Why Algebra?

Algebra is the Gateway

Algebra I is often called a gateway course since understanding of its content is fundamental to success in future math and science courses. The techniques and ideas in Algebra I pave the way to logical thinking which is essential for democratic citizenship, access to important careers, and perhaps even participation in The Great Conversation.

Is Algebra Appropriate for Eighth Grade?

All students should have a solid background in algebra by the end of eighth grade, whether or not it is through a formal course. Principles and Standards provides guidelines for introducing algebraic concepts early and then reinforcing and strengthening them throughout the grades. By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more sophisticated work in algebra in the middle grades and high school. Specifically, the Standards for grades 6–8 focus on algebra, as well as its connection to other important areas, such as geometry. By the end of eighth grade, students should have an algebraic background that enables them to enter substantive high school courses." Source: www.nctm.org/standards/faq.aspx "Any national mathematics curriculum must emphasize depth over breadth and must focus on the essential ideas and processes of mathematics. The generally long lists of state and local school district curricular expectations have led to teaching too much too quickly with far too little depth. The most important challenge for any mathematics curriculum is to be focused in scope and not simply a long list of disconnected expectations. An effective curriculum is focused, delves deeply into each topic and concept, and is coherent across grades." Source: www.nctm.org/standards/content.aspx?id=23273

"All students should have a solid background in algebra by the end of eighth grade, whether or not it is through a formal course. Principles and Standards provides guidelines for introducing algebraic concepts early and then reinforcing and strengthening them throughout the grades.

By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for more sophisticated work in algebra in the middle grades and high school.

Specifically, the Standards for grades 6–8 focus on algebra, as well as its connection to other important areas, such as geometry. By the end of eighth grade, students should have an algebraic background that enables them to enter substantive high school courses."

Source:  www.nctm.org/standards/faq.aspx

"Any national mathematics curriculum must emphasize depth over breadth and must focus on the essential ideas and processes of mathematics. The generally long lists of state and local school district curricular expectations have led to teaching too much too quickly with far too little depth. The most important challenge for any mathematics curriculum is to be focused in scope and not simply a long list of disconnected expectations. An effective curriculum is focused, delves deeply into each topic and concept, and is coherent across grades."

Source: www.nctm.org/standards/content.aspx?id=23273

The main concern most educators have over eighth grade algebra requirements is that not all eighth graders are developmentally ready for such abstract concepts. This had nothing to do with intelligence or motivation. Complex abstract thinking is a capacity that develops very slowly and at different rates in each individual. This natural development can be neither mandated nor accelerated.

It is possible to drum basic skills into students and test their ability to successfully repeat these processes on similar problems. Such tactics can look good on standardized tests, but have little relevance to actual algebraic thinking, which requires imagination and creativity as well as flexible and critical thinking.

Despite these caveats, a foundation in Algebra can be successfully taught to many eighth grade students provided the curriculum is inspiring and developmentally appropriate. This is an excellent age to awaken growing capacities for discrimination and reasoning. Eighth graders are shakily balanced on the divide between childhood and adulthood. They seek direction, balance, consistency, and truth. Algebra can provide an important foundation for this search.

The language of mathematics remains one of humanity's best attempts to provide a stable framework for ideas, be they religious, scientific, technical or artistic. An appreciation for the ancient underpinnings and soaring achievements of mathematics can provide an inner foundation for adolescents who are buffeted by seething emotions, novelty-based media, glittering gadgets, accelerating cultural transition, and Pentagon-funded video games as recruitment tools.

Education vs. Training

There is substantial evidence of efforts to dumb down education in order to train "workers, not thinkers." See for example various accounts of Rockefeller's philanthropic activities while funding the first Boards of Education.Whether or not this is the cause, many of us may have been trained to dislike algebra because we did not have sufficient opportunity for two essential experiences:

• The chance to invent, discover, explore, and test our own mathematical ideas in an open, experimental, creative, and judgment-free environment.
• The chance to discover the power, beauty and mystery of the ocean of universal patterns that surround and shape us. 

In this class, I clearly distinguish between the memorization of skills (training) and the development of mathematical thinking (education). Each has its place, and this class includes both. I hope that the students will learn to appreciate the difference as awareness of this distinction will serve them well as they enter the propaganda drenched adult world.

"While training narrows a person's perspective, education broadens it.
While training is about learning the right answers, education is about
learning how to ask the right question so that one can be
innovative, creative and responsive to change." 

-- Horace Jeffery Hodges

The Elements of Algebraic Thinking

Algebraic thinking is based on several key attitudes and insights:

1. Learn to Question

The great discoverers have all been great questioners. Their insights resulted from asking the questions that others failed to consider. An emphasis on developing good questioning skills awakens an awareness of the patterns upon which possible answers can be based.

2. Develop Pattern Awareness

The discovery of universally valid patterns is the fundamental insight that makes algebraic methods reasonable and possible. If there were no predictable patterns, there would be no need for algebra.

3. Use Standardized Methods

Once a significant pattern is found, short cuts and tricks can be used each time the same pattern is encountered. Such repeatable methods become generally accepted techniques to be used again and again by future generations. The memorization of such techniques makes up the bulk of most standardized, fact-oriented math classes. This is where many were taught to believe the study of mathematics ends. Actually, this is only the past or better stated, only the beginning...

One famous example is the Pythagorean theorem, which can be described in words, but more elegantly in Algebra and Geometry.

"In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle)."

4. Developing Faith in Predictability

When a scientific theory will not conform to predictable patterns, we usually decide the theory is false and abandon it. Unfortunately, patterns discovered by open and honest exploration do not always conform to our most cherished beliefs.

The ongoing discovery of universal patterns has been a long and difficult project for humanity, often coming into direct conflict with deeply held and passionately defended beliefs. Those who truly believe that predictable patterns tell us something important may have an easier time letting go of old beliefs, but...

An oft ignored fact is that the modern scientific method is based on a single, unprovable, and absolutely massive act of faith. That is the belief that the patterns we seem to see are in fact mathematically predictable, and can therefore be relied upon as the basis of all empirical knowledge.

The reason we can't prove predictability is that predictability itself is what we use for our proofs. If our faith in predictability is mistaken, then all empirical knowledge is an illusion (and you are probably not really reading this).

It's not only the scientists who rely on a fundamental faith in predictability. We all rely on evidence from predictable patterns to support our common--and not so common--beliefs. To do otherwise often seems insane.

All this talk of predictability may seem obvious and self-evident. Clearly it's easy to predict that opening  a door before walking through it is usually easier. Unfortunately, the plot gets thicker. Modern science is pushing at the edge of it's own faith in predictability. The discoveries of quantum physics challenge the basis of all previous scientific understanding. See for example, The Heisenberg Uncertainty Principle. Perhaps not surprisingly, even this challenge to all empirical knowledge can be stated mathematically:

5. Face the Unknown

When the patterns we seek do not appear, we may call this Chaos, Noise, Randomness, The Void, or The Unknown. We seem to have an inborn dislike for such situations. For example, music--one of our finest arts--is essentially an attempt to convert random noise into pleasingly predictable and rhythmic sequences of tones. No one knows exactly why we like music; we just know that we do. Humans seem to seek harmony and beauty in all it's forms. It's hard to describe harmony, but we know it when we see, feel, or hear it. More often than not the harmony we seek has a mathematical basis. See for example The Golden Ratio.

6. Seek Ultimate Truth

Scientific "knowledge" is always open to debate, testing and correction. Most scientists and mathematicians are well aware that scientists actually know very little. Science's greatest achievement has been in highlighting just how much we don't know.

When an important scientific theory seems to always conform to predictable patterns, it may be elevated to the status of a Law of Nature. Scientists work exceedingly hard to reduce such laws to a single, clear mathematical equation. Surprisingly, we have discovered very few such laws. For example we have the Laws of Thermodynamics, but only a Theory of Relativity.

One example of cutting-edge inquiry is String Theory. I won't attempt to define it. That task is left to future generations. But a rough description of the current state string theory will illustrate just how far real mathematics is from the boring factoids taught in many classrooms:

"Since the string theory is widely believed to be a consistent theory of quantum gravity, many hope that it correctly describes our universe, making it a theory of everything. There are known configurations which describe all the observed fundamental forces and matter but with a zero cosmological constant and some new fields. There are other configurations with different values of the cosmological constant, which are metastable but long-lived. This leads many to believe that there is at least one metastable solution which is quantitatively identical with the standard model, with a small cosmological constant, which contains dark matter and a plausible mechanism for inflation. It is not yet known whether string theory has such a solution, nor how much freedom the theory allows to choose the details."

Source: http://en.wikipedia.org/wiki/String_theory

If you found that interesting, you may be interested in other unsolved problems. There's plenty to do! More career-oriented folk may want to seriously consider the Millennium Prize Problems. Of the original ten, six still remain to be solved. Each solution is worth US$1,000,000.

7. Predict the Unknown

Another stage of mathematical discovery is the insight that methods developed long ago to solve an earlier set of problems are often useful for solving completely unforeseen problems. Besides enabling us to model the known world, mathematics is amazingly successful at modeling the infinitely larger world of the not-yet-even-imagined. As far as mathematicians are aware, there is no obvious reason why this is should be so. This almost miraculous seeming situation has lead many a philosopher to seriously ponder the Unreasonable Effectiveness of  Mathematics.

As humanity gradually learns to ask deeper and deeper questions about our world, mathematics again and again proves to be the preeminent language for accurately modeling our evolving understanding.

This leads to a more sophisticated appreciation for the apparently deep mathematical basis of our shared reality. The consequences of these discoveries can be shattering, liberating, terrifying, and awe-inspiring. Powerful, long established worldviews have fallen or been transformed by these discoveries. Through it all mathematics--our universal language--has survived and grown.

In Summary

Algebraic reasoning begins with the above insights, and its further development is inspired by them. As these insights are integrated, algebra becomes elegant, mysterious, and beautiful, much like the awesome world it seems to model so well.