Posted in Algebra, Calculus

8 Different Ways to Think of the Derivative

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, Math Lessons

Ten problem solving and geometric construction tasks

I’ve written a number of posts the last couple of months which I published in other sites. They are problem solving tasks mostly in geometry using GeoGebra and a few on function, trigonometry and calculus. May I share 10 of them here. The first six are teaching resources which I posted in AgIMat, a site about science and math teaching resources. The last four problems are in Math Problems for K-12 to help students in their revision.  Both sites are new ones. I hope you subscribe and promote them in your social networks. Thank you.

  1. Problem solving on congruent segments
  2. Square and triangle problem
  3. Triangle Congruence by ASA
  4. Angle bisector – two definitions
  5. Constructing the perpendicular bisector
  6. Exponential function and its inverse
  7. How to sketch the graph of the derivative of a function
  8. Ratio and probability problem
  9. Trigonometric equations and their graphs
  10. Proving trigonometric identities #1

 

Posted in Trigonometry

Slopes of tangent lines

One of the most difficult items for the Philippine sample in the Trends and Issues in Science and Mathematics Education Study (TIMSS) for Advanced Mathematics and Science students conducted in 2008, is about comparing the slopes of the tangent at a point on a curve. The question is constructed so that it assesses not only the students understanding of tangent lines to the graph of a trigonometric function but also students’ skill to use mathematics to explain their thinking. The question is one of the released items of TIMSS Advanced 2008 so I can share it here. The graph actually extends beyond point B in the original item.

Sophia is studying the graph of the function y=x+cos x. She says that the slope at point A is the same as the slope at point B. Explain why she is correct.

I don’t have information  if the students’ difficulty has to do with their mathematical understanding or it is the way the question is asked. I have a feeling that had the question been ‘What is the derivative of the function y = x + cos x?’, the students would have been able to answer it. But of course, the item is also assessing students’ conceptual understanding of derivative as the slope of the tangent line at a point on a curve.

The TIMSS Advanced tests were given to Year 11/12 populations. Because the country does not have senior high schools, the Philippines sample were Year 10 students from Science High Schools where calculus is a required subject. The group of teachers we were discussing this question with said they are only able to cover up to the derivative of polynomial functions although the syllabus cover derivative of trigonometric functions. Indeed, the problem should not be difficult to those who have taken calculus or at least have reached the topic about the derivative of trigonometric functions. The solution is pretty straight forward. The derivative of the function y = x + cos x is 1+-sin x so the slope of the tangent  at ? and 2? is 1.

Covering the syllabus is really a problem because of lack of time. Even if the students are well selected, I think it is still a tall order to cover topics what other countries would cover with an additional two years in high school. Quality of teaching suffers when teachers will teach math at lightning speed. One is forced to do chalk and talk.

The TIMSS item shown above can still be solved with basic knowledge of trigonometric function and slopes of tangent lines. The function y = x + cos x is a sum of the function y = x and y = cos x. The slope of y = x is 1. That slope is constant. The function y = cos x has turning points at ? and 2? hence the slope of the tangents at these points is 0. So, Sophia is correct in saying that the slope of the tangents at ? and 2? in y = x + cos x are the same. Students are more likely to analyze the problem this way if they have a conceptual understanding of the functions under consideration and if they are exposed to similar way of thinking, especially of expressing representations in equivalent and more familiar form. This way of thinking need to be developed early on. For example, learners need to be exposed to tasks such

1) Find as many ways of  expressing the number 8.

2) What number goes to the blanks in 14 + ___ = 15 + ____.

3) Solve 3x = 2x – 1 graphically.

You may want to read my other posts to items based on TIMSS framework here and proposed framework for analysing understanding of function in equation form and sample problem on sketching the graph of the derivative function.

 

References for understanding the idea of derivatives

1.Students’ conceptual understanding of a function and its derivative in an experimental calculus course [An article from: Journal of Mathematical Behavior]
2. Calculus: An Intuitive and Physical Approach (Second Edition)