20160417 UNEQUAL SIGN:

How teaching impedes learning.

Talking to Kids

Watching the victims of education as a parent and remembering me being one of them, I realized that transition from arithmetic to algebra (as a school subject) is greatly complicated by the equal sign. If educators had any interest in learning, they would have never done such thing.

Elementary education teaches children that = means "go figure". Electronic calculator designers carved this meaning in plastic.

My last student, the little girl TLG, is a kindergartener and a native American English speaker. Five days a week, I hear her saying thing like

(a1 + a2)×M

equals

a1×M + a2×M

Not being a native American English speaker and a kindergartener, I always say is equal to.

I don't correct TLG because I want her to pass for a local at school. Wait a moment… thank you! I just checked with the online dictionaries. Only Cambridge explicitly had the elementary school meaning. The other five allowed to choose between "to be equal to" and "to make or produce something equal to". Well, as long as TLG uses the first definition where appropriate (and I see she does), her English is OK.

Mathematical expressions are stronger than words. Looking up online resources for first graders, I found tons of single digit additions in the classical form:

7 + 5 =

If = means "is equal to", they make no sense because 7+5 is not equal to nothing.

At home, TLG has never practiced single digit additions. A true stereo learner, she learned long addition first, and started adding 10 digit numbers in standard decimal Hindu-Arabic notation before she turned five. Yet I've seen the damage caused by "go figure" meaning in every kid I taught before.

Instructions like "find a mystery number X such that X=7+5" would be better, but if we want them to add, why don't we just say "add this numbers"? And let the student answer: =12. I think, the sum is equal to 12.

Many websites are using columnar notation for single digit additions. This might help to pass the tests. Have you seen this method elsewhere?

The equal sign was introduced by Welsh mathematician and the first British algebraist Robert Recorde in his book The Whetstone of Witte published in 1557. Recorde stated his intentions very clearly: He was going to use = instead of writing "is equal to" every time. Before Recorde, people were using a different shorthand.

Equality is a relation. I prefer to explain it as a statement, which, like any statement, can be true or false. "Is equal to" really means "are equal" because if a=b then b=a.

Talking to Machines

Learning the artificial languages created to tell computers what to do, children face new challenges. Instead of familiar "go figure", in computer programming they find assignment.

Using a simple calculator, we press = key without thinking where the figured out value must go. It always goes to the display (which means, to its memory, because the display only shows what its memory holds).

In computers, the value goes to memory. It's huge and invisible, but numbered. To keep it manageable, computer programmers use variable names. Naming places in memory is a service, which every programming language offers nowadays. They maintain a table of variable names and memory locations.

Assignment makes the value in memory available through a variable name. In programing language, you may see:

a = 7

Which means, assign 7 to the variable name a (or make it point to 7).

Then the value of 5 may be assigned to the variable name b

b = 5

The third variable name c may receive the value of 12

c = a + b

How? OK, computer will replace the variable names with the values assigned to them and calculate the sum. In other words, the right side of so-called assignment is "go figure", scientifically known as evaluation.

Roughly speaking, there are two kinds of programming languages: one for the egg-heads, another one for those who learned their math in elementary school. The second kind is prospering.

You can instantly tell the languages of the second kind because they use = for assignment. Obviously, their creators wanted to be elementary school friendly, but they scare fellow graduates off talking about assignment instead of familiar "go figure".

Elementary school friendly computer languages do have the ways to express "is equal to". Usually it's == (two equal signs one after another). The trouble is, they treat them as questions. Expression like

a == b

are getting evaluated on par with arithmetical expressions. Computer languages answer the questions with boolean values true or false. Some even understand === for more specific purposes. Well, Robert Recorde drew his equal sign many times longer than we do today.

This leads me to Boolean algebra, but I am not going to follow. Boolean algebra is very accessible - a child of any age can understand the basics - and very useful. It fakes our thinking and makes it better. It's just a different topic.

Are Kids Machines?

When elementary school students meet variables, they demand the numerical values, and it's not out of stupidity. Kids simply do what they were trained to do. They said "go figure", but I need the figures for this!

Remarkably, that's how the general purpose programming languages react to variables names with no values to substitute. A program handling your input can ruthlessly points out that the variable is unknown, or its value have never been assigned.

Elementary school teaches human kids to act like simple computer programs, but kids have human emotions and attitudes. Algebra makes them suffer. Mandatory education does not care about students' sufferings. The business flourishes on their tears.

A teaching parent can easily prevent and counter such computerization. It's utterly important not to keep kids on steady diet of numbers. Young children, who learn to name things, easily pick up the idea of a variable. Optionally, it can be introduced through colors. From time to time, your student will scold you for teaching the stuff that normal American kids are not allowed to learn. Be prepared and don't give in.

Teaching, especially teaching a developing brain, is not enough. Every useful piece of knowledge must be maintained, and variables are very useful. Nine months ago, I went through a very difficult period. TLG got a live bird, and she forgot long subtraction. I re-taught it as a part of her mechanical "go figure" skills, but then I wanted her to learn to use it. Suppose you ate 223 candies out of 545. How many are left? No clue.

As usual, generalization helped. I told her that every known addition implies two other additions. If we know that any of the following statements is true, the other two are true too:

c = a + b
b = c - a
a = c - b

She knew this for years - for two years at least - but only with small numbers. Understandably, this informal knowledge was not even scalable. When she understood the abstract pattern, she started solving the problems with big numbers and without numbers at all.

Teaching equations wold be too distractive. I only wanted TLG to learn and remember how the quantities are related. Here they are in the stick notation:

Next are the stakes:

Soon TLG performed one of her routine long subtraction:

 4456028506
-3548862305
  907166201

I asked how she could verify the result. She said she could add the third number to the second. This was not new. I asked how else she could do it. To my astonishment, she suggested to subtract the third number from the first. With 10 digit, she could only infer this from the rules.

It was encouraging, but not enough to pass for a human by modern standards. Computer programs can do such things and more. Still, with "go figure" even such reasoning would be impossible. If I told a "go figure" student that

c = a + b

she or he would rightfully ask me what a and b were equal to… what I am talking about: It's not a columnar notation.

In 1st grade, the teacher caught TLG's big sister doing long addition. She learned the lesson, and did not reveal her ability to do long subtraction and multiplication prematurely. In 2nd grade, however, she was caught verifying small subtractions with additions. To her, it was natural because at home she used the chart of addition both ways. I quizzed her: What's five plus seven? Twelve minus five? Seven plus five? Five minus seven? Seven minus twelve, etc.

The teacher did not find this natural at all. She said she's never seen such behavior. Verifying subtraction with addition was another secret recipe, which school did not disclose yet. Her students were not supposed to understand it.

20160404 STICK AND STAKE:

Measuring as counting.

When I mentioned sticks and stakes in MINUS, MINUS, MINUS-MINUS, I knew I had to write an article about them. Mostly, to explain them better, but also to provide an entry point because they are very useful. This said, please, read the old chapter first.

To numeralize space, we tile the distances with a smaller distance called unit of measurement, and we count the tilings. The actual unit can be only one. We can move it forward, applying its beginning to where its end was.

While doing this, we count in one of two ways. First, we can count the units - the measuring sticks - like I did below.

This is what I call the stick model. It works best, if we can view the whole scene of measurement at once. This means, we must be big enough or soar high above to see the beginnings and the ends of the sticks.

Another way to count emerges where the distances are big, and we are small. The first example of it could have been the milestones of Ancient Rome. Numbering the middles of the miles would not be very practical. The milestones marked the beginnings and the ends of the miles. Standing at a milestone, the travelers knew how far away from Rome they were. I imagine a measuring rope with two stakes. The holes left where the stakes were stuck in the ground are points, and we count them. That's what I call the stake model of distance.

I don't supply the picture for the stake model. It's just a number line. I drew plenty of them for MINUS, MINUS, MINUS-MINUS.

On a small scale, the stake model is awfully problematic because we see too much. Counting the distances from zero, kids must exclude the zero point. For those who are not trained in sticks and stakes, zero is a point like any other. Why don't we count it then? Because zero is nothing? OK, but if we count through zero, we count zero point and exclude some other point, at which we begin.

On a line, we always have one extra point. If we join the endpoints of a line segment to make a contour, they become a single point. Think about a clock face. Oh, no, don't think about a clock face. The English AM/PM system is insane.

The intuition behind the sticks is much more solid. Fortunately, the pieces of paper, on which teachers offer their problems, are always smaller than the students, so the stick model is the best. Like probably every parent, I tell my kids to convert stakes to sticks. If in doubt, draw arcs and count them.

I can imagine an overly smart child drawing arcs between the middles of the sticks and counting them. The rule was explained in MINUS, MINUS, MINUS-MINUS: only distances count. It's a metric space.

In the stick model, there is no zero stick. Zero means no sticks. There was no zero in Ancient Rome too, but they did have so-called Golden Milestone, from which every road was supposed to be measured.

A good method to make stakes palatable involves another heresy: a unary operation of increment. Vitruvius described (but not invented) an odometer, a wheel-based mechanism dropping a pebble preloaded in a hole once a mile. In Ancient Rome, they called pebbles calculi, so the distances could be calculated.

Starting from empty (which was exactly 0), the system would increment the number of the calculi in the receiving compartment. One thought of such a machine running along the number line must be enough to answer every question and obliterate any doubt. Anybody knows where to buy a cheap plastic rendition by any chance? OK then, teach to use increment rather than addition. It's even cheaper.

Drawing one-dimensional sticks and stakes, we have an extra dimension for numbers and letters. In two dimensions, sticks become squares, and we can place our marks inside them. Such squares are handy for measuring areas.

Two-dimensional stakes remain points, and we often use grids to highlight their spatial order. The third dimension is not readily available for drawing. This tempts us to draw points (stakes) as squares (sticks).

A chart of addition on this animated page from STEREOLEARNING is made out of squares. The numbers are inside them. The chart looks like a stick model, but it isn't. Invisible stakes are in the centers of the squares. Zero is in place, and a navigator must skip the first number when counting.

Such transformations are frequent and difficult to understand. Many tables have their actual grids passing through the cells and intersecting inside them.

We must specifically train our kids to understand sticks and stakes. Allow me to give you one more reason why: The chart of multiplication available through this link is a stick model. There is no zero, and we count from the first number.

I don't explicitely teach TLG sticks and stakes yet, but I point out the differences and explain them. From time to time I even offer her problems. Occasionally, I still see her making mistakes trying to use a times table as an addition table.