate Teaching Elementary School Math and Science course this semester. I want my students to be able to see the relevance of school mathematics. I want them to see that it isn't just a means of balancing your check book or working out how to solve mathematical problems. I want them to go beyond the "do the math" syndrome that our culture seems to have fallen into when it comes to any form of quantitative experience. I want them to see the poetry and creative writing equivalents in mathemtics.

I want my students to appreciate the aesthetics of math in the same way that Vi Hart does. I want them to see how wonderful it is when you add two successive traingular numbers such as 3 and 6 and get a square number. Imagine three small wooden blocks next to each other so that a stack of 2 is next to 1; like a step. Now add a third stack of 3 to make another step and a total of 6. If you take another 3-step and invert it, it fits with the 6-step to make 9; two successive triangular numbers make a square number. You can make triangular numbers just by adding an additional step. 6 plus a stack of 4 is 10. Just like this: invert the 15 and it will fit with the 21 to make 36, a square number.

Isn't that just way cool? 

Fourth Place in the Olympics

Go forth and multiply, come fourth and get nothing.

I would imagine it must be quite devastating to come fourth in an Olympic event especially if it is by the merest of a hundredth of a second or fraction of a centimeter. No medal, no standing on the podium, no name in the record books, nothing to take home and share with friends and coaches, and nothing to bite on.

On the other hand coming fourth means that you are the fourth best in the world at something. Of all the people on the earth only three can do that particular thing better than you can. This is the problem with ordinal numbers, they say so much and so little all at the same time. They also often tend to turn into nominal, or naming numbers, once they have been identified, such as in dates. Although July 4th is the fourth day in July it is more widely used to identify or name a particular day.

Of all the different uses of number, the ordinal use is the one we use when we want to compare things numerically. They are the type of number that seem to carry the greatest social meaning. When we are in elementary school it is way cooler to be a fourth grader than a first grader and it's probably better to live on fifth avenue or ninth street in New York rather than 12th Avenue or 42nd street.

Floors of buildings are identified first as ordinals but this can get really confusing. In most western cultures there is no 13th floor while in the east there is no 4th floor; each number being consider unlucky in their respective cultures. Then there is the confusion created in Britain where the first floor is above the ground floor making every high-rise building in the UK actually one floor higher than anywhere else in the world.

The Olympics really have been incredible to watch and a credit to London, my home town. I certainly wouldn't  mind coming fourth in any Olympic event.