What is Traditional Math? Part 3 – Education Rickshaw

This is Part 3 of an interview series with Barry Garelick and J.R. Wilson about Traditional Math: An Effective Strategy that Teachers Feel Guilty Using. You can read Parts 1 & 2, and the rest of the saga as it comes out, here.

Q3: How does a traditional approach to math teaching offer opportunities for students to learn from their errors and failures? And how does it address struggle?

Barry: Students will make mistakes in math class; that’s unavoidable. Mistakes offer an opportunity to identify and correct misconceptions and common errors, as well as providing students a closer look at what is going on. It also benefits teachers, by letting us know where students need more practice and making us think of how certain concepts and procedures could be presented to minimize misconceptions in the future.

In the course of a lesson I ask students to do problems in their notebooks as I go around and check to see what they’re doing. Teachers know who the students are who have the most difficulty in class, enabling them to focus where mistakes are likely to be made.

Here are some common mistakes I see in seventh grade: The problem asks “What is 8% of 250?” A student may erroneously write 0.8 x 250 rather than 0.08 x 250. I might ask “How do I represent 8% as a decimal?” If they repeat the mistake and say 0.8, I’ll ask how do we write 8% as a fraction. They will then (usually) see that 8/100 is expressed as 0.08.

In algebra classes common mistakes include writing x^2+x^2=x^4 or sometimes as 2x^4. I might ask what is x+x and hearing 2x will remind them about combining “like terms” (which they have learned and used), and that exponents are not added unless power of the same base are multiplied. So x^2∙x^2 is x^4 but x^2+x^2 is 2x^2.

The above are mistakes that arise from mix-ups or forgetting a rule. With practice and repetition students will learn to recognize these type of problems. It is therefore important to continue to include such problems throughout the year to reinforce their learning.

Another and perhaps more significant source of mistakes come from misconceptions. And in this respect, it is often critical to find out how a student obtained the answer they did. For example, while students might correctly simplify xy/x by cancelling the x’s and getting y, I would sometimes see x/xy simplified to zero. The first time I saw this I asked the student how they obtained zero for the answer. They thought that in the case of xy/x, since cancelling the x’s eliminates them, then they must be equal to zero, since zero represents nothing.

This is an important mistake—both for me as a teacher to see how a misconception can arise, and for the student to be disabused of the notion. I often will use numbers and encourage students to do so to see what is going on in an algebraic expression. I substitute 3 and 5 in place of x and y in the above examples. Seeing (3∙5)/3 example, the student sees that 3 is divided by 3 which of course simplifies to (1∙5)/1 or 5. The 3’s don’t disappear, we are rewriting 3/3 as 1/1. Similarly for x/x and (x+y)/(x+y). All equal 1. If there is doubt about this, again, I advise students to substitute numbers in their place such as (3+5)/(3+5) .

Now looking at x/xy I will show the cancelling as 1/1y which we write as 1/y. The vanishing act of x is thus explained: the x’s are replaced by 1.

Even with clarification of misconceptions and mistakes, they will continue to surface. I will in future classes include problems in warm-ups, that repeat the problems on which students frequently make mistakes.

Let me turn now to how struggle is addressed. In the introduction to our book, I state that “I try to stem struggling with a problem and aim to have students be successful. My guiding principle is that struggling to learn the breaststroke is not the same as struggling to keep from drowning. The latter doesn’t teach you how to swim.”

Learning anything new involves some struggle. The initial stages of learning a new procedure or problem-solving technique usually involves imitation. And as anyone knows who has learned a skill through imitation—e.g., learning a dance step, bowling, golfing, playing an instrument—what looks easy is often more complicated than it looks. So too with math. Students will struggle to get the procedure down correctly at first. As they get better at it, I introduce some complexity to the problems. This requires students to make small but significant leaps of reasoning.

Let me give you an example, again from algebra. I will provide basic and explicit instruction on a type of distance/rate problem, using diagrams and other techniques. At first I might ask “Two cars go in opposite directions from the same spot; one going 60 mph and the other 70 mph. How far apart will they be in 3 hours?” Students have been given the distance = rate x time equation, and they learn how to solve such problems—it is relatively straightforward.

After a few like this I will give a similarly structured problem, but this time a different part of the problem is missing. Specifically: “Two cars go in opposite directions from the same spot; one going 60 mph and the other 80 mph. How long will it take for them to be 420 miles apart?” Students will definitely struggle with this but with some prompts from the teacher, they are able to build on the previous problem.

I might ask “What are we trying to find? How did we solve the previous problems? Can you set up a similar equation? What are you going to let x represent?” and so forth. Amanda Vanderheyden talks about this in her interview with Anna Stokke. In that interview Amanda talks about “acquisition instruction” which is that stage at which students are learning and understanding new things and building fluency with what they have just learned. In the beginning stages the acquisition tends to be difficult so it is especially essential to provide appropriate guidance. Leaving them entirely on their own might work for some, but for most, too much struggle will result in labored responses. It is not “teaching them to swim”. Ultimately, students set up the equation as 60x + 80x = 420, and they will get x = 3 hours.

Once students have reached mastery in a particular aspect of math, and are in what Vanderheyden refers to as the generalization and adaptation stage of learning, it is then appropriate to have them work on variations of these problems that require more work. Thus, problems in distance and rate increase with complexity as students increase their knowledge of tactics and can generalize and adopt them to other similar problems.

Here is an example of a problem that I might give as a challenge. This problem is not related to distance/rate as the problems above are. It deals with distance and rate of gasoline consumption. I have included it here as something that is a challenge to both seventh and eighth graders in the generalization and adaptation stage of learning for the tactics involved in the problem. It is a multi-step problem that can be solved without using algebra. (It is taken from an AMC-8 competition.)

“Karl’s car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full tank of gas, drove 350 miles, bought 8 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?”

This problem will likely cause students to struggle but they have the tactics and prior knowledge to solve the problem. They may very well need some guidance—and I do not hesitate to provide it. But it will likely be reminders and hints of tactics they learned and mastered at the acquisition stage.

J.R.: Barry addresses this question well. I’m going to piggy back on his statement from the introduction of the book. “I try to stem struggling with a problem and aim to have students be successful. My guiding principle is that struggling to learn the breaststroke is not the same as struggling to keep from drowning. The latter doesn’t teach you how to swim.” My efforts work towards helping students be successful and not setting them up for struggle. Learning to swim implies being taught or guided through the process with help, or self-correction, in developing the appropriate and efficient technique. I choose not to set students up for struggle as it may lead to an ongoing fear of water or possibly drowning. With math, it may lead to a strong dislike and avoidance of math. I would rather optimize the opportunity for student success than provide opportunity for students to struggle. Say no to struggle and yes to challenge.

At the beginning of the year, most of my incoming sixth graders didn’t seem to take advantage of opportunities to learn from their errors. I would have students check their own papers. If they had an incorrect answer, I noticed they would hurriedly erase or cross it out and write the correct answer in to make it appear they obtained the correct answer without any interest in why or how they came up with an incorrect answer. It was as if they wanted to please the teacher. Typically, there were no takers when I would ask if anyone wanted me to work one of the problems for the class. After doing this for a while, I began to see more students take responsibility for their learning and striving to be successful. If they missed a problem, they began asking me to work it for the class so they could figure out where they made their error if they couldn’t figure it out for themselves. Often, students were able to go back through the steps of their work and find where they made an error.

In the You Do phase of I Do, We Do, You Do, students would be encouraged to learn from errors. The class would be given a problem to work and then I would work through the problem for the class. Students would self-check their work; if they didn’t correctly work a problem they received immediate feedback in a non-judgmental way before attempting another similar problem. I saw students become curious about why they would miss a problem so they wouldn’t make the same kind of errors on similar problems. They took advantage of the opportunity to learn from their errors—without the possibility of drowning.

Students can also learn from the errors of others. For this reason, with certain kinds of problems I would point out common errors people make and how to avoid them. One common error I would point out to students has to do with the order of operations. Student may have a tendency to work multiplication before division even if the division step precedes the multiplication in a particular problem; same when the addition step occurs prior to subtraction comes first. I would emphasize that multiplication and division as well as addition and subtraction have the same status and to perform whichever operation comes first; i.e., we operate from left to right in such instances. In the problem 8 – 2 + 5, if the addition is worked first, it becomes 8 – 7 = 1. This is incorrect since the subtraction step comes before the addition step in this problem. Doing the subtraction first, it becomes 6 + 5 = 11 which is correct.

Another common error I would see students make is not lining up the decimals when adding or subtracting decimal numbers. But while lining up decimals is essential for such operations, it is not for multiplication. I would see some students lining the decimals up when starting a multiplication problem. I was prepared for such errors, and often, while giving instruction and working examples, I would point out common types of errors and how to avoid them. Seeing a student make a common type of error would provide a teachable moment. I would always tell students the first thing to do if they have an incorrect answer is to check and see if they miscopied the problem, with respect to placement of decimals.

In the examples provided here, we see that the traditional approach to teaching math does provide students with an opportunity to learn from their errors and failures.

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