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assignment 3-4

July 29, 2010

Below is my response to the posting ” the elements of algebra” on the math teach forum.  After I posted I was given a message that said it would have to be approved,  since I do not know how long that will take I figured I would post the original thread and my response here.

the original post

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“Algebra is essentially the study of and ability to manipulate algebraic expressions which represent mathematical relationships between variables. To successfully apply algebra one must be successful with two tasks.

1. Reduce a problem to one or more algebraic relationships (expressions) between variables.
2. Manipulate those expressions to a mathematical goal (the solution) while adhering to the basic principles that govern valid algebraic manipulations.

The application of algebra is like solving a puzzle. The first step is to state the problem as an (algebraic) puzzle and the second step is to solve that puzzle (using algebra). In the following two lists I repeat the theme above, first in terms of understanding and second in terms of ability.

To understand algebra one must …

1. Understand what algebraic relationships (expressions) between variables are and how they relate to problems.
2. Understand the algebraic principles that govern valid manipulations of these expressions and how, in the right sequence, they “solve” the problem.

To be successful in applying algebra one must …

1. Acquire the ability to “see” the algebra in a problem and state the problem algebraically, as one or more algebraic relationships (expressions) between variables.
2. Acquire the ability to determine the sequence of valid manipulations that will result in the intended mathematical goal (solution). Or in other words, the ability to solve the problem.

In a nut shell, that pretty much covers what algebra is. Its’ particular symbolic reasoning, notation and technique are well established.

Prerequisites

The prerequisites for algebra are actually not many. It is ridiculously simple to list the topics but very difficult to describe their relationship to a student’s success in algebra and the reason for this difficulty is that the “list of topics” says nothing of the nature and condition of the student. The following (very short list) is essentially the “math before algebra”. This is not a list of topics where one topic ends and the next topic begins. It is more like a list of domains that develop continuously and in unison. However, the order in which I have presented them here is the (chronological) order in which they each take on more focus during the curriculum. For example, in the beginning one must establish basic numeracy before we can begin to develop the notion of operation. Likewise, while problems play an important role throughout, midway they share more of their space with the development of operation (arithmetic).

1. Numeracy – Counting and Numbers. This includes everything related to labeling the numbers on a number line including all syntactical forms such as decimals and fractions.
2. Operation – The four basic operations and their ability to map (in a many to one fashion) numbers to numbers.
3. Problems – Exercises and problems. This includes the entire spectrum of problems from the simple “count the objects” problems in the beginning to the more complex “word problems” in the middle and end.

These are essentially the prerequisites of algebra or maybe better put as the “math before algebra.” I like the second phrase better because as I said earlier, these topics are ridiculously simple to list yet difficult to relate to success or failure in algebra. But they are always there when success is at hand and if not cause for success in algebra they are signals of some other success that is itself the cause for success in algebra.

The Topics of Algebra

In the beginning of this article I stated two basic understandings and abilities that define algebra and we could I guess repeat the table of contents of any good algebra book but the list still boils down to these few areas.

1. Algebraic Expressions – What are algebraic expressions and variables and how do they relate to (mathematical) problems.
2. Manipulations – What does it mean to “manipulate” these expressions, why can we manipulate them and why do we manipulate them. What is valid and what is invalid.
3. Algebraic Technique – After you get those first two under your belt you then expand the repertoire.

It seems that defining “algebra” is rather simple (ok it took me 6 months of hashing). But since algebra (and the math before algebra) boils down to such a short list I seriously do not think the answer to success is in the list at all. I think it is in the student and I think it has to do with “purpose”. Not with the “purpose of algebra” like many educators try to believe, but with “purpose of mind” itself. And I do not think this is something that all of the sudden appears with algebra, I think it is there all along. I don’t know if you can successfully change something as innate as someone’s purpose of mind. It seems like you would have better luck changing their personality which itself seems impossible to me. I can say one thing though. Thank god educators didn’t go insane over spelling as some holy grail of knowledge or I would be in as bad a boat as all these poor kids being forced through years of math that they fail to get.”

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my response

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As a high school math teacher I would like to congratulate you on your ability to break down algebra into its basics forms. While it may seem simplistic to others I think you have hit the nail on the head. There are a few things I would like to add to the discussion, however. If we are talking about pre-requisite algebra for higher level classes then I think linear functions and their graphs should be included in your discussion. I know that reducing algebraic relationships was mentioned but the value of multiple representations can not be underestimated. In order to get to the roots of Algebra we must focus not just on the solving of equations but also what different things we can do with the equations to make the information more accessible to everybody.

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2 comments

  1. Peter,

    The purpose of mind that you point out is either in there or not, is the inate desire to know and understand. It is having more curiosity than fear when exposed to a problem one has no exposure to solving. I believe we are all very curious beings. How else can we explain our children asking “But Daddy, why?” time and time again? 🙂

    Something happens to students that changes their mix of fear and curiosity with respect to math. Now, I grant, not everyone is “gifted” in math. You can define this “gift” in many ways. However, the one I use is the ability to delve into the problems using the skills one has been taught combined with their own intiution and understanding. As a result, I believe almost all children can be “gifted” in math, because there is a component that one can learn.

    Let me get back to my point about your purpose of mind, I believe we can find this purpose buried within our students by using real world problems to motivate our math lessons. Did you look at the Prezi slide show in this unit?

    It posited a great question. Find the rectangles whose area and perimeter are the same. I couldn’t let that pass. I was immediately grabbed and started jotting things down. Then, when I was working on the Wolfram Alpha assignment, I used it’s power to recast the problem into a parametric equation. Great fun!! My point is, the question sparked the purpose of mind. And we should seek to do the same!


    • @prmarcadia – do you have the W|A equation(s) and the problem somewhere? You need to link it from that Prezi page’s comments! The teacher who made the presentation, Alison Blank, would love to hear that story.

      @Peter – advocating for fundamental notions (like linearity) and multiple representations is something I find quite meaningful, too!



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