Tag: what is algebra

Posted in Algebra

What is algebra?

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Didn’t we say in our algebra class that in the grades they study about numbers and so now they will be studying letters instead? Didn’t we say that in algebra we now use x instead of box (in 3 + ___ = 15, we now write 3 + x = 15)? And isn’t it that since this announcement our algebra class activity has been about finding that 24th letter?

Well, we reap what we sow.

what is algebra

Just  a friendly reminder to take the teaching of variables and unknown quantities with meaning.

For a serious discussion about what algebra is, I  suggest the following articles.

1. What is Algebra? by Prof. Keith Devlin

2. Algebra vs. Arithmetic

Posted in Algebra

History of algebra as framework for teaching it?

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In many history texts, algebra is considered to have three stages in its historical development:

  1. The rhetorical stage –  the stage where are all statements and arguments are made in words and sentences
  2. The syncopated stage – the stage where some abbreviations are used when dealing with algebraic expressions.
  3. The symbolic stage – the stage where there is total symbolization – all numbers, operations, relationships are expressed through a set of easily recognized symbols, and manipulations on the symbols take place according to well-understood rules.

These stages  are reflected in some textbooks and in our own lesson. For example in in pattern-searching activities that we ask our students to express the patterns and relationships observed using words initially. From the students’ statements we can highlight the key words (the quantities and the mathematical relationships) which we shall later ask the students to represent sometimes in diagrams first and then in symbols. I have used this technique many times and it does seem to work. But I have also seen lessons which goes the other way around, starting from the symbolic stage!

Apart from the three stages, another way of looking at algebra is as proposed by Victor Katz in his paper Stages in the History of Algebra and some Implications for Teaching. Katz argued that besides these three stages of expressing algebraic ideas, there are four conceptual stages that have happened along side of these changes in expressions. These conceptual stages are

  1. The geometric stage, where most of the concepts of algebra are geometric;
  2. The static equation-solving stage, where the goal is to find numbers satisfying certain relationships;
  3. The dynamic function stage, where motion seems to be an underlying idea; and finally
  4. The abstract stage, where structure is the goal.

Katz made it clear that naturally, neither these stages nor the earlier three are disjoint from one another and that there is always some overlap. These four stages are of course about the evolution of algebra but I think it can also be used as framework for designing instruction. For example in Visual representations of the difference of two squares, I started with geometric representations. Using the stages as framework, the next lesson should be about giving numerical value to the area so that students can generate values for x and y. Depending on your topic you can stretch the lesson to teach about functional relationship between x and y and then focus on the structure of the expression of the difference of two squares.

I always like teaching algebra using geometry as context so geometric stage should be first indeed. But I think Katz stages 2 and 3 can be switched depending on the topic. The abstraction part of course should always be last.

You may want to read Should historical evolution of math concepts inform teaching? In that post I cited some studies that supports the approach of taking into consideration the evolution of the concept in designing instruction.

For your reading leisure – Unknown Quantity: A Real and Imaginary History of Algebra.

For serious reading Classical Algebra: Its Nature, Origins, and Uses and of course Victor Katz book History of Mathematics: Brief Version.

 

Posted in Algebra, Curriculum Reform

What is algebra? Why study it?

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I’m doing some  literature review for my research and I came across this article by L.A Steen in Middle Matters. He was arguing about the Algebra for All standard in the US and part of the article includes description of what is algebra. I thought I should share them in this blog because it is something very important teachers should be aware of when they teach algebra or what they conceive what algebra is and for. Oftentimes, when students ask what algebra is and what they are going to need it for, teachers lazy answer is “Algebra is just like your math in the grades only that this time you work with letters instead of numbers!”

  1. Algebra is the language of mathematics, which itself is the language of the information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995).
  2. It is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education.
  3. As a language, algebra is better learned earlier and harder, when learned later.
  4. In the Middle Ages, algebra meant calculating by rules (algorithms). During the Renaissance, it came to mean calculation with signs and symbols–using x‘s and y‘s instead of numbers. (Even today, lay persons tend to judge algebra books by the symbols they contain: they believe that more symbols mean more algebra, more words, less.) I think that many algebra classes still promote this view.
  5. In subsequent centuries, algebra came to be primarily about solving equations and determining unknowns. School algebra still focuses on these three aspects: employing letters, following procedures, and solving equations. This is still very true. You can tell by the test items and exercises used in classes.
  6. In the twentieth century algebra moved rapidly and powerfully beyond its historical roots. First it became what we might call the science of arithmetic–the abstract study of the operations of arithmetic (addition, subtraction, multiplication, etc.). As the power of this “abstract algebra” became evident in such diverse fields as economics and quantum mechanics, algebra evolved into the study of all operations, not just the four found in arithmetic.
  7. Algebra is said to be the great gatekeeper because knowledge and understanding of which can let people into rewarding careers.
  8. Algebra is the new civil right (Robert Moses). It means access. It means success. It unlocks doors to productive careers and gives everyone access to big ideas.

And I like the education battle cry Algebra for All. Of course not everyone is very happy about this. Steen for example wrote in 1999:

No doubt about it: algebra for all is a wise educational goal. The challenge for educators is to find means of achieving this goal that are equally wise. Algebra for all in eighth grade is clearly not one of them–at least not at this time, in this nation, under these circumstances. The impediments are virtually insurmountable:

  1. Relatively few students finish seventh grade prepared to study algebra. At this age students’ readiness for algebra–their maturity, motivation, and preparation–is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
  2. Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students’ mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
  3. Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
  4. Most teachers don’t believe that all students can learn algebra in eighth grade. Many studies show that teachers’ beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it. (all italics, mine)

I shared these here because in my part of the globe  the state of algebra education is very much like what Steen described. You may also want to read about the expressions and equations that makes algebra a little more complicated to students.

L.A Steen is the editor of the book On the Shoulder of Giants, New Approach to Numeracy, a must read for teachers and curriculum developers. The book is published by Mathematical Sciences Education Board and National Research Council.

Posted in Number Sense

Teaching algebraic thinking without the x’s

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Students need not wait till they meet Mr. x to learn about algebra. In fact, the best way to learn about algebra is to learn it while there aren’t x’s yet; when all the learners need to deal with are concepts that still make sense to them. Here is a list of tips and ways for teaching algebraic thinking as pupils learn about numbers and number operations.

1. Vary the “orientations” of the way you write number sentences.

For example, 5 + 20 = 25 can be written as 25 = 5 + 20. The first expression is about ‘doing math’, the second engages students about ‘thinking about the math’, the different representations of the number 25. The thinking involved in the second one is ‘algebraic’.

2. Be mindful of the meaning of equal sign

If you want to ask your learners to find, for example, the sum of 15 plus 6, do not write 15 + 6 =___. It’s a recipe for misconception of the meaning of equal sign. I recommend: What numbers is the same as (or equal to) 15 + 6? Better, What number phrases are the same as (or equal to) 15 + 6? This last one promotes algebraic thinking.

3. Encourage learners to generalize. Continue reading “Teaching algebraic thinking without the x’s” →