Diane Ravitch’s blog recently posted a piece by a math teacher named Stephanie Sawyer who complains about the new Common Core math standards.
I certainly have my own concerns about the Common Core (probably a topic for a separate post), but I very much disagree with much of this teacher’s criticism that the Common Core doesn’t teach the standard math calculation algorithms early enough.
Actually, I think that some of the basic algorithms–long division, for example–probably should NEVER be taught, let alone in elementary school! (I’m somewhat less opposed to alternate algorithms like partial quotients.)
Despite disagreeing with the premise of this piece, she made one particularly thought-provoking point mid-way through:
I can ask adults over 35 how to add fractions and most can tell me. And do it. And I’m fairly certain they get the concept. There is something to be said for “traditional” methods and curriculum when looked at from this perspective.
First of all, she seems right: when interacting with adults, I assume that they have a basic number sense and a sufficient understanding of fractions. They know that of something is more than , they know that of something plus another is .
Many, but probably a smaller number, might be able to determine that .
Many adults can add and subtract large numbers by hand. Many can probably even explain the concepts of place value, carrying, regrouping, etc. (However, while these people do probably have some understanding of these concepts, I’m less certain that they understand deeply enough to be able to apply these in a new situation–maybe numerical bases besides base 10.)
Many people can probably also divide somewhat large numbers by hand (or at least divide a large number by a small number), but I am sure that a very small percent of adults understand why long division actually works.
Sawyer seems very concerned that her 4th grade son can’t divide by hand:
When asked to convert 568 inches to feet, he told me he needed to divide by 12, since he had to split the 568 into groups of 12. Yippee. He gets the concept. So I said to him, well, do it already! He explained that he couldn’t, since he only knew up to 12 times 12. But he did, after 7 agonizing minutes of developing his own iterated-subtraction-while-tallying system, tell me that 568 inches was 47 feet, 4 inches. Well, he got it right. But to be honest, I was mad; he could’ve done in a minute what ended up taking 7. And he already got the concept, since he knew he had to divide; he just needed to know how to actually do it.
Is her complaint really that “he could’ve done in a minute what ended up taking 7?” If this is really her concern, than just buy the kid a calculator! He can do it in 5 seconds then!
To me, this story illustrates a HUGE educational success. First, the student knew to DIVIDE by 12 to convert inches to feet. A very common misconception among students (even many quite a bit older than 4th grade) is to try to multiply by 12 in this situation since feet are bigger than inches.
Second, he realized that multiplication and division are opposite of each other, so that if he wanted to divide 568 by 12, he just needed to figure out what could be multiplied by 12 to give 568. He also had the self-awareness to realize that he didn’t currently have the information he needed, since he only knew multiples of 12 up to 12 x 12.
Third, he CREATED HIS OWN CORRECT METHOD FOR SOLVING THE PROBLEM! In my mind, this is the holy grail of math education (or, frankly, any kind of education). He understood the problem deeply enough to come up with a way (albeit a slow way) of getting the result he needed.
If he does eventually learn some manual division algorithm, that third point in particular puts him in an excellent position to be more successful at understanding why breaking a division problem into smaller division problems (which all division algorithms do) is actually helpful and valid.
Moreover, having done and understood “the long way” that he created, he is in a much better position to actually appreciate the thinking that goes into creating a better/faster way. Think of how this student’s experience later learning an efficient manual division algorithm will differ from that of a student whose teacher simply walks to the front of the classroom and says, “OK students, today we are dividing, here’s how you do it.”
So, Ms. Sawyer, if you want your son to be able to calculate things by hand without a calculator in the quickest possible way, then yes, his education is lacking. Alternatively, if you want your son to understand and appreciate the beautiful thinking that goes into mathematical algorithms, have a deep understanding of how and why they work, be able to apply them to new situations, and COME UP WITH HIS OWN ORIGINAL AND CORRECT SOLUTIONS TO PROBLEMS, then your son seems to currently be experiencing an excellent education!
I’ve lately come to the opinion that we are just too ambitious. We should actually cut back on the mathematics instruction for the great majority of students. Only the STEM graduates will ever need mathematics instruction beyond arithmetic. They will never need to graph a function, solve a polynomial equation, determine a mean or median, or estimate the value of a complicated expression. Even most STEM graduates only need understand to the extent necessary to look up the correct formula and an example of its use.
Rather than employ enormous resources to teach material that will be of little consequence to students, we should rather at an early age (first year of high school?) recognize those students whose aptitude and inclination will lead them to a future that involves mathematics. Those will require advanced mathematics instruction, but only those. The remainder (pun intended) can devote their instructional time to subjects to which they find more affinity.
My research is in how secondary teachers and calculus students understand division. Honestly, the majority of the people I talk to are unable to draw a picture to explain what division means. That mom is my opinion is completely wrong. I’m not suggesting that kids never learn algorithms, but when I’m faced with high school math teachers and Calculus students who can’t draw a picture of division or write a story problem that would require division yet can calculate just fine I think that is signs of a huge problem.
Some of what adults can and cannot do well is a matter of practice. I used to be able to deal with bases other than ten fairly well, but not having needed to for nearly fifty years, it’s kind of atrophied.
Regarding whether to multiply or divide by twelve to convert inches to feet, when I was about nine, my father showed me how to do this by keeping the units and treating them like numbers – cancelling the ones in the numerator against those in the denominator. I didn’t know what he was talking about since in any case most of these exercises were intuitive – do you expect to get a larger number or a smaller one.
But when I got to high school chemistry, the teacher used the same method for her chemical equations. There intuitive doesn’t work, at least not for a student. She even had a name for it, “the factor label method.” For quite awhile I was the only person in the class who had a clue what she was talking about.
In my professional life, I have had two contexts for doing this. The first was currency conversions. The other was as a mining economist where I was the only person able to follow the presentations of the mining director when he would use this method in his presentations to management.