No Cursive; no hand calculations?

I recently read this neat Huffington Post piece about the demise of cursive handwriting. Well, demise maybe premature as the conclusion in the article was that cursive writing now belongs as part of the elementary school art curriculum.

So, if we can cast aside the century-old idea of creative handwriting as a mainstay of the elementary school curriculum then it's time we cast out the four algorithms, hand calculations, used to perform addition, subtraction, multiplication and division. I hasten to add that I am referring to the vertical form of algorithm where you put one number above the other with a line below and follow a procedure. We must, of course, still teach the concepts of addition, subtraction, multiplication and division in all their forms but it's time we used technology to do the tedious part, so to speak. Learning the algorithmic procedure does nothing to enhance students' mathematical skills. But, there it is in the Common Core Math Standards (CCMS) due to be implemented in 2014. 
 
Conrad Wolfram suggests we should be using computers to take the drudgery out of hand calculating in maths by allowing students to use calculators and computers to perform these tedious hand calculations. 

I was beginning to go along with this idea until I read this piece by Karen Fuson and Sybilla Beckman about the inclusion  of the "standard algorithm" in the CCMSs. Their argument is persuasive and cites the practice students gain in developing their understanding of place value and base ten when they are involved in the completion of hand calculations. It is part of the process of mathematization that students go through when they are learning elementary school maths. The use of manipulative materials and technology are important in developing ideas but it is what happens in the student's mind that is the most important aspect of learning math; the ability to abstract and manipulate ideas mentally.

Standardized Testing out of Control

As I was watching the Harry Potter movie, The Order of the Phoenix, with my son Andrew this afternoon it occurred to me that a passage from this movie is the perfect allegory for what continues to happen to education in the US.

Ever since the implementation of NCLB we have become slaves to the insidious blight upon our education system that comprises our constant need to test students from their earliest school experiences. Why do we do this? Why do we place such  incredible value on a momentary glimpse of what a student knows, usually through some spurious paper and pencil medium.

The Harry Potter movie made me think of this because when Professor Umbridge replaces Dumbledor as Head of Hogwarts she implements strict testing procedures for all the students. Learning goes out of the window and is replaced by teaching for the test in just the same way that "high stakes" testing occurs in all schools now in the US. Schools are defined as failing schools based on test scores on tests  which measure the narrowest, and often the least worthwhile, aspects of learning.

I am not the only one feeling this way. An AP item in the local press today describes how more parents are opting their kids out of standardized testing. 

There are so many reasons why standardized testing is so innocuous for just about everyone directly involved. It tends to only measure those things which are easy to "measure" such as recall and recognition. There are few tests that can effectively measure understanding and sense making.   The worst thing, however, is when it is used to measure teacher performance. This opens up the whole education system to all kinds of deviousness such as the Atlanta debacle of several months ago.

Of course we have to assess students to find out what they know and understand but there are so many better ways of doing this that through "easy to administer and score" tests. A I read the paper this morning it occurred to me that for my entire professional life, almost 45 years, society has been generally displeased with teachers and the field of education. Why is that? No other aspect of human endeavor has had to put up with such constant confrontation or complaint about lack of performance; not doctors, nor lawyers, nor automakers, nor plumbers, nor electricians, nor librarians, nor bakers nor candlestick makers.

What does our culture really want from the education system?      
  

Thinking Fractions

If you think about what you are doing when working with fractions everything becomes crystal clear. The worst thing to do is to visualize the algorithm you learned in elementary school; something like 1/2 + 2/3 or 1/4 x 4/5 or, heaven forbid, 1/2 divided by 1/4. If we do this then we are confined to thinking instrumentally procedurally about something that is comprised of only procedural knowledge. We fall back on senseless rules like "find the common denominator" or "cross multiply" or "change the sign and flip the second fraction". learning tricks like this will probably get you a better score on a traditional math test (like those used to compile the TIMMS report)  but will do little to help you understand what fractions are all about.

So for 1/2 + 2/3 think about different ways you could make 1/2. You could say it's the same as 2/4, 3/6, 4/8 and so on. Now do the same with 2/3. It could be 4/6. Aha, no need to go any further if you know that you can add fractions that have the same denominator. So 3 sixths plus 4 sixths is 7 sixths. We can count sixths just like anything else. Now you can see how you can get sixths by multiplying 2 by 3 if you need to do more difficult ones. But at least you now understand why.

1/4 x 4/5 is more challenging because the x (multiplication symbol) and the whole concept of multiplication of fractions can be different from using it with whole numbers. Visualize 4/5 using the model in the picture above. Now take 1/4 of that 4/5 and you end up with 1/5. The key to understanding this is seeing how the size of the 1 to which each fraction refers changes. The 1 of the 4/5 is the red 1 above whereas the 1 being referred to by the 1/4 is the 4/5. Really we're asking what is 1/4 of 4/5? Once you start to see the pattern, the relationship between the numerator and denominator then things get easier. Try 1/3 x 3/5 or 1/2 x 2/7 or 1/4 x 4/9. If there is not relationship between the numerator and denominator you can do it another way. For example 1/2 x 3/8 is 3/16. This can be done conceptually by halving the size of the fractional pieces. 1/16 is half of 1/8 so 2/16 is 1/2 of 3/8.

1/2 divided by 1/4 is really asking how many 1/4s are there in a 1/2. This is easily conceptualized by thinking about a football game; how many 1/4s are there in the first 1/2? Clearly there are 2.

3/8 divided by 1/4 is a little more difficult because the referent of the fractions changes. Again, think how many 1/4s are there in 3/8 by visualizing the fractional pieces. Compare 1/4 (a yellow piece above) with 3/8 (three dark blue pieces and you'll see there are 1 1/2 quarters in 3/8. The referent for the 1 1/2 is the 1/4.

from this conceptual understanding it's easier to "mathematize", to use Bob Wright's term,  what is going on by working out how the procedural knowledge of operating with fractions works. 

Simple Algorithms are a Waste of Time?

As I start my sabbatical semester  I seem to be           generating more questions than answers as I explore ways in which children  learned math before immigrating to these shores. My goal is to interview as many recently arrived people as I can from Nepal, Vietnam, Bosnia and Somalia, the four countries  most widely represented in the Burlington, Vermont, school district. 

The more I learn about the maths in different countries the more amazed I am that we have assumed for so long that math is the same the world over. But that's another story.

The question I have been wrestling with for quite some time now is why we still teach the four basic algorithms as illustrated above. The procedures involved in teaching all four algorithms to the level expected by sixth grade still consumes a large part of the elementary school curriculum even though almost everyone these days has a personal electronic device that will do the calculation in a nanosecond. The saddest part of all is that so many students who are able to complete the procedures have little idea about when to use them and even worse, many forget how to do it because, in real life, they never have to do it. When was the last time you worked out with a pencil 398 divided by 72? 

Above is a traditional addition algorithm although for some inexplicable reason the smaller number is on top. (remember how you were told to put the largest one on top as you would need to do this in subtraction?) The example above also has the wonderfully anachronistic "carry" as a command of what to do with the ten ones which magically become just 1 (interestingly ten 1s become 1 ten which changes the word 'ten from a noun to an adjective - the linguistic essence of regrouping).

So think about those two numbers 15 and 29. In what relationship could they be?

How about two groups of people, one of 15 and one of 29 being joined together into one group. Maybe combining two college classes because a professor is unwell.

How about a large gathering of people, such as a school staff, in which you know there are 29 women and 15 men and you want to know the total number of people. There are no separate groups; this is what we call part-part-whole.

How about two bags of candies? One bag contains 29 candies and the other bag contains 15 more than the first bag? This time we start with two groups and one of the numbers is not a group at all but the difference. Or one bag contains 15 candies and the other contains 29 more?

How about wanting to know  how many  free tickets you started with when you have given 15 away and now only have 29. You can even use addition to find the answers to problems that solve separation.

We could so easily set up the relationship like this; 15 + 29 = ? This would help children develop the real sense of a numerical equation along with the most important thing in algebra; the idea that the = sign means "is the same as".

If only we could spend  our time developing children's  number sense and their ability to recognize  mathematical relationships then use a calculator for the drudgery part instead of the endless hours we waste with mindless, boring and tedious  algorithms.