In Which An English Major Proposes a Solution to a Common Core Math Problem
I’m sure by now many of you, dear readers, have seen the story about the Common Core math problem that not even a person with training in higher math could solve. In this post I’ll lay out some background involving the story, and then propose a simple solution for the problem in question. I’ll conclude with a brief note about why I think problems like this are important.
First off, a bit about my background (for new readers). I earned my bachelor degree in English Language and Literature from the University of Michigan, which required almost no mathematics courses. I did elect to take a number of science courses with complex math, as well as statistics courses, and communication studies classes that required an understanding of statistical analysis. It can hardly be argued that I am a math expert since I’ve never really taken much beyond calculus.
I have to admit that I was a bit surprised when I first came across the story not too long ago. According to Elise Sole of Yahoo’s Shine network, Jeff Severt, a frustrated parent who has a bachelor of science degree in electrical engineering, couldn’t solve his child’s homework problem. I think I can sympathize with that. Right around the time I hit fourth grade my parents failed to be of any help with my own math homework, and the frustration of that helplessness was pretty evident on my father’s face, especially.
I’d like to note that Elise uses words like “elaborate” to describe the method the fictional “Jack” character uses to solve the problem, which I believe is misleading. I think with some thought it’s actually quite simple, but that might be the training I’ve had in logic and critical thinking as an English major. I do have to admit, however, that the answer to this problem suggested itself to me fairly quickly.
Now, I’m not saying this makes me a better mathematician than Jeff Severt, and I’m not intending to write about my grand intellect (which, honestly, could probably be about as grand as an old saltine for all the good it does me–that was a bit of humor). I am saying that, just maybe, the issue here isn’t with the method that Common Core is using here. Indeed, I think the general goal of trying to teach different methods of solving math problems, as well as teaching the logic that underlies mathematics, is worthy. Perhaps, then, the issue is with critical thinking skills.
The problem has all the information one needs to solve it. It requires a bit of logical inference based on the information that is given and the ultimate goal you need to reach. I think this is a good way to teach people how to use critical thinking and logic to solve problems instead of just using the algorithm that Jeff uses to get it over and done with. Perhaps it’s just my inner-geek speaking, but I actually like problems that make me think like this one. The method outlined in this problem is not the quickest, certainly, but it does present a real problem that requires a bit of thinking to reach the solution.
Anyway, what is the solution?
The first thing you note is that Jack was trying to solve the problem 427-316 by using a number line to count out the numbers he was subtracting. Built into the question is the information that Jack reached the wrong answer, and on the number line you see that his erroneous answer is 121. To show the process of counting from one number to another, he uses arcs that sweep above the number line.
The first three arcs, starting from 427 and working backward, are groups of 100. Under the number line, after each arc, you see that Jack correctly notes the number that is reached when he works backward in groups of 100. The order is this: 427–>327–>227–>127. When the three groups of 100 are combined into 300, you should start to get a feel of where the logic is going in this solution. Jack is making groups of numbers that add up to 316, to make counting back from 427 to 111 simple when showed graphically on a number line.
With 300 numbers accounted for, and with the number 127 reached, Jack then needs to account for 16 more numbers to add up to 316. The number line shows that he starts to count by one with smaller arcs. There are six such small arcs that you can count, bringing the number down to 121. That leaves us with 306 of the 316 we need. This is where Jack makes his mistake. He stops at 121.
The answer to the question, then, is to show that Jack had the right strategy for this method of solving the problem, but he didn’t go far enough to reach the 316 he needed to get the right answer. A suggestion about fixing these problems in the future could be to add up the groups of numbers he counted off on the number line to make sure he has taken away the correct number from the starting total. The difference between 121 and 111 is 10, the exact amount needed to get from 306 to 316.
What I think that Jeff, and many who sympathize with his frustrations, miss is that this is a problem that wants children to explore different methods of thinking about numbers. Sure, when you write out the problem the way that Jeff did, it is trivially easy to solve. However, Jeff’s difficulty in solving the problem, especially with a background that suggests math intensive studies, underscores why we need to teach these methods.
I don’t have particularly strong feelings about Common Core one way or another, but I am a fan of teaching the logic of numbers and different ways of thinking about them. My training as an English major, which included critical thinking and logic (plus a bit of old-fashioned pattern recognition) helped me to see the solution to this problem relatively quickly with no headache involved.
I would caution those who are critical of these methods, and argue that this is making math too confusing and complex, to slow down and think about how we’re teaching our children to think. Teaching methods like this are relatively unfamiliar to a lot of people and will be prone to problems, but I think the long-term payoff will be greater than the bumps along the way. Critical thinking skills are vital to understanding the world and all of its complexities, and this certainly challenges the thinking skills of people used to simple algorithms to solve math problems.
So, gentle readers, I hope this was a useful exercise. I certainly found it an enlightening experience.