Computer-algorithm Methodology in Mathematics Instruction Hansheng Yang Bin Lu Southwest University of Science and Technology, Mianyang, Sichuan, 621010, P.R.China California State University at Sacramento, Sacramento, California, 95819, USA

Outline 1. Introduction. 2. Flowchart in instruction. 3. Examples-lecture design. 4. Comments of students and experts.

1. Introduction The computer-algorithm methodology in mathematics instruction is a type of teaching method that uses flowchart or directed graph to help understand and analyze a mathematics problem, and design a procedure to solve the problem. Teaching mathematics courses using this type of instruction has recently produced many positive results at Southwestern University of Science & Technology and other universities in China such as UESTC, Southwest University,etc.

In the following sections, we will give a brief discussion on this instruction method, and three sample examples on how to design a lecture with flowcharts. We believe there are many other topics in college mathematics course that can be easily adapted using this style of instruction, so to achieve our ultimate goal, teaching mathematics effectively.

2. Flowchart in instruction The computer-algorithm methodology in classroom instruction,originated in 70s [1]for non-math teaching, the main idea is to design each lecture as flowchart, or directed graph, to highlight the relationships among various concepts, logical derivations, and procedures in a problem-solving,to strengthen students’ understan- ding and enhance their ability of transforming mathe-matics problems into algorithmic flowcharts, and ability of mathematics modeling.

Teaching, using the flowchart or directed graphs, originated in the 70s [1] etc, in various disciplines such as law major, less in higher education for mathematics. In 1980s this method of teaching was used in several universities in China in non-mathematics disciplines. Starting 80s, Yang [4, 5] was first one to propose the flowchart method in college mathematics instruction, in particular, in calculus class, and has been using this method to teach college mathematics course. The cored idea is how to find out “the smarter even most concise process” and how much ways existed in solving a mathematical problem.

Yang aware, independently, that it is much inviting, smart and concise teaching and learning method in mathematics even science to make use of the computer-flowchart, directed graph or framework- structure graph [5]. Based on the technique of designing flowchart, a process of solving a mathematics problem is constructed as an algorithmic procedure, moreover, the relationship among various concepts and logic derivations involved are illustrated, which further enhance the classroom effectiveness.

From these, students can observe that various concepts and theorems related to a problem are not isolated, but logically interconnected. The flowchart of the problem will make the logic connection and derivation more visible and more comprehensive as well.  We believe this way of instruction will enhance students’ ability of understanding, analyzing and formulating a mathematical problem, ultimately these will help their ability of problem- solving. The major emphasis of this pedagogy is that students explore more, in contrast to more traditional more proofs.

3. Examples-lecture design Lecture 1. Exploration on the Mean Value Theorems. Consider a function f continuous everywhere on a closed interval [a,b] and differentiable on the open interval(a,b). Immediately we have the Rolle’s theorem:   Rolle’s Theorem. Let f be continuous on [a,b] and differentiable on (a,b) . Also, assume that f(a)=f(b) Then there is at least one point in the open interval (a,b) such that

In the Rolle’s theorem there is a special condition and which makes the derivative of f(x) have at least one zero point in (a,b). It is natural to ask what conclusion may be drawn when . We suppose naturally that in this case the conclusion should be the following: there is at least one point in (a, b) such that where c is a non-zero constant to be determined. Now consider a new function ...(1.1) on [a,b] , and ask that:

Furthermore, if we replace in (1.1) by g (x): which value of c will satisfy the condition required by Rolle’s Theorem? From we can very easily determine that the constant : . Then we obtain or discover the new theorem:  Lagrange’s Mean Value Theorem. If f is continuous on [a,b] and differentiable on (a,b) then there is at least one point in (a,b) such that  Furthermore, if we replace in (1.1) by g (x): ...(1.2)

where g is continuous on [a,b] and differentiable on (a,b) where g is continuous on [a,b] and differentiable on (a,b) . Likewise, we need to find the value c satisfying ? Thus, by (1.2), we can verify that where we require that , or everywhere in (a,b) . From this we obtain another result as following

Cauchy’s Mean Value Theorem Cauchy’s Mean Value Theorem. If f and g are continuous on [a, b], differentiable on (a, b) and everywhere in (a, b), then there is at least one point in (a, b) such that   Now to summarize the explorations and thinking process, we give the flowchart for thinking and discovering some differentiable mean value theorems as following.

4.Comments of students & experts.

Comments from Colleagues “The flow-chart instruction of mathematics is instrumental in helping the logic relations with a notion or a method, it definitely is helpful in enhancing teaching effectiveness, and students’ capability of logic thinking. ’’ ----Zhang Jingzhong, Academician of CAS, Chairman of the subcommittee for mathematics of Higher Education Committee

“This methodology is useful in guiding instructors and students to systematically study mathematics. It is worthwhile to disseminate….” Professor Yu Bingjun, Board Member of Mathematics Society of Sichuan Province

Comments from students “…His method of teaching with the flowchart makes inter-relation among the isolated concepts clear…” “…I did not like mathematics before, now I begin to like mathematics, due to his unique teaching…” “…His teaching is the best I have seen, I strongly recommend to take his class…”

“This method is helpful in increasing students’ capability of mathematical modeling, and computer algorithm design for a real world problem…” ----- Professor Xu Daoyi, Sichuan University

We wish to thank Professor Deborah Hughes Hallett, for her guide and help, and Southwest University of Science & Technology, for its support.

References 1. Ruth M. Beard and James Hartley, Teaching and learning in higher education, Harper & Row, Publishers, London, 1984 2. Yang Hansheng, A Math-teaching Reform in Information Age, Journal of Mathematics Education, Vol.8 No.1 (1999) 34-37 3.Yang Hansheng, Computer-Algorithm-Type Teaching Methodology, A Teaching Reform Facing the Information Era, Mathematics Education in Universities, Selection of

Dissertation Abstracts, International Conference of the Reform of Mathematics Curriculum and its Education in the 21st Century, ICM 2002 Satellite Conference, Chongqing, China (2002) 33-34 4.Yang Hansheng, Teaching method in knowledge framework and system of engineering mathematics (awarded “National Prize of Excellent Teaching Achievement in Universities of China”, 1990). 5.Yang Hansheng, Framework and structure of Advanced Mathematics, Publishing Press of Science & Technology of Sichuan, 1985.

Thank you very much