Counting and Why It Matters
There are lessons I learned from Crewton Ramone and Jerry Mortensen that will stick with me forever; chief among them is counting. Crewton talks about the confusion that happens when both teachers and students don't understand what we are counting. This seems to be a similar finding of Lipping Ma in Knowing and Teaching Elementary Mathematics. From Jerry Mortensen:
- Math is the study of numbers.
- All we can do with numbers is count.
Lesson learned: Know what you are counting.
I was going to blog about this anyway, as it is foundational to understanding how we write things and how we talk about them, but then Twitter happened this week. Being new to twitter, I'm not a good tweeter nor conversation follower. They use shorthand I don't understand. But I did understand this:
We should not teach "keyword" or "clue word" strategies in math.
— Tracy Johnston Zager (@TracyZager) March 1, 2016
My first reaction was to retweet it with this snarky comment attached, "How about just teaching them math?" Or something of the sort. I am unsure how this retweet etiquette works so I promptly deleted the tweet and decided to write this blog post instead. Tracy had the same idea, which is why she tweeted it. So, instead of a rant let's talk about counting and language.
Math is a Language
Math is the language we use when we are talking about quantity. Most the of the confusion we experience, for both parents and students, is that we don't understand the language. We don't understand its symbols or what they represent, or we have only a vague notion of what they mean. That can be remedied. That is also something I learned from Jerry Mortensen and Crewton Ramone. This is not hard stuff. Every parent can become fluent in mathematics and so can every student. It is the job of the teacher (homeschool parents are the teacher) to make that happen; to provide a math rich environment where a child can be both taught by direct instruction when needed, and left to inquire and explore. Even if your math experience has been fear and tears, you can provide the kind of instruction that produces a math literate student.
Counting: Not as Easy as It Sounds
In a previous post series, I discussed the relationships found in the blocks. We examined the relationships you can "see" and what you can teach your children to "see" in the blocks. I got quite a bit of positive feedback from parents. But it's clear we should step back even further and discuss counting in general.
How we write math and the words we use to describe math depends on what we are counting and what we want to know about what we are counting. Sometimes we are counting a thing, sometimes we are describing a relationship. Once you understand the language, presenting it to your students is super easy.
At some point you say, "This looks like we are repeating the same thing over and over." That might be because it is the same stuff over and over, we just change the numbers.
What Are We Counting Anyway?
In the above image, what are we counting? We don't know. It could be a half a dozen different things. But, the word to the right of the image mentions addition, so we will count red rods. What do we want to know about what we are counting? We want to know which block makes same with one red rod plus another red rod. Answer: p
You'll notice that following images are exactly the same. It is all same stuff. What we call it and how we write it all depends on what we are counting and/or what relationships we are describing. It depends on the information we want to know.
In the above image, I moved one of the red blocks slightly below the other for demonstration purposes. I would do the same for a newer/younger student as well. This time, we want to know the difference between the red and the purple block. If we have the purple block and someone took a red what would be left? Or if we had a red block and we needed to get to the purple block how much more would we need? Either way, we need to count the difference between the two blocks.
We are counting r's again. If we have 2 of the red kind or 2r which block does that make same with? Answer: p.
For division, we are counting red rods again. However, we are not looking to know the same thing about the red rods as we did with multiplication. What do we want to know? We want to know many red rods are contained in a purple rod. Answer: 2
Fractions are interesting little buggers when it comes to counting. If we multiply fractions the number seems to get smaller. When we divide with fractions, the numbers seem to get bigger. This causes students much consternation and confusion. No doubt the same is true for parents.
Let's say the purple rod = 10. We can mentally cut it in half we get 5. That seems an awful lot like division. But we write that as ½ × 10 = 5. Similarly, ½ × p = r. Now let's say we want to divide 10 by ½. We write that as 10 ÷ ½ and our answer is 20. At first glance, that seems an awful lot like multiplication.
So let's try this instead. I have a jar with ten ½ dollar coins in it. How much money do I have? $5. Multiplication, like addition, is commutative. It doesn't matter what order we multiply things. I can say ½ x 10 or 10 x ½, I am going to get the same answer. 10 counted 1/2 a time or 1/2 counted 10 times still gives me 5. What are we counting? ½. What do I want to know about what I am counting? How much is ½ added 10 times? What number makes same with ten of the 1/2 kind.
Let's look at division and fractions. 10 ÷ ½ means how many halves are contained in 10? Each whole number contains two halves. That must mean that 10 ÷ ½ = 20. If we take 10 dollars and trade that in for half dollar coins there will be 20 half dollar coins in 10 dollars. What are we counting? ½. What do we want to know about what we are counting? I want to know how many ½'s are contained in the number 10.
Back to our picture.
1/2 is one part of two total parts. The rod that will give us the two-part kind is a red. I wrote above that fractions are a relationship between two numbers. If ½ represents the relationship of the red rod to the purple rod then the inverse is also true. The purple rod to the red rod is 2 × r. After working for some time with the blocks the student will come to understand that ½ means that one number is contained in another 2 times. ¼ would mean that one number is contained in another 4 times.
"Wait," you say, "Isn't that just like division?"
Indeed! It just depends on what you are counting and what you want to know about what you are counting.
With division I am counting the total number of times one number is contained in another. With fractions, I am counting parts. ¼ is one part of 4 total parts.
What symbols we use, how we describe things, all depends on what we are counting and what we want to know about what we are counting. If we teach children to count and count well, they will not need to rely on keyword strategies.