Category: Algebra

Posted in Algebra, Calculus

8 Different Ways to Think of the Derivative

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In his paper The Transition to Formal Thinking in Mathematics, David Tall presents W.P. Thurston’s seven different ways to think of the derivative:

  1. Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
  2. Symbolic: the derivative of x^n is nx^n?1, the derivative of sin(x) is cos(x), the derivative of f ? g is f ? ? g ? g? , etc.
  3. Logical: f ?(x) = d if and only if for every ? there is a ? such that when 0 <|?x|< ? , then slope
  4. Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
  5. Rate: the instantaneous speed of f (t) , when t is time.
  6. Approximation: The derivative of a function is the best linear approximation to the function near a point.
  7. Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power. (Thurston, 1994.)

David Tall argued that the list excluded the global concept of local straightness so he added added the eighth that he believes that other 7 can be built.

8. Embodied: the (changing) slope of the graph itself.

In the same paper, David Tall presents a learning framework for derivative based on his Three Worlds of Mathematics Framework.

understanding derivative

Posted in Algebra

What are the Three Worlds of Mathematics?

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There are three worlds of mathematics according to David Tall: the world of conceptual embodiment, the world of symbolic calculation and manipulation, and the world of axiomatic formalism. This classification is based on how mathematical concepts/objects developed. It is important for us teachers to be at least aware of these three worlds.  It tells us that math ideas are not formed in the same way therefore we can’t teach all math topics in the same way. The use of real-life contexts, the use of concrete materials, may afford learning of some concepts but may hinder the learning of  others.

World of Conceptual Embodiment

According to Tall, the world of conceptual embodiment grows out of our perceptions of the world and consists of our thinking about things that we perceive and sense, not only in the physical world, but in our own mental world of meaning. The world includes the conceptual development of Euclidean geometry and other geometries that can be conceptually embodied such as non-Euclidean geometries and any other mathematical concept that is conceived in visuo- spatial and other sensory ways. A large part of arithmetical concepts also developed via conceptual embodiment (see Figure below).

World of Symbolic Calculation

The second world is the world of symbols that is used for calculation and manipulation in arithmetic, algebra, calculus and so on. The ‘development’ of the objects of this world begin with actions (such as pointing and counting) that are encapsulated as concepts by using symbol that allow us to switch effortlessly from processes to do mathematics to concepts to think about.  But the focus on the properties of the symbols and the relationship between them moves away from the physical meaning to a symbolic activity in arithmetic. My post Levels of understanding of function in equation form describes the development of the idea of equation from action to object conception.

World of axiomatic formalism

The third world is based on properties, expressed in terms of formal definitions that are used as axioms to specify mathematical structures (such as ‘group’, ‘field’, ‘vector space’, ‘topological space’ and so on).  It turns previous experiences on their heads, working not with familiar objects of experience, but with axioms that are carefully formulated to define mathematical structures in terms of specified properties. Other properties are then deduced by formal proof to build a sequence of theorems. The formal world arises from a combination of embodied conceptions and symbolic manipulation, but the reverse can, and does, happen.

development of mathematics

 

Read the full paper Introducing the Three Worlds of Mathematics by David Tall.

Posted in Algebra, Mathematics education

Develop your ‘Variable Sense’

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Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).” – NCTM

If there’s number sense, there must also be also ‘variable sense’. Number sense is associated with arithmetic and basic to numeracy while variable sense (also called function sense) is associated with algebra. I collected the following set of tasks I believe will develop variable sense. Having a sense or feel of variables helps develop algebraic thinking and functional thinking.

Task 1

What must be true about the numbers in the blanks so that the following equation is always true?

____ + -2 = ____ + -4

Task 2

The following integers are arranged from lowest to highest:

n+1, 2n, n^2.

Do you agree? Explain why.

Task 3

What is the effect of increasing a on the value of x in each of the following equations?

1) x ? a = 0

2) ax = 1

3) ax = a

4) x/a = 1

Reason without solving.

Task 4

Drag the red point. Describe the relationship among the lengths of the line segments in each of the figure below. It would be nice if you can come up with an equation for each.

[iframe https://math4teaching.com/wp-content/uploads/2012/10/meaning_of_variable.html 600 360]

Tasks 1 and 2 are common problems. Task 3 is from a research paper I read more than five years ago. I could not anymore trace the paper and its author. Task 4 is  from Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics by Ann Watson and Liz Bills from the book Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education). I just made it dynamic using GeoGebra.

Posted in Algebra

How to derive the quadratic formula

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As I wrote in my  earlier post about solving quadratic equations, introducing the quadratic formula in solving for the roots of a quadratic equation is not advisable because it does not promote conceptual understanding. All the students learn in using the formula is to substitute the values and evaluate the resulting numerical expression. I have seen test questions like “In 2x^2-4x+4=0, what is the value of a, b and c?”  Where is the mathematics in this item?

Not teaching the quadratic formula in solving for the roots of a quadratic equation does not mean that the quadratic formula will not be part of the algebra lesson. It would be a good exercise at the end of the unit to ask students to derive a formula for finding the roots of ax^2+bx+c=0 because you will be talking about vertex and discriminant (if you think they need to know what a discriminant is as this will just add to the terms they need to memorize) in later lessons. However I suggest that you ask the students ‘to solve for x’than ‘derive the formula’.

Problem

Solve the equation ax^2+bx+c=0.

Solution

 

 

 

 

 

 

Express the left hand side as product: x(x+ \frac {b}{a}) = \frac {-c}{a}.

Complete the square:

The rest as they say is pure algebra:

As you can see, deriving the quadratic formula is a beauty. Using it is not. Completing the square and factoring will do for students solving quadratic equations for the first time (ninth grade, for most countries). What is needed at this point is exposure to different problem solving context requiring representations of and solving quadratic equations.

Coming up in the next post is the meaning of this in graphs.