Pop Quiz!(No Googling or peeking at other people's answers before responding, please - this is a closed-book quiz. Also: this will not go on your permanent record.)
With reference to the long multiplication problem above, please answer the following questions.
Q1) What does the "4" that is circled mean?
Q2) What does that zero in the second row mean (the one that is circled)?
We'll come back to this later.
Common Core Math Standards: What's the Point?Our topic today is the Common Core Math Standards. They seem to have some people in a tizzy.
The picture at left, for example, has been making the rounds on the internet. Evidently someone snapped this picture of their child's math homework because they were enraged by it, and lots of people are hopping on the bandwagon.
This is curious to me.
The intent here should be pretty clear - the point is to develop the child's conceptual understanding of subtraction. This happens in parallel to developing their fluency in the standard subtraction algorithm that we all learned as children (not shown on this page, but also covered in the Common Core standards). The standard subtraction algorithm is efficient for calculating, but doesn't support understanding.
Do We Really Need to Change the Way We Teach Math?Why would we want to teach a child to understand the concepts behind the algorithms? Isn't that just a waste of time? We all did fine just learning the algorithms by rote, didn't we?
Well, no. It turns out adults in the U.S. aren't very good at math.
Speaking of which - let's take a moment to reflect on the quiz above. How did you do on it? How quickly could you answer? Did you step away to search the internet for some clues or look at the poll results before submitting your own answers? (Naughty monkey!) How confident were you in your responses?
The simple fact is we are failing our kids in math education, and we have been for generations. Here are some more fun facts to illustrate the point:
- According to the National Assessment of Educational Progress, only about two in five U.S. 4th graders are proficient in math. That drops to about one in three students by 8th grade.
- According to the Programme for International Student Assessment, the U.S. ranks below the international average on tests of mathematics. This is despite the fact that the U.S. spends more per pupil than other countries.
It's pretty clear we're doing something wrong. Solving big problems like this requires big changes.
How Can We Do Better?If we look at some of the top performing countries, like Singapore, and ask what they are doing that we aren't, the most obvious difference is that they focus on conceptual understanding a whole lot more than we do, and they spend a lot less time teaching things by rote. They teach the kids several models and methods for thinking about numbers and operations on numbers, for example - methods like the "Counting Up" method represented in the picture above. It's not a secret. We have just stubbornly refused to do it (until recently, anyway, with the arrival of the Common Core math standards).
But Aren't We Just Complicating Things?
Teaching concepts may take more time than memorizing a few recipes for calculating without understanding (at least initially), and some people seem to object to spending the extra time.
This is extremely short-sighted. There may be more time spent initially developing understanding, but that investment will pay dividends many times over across the student's time in school, from kindergarten through high school (and beyond). Understanding the concepts makes later learning far more efficient - and effective. By spending more time developing foundational understanding, we could actually get better outcomes while spending less time on the math curriculum overall. Not only that, but students who understand what's going on have a much better experience, are more engaged, and are more confident. All good.
Why Can't We Just Do What We've Always Done?What does it look like when we fail to teach children conceptual understanding? The video below shows one example of a second grader working on some grade-level math problems.
Notice that she knows the numbers and she knows how to count - these are typically learned by rote. Her conceptual understanding is very weak, however, and as a result she has to go through a laborious process of counting up from zero to answer simple questions like "How much do I have if I add 10 to 35?" And then she gets the wrong answer. Repeatedly.
Time invested in developing authentic understanding is not a waste of time. Quite the contrary. The real waste is time spent teaching without developing understanding - which produces the kind of disjointed, brittle, and tentative knowledge shown in the video - which is ultimately quite useless and will likely fade away rapidly.
This is all too common. This could well be any of our children. The maddening thing is that there's every reason to believe this child - and virtually every child - is completely capable of understanding the concepts she would need to reason fluently about the questions being asked of her in this video. She's not failing at math. Our education system is failing her. The same way it has failed generations before her. In large part by teaching math by rote, without conceptual understanding. (Don't scapegoat the teachers, by the way - the root problems here are systemic.)
We could try to press ahead as some people are advocating and just teach this child the algorithms for long addition, multiplication, and division as we have always done. But with such a weak foundation of understanding, what would we really expect to achieve that way? Her performance on those would quickly come to look like her performance here - labored, uncertain, and error-prone. Eventually she could well stop using the algorithms for lack of confidence, or even forget them altogether.
This is the sort of large scale, systemic problem the Common Core Math Standards are meant to rectify.
Could the explanation in the textbook shown in the picture above be edited for clarity? Sure it could.
Is that evidence that the Common Core is a failure and should be trashed? Far from it. We've been doing math education wrong for a very long time. The Common Core Math Standards represent a big step in the right direction - in the direction of what Singapore and other top-performing countries do, in fact.
Some people seem to think the textbook image above is crazy. What's really crazy is recognizing that the status quo is not acceptable while repeating the same educational processes generation after generation and expecting a better result.
Come to think of it, wasn't that literally Einstein's definition of insanity?