Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind

September 2005 Mathematics Learning Log


An Example of a "Learning Process" Journal

Saturday October 15, 2005
Learning Log Number 22

7:35 am Lethbridge, Alberta

I continue to struggle with my approach.

The first priority is for me to re-establish my focus. I want to genuinely understand calculus.

The next step is review my resources. I have a number of books on calculus. The substantive textbooks are:

  • Calculus. 6th ed. (1998). Larson, R. E., Hostetler, R. P. & Edwards, B. H.
  • Calculus. An Intuitive and Physical Approach. 2nd ed. (1977). Kline, M.
  • Calculus. 3rd ed. (1994). Spivak, M.
  • What is Mathematics? 2nd ed. (1996). Courant, R. & Robbins, H.
  • The Calculus Tutoring Book. (1986). Ash, C. & Ash, R. B.

One of the relevant factors at the moment is the weight of these books as I will be taking some of them with me on my study leave.

A review of these books leads me to select the Spivak book as being the most rigorous and the Larson book as being the most modern in terms of its use of the web and packages such as Mathematica.

I also have two books on mathematical software: Mathematica and Scientific Notebook. This complicates matters as I now have to Learn two additional notational conventions to supplement the manual notation.

This is a good start. I have pruned my topics, putting both Number Theory and Recreational Mathematics on the back burner for the moment.

My table of books is now much smaller:

Calculus. (1994). Michael Spivak      
Calculus. 6th ed. (1998). Roland Larson, Robert Hostetler & Bruce Edwards      
Mathematica Navigator. 2nd ed. (2004). Heikki Ruskeepaa      
Doing Mathematics with Scientific WorkPlace & Scientific Notebook. Version 5. (2003).Darel Hardy & Carol Walker.      
Calculus (1994) Michael Spivak      
Read chap 1 Basic properties of numbers [p. 3 - 20]      
Notes for chap 1      
Exercises for chap 1 [p. 13 - 20]      

I made a few brief notes yesterday in my pocket notepad:

Learning mathematics involves the following important principles:

  • repetition
  • review
  • establishing a regular time and place
  • knowing when one really "has it".

This leads to the inescapable conclusion that I must set up a Learning timetable, and stick to it. I think that 10 hours/week is realistic. This could be scheduled as a 2-hour session each weekday morning. Let's try inserting that into my electronic PIM software. Done. I have this set up for 6 - 8 am for the next two weeks.

I checked out the Houghton Mifflin web site for the Larson book:


This led to chapter summaries. Here is the summary for chapter 1


These look like an excellent set of objectives.


1.1.1 Understand what calculus is and how it compares to precalculus. [p. 41 - 43]

Calculus is the mathematics of change. Important topics include tangent lines, slopes, areas, volumes, arc lengths, centroids and curvatures.

Calculus is an extension of discrete mathematics to include the idea of a limit process.

1.1.2 Understand that the tangent line problem is basic to calculus. [p. 44]

A secant line is the straight line joining two points on a curve. As the two points approach each other, the secant line (extended in both directions) approaches the tangent line. The tangent line is the limit of the slope of the secant line at the specified point.

1.1.3 Understand that the area problem is also basic to calculus. [p. 45]

The area under a curve can be approximated by a series of rectangles that approximates the curve. As the number of rectangles increases the area of these rectangles tends to become better and better and in the limiting case of an infinite number of rectangles is defined as the area under the curve.

1.2.1 Estimate a limit using a numerical or graphical approach. [p. 47 - 48]

The numerical approach involves creating a table of values as the independent variable gets increasingly close to the specified value from both sides. One can then plot these values to obtain a graph of the situation. Alternatively one can use computer software (Mathematica, SNB) to obtain the graph.

Here is the result using Mathematica (web, Mathematica). It took a few trials to get the notation right. I must review some of the parameters for controlling the graph (eg. color, scale of axes). Mathematica fails to note the fact that the function is undefined when x = 1. Then again, maybe there is a way to tell it this, but I don't know this at the moment.

I need to have a close look at SNB and see if I can figure out how to get the graph with this software.

This objective breaks into 3 subobjectives: pencil, Mathematica and SNB.

I have been at this for 2 hours and it is time for a break. This has been a good morning.

My next goal is to look at both SNB and Mathematica and see if I can figure out how they handle graphs like the one I tried above.

9:35 am

Total elapsed time for the day: 2 hr. 20 min