## Tuesday, October 30, 2007

### Ideas for Math Pojects

I've included a few ideas for math projects that allows students to investigate a problem. These problems are not run of mill and hence they will provide students a glimpse of how mathematical research can is done. This list is certainly not exhaustive. It is targeted to juniors only. The ideas are grouped according to major topics in mathematics.

Arithmetic/Number Theory
There are many sets of interesting numbers. Examples include polite numbers, amicable numbers and frugal numbers. Come up with your own set of numbers with some special properties and investigate.

Combinatorics
How many latin squares can be formed using a 3 by 3 grid? What about an n by n grid?
What is the answer to the 36 officer problem?

Geometry
Investigate the tiling of polygons. What polygons can be tiled and which can't?
Investigate the ways of estimating the volume/surface area of irregularly shaped object

Applied mathematics
What is the most efficient way to insert people in a duty roster whom have their own preferences?
Do you get more wet when you run in the rain or when you walk?
What is the fastest and cheapest way to get to a location in Singapore (for instance home to school) using MRT, Train and Taxi?
How do we program a lift so that it can transport people efficiently?
How can we estimate the volume of the Earth?
Why do buses sometimes come in pairs (or threes)?

## Saturday, October 13, 2007

### Nurturing the Mathematical Mind

We use mathematics in the course of their lives. However, we’ll possibly only need to use everyday math like arithmetic, percentages, ratios, etc. These are the content taught at the primary school as well as in the secondary 1 and 2 levels. But what is mathematics education about really? Surely, most of the students will not become mathematicians. How many of us are going to use logarithms in our everyday lives? How many of us would apply trigonometry identities to solve problems? And how many of us need to apply techniques in integration when many calculus problems can be solved readily with the use of computers? Mathematics education should not just be about learning content but about the thinking process. This article will discuss the many dimensions of mathematical thinking and how it applies to everyone.

Identifying Patterns

Identifying patterns is what mathematicians do. Looking at a set of mathematical objects, we can classify them according to certain attributes. There might be patterns involved that interest us and these patterns may be used to draw conclusions. At the school level, students come across many patterns in mathematics lessons. However they may not be aware of it.

Given a number pattern, what comes next?

What pattern do you observe regarding sine, cosine and tangent ratios?

What pattern do you observe in the y-values of a quadratic graph?

Observing a pattern is one thing, interpreting and drawing conclusions is another problem. For instance, given a number pattern, can we formulate a general formula for the nth term? Given a quadratic graph, how and why do the values of y change? All these questions relate to patterns.

We all see patterns in real-life. For example, stock market analysts try to identify trends in graphs to establish times to buy and sell shares. Biologists study patterns in the spread of epidemics to establish the source of the infection. Business executives study trends in the use of products in order to decide on which product to market. All these situations require the mathematical mind.

Recognizing Relationships

A mathematical mind does not only observe patterns, it is also able to establish relationships between different areas. In the classroom, students have to recognize relationships too. For instance, how does the sum of interior angles of a polygon relate the number of sides? Does changing the number of sides of a polygon change the sum of exterior angles? In medicine, a doctor might relate chocolates to the elevation of depression in his patients. How does he arrive at this claim and how does he test his hypothesis? An anthropologist studies the height of individuals over the past 100 years. He relates the increasing height of individuals to the better standard of living the world now has. All these relationships are identified via the mathematical mind.

Analyzing arguments with rigor

Mathematics as a subject is not without rigor. Proofs must be presented with absolutely no loopholes. For instance, in the classroom students learn geometry. It is apparent that when proving geometrical theorems, we cannot exemplify proofs using concrete examples but also proof in a mathematical logical and general way. This is done either by induction or deduction. Now for instance, we want to prove that the sum of interior angles of a triangle is 180 degrees, we cannot just draw one triangle, measure each angle and ascertain our result. This is an example of a fallacy in analyzing statements. One must prove in general. One way of course is to use geometrical axioms to help us establish this fact. Starting from the foundations of geometry to help us build the proof with no fallacies. When we are outside of the classroom, we should also analyze statements, assertions with our mathematical mind. For instance, there are always many reports of health issues. Should we believe them all? For example, does eating aspirin reduce the risk of coronary disease? We should use our mathematical mind to analyze and scrutinize statements to better enable us to make the right (or rather better) decisions in everyday life. The mathematics student will be able to identify the following fallacies in thinking:

• Cyclical reasoning
• Linking casual relationships of cause and effect
• Using particular examples to draw general conclusions
• Using obvious results to draw conclusions
• Misinterpreting data

Synthesizing arguments with rigor

We should not only be satisfied with only analyzing statements made by others. The mathematical mind is able to synthesize arguments as well. By avoiding the pitfalls in reasoning as mentioned above, one can formulate statements, prove assertions, draw conclusions in our everyday lives and hence make the right (better) decision.

In summary, mathematics education is not just about learning mathematics concepts and techniques. It is about nurturing the mathematical mind which lies dormant in all of us. By learning mathematics, we learn to awake the mathematical mind and help us identify patterns, recognize relationships, analyze and synthesize arguments.