Educational Designer

This opinion paper explores and questions why numerical (iterative) methods, so widely used in business and scientific endeavours, are largely underrepresented in school mathematics curricula. Mathematics topics such as solving quadratic equations by using the quadratic formula or using algebraic procedures to solve problems in measurement, geometry or probability are deeply entrenched in the teaching strategies employed by teachers and in the resources developed to support those behaviours. Iterative approaches to the same topics offer an alternative, and in many instances more efficient, approach. They are often more suited to struggling students and can lead to greater success and deeper understanding of the underlying mathematical concepts. Classroom examples are offered, the potential advantages discussed, and finally some thoughts are given on why in this digital STEM age numerical approaches have not penetrated school curricula to the extent they could or should.

Am I so out of touch with prevailing realities, conventions, and purposes of maths programs? Or are we all so trapped in some methods, procedures, and traditions long past their use-by-date that they are nigh on impossible to be re-evaluated?

We build curriculum to provide students with all the tools they need for their mathematical journeys. Some years ago, I studied a university course titled ‘Numerical methods that (usually) work’ based on Acton (1970). It was about the power of empirical approaches to solving problems as an alternative – or additional – tool to go alongside formulaic approaches. I was entranced by the power of some of these approaches, so suited to this digital age, so suited to a wide range of student abilities, and I became convinced that within a few years they would feature prominently in school mathematics courses at all levels. That this has not happened is, and remains, a mystery.

In this article, which I offer as a highly personal commentary, I outline some samples of the approaches, and then discuss some various general advantageous features. Finally, I ponder why numerical methods have failed to penetrate school curriculum in any substantive way.

In my various roles supporting teachers to build or enhance their personal repertoires of teaching and learning skills, I have become a lesson collector - but I collect for a reason. The lessons I collect I argue are ‘interesting’. By this I mean professionally worthy of serious discussion as they illustrate how various current interest areas within maths education are being or might be interpreted at the classroom level. Their purpose therefore is as practical vehicles to generate professional learning discussions at the school level around various current issues and interest areas within school maths programs.

The lessons that I find, collect and codify have titles such as Multo, The Architect’s Puzzle, Snakes and Ladders, Heads and Legs, Maths in Motion, Algebra Walk, Licorice Factory, and Fraction Estimation. They all currently stored within the Australian Association of Mathematics Teachers Maths300 project (AAMT, n.d.). They are available to professional development leaders anywhere to plug them into their networks to generate discussions about notions of problem solving and reasoning, meaningful contexts, genuine understandings, effective use of technology, equity and differentiation, pedagogical strategies, assessment techniques, explicit teaching, learning theories, interconnectedness, and the ever present balancing act between skill development and open-ended investigative approaches. In other words, all the big ideas that can feed into teacher design of classroom lessons. By and large I feel moderately satisfied that the professional discussions have been productive as teachers constantly seek to expand the quality of their personal teaching repertoire and to develop rich and balanced programs.

When I reflect on lessons that have had an impact and those that haven’t, there is one particular lesson type that stands out. It disturbs me. It has not had the impact I think it deserves, and as mentioned above, that mystifies me as to why. The university course I studied and subsequently explored in classrooms used iterative empirical approaches very different from traditional formal analytic approaches. I was impressed and convinced that these methods could and should become commonplace and replace much of traditional formula methods. Indeed, these approaches are very evident in mathematical modelling in all sorts of business and scientific contexts such as climate change, economic theories, weather forecasting, population projections and town planning. Yet school mathematics structures have largely ignored this reality.

To illustrate this lesson type, I offer the two classroom lessons below. The major idea they both highlight and promote is an empirical iterative approach to problem solving. I hope the reader will indulge my argument by way of personal experiences.

As my students progressed, a few interesting and important learning features became prominent. Although these are informal observations from myself and other trial teachers, each of these seem worthy of serious discussion and analysis with teachers as they explore whether these numerical methods have long term potential at both a personal and systemic level.

1. Mathematics. This iteration is an utterly legitimate mathematical process. Indeed, in this electronic digital age many, if not a majority, of real industrial economic and scientific problems are being solved using iterative techniques.
2. Understanding. There is a strong focus on the concept of the iterative process through what is essentially a first principles approach. Nearly every student could articulate to me what they were doing and why – at any stage of the process.
3. Access, Differentiation, Mixed Ability. All students could make a guess and get started. The formula approach inhibits access unless the algebraic facility has been well established.
4. Language. The presentation and discussion highlight the connection between everyday language and the language of mathematics.
5. Does iteration take too long? My students were solving a problem in about 3 minutes. That’s about the same time the traditional approach takes, so it is not less efficient in terms of time.
6. Connections. Because it is closer to a first principles approach than just using the formula, students were actively interconnecting ideas. For example, when finding the area of a circle, students connect ideas of area, algebra, and decimals.
7. Technology. The effective and extensive use of calculators in this current era of STEM is to be encouraged.
8. Estimation. As students began to achieve success, they became much better at making a ‘next guess’ estimate.
9. Mistakes. You can make a mistake and recover. For example, making an estimate that is ‘worse’ than the previous one is soon noticed, and a better guess created.
10. Discussion. The process seemed to encourage beneficial small group work and associated discussion in a manner that does not occur in the formula approach.
11. Level of Accuracy. Several of my students wanted to get the answers exact and worked out answers to 4 and 5 places (and then were disappointed when the calculator of the day could not ‘go further’).
12. Confidence. Students became faster and more confident as they started to ‘churn out’ correct answers.

Over years of successfully exploring and enjoying these approaches, I have had much cause to ponder why they have not become more widespread, particularly at systemic policy levels.

Are numerical methods currently recognised and promoted within school policy and scope and sequence documents. It seems not. The current (2022) page of Wikipedia on quadratics (“Quadratic equation”, 2022) has comprehensive coverage of solving quadratics but it reveals not a single word about numerical methods. Similarly, other web searches such as ‘What is the fastest way to solve quadratics?’ or ‘What is the best way to solve quadratics?’ all mention factoring, completing the square, using the classic formula, and graphing but not iterative techniques. Graphing calculators use iterative software, but this is often automated and hence the logic of the approach is often masked and not made explicit to students.

A second obstacle to adoption may be that there is some confusion of purpose for these tasks. For example, is the Friday test’s area task (finding the radius for circle area 40 cm²):

1. primarily about area and providing students with an algebraic tool to assist with the calculations?
2. primarily about building algebraic manipulation skills so the topic of area is merely a convenient context in the service of that algebra?
3. primarily to showcase the power and beauty of the number π?

It is probably parts of all three, but I contend that for struggling Grade 8 students these three are all largely unattainable. If the purpose is about area-related calculation then it seems to me a ‘no-brainer’ – many more students will benefit and be empowered by the iterative approach, gaining understanding and success. If the purpose is about the undoubted extraordinary beauty and usefulness of π, then Grade 8 is arguably not the place for this to happen for a majority of students.

Third, whilst ‘business is booming’, curriculum is not. The Australian Curriculum: Mathematics (v.9) (ACARA, 2022) is the formal policy provided to Australian schools to guide courses. A search reveals very little mention of iterative problem solving techniques anywhere in the Grade 8 to 10 guidelines and the two current textbooks I consulted did not present it in any way. While largely absent from school curriculum, it is not so in business and scientific endeavours. Any search will yield multiple references and scenarios to the widespread use of numerical algorithms. Sundaram (1998), for example, commented:

Optimization methods using differential calculus can be applied to solve certain problems but as the problem becomes too cumbersome then classical methods get replaced by iterative techniques. This process of finding solution iteratively involves extensive computations and there are several algorithms on iteration. (p. 1)

Just recently, while preparing to finish this article a neighbourhood student sought my assistance in the following word problem which had been set for homework.

A rectangular floor vent is 12cm long and 6cm wide. It is to be enlarged by increasing the length and the width by $x$ centimetres.

1. What is the new length of the floor vent (in terms of $x$)?
2. Show that the area, $A$, of the new vent is given by $A=x^2+18x+72$.
3. The area of the new vent must be 50% more than the original area. Find the dimensions (to the nearest centimetre) of the new vent.

I found the task as presented to be rather depressing in two fronts.

First consider the context. Word problems are often used to provide contexts or scenarios to illustrate how mathematics is applied to ‘real world’ problems. But the scenarios are often spurious and the main purpose being to practise use of formulaic procedures. Here, no effort is made to establish a meaningful context. Presumably the question intends a vent for heating. For example, asking students to check their household to see if there are vents and of what size might establish the context as meaningful. The most common vent in house heating here is 30cm by 10cm which presumably is sufficient in size to provide appropriate volumes of air. A size of 12cm by 6cm is not at all common and probably insufficient for heating. Why does the author not account for this? The numbers are misleading and clearly contrived solely for the quadratic algebraic purpose, not for any meaningful problem to be solved. Pitman and Pateman (1985) in their report to the Australian National Mathematics Curriculum and Teaching Program produced an overview of ‘What’s wrong with school mathematics?’ by surveying teachers, students, parents, universities, and the wider community. At the top of their list was ‘boring’ closely followed by ‘irrelevant’. The above type of word problem does little to alleviate these criticisms.

Second, consider the solution method. The expected explicit solution method is to generate a quadratic equation, to recognise the three coefficients, and then apply the quadratic formula. If the purpose was to solve the problem then this quadratic pathway is a very poor choice when the problem is solved numerically in a fraction of the time. If the purpose is to solve the problem (a big ‘if’), then it yields quickly, efficiently, and logically using an empirical approach.

Students need to be building a toolbox of solution methods and strategies and to learn to choose the appropriate tool for the task. In many tasks, such as those discussed here, the formulaic methods seem to be wholly inappropriate and the empirical method vastly superior. I finish with a hope embedded in a question. What needs to happen in curriculum planning to recognise the benefits of numerical approaches and to enshrine these in an appropriate manner into general systemic curriculum policy?