I would have never thought that after 39 years of having no clue about math, about why we do what we do when we subtract, or not having a remote idea of how I’d be teaching things like long division, I’d be reading math books like others read novels, and demystifying arcane principles and bias of all sort my traditional math education had formed over my life.
In my defense I would say that despite of being a pure language type of girl who took Latin and Greek in high school to avoid even simple math, and who only flunked one term math test in her whole life. (I think the teacher ended up giving me a decent grade since I was straight 10 in everything else – In Spain we are given grades from 0 to 10). In spite of that my husband and I have good care of our finances, no debt, we meet our ends with no problems, and we save money and plan a trip to Europe every 3 years with one salary. So, have I needed my precarious knowledge of calculus so far? no. I even worked as a cashier for a bank in Spain for three years.
After teaching six years, and teaching math for k-1st students, I’ve realized I never understood math. It’s not accurate to think that math for first graders is not sophisticated. Any mathematician or good math teacher will tell you that teaching math right from start takes not a gene or a special talent some have some don’t have, but hard work, preparation, and a good foundation.
Knowing and teaching elementary mathematics, by Liping Ma is the third book about math I’m reading from the LIVING MATH book list. I’m only in the second chapter of this concise 7 chapter book, and I will post again about this book, I’m sure. I wanted to write about what I already got out of this book before I forget it, as a type of ‘narration’ of what’s been disclosed so far.
It happens that in America (and many other countries, because I can relate to this) most of the teachers teach procedures, or give false conceptual explanations about what they are teaching. In case you are wondering what this means, it’s when we for example teach children to subtract 15 from 31 and we talk about ‘borrowing’, which is inexact and doesn’t explain anything about the ‘concept’ of subtraction. In China by contrast, teachers think about what mathematical knowledge students have to understand to be familiar with those subtractions. Here teachers allow students to use manipulatives and don’t know how that relates to what they are doing, or find ‘formulas’ or tricks to teach procedures. In China children have to resolve this type of subtraction with manipulatives and always talk about what they have done. The teacher is aware of many ways of solving that problem, and they call this type of subtractions “decomposing a higher valued unit”, as addition is “composing”, they won’t teach the fact: one ten is ten units, but they’d talk to the kids about ‘ratio’, a ten ‘ratio’ is what’s at stake, so ten of the inferior value compose one of the immediately higher value, children then learn not only that 1 ten is 10 units, but 10 tens is 1 hundred and so forth. They won’t say wrong statements such as “you can’t substract 7 from 3, because 3 is smaller”, which is incorrect, or you won’t say things as you borrow “1” , but you will decompose (break apart) the number to be able to perform the calculation. And they explore different forms of decomposing the number, for example, 27 can be 10 + 10 + 3 + 4, or 10+10+7…they allow the students to use manipulatives and write their findings and present them to the class.
Good math teachers know that the knowledge under this type of calculation requires knowledge about place value, and Chinese teachers find calculations with numbers up to 20 to be not as simpleton calculations to be learned by flashcards and plain memorization. Mathematical reasoning, proper understanding of terms, explanation of principles and ways to solve a problem are encouraged from day one.
Another example in chapter 2 is the common mistake by 6th graders here in America when they multiply 123×645, they line the results like this:
When American teachers were asked how to solve this mistake, most of them addressed the procedure, saying how they’d guide the students into how to remember what column to start to write the result. Some talked about the “place value”, but not as a concept, but to simple refer to the “space” that has to be left because that’s the way we do it. Not many discussed it as the Chinese teachers who said that when we multiply by “4” in 645, we are truly multiplying by “40”, and you could decompose this into three multiplications, by 5, by 40, by 600, and add the result.
We need to look for ways to point to the why and how, and reason the computation, versus memorizing a set of tricks and rules that get mixed up and we can’t recreate if we forget them because we don’t know what we are doing. That’s what conceptual means, versus procedural which will be to ONLY use the memorizing strategy. After you taught the conceptual part of a calculation, you teach the procedural too, but the retention is better and in case you forget, you can figure it out too if you know what you are doing, but it doesn’t work vice versa, that’s why students are making these mistakes and not knowing.
My conclusion is simple, less paper and pencil, less drill and old trick math, and more mental math, more games and exploration, conversations and true going to the bone of what we are doing, more initial attention to the process, and less to the result. If we are sure about what we are doing, then, by practice, we’ll get it right. If we don’t know what we are doing, where we are going with it, how we got there, practice will give us the right answers for today, it will be forlorn and forgotten later. (And sadly I am noticing all my little experience with math was procedural, and consequently, I’ve forgotten all those tricks and rules, so I’m starting from the very bottom up, and have to learn it again. But I’ll learn it right, even if I don’t make it to physics LOL).
To a meaningful math life, and to a future math teaching and learning that is not tinted by old biases. This time I won’t run and hide every time the teacher says math time, because I’ll be the teacher LOL.