Ed Tech 4 Math

A university professor once gave an assignment to his students at a major university in the United States in which the students had to write an essay addressing the following question: “What is risk?” One of the students responded with just one sentence and his name on a blank piece of paper: “This is risk!!!” The professor was dazzled and gave an A+ to this three word essay.

Next year, the same professor gave the same assignment to a different group of students. One of the students, familiar with what happened the last year, submitted a similar essay which received an F along with the following comment made by the professor: “It is foolish to take the same risk twice.”

Learning from experience is an important issue for everyone. No matter how many times you tell a child not to approach a hot stove, he will learn this only after he gets burned once; a Turkish proverb goes as follows: “One experience is worth a thousand warnings.”

In fact, learning from experience is absolutely essential sometimes, especially while we are teaching mathematics. Every so often, we want our students to make some very critical mistakes and we create questions with a variety of traps that our students will very likely fall into. The reason is simple. When our students make such mistakes, we will have the opportunity to bombard them with a heavy load of knowledge.

Experience does not only involve learning from mistakes; it also comprises learning from experimenting. At this point you might start arguing that mathematics is not an experimental branch of science. With all due respect, this is not true. We can conduct experiments and we are very fortunate that our experiments only require a pen, a few papers, and our brains.

Follow the link for a more thorough exploration of this idea.

Here is an interesting contest problem: x is a 99 digit positive integer where each digit is 9. What will be the sum of the digits of the result when x is squared?

Let us conduct a few experiments:

9^2 = 81

99^2 = 9801

999^2 = 998001

9999^2 = 99980001

99999^2 = 9999800001

999999^2 = 999998000001

We see that in each operation, the digits of the result sum up to 9 times the number of digits of the original number. So the solution to our original problem, by the help of this little experiment, is 9 times 99 which is equal to 891. Please note that students are allowed to carry out each of the above operations with their handheld calculators.

Here is another interesting problem (again, being solved with the assistance of technology, in this case, a graphing calculator): How many real solutions does the equation given by 2^x = x^2 + 1 have?

When we ask this question in a precalculus class, our students will be inclined to try out the trivial numbers of 0 and 1 first. Luckily, each of these is a solution to this equation. Now, let us graph y = 2^x and y = x^2 + 1 on the same coordinate plane. How many points of intersection do we see? Not 2, but 3; therefore there are 3 real solutions to this equation and we have come up with the correct answer using technology. Some of our colleagues may argue that we do not need technology to graph the curves y = 2^x and y = x^2 + 1; in an algebra 2 or precalculus class, many (if not all) students gain the necessary knowledge of the subject matter in order to graph these curves by hand, I definitely agree. However, there may be other cases where the curves are harder to sketch, and in such cases they will have to use technology.

We can use technology in a variety of ways. We can make the students perform investigations by experimenting with simple cases to make deductions and arrive at generalizations and theories by themselves instead of bombarding them with definitions and rules. We have the computers, handheld devices (including scientific and/or graphing calculators), the Internet, and brilliantly engineered applications that are already there to ease the practice. So why not employ them to make the process easier, more fun and durable? Remember the Turkish proverb? “One experience is worth a thousand warnings.”

This entry was posted on Friday, August 15th, 2008 at 1:18 pm and is filed under Education Research, Educational Technology, High School Math, Instructional Leadership. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.