Harriss; a Fractal for St. Patrick

Here's something you don"t see every day; a new fractal named after the creator, Edmund Harris, a professor of mathematics at the University of Arkansas. It also seems like a very appropriate fractal for St. Patrick's day as it has a certain Celtic feel about it.

As remarkable as it might seem this fractal is based on the Golden Ratio rectangle as explained in this very informative article in the Guardian. You can  see similarities between this fractal and the one created by the squares of the number s in the Fibonacci sequence which, of course, also follows the golden ratio.

The patterns created by fractals are based on numerical  relationships and provide us with an incredible insight into the structure of mathematics and our number system, not to mention the mathematical structure of the natural world. I think this is what the creators of the Common Core state Standards for Mathematics had in mind when they came up  with the wonderful math practice standards, especially numbers 6,7, and 8.

Friday the 13th and Pi - Day

So we have two interesting mathematical days coming up; Friday 13th and Pi Day.

Friday 13th is a day of dread and fear for many especially those who suffer from Triskaidekaphobia, or the fear of 13. Strangely enough not everyone in the world thinks 13 is an unlucky number. For example, in many Asian countries 4 is the number that is the unlucky one because the Chinese word, for example, for 4 is similar to the word for death. Here are some other unlucky numbers from around the world.

So  Pi-Day this weekend is not just any old Pi Day. It's Pi Day of the Century because 3.1415, the first four decimal places of Pi is also the complete date 3/14/15. This has set off great excitement in the profession of math education There are all sorts of things you can do to find out about Pi of the century. Here's some really interesting info about Pi. Here's Pi at MOMATH  and here is Pi Day in Chicago. And, of course, Pi Day of the Century at the NY Times.

Something most people in the US don't realize is that Pi Day is only celebrated in the US because in the rest of the world the date is written differently.  

And of course, the most important thing to remember about Pi is that it is a ratio between the circumference and the diameter of a circle. The circumference of every circle is just over 3 times the distance across the middle. If the diameter is 7 the circumference is 22.

Diane Ravitch Calls It Like It Is

One of the remarkable things about Diane Ravitch is that she has been a staunch advocate from both sides of the great abyss. The fact that she now supports genuine, authentic and principled education gives hope to many of us in the face of the tsunami of testing, "accountability" and political meddling that is at the heart of the dismantling of the system of  education as we know it. Her recent blogpost about the heavy hand of the Federal Government regarding testing in Vermont is both comprehensive and eloquent and serves as a dire warning to anyone interested in the salvation of  our education system as a place where children learn, grow, flourish and blossom as individuals within a caring society.

The impending, inevitable  public outcry of perceived declining standards illustrated in the soon-to-be-released SBAC scores will only serve to highlight the wisdom of Ravitch's words as teachers and the education system are subjected to yet more abuse from an ill-informed public that seizes upon the simple but misguided notion that increasing levels of evaluation will improve the quality of education.

One of my colleagues at SMC, Beth Peterson, quoted Gary Oldfield when responding to Ravitch's blogpost with this brilliant observation " Setting absurd standards and then announcing massive failures has undermined public support for public schools . . . We are dismantling public school systems whose problems are basically the problems of racial and economic polarization, segregation and economic disinvestment.” (Educational Researcher, August/September 2014, p.286).

The SBACS, the tests in English language arts and math, adopted by many States as the  compulsory evaluation tool for the new Common Core State Standards will be an  unmitigated disaster. They are difficult to administer, difficult to take by students, and are scored on the other side of the country in Washington State. Costing around $27 per student they will undoubtedly make someone financially rich and the rest of us who have to suffer them culturally poor.  

Perhaps parents should be asked to take the SBACS to see what their children are being subjected to.

   

Teaching Fractions and the Referent Unit

Seeing his wife hard at work in the kitchen, her husband decides to give her a hand and asks if he can help. "It would be wonderful if you could peal half these potatoes for me", she replies handing him a bag of potatoes.  This is what he did.

Whenever we use a fractional term, either in he classroom or in everyday life, we have to be clear about the referent, or the one, to which the fraction refers. Sadly, this is a key concept in teaching fractions which is almost never taught.

Ask a group of fourth graders if they would rather have half the money in your left hand or a quarter of the money in your right hand and they will nearly always say they would prefer half the money in your left hand. When you open you hands to reveal a quarter in your left hand and a $20 bill in your right hand the look on their faces is priceless.

When we add or subtract fractions this is a fairly straight forward activity if we have the one to which the fractions refer at hand. If we are just doing this procedurally without manipulatives then we need to make sure that children know that both fractions refer to the same one.

When we multiply or divide with fractions, however, things get significantly more difficult. In multiplication, for example, 1/2 x 1/4 can be better understood if we make sure the one to which each fraction is known. A half of a quarter of a piece of paper is an eight of the piece of paper. The referent for  the quarter and the eighth is the piece of paper but the referent for the half is the quarter piece of paper.

A similar situation is true in division such as in 1/2 ÷ 1/4.  How many quarters  are there in the first half of a football game, There are two. Here, the referent for the 1/2 and 1/4 is football game and the referent for the 2 is the 1/4 of a football game. This always seems to be such a much more important thing to teach than "change the sign and invert the second fraction"

 

How has "borrow" survived for 75 years?

Sometimes it feels like my professional life has been one of failure and futility  especially when I think if one particular aspect of math education; the use of the word "borrow" when completing the subtraction algorithm.

For around 75 years in the US we have been using the word in a context which makes absolutely no sense at all. We have corrupted the use of the word "borrow" to become a synonym of steal since when we borrow in the subtraction algorithm we never pack back.

This wasn't always true of course. Up until around the early 1940s we used a subtraction algorithm that was significantly different from the standard one taught in elementary schools today. It was called the equal addition method of subtraction and involved adding ten ones to the top number and one ten to the bottom number when the bottom number in, say, the ones or tens place,  was larger than the top number.

In the example above the 3 is larger than the 2 in the top number in the tens place  so you borrow ten tens (you actually just pluck them out of the air) and then you pay them back as one hundred in the bottom number (again, just literally plucked out of the air).  So, this is done to the words "borrow one and pay it back"; all very ethical. This method is still used in many places around the world such as Bosnia.

Around the early 1940s we changed the method of subtraction in the US to the decomposition method where you decompose the number by regrouping the ones and tens and so on. In this example, you cannot 

 take 3 from 2 in the tens place so you regroup 600 and 500 and ten tens, adding the ten tens to the 2 tens so that you now have 12 tens from which you can take 3 tens resulting in the 9 tens or 90 in the answer.

This method was far more logical and much easier to teach especially if you used the concept derived term "regroup".

Unfortunately, for some strange, bizarre, odd, curious, quirky, irrational and totally mystifying breach of logic, the word "borrow" survived as a metaphor.

This is almost as illogical as ma and Pa Kettle's explanation of 25 divided by 5 equaling 14.

Math Around the World

Every semester I invite a group of International students studying at St. Mike's to come to my math education class to be interviewed by my students. The purpose of the activity is for my students to realize that math is not the same the world over and that they will be faced with procedural math problems that they might not, at first, understand.

This semester we met with a group of students from China, Japan, Congo, Saudi Arabia and Spain. Using the Multicultural math interview i have developed over the years for my research my students got to know the math of a variety of countries from around the world. There was much laughter from both my students and the International students when they showed each how they did, for example,  subtraction problems or multiplication problems. They were also fascinated by the numbers that have significance in each culture. The international students from China had no idea that 13 was unlucky in the the US and the US students had no idea that 4 was an unlucky number in most Asian languages because the word for four sounds like the word for death.

One of the great bi-products of this experience is that the international students get to meet more SMC students on campus which helps them feel more at home as they go about their studies. I was once an international student, in 1977, and know what it's like to be i a new land with new methods, social expectations and, yes, a different language. It was George Bernard Shaw who said that the UK and US are two countries separated by a common language. You should have seen the look on the students faces when I asked where the rubber was so I could erase the chalk board!

Math Fact Fuency

Nicky Morgan, the British Minister or Education, recently announced that "all pupils must know, by heart, their times tables up to 12 x 12". Apart from the use of the archaic term "times tables", which one would expect from a Tory, the whole issue of fact fluency, to give it its current term, is a really interesting one.

I have always believed that fact recall, but up to 10 x 10, is an important part of the mathematization process children go through. It's the math equivalent of being able to spell words, but the facts are by no means the "basics" of mathematics. The basics are everything that is included in the field of numeracy; being able to count, to recognize number patterns, to subitize, to see numerical relationships and so on. Remembering the math facts makes math easier and more efficient.

Memory, remembering things, is a crucial part of education, but it is pretty useless when we memorize things with absolutely no understanding of what we are memorizing. Memorize this list of words; Arun, Ouse, Rother, Stour, Medway, Darnet, Mole and Wey. Now use any of these words during a conversation you have with someone  over the next few days.

If we are going to require students to remember their math facts they must understand what they mean. The multiplication facts for example can mean 'groups of' as in 5 groups of 4 people are 20 people. They can mean area as in a carpet 5 yards by 4 yards has an area of 20 square yards. They can also mean the muliplicative comparison as in "I have 20 Hotwheel cars which is 4 times as many as you have if you have 5".  Each of these concepts of multiplication is different but each can be solved with recalling the fact 4 x 5. Or is  it 5 x 4?

Talking of which, when you see the fact written 4 x 5 do you read it as 4 groups of 5, or four 5 times? I asked my grad class this the other day and half saw it one way and half the other way.

For a great read on the topic of fluency read Jo Boalers incredible article Fluency Without Fear which includes a very relevant criticism of EngageNY's approach to fluency.

The names above, by the way, are the Rivers that flow out of Southeast England.