Tag: understanding mathematics

Posted in Number Sense

What can the representations of numbers tell us?

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Numbers can be represented in different ways. The kind of representation we choose can highlight or de-emphasise the properties of the numbers.

Studies about understanding mathematics discuss about two kinds of representations of a mathematical idea: (1) transparent representations and (2) opaque representations. A transparent representation has no more and no less meaning than the represented idea(s) or structure(s). An opaque representation emphasizes some aspects of the ideas or structures and de-emphasizes others.

Examples:

  1. Representing  the number 784 as 28^2 emphasizes – makes transparent – that it is a perfect square, but de-emphasizes – leaves opaque – that it is divisible by 98.
  2. Representing the 784 as 13×60+4 makes it transparent that the remainder of 784 on dividing by 13 is 4, but leaves opaque its property of being a perfect square
  3. For a whole number k, 17k is a transparent representation for a multiple of 17, as this property is embedded or ‘can be seen’ in this form of the representation. However, it is impossible to determine whether 17k is a multiple of 3 by considering the representation alone. In this case we say that the representation is opaque with respect to divisibility by 3.
  4. An infinite non-repeating decimal representation (such as 0.010011000111. . .) is a transparent representation of an irrational number (that is, irrationality can be derived from this representation if the definition adopted is its being non-repeating, non-terminating decimal; It becomes an opaque representation for the definition of irrationals as numbers that cannot be expressed as quotient of two integers.)
  5. 2k+1 and 2k are transparent representations of odd and even numbers, respectively.

But what about prime numbers and irrational numbers in general? What are their representations? P for prime is not a representation.  In the article Representing numbers: prime and irrational, Rina Zaskis argued these two numbers have something in common: they both cannot be represented. Don’t we say irrational numbers are those that cannot be represented as a quotient and prime numbers are those that cannot be represented as a product? The examples I listed above were from the same paper. The author used them to argue the importance of representations and how the absence of it can become a cognitive obstacle to understand the concept.

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

 

Posted in What is mathematics

Bedrock principles of math and what it means to understand math

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In one of my LinkedIn group, someone started a discussion with this question What  is the bedrock principle of mathematics? May I share some of the answers.

1. “The bedrock principle of mathematics is the axiomatic system. The realization that there are propositions that must be taken for granted in order to have something to build upon.”

2. “Speaking about foundations, in my opinion the bedrock should be enlarged at least as follows:

  • discerning that two things are different;
  • identifying two things which share same property;
  • discovering relations among properties.”

3. “I would say the bedrock principles of mathematics are:

  • The ability to differentiate two things
  • The ability to rank two things (as to most value, shortest route, least danger, etc.)
  • The ability to expand the above to more than two things”
Bedrock principles of any discipline of course can’t tell us how one can know if he or she understands a piece of that discipline. So I asked How can one tell if he/she understands a piece of mathematics?
According to Peter Alfeld you understand a piece of mathematics if you can do all of the following:
  • Explain mathematical concepts and facts in terms of simpler concepts and facts.
  • Easily make logical connections between different facts and concepts.
  • Recognize the connection when you encounter something new (inside or outside of mathematics) that’s close to the mathematics you understand.
  • Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

You may also want to read my post To understand is to Make Connection.

The challenge of course will still be this question: What does all these imply about teaching mathematics?