Tuesday, March 3, 2015

Why does the use of "borrow" persist in subtraction?

Sometimes it feels like our goal of improving math education  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. At least 75% of the students I work with use the term "borrow" when completing subtraction algorithms.

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.

Wednesday, February 11, 2015

Difficult Conversations 101

Marshall Memo 573 
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            In this New York Times article, Alina Tugend says many people aren’t clear what is meant by feedback and tend to conflate three quite different things:
-    Appreciation – That was especially effective; What you do matters;
-    Coaching – Helping someone get better through advice and mentoring;
-    Evaluation – Being rated or ranked against a set of standards.
“When we say we want feedback,” says Tugend, “most of us desire appreciation, dread evaluation, and forget about the most important part, which is the coaching.” Evaluation doesn’t happen very frequently, but people should be getting appreciation and coaching on a regular basis. 
            The problem with evaluative feedback is that it tends to blot out other messages. “It’s like getting a paper back in high school,” says author/Harvard Law School lecturer Sheila Heen. “The first thing you do is look at the grade, not the comments on the paper. The evaluation tends to drown out everything else… If you’re clear you’re coaching and not evaluating, it lowers the stakes.”
            Many managers aren’t comfortable delivering criticism and put off speaking up about concerns they have. “I don’t think there’s any way to get rid of the pain or defensiveness,” says Heen. “But we can get better at understanding our reactions so we can move on faster to finding the value.” Graham Bradley, a psychology professor at Griffith University in Australia, has the following advice for such conversations:
-    Prepare thoroughly.
-    Give specifics and express exactly what behavior you want to address.
-    Allow the other person to speak, listen carefully and empathetically, and seek to understand the full context.
-    Provide direction and goals to change the behavior.
-    Ensure that your message is received and the employee takes responsibility.
-    Keep a record of the conversation.
A key point is making sure that at the end of the conversation, you are on the same page with the employee. When wrapping up, says Nancy Alarcon of the University of Washington, “I try to revisit some key points and write them down. I also say, ‘Why don’t you recast that for me one more time,’ and ‘What are we going to have to work on next or what do we have to do to effect the change needed?’”

            Tugend also suggests that employees regularly solicit feedback from their superiors. “Most of us would rather float along with the idea that no news is good news,” she says. “The trouble is, it may turn out that no news is bad news, but we won’t know that until it’s too late.” Don’t ask your boss global questions like, “How am I doing?” or “I was wondering if you have any thoughts about my work.” Instead, ask, “What’s one thing I’m doing or failing to do that’s getting in my way?” or “What specific change could I make to improve my work?”

Monday, February 2, 2015

Understanding Math and the Common Core

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.    

CCSS Math Practices

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. 

Math fact Fluency

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 it's 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 math 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. 

Wednesday, January 21, 2015

Mathematization 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 materials.

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 student 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, of a mathematical relationship from an idea or concept. 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. This is what the Common Core State Standards for Mathematics is trying to achieve.

Thursday, January 8, 2015

Happy New Year!


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