Lessons from CAS
Introducing Computer Algebra Systems (CAS) into mathematics classrooms is a lot more involved that simply identifying what buttons to press and what platform to use. Enormous pedagogical and paradigm shifts may need to take place. It can also be a challenging time for students, depending on the stage of their mathematical journey. Parents may also contribute to the equation. Then there’s assessment! This document refers to aspects of an 18 year journey through the continued introduction of CAS at the student, institutional, state and national levels. Specific interactions with students and educators will be used as waypoints along this journey.
Cassie’s Story – 2003 [Year 9: 14 to 15 years of age]
The first topic was Pythagoras’s theorem. Cassie didn’t complete the problem solving task, investigation and performed poorly on her test. The second topic was algebra, solving linear equations. It was very clear that Cassie was already in trouble. Speaking with Cassie directly, she mentioned that she had failed mathematics in Primary school (Elementary), Year 7 and Year 8. “I’m just not good at maths” was her self-diagnosis. Fortunately I had a TI-92Plus calculator loaded with the Symbolic Maths Guide. I generated some questions, showed her how to use the calculator and left it with her. Cassie worked solidly for the remainder of the lesson. She was happy that she could ‘solve’ the problems, but commented “But I still won’t be able to do the test”. I advised Cassie that she could use the calculator on her test and in fact she could use the same technology on her state level exams should she do mathematics in Year 12. Other students were intrigued and commented that using this calculator was ‘cheating’, evidence of how students perceive mathematics already at this level.
“What’s the first step you would do Cassie”, I asked. “You could multiply both sides by 4” she replied. Cassie worked through the problem step by step as I (the teacher) worked just like the calculator. Cassie solved the problem much to the dismay of her fellow students. Cassie went on to score more than 90% in her test, the first time she has passed a mathematics test in more than 5 years! She tackled mathematics with a new found confidence and optimism.
CAS can be used to help support student learning by shifting focus and providing supportive scaffolds that catch incorrect algebraic manipulation.
Chelsea’s Story – 2003 [Year 11: 16 to 17 years of age]
Four of the girls in this class would always sit together of which Chelsea was one. The group had been in the same class for the past four years and were all very studious. They were happy to do every question from the textbook and their exercise books were immaculate. These girls seemed to derive pleasure from getting questions correct and essentially racing one another through each exercise. The girls however were part of one of the first “CAS enabled” senior mathematics classes. Some of the students in the class did not like using the CAS, it was ‘cheating’, initially, Chelsea held this belief. As the course continued students started learning calculus. Most classes would start with a ‘problem to solve’, often an opportunity to discuss important concepts.
Students started working through the problem, books open, calculators on and heads down. Chelsea was just sitting there, staring at the board where the question was written. I asked Chelsea “Do you need some help?” “No, I’m done” she replied. I could see her book wasn’t open and her calculator wasn’t even switched on. “Okay, can you show everyone how you solved the problem?” “Sure” said Chelsea. She took her place at the board, drew a well labelled diagram, as you would expect from Chelsea.
Then she proceeded to announce: “I’ve drawn a line between the origin and a point P (x, y) on the curve. I would use the distance formula to work out an equation involving d, x and y; then substitute the equation for y. Now I have an equation involving just d and x which I’ll define as d(x). Now I can differentiate, as I need to find the minimum value for d, I’ll solve the derivative equal to zero; that would give me a value for x. Now I can substitute that value into d(x) to get the distance.” I was impressed! “So, what did you get for your answer?” I asked. “Do you want me to press the buttons?” she replied.
You can change student’s perceptions of mathematics.
Lizzie’s Story – 2004 [Year 12: 16 to 17 years of age]
A delightful mathematics problem involves paper folding. An A4 piece of paper oriented in ‘landscape’ mode is folded such that the corner of the page just touches the base of the page forming a triangular region.(Shown) Determine the location of the fold and the corresponding maximum area of the triangle.
The physical act of folding the paper and collecting the data is a great way to start this problem. Students generally let the ‘height’ of the triangle be represented by x. They can see that the hypotenuse of the triangle is therefore 21 – x. This gives them two sides of a right angled triangle so they are able to generate an expression for the base of the triangle and subsequently an expression for the area in terms of x. Using calculus students find that the maximum area is generated when x = 7.
Lizzie was very astute and asked “Will this always be the case? The optimum height is 1/3 of the height of the paper.” The beauty of CAS in this case is that students can work through the same problem but swap out the 21 (page height) for h and see the solution occurs when x = h/3 . For me, this is a beautiful relationship. Lizzie however followed up with the question: “Why?” “What do you mean?” I asked. Lizzie continued, I can see that now, but why? She could see the general solution. Even when the shape of the original paper changed, Lizzie could see that, within certain limitations, the optimal solution would be when the x = 1/3 the height of the paper. She wanted to know, why 1/3?
When you focus on conceptual understanding, students can ask some really tough questions!
Robbie’s Story – 2005 [Year 12: 17 to 18 years of age]
Dynamic environments provide for some amazing visuals that include the option for students to explore, collect data and generalise. One of the most common introductory calculus questions is identifying the maximum volume of a box cut from a rectangular piece of card.
Students were provided with pieces of card, scissors and rulers. As students created their open boxes they would measure the dimensions and volume and write their results on the board.
Robbie asked the question: “What height would you like me to measure?” I didn’t really understand the question; I thought it was obvious; the box has an open top. “Place your box on the table, with the open section facing up and measure the height. Make sure you ruler starts at zero or you will have to move your box to the edge of the table.” Robbie had already done this apparently, but didn’t know which ‘height’ to measure. I went over to look at Robbie’s box, he had cut rectangles from the corners of his original card; the ends of his box were higher than the sides! I sorted this out and made some mental notes. In subsequent years I would give students some dotted lines to cut along so they didn’t make the same mistake. I later decided it was okay if students made this mistake. What if they didn’t understand the seemingly obvious requirements for the original cut out?
It’s okay to make mistakes; it’s part of the learning process. Sometime dynamic diagrams can deprive students of the opportunity to make mistakes!
Teacher Development – 2006 [National Teacher Training]
I was providing teacher training in a country where they were piloting CAS. This particular country decided that they would introduce CAS at the start of student’s high school journey. The justification was based on a number of factors, mostly on the notion that it would not interfere with ‘high stakes’ exams, therefore teachers would be more likely to take ‘risks’ and be willing to explore. One of the problems was that many of the teachers involved in the pilot were not mathematics trained! The shortage of mathematics teachers in the education system resulted in a lot of out of field teachers taking junior level mathematics classes. These teachers could step students through algorithms, follow textbook examples and mark student work, but they didn’t really have a strong understanding of mathematics.
Activities were written to assist teachers, but teachers struggled themselves with some of the content. For example, a ‘traditional’ approach to helping students build algebraic expressions is to provide them with blocks, build specific constructions, record the quantity of blocks and identify a rule relating to the number of blocks and the build.
Problems similar to the ones above are included in textbooks and are relatively straight forward. Students write rules relating the number of blocks (squares) to the pattern number and then use the rule to predict how many blocks are required for pattern 100. What happens when you reverse this scenario?
Some teachers struggled with relatively simple variations. More conceptually demanding questions posed an even greater challenge for the teachers to solve.
Teacher’s knowledge of mathematics needs to extend beyond procedural.
Teacher Development – 2007 [Assessment Items]
As the pilot study continued, some national assessment items were being compiled. I worked with teachers and the relevant curriculum authority to help write examination questions. Teachers were struggling to write assessment items that could not be trivialised through the use of CAS. I suggested teachers look through past examinations and identify any questions that required ‘something more’ than an algorithm to solve, something that challenged conceptual understanding. Three years of papers and nothing surfaced, other than the realisation that assessment had focused very much on procedural knowledge rather than mathematical understanding.
Understanding this change is very challenging. Even textbook companies have struggled with the idea of CAS. One textbook company tried to encourage me (my school) to ‘update’ to their latest version of their textbook. I sat down with the company representative and we checked over the questions. Previous question items were all the same as their new book (CAS enabled). “What is the difference?” I asked. “We’ve put screen shots and calculator instructions in the new series” was the response. Some questions that were previously instructed to solve ‘by hand’ now stated ‘use CAS’; but the question remained the same.
A range of categories exist for classifying questions in a CAS enabled environment:
- CAS enabled
- CAS neutral
Trivial questions are those where a student should be able to solve the problem without CAS or that CAS provides no benefit due to the simplicity of the question. CAS enabled questions are such that appropriate use of CAS provides a significant benefit or makes them more accessible. CAS neutral means that the CAS provides no benefit as attempts to ‘solve’ reveal little or no information about the solution.
Assessment in CAS enabled environments forces us to reflect more on the actual purpose of a specific question. What is it trying to measure? CAS may be driving higher quality assessment items.
Martin’s Story – 2015 [Mathematical Accuracy]
The very nature of computer input is that it should avoid ambiguity. Similarly, the output is generally very precise. CAS can act as a mathematical litmus test.
These are just some of the stories that have contributed to my experience with CAS. No doubt there is much more to learn. One of the biggest challenges is to ensure that the learning continues and permeates into all classrooms so that it becomes a transformative tool rather than a misused one.