## Tuesday, July 30, 2013

Having read the National Center for Teacher Quality (NCTQ)  report on teacher preparation throughout the country I was eager to see what they considered to be model programs. The report was so damming of such a large percentage of the 2000 or so teacher education programs reviewed that  the few programs they  held up as model programs must be remarkably good.

So, being a math educator I went straight to the Louisiana State University video to see what NCTQ thought was a model approach to teaching elementary school mathematics. The commentary by the professors, students, teachers and administrators had been clearly carefully scritped to include what we all know is good about math education as well as the importance of having close ties with mathematicians in the university math department. The video clip began with a college professor filling a blackboard with equations very much like he could have done in the 1950s or 60s.

This remarkable anachronism, the use of chalk and a chalkboard,raises serious questions about the LSU program. I haven't used chalk or a blackboard for at least 10 years since we discovered that chalk dust and computers do not live well together.There was a SMARTboard being used in a public school classroom later in the video but it looks as though LSU is still using blackboards in their university classrooms. This would imply that prospective students cannot use their laptops in the college classroom, that there is no instruction in the use of a SMARTboard and that students are not taught how to teach math through the use of technology such as iPads and interactive on-line activities.

One of the other videos at one of the other colleges also showed the extensive use of a blackboard.
I find this quite remarkable.

## Monday, July 29, 2013

### Mathematical Perspectives

One of the neat things about maths that makes it so exciting is the way you can look at the same thing from a different perspective. You can see 4 as 2+2. But, and this is what is so neat, you can see 4 from an infinite number of different perspectives. You can see it as 3 + 1 or 19 - 15 or as the square root of 16 or as the square of 2. You can see it as 4,326, 296 - 4,326,292 or as a square of 2 inch sides. It's also a Lucas number and a quartet such as the Beatles. It's a golf foursome or two couples out for dinner. Tetra and quad also mean 4 as do quatre and vier. Here's even more stuff about 4 .You can see it as 11 in base three or as 100 in base four. Isn't that neat!

I see math everywhere because I think that way. I get excited when I see an array of numbers that are similar or are sequenced. I like it when patterns appear out of nowhere like the 7272727 in my last post. Perhaps I have a well formed logical-mathemical intelligence ala  Howard Gardener's multiple intelligence theory . I am by no means a mathematician and I would struggle to complete some upper level  H.S. math classes but I see the fundamental constructs of mathematics with great clarity and in great depth.

I believe that these basic constructs such as the idea of part-part-whole are what we should be teaching young children so that they can see the world through a mathematical lens. For far too long we have clung to what the Victorians identified as "basic math" defined by the 4 operations of addition, subtraction, multiplication and division. These are no more basic to mathematics than declarative sentences and conjunctive verbs are basic to learning to read.

## Saturday, July 27, 2013

### Numercial coincidences are cool.

I love it when a numerical coincidence occurs. It's neat when you see several numbers the same or a sequence of numbers crop up in a series of unrelated things. It's like a quantitative harmonic convergence or QHC.

I had a neat QHC this morning when I was writing the check to pay my electricity bill. My eyes were thankfully distracted from the large dollar sum incurred through the extensive use of AC during the past month by the check number, 7272.

As I recorded the number in my check book register I wrote the check number followed by the date which turned out to be 7272 7/27 or 7272727. Not only is it a wonderful sequence but it is also a palindrome. Isn't that way cool.

The mathematical equation above is also a remarkable QHC. It's even more amazing because, as far as I know, this is the only time this happens. It's even more amazing still because the total of each side of the equation is 365, the number of days in a year.

12/12/12 was probably the greatest QHC that most of us will experience in our lifetimes. My birth-date turned out to be a neat QHC after i emigrated to the US. In the UK it was 19-12-46 which was depressingly boring and unspectacular. Now, it is 12 -19 - 46 which is really neat because sometimes I say "twelve nineteen forty six" as if it were 12-1946 and people think I have left out the day.

Here's another neat QHC. Have you ever tried multiplying a number by 9 and adding the digits in the product as in 2 x 9? It equals 18 which, if you add the 1 and 8 you get 9. Now multiply 18 by 9 and you get  162. Again, add the digits and you get 8. Just keep multiplying by 9 and the digits will always add up to 9.

## Wednesday, July 24, 2013

### Intrinsically Interesting CCMS

At last, I've found an organization that believes mathematics education has to be something other than rigorous, precise, challenging and boring. Hats off to the Wisconsin Department of Public Instruction who believes that math should be not only "intrinsically interesting" but should also involve "collaboration, discourse and reflection" and be "meaningful and engaging". The image from their website even shows students smiling, engaged and clearly collaborating as they, presumably, explore meaningful math activities on their computers.

I came upon this refreshing web-site by "Googling" Common Core Math Standards Resources, a search that yielded 2,640,000 relevant hits in 29 seconds. When I "Binged" the same phrase I got a staggering 49,900,000 hits. Many of the resources are authored by States (e.g. North Carolina) , Foundations (e.g.The Noyce Foundation)  or professional organizations (e.g. NCTM )  but by far the greatest number are from  commercial organizations (e.g.Office Max)  hoping to cash in on the re-education of the nation's teaching force.

Having clearly only scratched the surface of this vast ocean of resources it is becoming clear that some of the authors really do understand the intent of the writers of the CCMS while others don't. Becoming a connoisseur of such resources is probably as important as constructing a meaningful understanding of what the CCMS are all about.

There are some excellent resources at Ve2,  The Vermont Educational Exchange. The Phoenix Rising article, in particular, by  Hung-Hsi Wu  is a really good read.

## Tuesday, July 23, 2013

### Fractals from Pennies

Here's a great activity for helping students see the beauty of math. These are fractals made of pennies (cents) stuck to the wall outside my office. There's a Sierpinski Triangle, a Koch Curve and a Koch Snowflake. They are very easy to make and use the iteration of a simple pattern based on three pennies. There are many mathematical relationships in terms of both a numerical and spatial sense that you can  develop all the way from kindergarten to high school algebra.

Here's my student Lydia Koch's wonderful Math eNotebook assignment on pennies which started the penny fractal idea.  Isn't it an amazing coincidence that her last name is the same as two of the fractals!
There are so many things ot do with fractals when learning math. Here's the Cool Math fractal page , and here's the NCTM fractal generator.

Of course, the sky is the limit when it comes to fractals so here are a few more really cool fractal links.

The Fractal Foundation,
Vi Hart's Fractal fractions.
Fractals in Nature

## Monday, July 22, 2013

### Two Additional Math Practice Standards

So here are the 8 good but somewhat boring and dull math practice standards from the new Common Core that  I mentioned in my last post.

Here are math practice standards 9 and 10 which I think should be added so that math is a little bit more alive and connected with goes on is schools in the twenty-first century. They are still in draft stage so feel free to provide feedback.

9. Enjoy and celebrate mathematics

Mathematically proficient students enjoy and appreciate the aesthetics of quantitative and spatial relationships. They are captivated by the challenges of resolving mathematical problems and are able to use their mathematical understanding in creative and novel ways. They will demonstrate genuine curiosity when faced with novel mathematical situations. Younger students will share their excitement about finding several different ways of making 6, of understanding why a square number is so called and that pi is a ratio between the circumference and diameter of a circle and not just a number that goes on forever.  Older students will recognize and celebrate the artistic elements associated with fractals and the aesthetic characteristics of algebraic relationships.

10. Recognize linguistic and cultural diversity in mathematics

Mathematically proficient students will recognize that math is not the same the world over. Living in diverse communities students will recognize that there are differences in mathematics and the ways we learn mathematics based on local and global cultural differences. As they work with students from different cultures they will be aware of the ways language development, as well as the language used in  mathematics, are major factors in learning math for all students as well as those who are English Learners.

### Common Core Math Standards Lack Joy and Creativitiy

For years now I have anticipated the coming of the Common Core Math Standards due to be implemented next year. For the past year I have read and reread them as I adapt my courses to address both the math content and practice standards contained in the Common Core Standards document.

The math content required of elementary school students has been pared down so that it is no longer quite "a mile wide and inch deep". Perhaps it is close to being a kilometer wide and a meter deep! The Math Practice Standards too contain reference to dispositions, pattern and structure as well as the less exciting perseverance and precision.

But, and this is a really big but, they are deathly dull and uninspiring. Each of the eight standards begins with the phrase "Mathematically proficient students...." and most contain words like 'regularity', 'careful', 'precise', 'fluent', 'efficient', 'sensible', 'worthwhile', 'diligent', 'efficacy', and 'accurate'. There is absolutely no reference to the joy of discovering quantitative or spatial  relationships, the existence of mathematics in the natural world such as fractals, number patterns and geometric shapes.

There is no reference to anything creative such as the relationship  between music and math or art and math, or using either of these disciplines to develop mathematical thinking in a numerical or spatial sense. There is no reference to the role of language in mathematics or the fact that mathematics can be defined as  a language with semantics, syntax, and structure. There is no creative math equivalent to the poetry and creative writing identified in the English Language Arts Common Core Standards.
Imagine learning to speak, read and write without those creative components that have always been an integral part of that discipline.

The Common Core Math Standards are a great step forward in the identification of what and how students should learn mathematics. It would have been so much better if they could have been a great leap forward. It would have been so much better if they could have addressed the issue of why so many students don't enjoy math. It really doesn't have to be that way.

## Friday, July 19, 2013

### When is Three a Half of Eight?

I spent the past two days at a meeting at the Vermont Agency of Education discussing the new Elementary Education Licensure requirements that are due out in a couple of years. The two days of meetings went by quickly and although I don't think we achieved all our goals there was some great discussion and some meeting up with old friends. It was particularly good to see Casey Murrow again after so many years and to know that he is still publishing his wonderful Connect magazine at Synergy Learning.

It was also good to meet up with some folks from the Vermont Math Institute as we always have really good mathematical discussions. One interesting question that caught my ear that I hadn't  heard before was "When is three half of eight?". Always up to the challenge I thought about the question from every angle I could. I thought about different bases and how the 8-ball is significant in pool. I tried fractions and finally realized that I was getting nowhere. If using 8 as a cardinal number doesn't work is there something else about it that I could use? Perhaps the answer lay in the physical quality of the numeral 8. Bingo! If you slice the numeral 8 in half vertically the half on the right side is 3 and the half on the left side is a backward 3.

This is quite e neat mathematical problem because a fraction only really has value when you know what the one is to which it refers. As long as I kept thinking about 8 as a cardinal number, the identification of a group of eight, I was getting nowhere in solving the problem. If, however, I found a different 'one' to which the 'half' referred perhaps my problem would be solved. Once I had selected the numeral 8 itself it was then only a short step to see the 3 as the right side of the 8.

Now would you rather have half the money in my left hand or a quarter of the money in my right hand?