The Use of Computer Algebra Systems (CAS) in Senior Mathematics
I didn’t introduce CAS to generate better results; I did it to prepare my students for the world as it is, not as it was.
For the past seven years, St. John’s Anglican College has utilized Wolfram’s Mathematica, a computer-based computer algebra system (CAS), as part of its mathematics classes, especially in senior Mathematics B and C. During this time, it has become clear to me through many conversations, occurrences, and self-reflection that teaching students mathematics without due regard for the computational technologies that they have available to them is limiting their engagement with and understanding of mathematics.
I want to be clear right up front that I still teach students how to solve problems using pen and paper as they are valid tools to utilize in many situations. What I am concerned about though is that I believe we, as mathematics teachers, give those particular technologies (the pen invented c.1880 and paper invented c.200BC) far too much credence and may have even fallen into the fallacious belief that using these technologies actually shows greater understanding. I hope to illustrate through a series of examples in current mathematics classrooms how we can see the double standards we apply in different contexts and how maybe we can better equip our students for mathematical success through the use of CAS technologies.
My first example highlights our excessive dependence on hand-written working out. I was helping a student I tutored from another school solve the following integration in preparation for her final Year 12 Mathematics C exam:
To solve this by hand involves converting the expression into partial fractions and then integrating each one. For this expression we’d equate it to the following and solve for A, B, C, D, E, F & G:
This type of question is regularly done in integral calculus courses, but the actual integration happens after the conversion. All the time and effort goes into getting the expression into a form that can be integrated; and a problem like this one takes a lot of time and effort.
After a long, arduous and repetitive process including correcting a number of errors we got to an answer. This is a difficult problem — well, it is if we insist on doing it by hand. But using Mathematica or a graphics calculator, we’d solve it like this:
But by doing it this way, haven’t we cheated? Critics say,“The students don’t understand how the method works; the machine does the thinking for them; you’re dumbing down their learning.” Yes, we’ve made what was a difficult, complex question into a simple one. But did that question ever really deserve to be held up as an example of mathematical complexity? Have we learned anything new about the concept of integration by doing it by hand? Does a student who solves this problem demonstrate any understanding of when or why to integrate?
Why is it acceptable to have a machine solve [some] problems but not others?
Integration by partial fractions is a method we employ to solve a specific type of integration problem. Conceptually, there’s no good reason to teach partial fractions in an integration topic. It’s a simplification trick for someone working with pen and paper so they can do the integration with pen and paper. If it happened that we could integrate the expression without needing to simplify, we wouldn’t need to cover partial fractions in the course at all.
The next two examples I’d like to consider are areas where we happily let a computer do the computation and I don’t know of anyone advocating that it is dumbing down the content or degrading students’ understanding of the concepts.
Then we convert 40° into radians and apply the Taylor Series to calculate the sine of the angle. At this point, some readers may not understand at all what I’m talking about, so visit here.
Why is it acceptable to find the sine of an angle by putting it into a calculator, without performing the underlying calculations? To actually solve it by hand is a long repetitive process where students are likely to make mistakes and it requires understanding that isn’t really necessary to the problem at hand, such as radians and factorials).
Secondly, we can consider the example of finding the square root of a non-square number. There’s an iterative process that students can use to solve that too. Why shouldn’t they demonstrate their understanding of that process? “Well, obviously you don’t need to know the process to understand what the square root of a number is.” Exactly.
My argument isn’t that mathematics students shouldn’t learn written methods. But I believe that they are part of an available suite and we should be showing students all methods and reasonably expecting them to apply the appropriate one given the complexity and requirements of the task.
Does your students’ lack of knowledge of the pen and paper method restrict their understanding and application of the principle? Why aren’t these examples railed against for making it too easy, or dumbing down the process? Why is it acceptable to have a machine, such as a calculator, solve these problem but not others? We know students are quite capable of applying their understanding of the concepts of trigonometric ratios and square roots to a variety of situations without any understanding of the mechanics that were previously utilized to solve these values in the pre-calculator era.
Again to be clear, my argument isn’t that mathematics students shouldn’t learn written methods. But I believe that they are part of an available suite and we should be showing students all methods and reasonably expecting them to apply the appropriate one given the complexity and requirements of the task. We should value as part of their learning the role that technology can play in assisting a student’s understanding of a concept.
So won’t this make maths tests too easy? Let us think back to the integration example: why are we asking a student to integrate a function like that in the first place? What is being tested is the student’s ability to have memorized and then apply a repetitive computational process, like a computer does. They don’t need to have any understanding of what integration is or what it’s used for, they just reproduce the process and get an answer. If it’s the right answer, all I’ve done is proved that I’ve trained a student who is well positioned to be replaced by a computer. They’ve done nothing I can’t do faster and with more accuracy on a computer. The problem is with the question. I need a human to understand when and where to apply integration, to be able to break a question down and identify what concepts are involved and what methods need to be applied, so I should write questions that test these skills.
Victoria has had many years’ experience in incorporating CAS technologies into their VCAA external assessment program. Leigh-Lancaster (2010) found that on technology neutral items students at the top end, academically speaking, from both cohorts (CAS and non-CAS) performed essentially the same, while students in the middle consistently scored higher and students at the bottom still tended to score slightly higher. As these were non-technology items being considered, the results seem to support the notion that having access to CAS was not a crutch for the students and didn’t adversely affect their understanding of the concepts assessed. In fact, it would seem to indicate that for those students who typically don’t do as well, CAS environment students had a greater understanding than a traditional non-CAS environment would have allowed them to demonstrate. I believe that CAS allow students, particularly those for whom mathematics is not comfortable, to engage with concepts that previously they were excluded from due to the need to memorize and faithfully reproduce a given procedure to demonstrate any understanding.
In the seven years that we’ve been using Mathematica with our senior classes and on their exams, we haven’t seen any ground-breaking changes. Some kids have struggled with the need to engage with the technology and others have embraced it and found it a significant advantage. I didn’t introduce CAS to generate better results; I did it to prepare my students for the world as it is, not as it was. There are more people walking around with a computer in their pocket than a pen and paper. What world are you preparing your students for? The world of the future? The world of the present? The world of fifty years ago? Mathematics hasn’t changed, but the tools and techniques used to interact with it have. Can your students use the tools at their disposal to solve the problems they’ll face?
References
Leigh-Lancaster, D. (2010). The case of technology in senior secondary mathematics: Curriculum and assessment congruence? Melbourne: VCCA. Retrieved 27 Feb 2016 from http://www.vcaa.vic.edu.au/documents/vce/drdavidleigh-lancaster.pdf
About the blogger:
Miles Ford
Miles Ford integrates Mathematica and the Wolfram Language into every aspect of his teaching of the Queensland and Australian mathematics curriculums. This includes using them to teach concepts through notebooks and interactive models, as well as student use of their computational power and programming potential across the learning and assessment spectrum. While primarily focused on the mathematics classroom, he also works with other teachers to take advantage of Mathematica’s power across the science, technology, and humanities fields.
Miles is the Head of Mathematics at St John’s Anglican College and has been using Mathematica since 2011, when he introduced it to his senior mathematics programs. Over the last few years, he has expanded its use throughout the mathematics department across all secondary school levels and into other curriculum areas. He particularly enjoys using the Wolfram Language to solve novel problems and helping students to develop their own solutions using Mathematica. Miles presented at the Australian Wolfram Technology Conference 2015 on the process of embedding Mathematica into the mathematics program.
