## Mathematical Structure-

## Structure and Context over Process and Procedures!

### by Bill Reed

As I observe math classes, I see processes and procedures being taught! This worries me more and more! Please do not get me wrong, processes and procedures are important. However, if the processes and procedures we are teaching are not attached to any structure or context, it is just memorization! I do not know about you, but my memorization skills have their limitations.

According to an article in Scientific America only 2 to 10 percent of all children have a photographic memory. “Is There Such a Thing as a Photographic Memory? and If so, Can It Be Learned?” *Scientific American*, Scientific American, 12 Mar. 2007. Why do so many teachers show a process or procedure and never address the structure of how or why that process works? They never put the process or procedure in context and show how and where it works and how it can be useful to the student. Unless we are teaching the 2 to 10 percent of the students with photographic memories, we are setting ourselves up for frustration and our students for poor performance.

### Memorization?

I have a question for you all. Do you have at least 10 different phone numbers memorized for family and friends you could easily dial and call on your cell phone WITHOUT using your contacts or speed dial? If not, why not? You know these people and hopefully call them occasionally.

Yet, as teachers we expect students to memorize and retain facts, procedures, and processes on a daily basis. Sometimes we forget that our class is NOT the only class a student has throughout the day. I know this may come as a shock to some of you, but your class may not even be the *most important* class the student has all day! Expecting students to retain all the processes and procedures without showing them the structure is not realistic! Most students have difficulties memorizing all the facts they are expected to learn and know from all their classes.

### It’s about the CONTEXT!

I think showing the students the processes or procedures to solve a math problem are wonderful teaching tools. We must complete the teaching and student learning by then asking the students to explain the mathematical structure that the process or procedure is based upon or allows it to work! The processes and procedures we are teaching must be put in context! Why does the process or procedure work? When is the process or procedure applicable? What problems does the process or procedure work with and what are the problems where they do not work?

So many times, we leave out the second and, in my opinion, most important part of the teaching and student learning, by not addressing the mathematical structure. A great example of this is when teachers show the process of multiplying binomials using F.O.I.L. Just to be totally transparent and upfront, I very much dislike the teaching of the F.O.I.L. process! Teachers show and explain the F.O.I.L. process and expect students to use it.

After a teacher I was observing finished a good lesson where they taught the F.O.I.L. process, I asked a few students “How does the F.O.I.L. process work?” They looked at me like I had two heads. One even said; “Because the teacher showed it to us and told us how to do it!” The students had no understanding of how or why F.O.I.L. works. A couple of days later I was back in that same teacher’s classroom and the students were practicing multiplying binomials. Many students could not remember the F.O.I.L. process and were struggling with their assignments.

Please understand, I know that having students discuss and understand the mathematical structure behind the F.O.I.L. process will not get 100% of the students to understand and remember how to multiply binomials. Yet, it will increase the students’ long-term memory of the F.O.I.L. process and help the students understand why the F.O.I.L. process works. It puts the process they had been presented into context. The mathematical structure gives the students something for them to grasp and associate the learning with for future reference.

### Expanded Distribution Process

I would be remiss if I didn’t say, “Please STOP teaching the F.O.I.L. process! Teach the expanded distributive property instead!”. The expanded distributive property works for **ALL** multiplying polynomial problems. The F.O.I.L. process only works for multiplying a binomial by another binomial. The F.O.I.L. process is a pneumonic that has a very limited scope of problems for which it is applied. Teaching the expanded distributive property works on all problems the F.O.I.L. process work with and so many more. Why not teach one process that has so much more structure to it and applies to such a wider range of problems?

### Math is Like a Diamond

Teaching processes like F.O.I.L. reminds me of a presentation I attended at a California Mathematics Council North Conference many years ago. The keynote presentation was titled, “Math is like a Diamond”. I wish I could remember who presented this excellent talk. The presenter proceeded to explain what makes diamonds so valuable. It is the multi-faceted cuts that give diamonds their sparkle and luster. The more precise the facets are cut and arranged the more the diamond sparkles and the more the diamond is typically worth.

If you take a diamond and round off all the corners of the multi-faceted surface it loses its’ luster and most of its’ value. Yet that is what we do when we are teaching mathematics to our students. When we round off the corners and all the facets and structure of the mathematics the students are learning to “make it easier for them to grasp” we lose all its’ luster and all its’ value! That is exactly what happens when we just teach processes and procedures without any structure!

I understand teachers boil down to simple processes and procedures the mathematics they are expected to teach. It is their attempt to make mathematics more manageable and accessible to the students. It is also for expediency due to the quantity of material they are expected to cover throughout the year.

I also know that this is causing students to see mathematics as a set of rules, processes, and procedures they must follow that lacks meaning and context. This is exactly what worries me! Mathematics is so much more than a set of procedures, rules, and formulas! Do students see the beauty of Mathematics? Do they see how the structure allows them to do so many things easily? If not, we are failing our students and making mathematics lose all its’ luster and value.

### The Difference of Two Squares

I was recently presenting about the SAT Assessment, specifically the math concepts that are regularly found on the math portion of the SAT Assessment. I was sharing that the “Difference of two Squares” is one of the concepts that students should know for the SAT and understand in general to help them out with basic multiplication. A teacher asked how the difference of two squares helps with basic multiplication. I put the problem on the whiteboard. 37 x 43. Then I wrote 1600 – 9. The answer to 37 x 43 is 1591. Then I asked the teachers what I had done and why it worked.

One teacher said, “Oh my gosh, I never understood an application for the difference of two squares until now.” They had taught it for many years but had never put it in context. They had been teaching the process and procedure without any structure or context. The structure for the difference of two squares is its symmetry! In the case of 37 x 43, it is (40 – 3)(40+3). This allows one to easily multiple the two numbers 40 x 40 which is 1600 and 3 x 3 which is 9. Subtract the 9 from 1600 and you have the answer 1591. If there is symmetry in the multiplication you can work the problem much easier and many times in your head. Try it. Multiply 56 x 64 in your head. Did you think 3600 – 16 = 3584?

### Conclusion

As math teachers, we teach many processes and procedures. What I am suggesting to you is to make sure you associate the processes and procedures with the mathematical structures that make them work! Make sure you put the mathematics you are teaching into context for the students. Students must understand why, how, and when to use the mathematics they are learning. Do not just teach the processes or procedures where all value is lost.

You will find that students not only remember the processes and procedures much longer, but they will also have a much greater ability to apply what they are learning. Isn’t that what teaching math is all about? Getting the students to see the beauty of the math they are learning and apply the math to solve more complex problems should be our ultimate goal. I know this will not always happen, but we should at least try our best to facilitate this goal whenever possible.

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