WitCH 143: David and Kaye’s Excellent CAS Adventure

In 2022, David Leigh-Lancaster and Kaye Stacey had an article published:

Evolution over Two Decades of CAS-Active Senior Secondary Mathematics Curriculum and Assessment

I stumbled across the article a year ago, pondered for a year what to do with it, and I’ve finally decided: give up. Life is simply too short to address all that is annoying and wrong with the article, so I’m simply putting the thing here as a WitCH and people can do what they want. Or nothing. I don’t care. I’ve read the damn thing three times, which is four times too many, and I just don’t care.

In their article, David and Kaye take us on a twenty year tour of CAS in Victoria. It is an astonishing, monumental exercise in autohagiography. The authors couldn’t be more pleased with themselves. They go on and on and on, with never a doubt for the excellence of their CAS adventure. It is clearly just so, so wonderful what David and Kaye have done for Victorian education. There’s not more than a nanohint of a downside, not more than a nanohint that something big – solid skills and rigorous mathematical analysis, for instance – may have been lost. As to why we are pretty much the only place in the World that does this stuff to the extent that our students can’t now take a piss without using their CAS, that’s just a mystery.

David and Kaye realise, of course, that we shouldn’t simply be taking their word for things. It was important to also include the opinion of the teachers, the troops on the ground, in order to

broaden and test the authors’ views …

To this end, David and Kaye’s article includes extensive input from a random sample of four of their close mates. These mates are variously described in the article as,

four implementing teachers (the consultants)

four leading teachers

leading educators

[including David and Kaye] six expert long-term participants

four expert teachers

leading members of the Victorian mathematics education community

As to the value of the input of these leading expert teacher member participants, the article notes,

When reviewing this article, the consultants (and authors) almost always agreed with the comments attributed to others.

Golly. Will wonders never cease.

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Below is a subset of lines from the article that got up my nose. They are (not close to all of the) lines that I would have attacked if I had more time and less of a life. I’ve added no comments: this is a WitCH. The article is open access (which still makes it overpriced): people can check if I’m cherrypicking or out-of-contexting.

 

Abstract

The authors critically examined the strategic design decisions that were made for the initiative over time.

The place of CAS in senior mathematics is now widely accepted, partly because an examination component not allowing any technology has been implemented. Examination questions have become more general, which may have added difficulty, but more questions involve setting up a real situation mathematically.

 

1. Introduction

From the start, it was very clear that allowing CAS would have a much greater impact on examination questions and solutions than allowing graphics calculators and that teachers and students would need considerable support to use these more complicated devices well. The change would have to be handled carefully.

 

We will show how the deliberate alignment between curriculum, assessment, and pedagogy, has been critical to the successful adoption of a technology-active curriculum and assessment, in particular for the external examinations. This initiative needed a positive policy framework; strong and informed leadership; clear expression of intention with respect to curriculum and high-stakes assessment; a sound research basis with careful monitoring of teachers’ opinions; and substantial professional learning and resource support for pedagogy, all maintained over an extended period.

Expectations and practices change over time, and there has been, and continues to be, genuine critical discourse in Victoria by different stakeholders on what is reasonable, what is mathematically desirable, and what technology policies maintain a suitable balance for sub-groups of the student cohort, and for what pathways and purposes. This paper reflects various aspects of that journey from several perspectives, each variously concerned with what is best for students, making it all work, and progressing contemporary curriculum development.

 

2. Outline of VCE Mathematics Study Designs (2000–2022)

For mathematics, a stronger endorsement and expectation of technology use in Victoria is one of the main differences from other Australian states and territories.

 

School-based assessment involves students in mathematical investigation, modelling, or problem-solving on three extended tasks devised by teachers according to VCAA specifications. These provide opportunities for students to show how they are able to use the technology of their choice to model situations, represent problems or data, test what-if scenarios, and do computation (numerical, graphical, statistical, symbolic).

 

3. Strategic Management of Technology Change

High-stakes examinations, such as those for VCE Mathematics, play a dual role of leading change in schools as well as adapting to it, whilst always needing to maintain high public confidence.

 

It was also important that those who did two mathematics subjects were not required to own two calculators. Since 2011, this has been an important reason for assuming CAS in the elementary mathematics subject.

 

Hence, CAS became the assumed technology for all senior secondary mathematics subjects from 2010, … . This is an instance of successful strategic design implemented over a 10-year period [5].

 

3.1 A No-Technology Examination

Following consultation with teachers and universities, particularly in response to concerns (but not convincing evidence) that students using CAS may not develop adequate by-hand skills in algebra, from 2006 a new structure for examinations in advanced and intermediate mathematics was adopted. No technology was permitted in the 1-hour Examination 1 (40 marks), whilst the 2-hour Examination 2 (80 marks) assumed access to the designated technology. 

 

It is likely that without the two-component model, public confidence (including from the tertiary sector) that students are learning important aspects of mathematics may not have been adequately assured.

 

At the same time, the 1:2 overall mark weighting sends a strong signal that the curriculum recognizes that in professional use of mathematics, people have access to enabling technology and use it as applicable. Students should be prepared for this.

 

Research summarized in [3] showed that there was no evidence from the common ‘no technology’ Examination 1 of 2006–2009 that students using CAS developed weaker by-hand skills than students using graphics calculators (see also [16]).

 

Consultants 1, 3, and 4 strongly supported retaining the no-technology component and Consultant 2 stated that allowing technology in both examinations was equally viable, providing teachers were given sufficient support.

 

Consultant 1, who has been both a teacher and examiner, noted that it is possible to test understanding of algorithms in a technology-assumed examination, but such ‘show-that’ questions are very time consuming. She cited Specialist Mathematics Examination 1, 2021, Question 3c as a straightforward question that tested the procedure for finding a confidence interval but required no technology use or detailed calculations. … Consultant 1 commented that in contrast, in Examination 2, students could use the calculator to locate the confidence interval quickly, then move onto ‘something more interesting’.

 

4. Teaching, Learning and Examining Mathematics with CAS

The following main themes emerged, with general agreement across all four consultants. Examples that illustrate the main themes are provided in the following sections.

•  Examination questions have been adapted to the continuing expansion of the functionality of CAS calculators over time, in some part influenced by the Victorian curriculum and assessment. Various supplementary apps have been helpful, as students use them to streamline solving standard types of examination questions.

•  Students and teachers have increased visualisation through the use of sliders, dynamic displays, animations, and simulations to show or explore behaviour. CAS is often used by students to get a ‘preliminary idea/insight/explore the territory’ before proceeding with further work.

•  Teachers put more focus on efficiency and effectiveness in problem-solving approaches, so students will be more strategic about when and how to use the technology (think first, then do). Students give careful consideration as to how marks may be allocated as partial credit for constructed responses in examinations.

 

4.1 Adapting Questions to Expanded Functionality

… as Consultant 2 noted,

“students have also been making increasing use of pre-prepared files on their device for use in tests and exams. Many students go into exams with files that serve as organised templates for carrying out the necessary computations for standard questions likely to be on the examination. Some students have additional pre-prepared files and programs for solving particular types of questions. For TI-Nspire users for example, the under-appreciated ‘Notes’ application is ideal for this purpose.”

 

In the early years of the initiative, the examination paper was vetted by experts on each approved technology to check that there was no overall advantage to users of any one technology. Consultant 4 expressed a strong conviction that this aspect of equity extends beyond examination questions to having all models of devices well represented in textbooks and teacher support materials, so that all students can be taught about the range of capabilities of their own device.

 

As well as considering the hard-wired functionality in each CAS device, examiners have had to be aware of the ever-changing additional apps that students can upload to their devices (or perhaps only to certain brands or models). Some questions remaining may have limited mathematical value in the technology-assumed examination. 

 

Consultant 3 favorably identified that the greatest change that occurred in the Mathematics Methods examinations due to CAS use is that there are more questions that have a ‘problem-solving’ or an ‘explorative’ structure (e.g., 2016 MM Exam 2 Q4 and 2017 MM Exam 2 Q4). He noted that

“the inviting aspect of these questions is that they require a complete set-up of the question from development of the equation to solving via a CAS, providing a very efficient means for a solution”.

 

Consultant 3 also observed that the techniques of calculus now receive much less emphasis and feels that the mathematics behind the derivative is being lost. On the other hand, many questions now involve a deeper investigation of derivative-related mathematics; for example, questions about tangents and normals.

 

4.3.1. Increased Range of Methods Leading to a Deliberate Choice of Efficient Solutions

Some techniques to save time on examinations seem to have little intrinsic value outside of the examination room, but others are mathematically valuable.

 

Another mathematically beneficial side effect of using CAS is that students must become more versatile in recognizing mathematical expressions in a variety of formats. Every computer algebra system has a slightly different protocol for how expressions are presented, and these will be different sometimes to the standard output from by-hand algorithms, or other brands of CAS devices. Consultant 4 noted this as a positive effect on students, even though many students find it initially disconcerting.

 

4.3.2. Changing Curriculum Content

Consultant 1, for example, explained how she uses recursion functionality to give students at that level a better feel for concepts such as simple and compound interest and the difference between them (see Figure 7). She has less focus on the formulas and encourages these students to use recursion for relevant examination questions.

 

4.4.1. Increased Use for Visualiation [sic] by Students and Teachers

All the consultants noted that technology (both functionality and display) has improved over time and consequently the use of visualization has expanded. This expansion is by students exploring and solving problems, and also by teachers when developing students’ understanding of concepts. Sliders have been especially useful. Of course, not all teachers use this. Consultant 4, for example, commented that some teachers still choose to teach functions with the algebra and the graphing quite separate, perhaps especially if they do not have strong technology skills.

 

Figure 8 shows two examples used by Consultant 1 to get students to think and become more engaged in the learning process. She commented that good classroom discussion is supported by the slider visualization on the top row, when she asks questions such as what changes when a changes, and why are there two stationary points when a is negative and none when a is positive.

The second row shows two images in a series to teach that; when the sample size is large, the sample proportion from a binomial distribution is approximately normal. Students can observe and discuss the differences in the distribution of the sample proportion starting with the left-hand screen showing data from 100 samples of size 4 and gradually increasing to end at the right-hand screen showing 1000 samples of size 1000.

 

5. Discussion and Conclusions

Taking a wider perspective on the responses of the consultants, we see that they were in broad agreement with each other about the changes, challenges, and advantages that the introduction of CAS has brought to senior mathematics. … This gives us increased confidence that similar changes may have been noticed by a wide group of teachers, and that they do indeed arise from real changes in the examination questions and capabilities in technology rather than discourse in the community changing over time.

 

At the strategic level, the success of the initiative was supported by a bi-partisan forward-looking political environment; a positive policy framework within the VCAA; strong links with research; and long-term practical support from across the Victorian mathematics community in schools and universities, professional associations, and technology companies.

 

Whilst nearly all members of the Victorian mathematics community accepted that students should be well prepared for a world where mathematics is typically done in conjunction with technology, the initiative nevertheless challenged mathematical values and long-standing curriculum practices of both enthusiasts and skeptics, so it was essential to find a careful way forward.