Reverse Engineering the Basic Operations
Deconstructing the learning sequence needed to add
An adult is able to read 4+6= and immediately know both the question and the answer. The adult writes 10. Why? How? Let’s reverse engineer the thinking steps involved so that we can better understand how to help our students learn.
Decode to words
Like reading any sentence, we must decode the symbols into words. 4 becomes “four”. + becomes “plus”. This is one strong reason I encourage good reading skills before starting math.
Mathematically comprehend the words
It’s one thing to say “four”, but now we need to translate it to “four-ness”. Four is a quantity and can be represented by four things: ||||. Six becomes ||||||.
Plus is an operation which tells us to take the four objects and put them together with the six objects to become a bigger group of objects. ||||+|||||||=||||||||||
We can now reverse the process. |||||||||| becomes “ten”, which is 10, the answer.
Quantity and operation
This all pre-supposes an understand of “quantity”, which assumes the ability to count. Four is not only ||||, but it is also ****, and also @@@@, and zzzz, and so forth. A student could memorize sticks and numbers and appear to add, but they would just be spelling in a different language without understanding. Adding is counting, then counting some more. Two quantities become a bigger single quantity. This also goes beyond reading.
Why?
Now we must answer the (probably unspoken) question of the student, Why do I want to add little lines printed on a paper? Without motivation and relevance, learning cannot take place, no matter how finely detailed we breakdown the steps.
Our steps so far have involved abstract symbols on paper, but what are we really trying to teach? Is it not the point that our abstract symbols are general representations of many specific scenarios in real life? When we put sticks together on paper, we are representing placing sticks on the firewood stack on the side of the house, or putting books on the library shelf, or pulling dollar bills out of the wallet to pay for something.
Each one of those operations can be performed until a specified space is filled or height is reached. No counting would be required. Therefore, no concept of quantity would be needed. However we don’t want to draw lines for our kids all of their lives. Teaching them to count, then to add, helps them generalize a skill then apply it to many specific cases. We ask for ten dollars or ten sticks, then they figure it out, which saves us the effort of drawing lines and empowers them to solve many real life problems and helps them communicate more effectively.
What I have shown so far is why we want our children to add, but do they want to add? Are they asking the question, or are we asking it for them?
Comparing vs. counting
Very young children do not comprehend counting nor care to. They just want the same pile of candy in their hands as brother or sister. They want their ice cream cone as tall as the other persons. Loose comparisons of bigger, smaller, equal are good enough for them. Precise counting is a mysterious burden of details.
Yet, general comparison of real things is a step along the way to specific counting of abstract amounts.
Read that sentence again and notice the metamorphoses involved.
General comparison of real things is a step along the way to specific counting of abstract amounts.
Children know things. They only care about things. The quantity of things is an unknown mystery to them. They feel bigger versus smaller, but they don’t quantify it by counting. We adults, who have passed through the period of adolescent rule-keeping and enforcement with nit-picky justice for all, care about precision, but not children. When they become old enough to care, then they are old enough to learn.
What do we have so far?
Verbal comparison of objects precedes verbal counting of objects precedes adding of objects precedes symbolic counting precedes symbolic adding.
Is that sequence exact and ironclad? Of course not. It follows the general developmental stages of infant to child to youth, but we all know someone who has taken their own path. For example, adding of objects could easily be switched with symbolic counting. Yet, the dissection is helpful for parents and teachers. Our real problems begin when we try to enforce the sequence on kids according to age, rather than according to their capacity to care, to ask why and how.
You might think we are done at this point, but we actually need a step between the concrete object phase and the symbolic operational. Pictures help bridge the gap by being part real and part abstract. Why do we need them? How else can we show 2 subtract 5?
Pictures
2+5 is easy to perform with blocks, so is 5–2. However, how do you take 5 blocks away from 2 blocks? How do you hold –3 blocks in your hands? No matter what gimmicks or tricks you might use, abstraction is required.
The child dispenses with both the question and answer by saying, That’s ridiculous! Yet we know that in their future that general question is very practical when it comes to temperature, elevation, money, and other things. What we can’t produce in reality, we can draw in the virtual world. Pictures can demonstrate positive blocks and negative (ghost) blocks. Also, a number line can show it. It can show anything.
Number lines
You can draw 2 green blocks and 5 red blocks, then explain that red and green cancel each other, leaving 3 red/negative blocks. Or you can draw a number line with arrows that go up and down. 2 minus 5 gives an answer in negative territory.
We need not wait to introduce the number line until we introduce negative numbers. It can be used with regular adding to help establish the model in the eye and mind of the student.
Also, number lines and other illustrations make it easier to shift from the relatively more expensive, and storage space requirements of, objects to workbooks. 1,000 dots on a page can represents 1,000 beach balls with considerably less space!
One caution is that pictures in books or on computer screens can be overused and trap the student in the virtual world. This sends the message that math is only useful in the virtual world.
Summary
Objects to pictures to symbols; general comparisons to precise counting; real to abstract; this is the sequence in which we all learn in our own time, if we are motivated. Montessori and Singapore math methods work with this sequence with great results. Observe your student, analyze your methods, and you too can be a more effective math teacher!