Most students did not have proofs covered in geometry, so this is their first look at the idea of proof. This is changing, as geometry classes are again covering proofs but that isn't the case for the students in this class.
The issue is relevant because students are blindly solving problems without any justification or checking as to the reasonableness of their solutions. I used one problem that many students tackled on their portfolio problem as an illustration. Every one who tried this problem got it wrong and in the same way.
The problem was if you take the first 8 letters of the alphabet and require that the letter sequence ABC always appears, how many different arrangements of the letters can be made. Students stated the solution was 8C3 because the had 8 letters and they needed to keep the 3 letters (ABC) together. There was no checking, verification, or other justification. While this is a nice conjecture, it needs to be affirmed as to its reasonableness. It basically needs to be proved that the solution always works.
I reiterated that larger problems should be broken down to smaller entities so that the reasonableness of solutions can be checked. In this case, making a table of the number of additional letters and the number of arrangements quickly shows that nCr does not work.
I moved from this to the concept of proof. I referenced back to triangular numbers and the closed form of the formula. The first question posed was to get students thinking about how to demonstrate a formula always works. As expected, students thought you could plug in values to test the solution. I then asked how many values is enough? Do you need 10, 100, 1000? How do you know if the formula will still work on the 1001st term?
To get students thinking about what it takes, I then asked them what they would need to see in order to be convinced. Students struggled with this. There was some thinking about seeing a pattern that clearly repeated.
I conveyed the basic tenets of mathematical proofs:
- Proofs use accepted truths (called axioms)
- Proofs are a series of statements that are logically connected
- Proofs demonstrate that the proposition will always be true under the stated conditions
The last tenet emphasizes that the context of statement truth needs to be considered. For quadratic equations, all real solutions fall on the x-axis. However, this statement is not true over the field of complex numbers. The truth of the statement is within the confines of the stated conditions.
I asked students to consider why proofs are important. There were some good assessment of the need for proofs:
- They verify the validity of a statement
- They create a foundation to build upon
- They provide understanding of a problem
I then introduced mathematical induction as one way of proving statements or equations.
I presented the steps to conduct a proof by induction:
- Demonstrate a statement is true for a starting value.
- Demonstrate that from any given starting point (k) you can reach the next step (k+1)
and asked students why this would demonstrate that a formula always works?
Students were wrestling with the ideas and had no clear understanding of the logical basis for what was happening. I used the analogies of climbing a ladder and dominoes falling to try and make a concrete connection between the concept of induction and something that students could grasp.
I tried starting things off with a simple example. I think a different example needs to be used here, perhaps using the sum of the first n integers or the sum of the first n odd integers? For this class I used every positive even number can be written as 2n for n > 0. I started with a value of n=1. In this case 2 x 1 = 2, which is even. I then assumed that 2k was even. The next even number is 2k + 2 (having discussed the recursive formula that En = En-1 + 2. Then 2k + 2 = 2(k + 1) so the formula works for the next even number and by induction will be true for all positive integers.
Students were confused and, not surprisingly, many seemed indifferent to the idea. I asked them to try using the idea of induction to prove the triangular number result. Most got stuck and had to be led through the process. Some students managed to figure out how to demonstrate the result for n=1, but didn't know where to go from there.
We'll try working through one or two more before diving into number theory and cryptography, where proofs will show up again.
Although it was a struggle, I think it is worthwhile to convey the ideas of proof and to have students work through the reasoning and logic that proofs require. We'll continue to look at proofs throughout the semester. As a first lesson on proof, I think students started to understand this was an important, and perhaps difficult, task.
View the class summary for a student perspective and to see the lesson slides.
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