When I first started teaching, I learned that the best thing to have students do after factoring a trinomial was to have the students check by multiplying out the binomials. At the time, it naively made total sense - students don't even need me to be there to practice! They can do this on their own while sitting on the subway or waiting for the bus - whatever dead time they have. The students that need to practice factoring can do as much of this as they need until they can factor with some degree of automaticity.
Some (not all) students took my advice. Of those that did, I often saw stuff like this:
x² - 4 = (x - 2)(x - 2)
= x² - 2x + x(2) + 4 = x² - 4
This was a worse situation than how we started - not only were they factoring incorrectly, but their inability to multiply binomials was giving them the false idea that they were doing a good job of factoring! This frustrated me to no end - even if I did give students time during class to practice and develop these skills, what could I tell them to do to improve outside of class? One colleague considered stopping giving homework because he saw it repeatedly reinforcing student errors. I didn't go that far, but I did start grading homework to try to find mistakes.
The missing piece for these students is the lack of useful and correct feedback. Most of them learned the procedures, but made arithmetic or careless errors such as leaving out terms when simplifying. Without any correct data to make decisions on, these students were just going through a procedure and generating incorrect results, and using the incorrect results to validate an incorrect procedure. If they had a way to generate correct feedback, this experience would stop being worthless and instead become a useful method for developing student skills!
This is where CAS systems come in - Wolfram Alpha is nice, but Geogebra CAS is even better because of speed. I worked with a student that needed practice both in simplifying polynomial expressions and factoring polynomials completely. This is what I had him do while he sat with me:
- Make up a pair of binomials of the form (x - 5)(4x - 5), multiply them, and then find the quadratic and linear coefficients. When you are ready, use the Simplify command to check your answer.
- Make up a product of polynomials of the form 4x^2(x+5)(2x-5) . Multiply it out all the way on paper, and then check your result using the Simplify command.
After this step, we talked about how he could do this on his own and check his work. While we were sitting there, he made mistakes, but was able to catch them himself. He was the source of the problems, and was able to check and see if his final answers were correct. We then moved on to factoring practice:
- Write out 15 products of binomials (3x-1)(x+5). For some of them, add a monomial factor. Include a couple sum and difference polynomials as well. Multiply any three of them out manually and check using Simplify.
- Use Geogebra to multiply any ten of the the rest of them and write down the resulting polynomials on a separate sheet of paper.
- Eat dinner, watch TV, or something that has nothing to do with factoring.
- Return to the paper and factor the ten polynomials you wrote down completely. Use Factor to check and make sure your final answers match what Geogebra produces - if there are differences, check to see if you have actually factored completely or not. Make a note of any repeat mistakes.
There is a whole lot of extra busy work involved in this process, but part of that is because it's easy to factor a polynomial that you just generated moments before if you still remember the factors. For some students, this won't matter, but it helps ensure that the exercises generates are actually useful. This student was on fire during class today, even though we were looking at a different topic entirely. I should have asked him directly whether this is the case, but perhaps the boost of confidence going through this process gave him is part of the reason. I also really like that this method allows the student to simultaneously work on multiplying polynomials and factoring them. My method beforehand would have been to stick to multiplying, then factoring, and then mix them up - there's no reason to do this.
Computer algebra has been around for a while. The reason I think it's now to the point where it can be transformational is that it's easily accessible, easy to use, and almost instant. This idea of using technology (and particularly Geogebra) to help students develop their pencil and paper skills is one that really excites me. I'm excited to see if it works with the students that came in a bit behind but are willing to put in the time to catch up. I don't want my class time to be spent learning algorithms - that defeats my strong belief that we should focus on teaching mathematical thinking, modeling, and problem formulation instead of algorithms. That said, students do need to be able to develop their skills, and this offers a personalized way to help them do so on their own.