Tuesday, March 3, 2015

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.

Tuesday, February 17, 2015

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!

Sunday, February 1, 2015

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.

Monday, January 26, 2015

The Math Practice Standards

I've read a lot recently about the math practice standards as described in the Common Core State Standards and have come to the conclusion that if the CCSSM content standards are the 'what' of teaching math the the CCSSM practice standards are the 'how' of teaching math. In other words, teacher must know the math practice standards so well that they become an integral part of the teacher's interaction with the student.

Any behavioral changes we need to implement in our interactions with students takes time but I can feel myself slowly changing when I interact with my students during my math  education classes. My focus is also changing when I observe my student teachers teach math. This morning, for example, I observed a student teaching third graders all about multiplication facts She did a great job drawing the students' attention to the structure made by the facts on the multiplication table, a 10 x 10 square. She also had the students identify the regularity of counting by 4s or 5s. And, if this wasn't enough, she also gave the students some problems to model mathematically to demonstrate their understanding of the multiplication facts.

The key to the implementation of the math practice standards has to be in how we interact with students mathematically; what we ask them to do and how we ask them to do it. The activities themselves are not going to do this. We need to consistently and conscientiously ask, require, tell, suggest, model, demonstrate, and use any other appropriate verb that will, over time, help students integrate the math practice standards into their lives.

They really are dispositions by which to live our mathematical lives. 

Thursday, January 22, 2015

Understanding and the CCSSM

Every semester, at this time, I introduce my students to the idea of understanding maths. It's not an easy task especially for those graduate students who may not have experienced learning elementary school maths for many years. Even with some of my undergrads the process of having to understand something they have known by rote, "off by heart",  for some times seems like a pointless task.  But, we have to understand the maths we are planning to teach so we go through the topic carefully to make sure everyone is on board.

In every field of study or human endeavor there are giants; people whose ideas, or deeds, have stood the test of time and are as fresh and relevant today as they were back then. One  such person in math education is Richard Skemp.

In 1976, Richard Skemp, a British educator published a definitive article in the journal, Mathematics Teaching, in which he described two types of understanding of mathematics; instrumental understanding; "rules without reason", "rote learning", "pure memorization",  and relational understanding; or understanding the what, why, how, when, connected with,  and so on; the type of understanding to be found in the CCSSM. Having found that my students have a hard time remembering the terms I now call them fragile understanding and robust understanding which always seems somewhat onomatopoeic.   

The wonderful thing about robust understanding is that it not only makes problem solving so much easier, it makes the retention and recall of all those illuminating but tedious  math facts children have to learn infinitely more efficient and effective. If, for example you can relate the multiplication facts to the addition facts they suddenly have a new structure that gives them sense and aids in meaningful recall. 6 x 5 for example is 5 x 5 plus another 5. Then if you can count by multiples of  5 the mental structures you are creating makes forgetting almost impossible.

We all learn that Pi is  3.14 and just about everyone knows that it goes on forever without establishing a repeating pattern. Every high school hallway is adorned with it on pi-day on March 14. (July 22 in the UK).  But ask anyone what it means and very few can tell you that it's a ratio between the diameter and circumference of a circle. Every circle is just over 3 times further around the outside than across the center.

This is what the CCSSM are designed to do.    

Wednesday, January 21, 2015

Mathematize and the CCSSM

Some time ago when I was learning about the Math Recovery program and reading the wonderful books co-authored by Bob Wright I came across the verb 'mathematize' and its noun counterpart "mathematization". I always thought it was a wonderful way of describing what math education at the elementary school is all about.

In his book, Developing Number Knowledge,  Wright defines the term (p15) this way;
                  Mathematization means bringing a more mathematical approach
                  to some activity. For example, when a student pushes some
                  counters aside and solves an addition  task without them,
                  we say they are mathematizing, since it is mathematically 
                  important to reason about relations independent of concrete

Others define it as "reduction to mathematical form" (Merriam Webster), "to treat or regard mathematically" (The Free Dictionary) and "explaining mathematically" the Collins dictionary.

The really, really interesting thing about all these definitions is the idea of reduction or movement from real life, concrete situations such as that described by Wright, to the symbolic form of symbols and algorithms typically used in math. This is completely opposite to the way math has traditionally been taught and  how it is sadly still taught in poorly taught math classes.

A classic example occurred in my math class yesterday when I asked students what 1/2 ÷ 1/4 meant. No one knew. My hunch is that if you randomly asked 100 people on the street only a handful would be able to tell you that this meant how many quarters are in a half. What really makes this intriguing is that most people would tell you to change the sign, flip the second fraction, multiply and get the answer 2. 

In other words people have not gone through the process of  mathematization when they have learned this procedure. A real world, concrete idea has not been "reduced to a mathematical form". They learned the mathematical procedure without any sense of what it meant or connection to any concept or relationship. There was no derivation, if you like, from a concrete experience or idea  to a symbolic,  mathematical relationship. This happens all the time in math.

Students are taught a square number is the result of "a number times itself" instead of a number that makes a square.

They are taught a prime number is "a number divisible only by 1 and itself" instead of a number that can only make one rectangle (e.g 1 x 7 or 1 x 13).

Students are taught the symbolic mathematics first and not the idea so they cannot be mathematized. Instead, we should be mathematizing them from concrete experience  to symbolic representation. This is what the Common Core State Standards for Mathematics is trying to achieve.

Saturday, January 17, 2015

Confidence; the Forgotten CCSS Math Practice Standard

One of the interesting things about the Common Core State Standards for mathematics is that nowhere does the word 'confidence' appear. Yet it is, perhaps, the one thing that will help young children, and many adults too, develop their understanding and skills in the field of maths. All eight math practice standards, in some way, imply the development of confidence.

I'm always very aware of this concept at the beginning of the semester when I'm getting to know my new students. There are also activities I do in which I ask the students how confident they are in their responses. For example, we use a math manipulative called Unifix cubes which are colored plastic blocks that fit together. As they explore the materials I ask students to describe them so that they can get to know how to use them. I ask them how many different colors there are After a minute or two someone will say "ten" and I'll say "how confident are you". This seems to stop them in their tracks as if I am challenging their counting skills. They start looking through the box of cubes again to see if there are any more and after a few minutes they'll say they are sure there are ten.I then ask "on a scale of 1-10 how sure are you? The usual response is 9 or 9.5.
I then ask, How sure are you the sun will rise tomorrow morning?  "Aha 10" they say. So why not a 10 with the number of different colored Unifix cubes?  For some reason math seems to promote this lack of complete confidence.

I think confidence in maths comes from a variety of sources but primarily I think it comes from how well supported are the things we know and understand. If we can see something as part of a pattern, or part of a system, or part of a family, or part of a schema, or part of a series, or part of a predictable structure,  or related to something else we know or understand then our confidence ill be increased.
If we know or understand something in isolation, without these connections, then we have far less confidence.

So, taken together, the Common Core State Standards for Math Practices add up to the development  of a student's confidence on the content standards they are learning.