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, Graphs and Functions, High school mathematics

What is an inverse function?

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In mathematics, the inverse function is a function that undoes another function. For example,  given the function f(x) = 2x. If you input a into the function f, the output is 2a. The inverse function of  f(x) is the function g(x) such that if you input 2a into g(x) its output is a. Now what is g(x) equal to? How does its graph look like? Is the inverse of a function also a function? These are the basic questions students need to answer about inverse function.

How to teach the inverse function
Functions and their inverses

The idea of inverse function can be taught deductively by starting with its definition then asking students to find the equation of the inverse function by switching the x and y in the original function then expressing the equation in the form y = f(x). This is an approach I will not do of course as I always like my students to discover things for themselves and see and express relationships in all three representations: numerical (ordered pairs or table of values), geometrical (graphs) and symbolic (equation) representations.

In teaching the inverse function it is important for students to realize that not all function have an inverse that is also a function, that the graph of the inverse of a function is a reflection along the line y = x, and that the inverse function does not necessarily belong to the same family as the given function.

The concept of inverse function is usually taught to introduce the logarithmic function as inverse of exponential function. Important ideas about inverse function such as those I mentioned are not usually given much attention. Perhaps teachers are too excited to do the logarithmic functions.

I suggest the following sequence for teaching inverse. I’m sure many teachers and textbooks also do it this way. What I may just be pointing out is the reason behind the sequence. I also developed three worksheets using GeoGebra. The worksheet is interactive so that students will be able to make sense of inverse of function on their own.

Start with linear function. Its inverse is also a function and it’s easy for students to figure out that all they need to do is to switch the x‘s and y‘s then solve for y to find the equation. You may need to see the inverse of linear function activity so you can make sense of what I am saying.

The next activity should now involve a quadratic function. The purpose of this activity is to create cognitive conflict as it’s inverse is not a function. The domain needs to be restricted in order to get an inverse that is also a function. Depending on your class, the algebraic part (finding equation of the inverse) can be done later but it’s important for the students at this point to see the graph of the inverse of a quadratic to convince them that indeed it is not a function. Click the link to open the activity inverse of quadratic functions.

The third activity will be the inverse of exponential function. By this time students will be more careful in assuming that the inverse of a function is always a function. Except this time it is! It is also one-to-one just like linear, but it’s equation in y belong to a new family of function – the logarithmic function. Click the link for the activity on inverse of  exponential functions.

Teaching principles

There are at least three math teaching principles illustrated in the suggested lesson sequencing for teaching the inverse function and introducing logarithmic function.

  1. Connecting with previously learned concepts. Start with something that students can already do but in a different context. In the above examples they are already familiar with linear function and they already know how to find its equation.
  2. Creating cognitive conflict. The purpose is to challenge possible assumptions and expose possible misconceptions.
  3. Making connections. Mathematics is only understood and hence powerful when there is a rich and strong connections among related concepts, representations, and procedures.

You may find the Precalculus: Functions and Graphs a good reference.

Posted in Elementary School Math, Geometry

Angles are not that easy to see

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Like most numbers, geometric objects such as angles, are abstraction from properties of real objects and quantities. For example, the idea of “two-ness” can be abstracted from real objects such as two apples, two chairs, two goats, etc. It will not take along for a learner to figure out what the idea of two means. Abstracting angles from real objects this way is not as easy as one might think it is.

Look around you and find something that to you looks like an angle. Chances are you would identify corners as forming an angle.  That’s easy because you see two sides meeting at a corner. But doesn’t the door also forms an angle when you open it?  But where is the other side? How about turning the door knob? Doesn’t it form an angle also? But where are the two sides there? It doesn’t even have a corner!

Mitchelmore and White (2000) of Australian Catholic University conducted a study of 2nd, 4th, 6th and 8th grade students understanding and difficulty about angles.  They found that students do not readily incorporate ‘turning’ in their idea of angles. They found that it is the line (or arms) of angle  which are the key to students identifying angles in different physical situation. Their study showed the easiest angles for students to learn are the two-line angles. These are angles in which both arms are visible such as corners of geometrical figures, corners of rooms, blades of a pair of scissors. The second group of angles are the one-line angles. In these angles, only one arm is visible. The other line must be imagined or remembered. Examples are the angles formed by an opening in a door, a hand of a clock and sloping of roofs. The most difficult for the students to identify are the no-line angles in which neither arms of the angles are visible. Examples include the turning wheel and spinning ball.

One can be said to have an understanding of the concept of angle if he/she can recognize all these types of angles in physical objects and is able to see that they all share the same property: they all consist of two linear parts (even if they are not visible) and they cross or meet at a point and that the relative inclination of the two parts has some significance – it defines the sharpness of the corner or the their openness.

So what is the implication of these to teaching? The most obvious is the importance of exposing students to as many different physical situation that can be represented by angles. Starting with the definition an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle and then drawing the angle figure on the board is certainly the most ineffective strategy the teacher  can do to teach students about angles.

 

Posted in Algebra, Calculus

Teaching the derivative function without really trying

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New mathematical ideas are usually built on another mathematical idea or ideas. Because of this, the teaching of mathematics if it is to make sense to students, should reflect this ‘building on’ process. Students should be able to see how the new idea is connected to what they already know. Good teaching of mathematics also demand that this new knowledge be useful and connected to the mathematics that students will encounter later.

Here is an example of a lesson that teaches the idea of derivative without really teaching it yet. This means that you can introduce this in Year 9 or 10 in their lesson about graphs of second degree function. The only requirement is that they understand the function of the form f(x) = ax^2. The task requires determining the equation of linear function of the form y = 2ax, which happens to be the derivative of ax^2. Of course you will not introduce the term derivative at this year level. You are just planting the seed for this important concept which students will encounter later.

The lesson uses the applet below. Of course, much of the success of the lesson will still be in questions you will asked after students initial exploration of the applet. You can find my proposed questions for discussion below the applet. [iframe https://math4teaching.com/wp-content/uploads/2012/02/Deriving_function_from_ax_2.html 750 620]

Questions for discussion

  1. You can move point A but not point B. Point B moves with A. What does this imply?
  2. What do you notice about the position of B in relation to the position of A?
  3. What is the path (locus) of point B? Right click it and choose TRACE then move A to verify your conjecture.
  4. What is the same and what is different about the coordinates of points A and B?
  5. To what does the coordinates of B depends on?
  6. What is the equation of the line traced by B?
  7. Refresh the applet then use the slider to change the equation of the graph, say a=3. What is the equation of the line traced by B this time?
  8. What do you think will be the equation of the path of B if the graph is f(x) = ax^2

By the end of this lesson students should have the intuitive notion of derivative and can find its equation given the function f(x) = ax^2.

There are actually 8 ways to think of the derivative. If you want to know more about Calculus, here’s a good reference:

The Calculus Direct: An intuitively Obvious Approach to a Basic Understanding of the Calculus for the Casual Observer