I recently began to teach my S4 class about graphs of quadratic functions. Based on my previous experiences, this is a topic many pupils struggle to cope with. In this post I will offer some possible reasons for this and some suggestions on beginning to address them. A further post will seek to develop this approach.
I should say at the outset that most of what I recount here will reflect what many teachers are already doing. What has changed for me is that I feel I have been able to plan and deliver a more coherent set of lessons that seek to link procedural knowledge to conceptual understanding and to give my pupils a greater sense of ownership of their learning than I have managed in the past when teaching this topic. This would appear to be borne out by my informal observations and assessments of their progress to date.
The reasons pupils struggle with this topic are no doubt many and complex. However, I would conjecture that many learners fail to grasp the following concepts, and the ways in which they interact :
- the graph of any function is simply a visual representation of its algebraic form;
- there are correspondences between certain features of both the algebraic and graphical representations of a given function;
- there are a range of “families” of algebraic expressions: graphs of their associated functions – just like human families – have characteristic features that identify them as belonging to one family and distinct from others;
- the different algebraic representations of a quadratic function mean that pupils have to learn the ways in which each form links the algebraic and graphical features; this can overwhelm some learners.
The above points, along with prior and intended future learning, informed my planning for this topic. Below I describe the first lesson.
As pointed out by Mark Horley in a response to my previous post, context is crucial in understanding a learning episode. I shall, therefore, provide a brief sketch of relevant prior learning. The class had previous experience of multiplying together two linear expressions to generate a quadratic expression, along with the inverse operation of factorisation. In addition, they had learned to transcribe a quadratic expression from standard to completed square form and vice-versa. These three skills, in fact, formed the basis of the opening activity for the lesson. My learners had not yet encountered solving a quadratic equation. I taught them function notation as a prelude to the current topic, alongside finding the value of the function for a given argument, and vice-versa. In addition, they had a thorough previous grounding in linear expressions and their associated graphs (we did not use the term function at that point).
As I’m sure a great many teachers would, I started with the graph of
y = x2
by having pupils compile a table of values from which coordinate points can be extracted and plotted. My rationale here was to allow learners to see that the shape of the graph is a direct consequence of the algebra: for a given point on the graph the value of the y-coordinate is the square of the x-coordinate. While I made extensive use of graphing software (Desmos) later in this lesson and throughout the topic, I felt it was important in this early phase that learners experience the whole process by which the algebraic representation generates the graphical one. While not quite concrete, I do feel that this exercise is somewhat enactive in character (to use Bruner’s terminology): it certainly puts the onus for generating the correct graph on the learner rather than on the “black box” that technology can be, at least when used indiscriminately.
From the resultant graph I then pointed out the following features, introducing appropriate terminology and linking back to the squaring operation to emphasise the connections between the algebra, the coordinates and the various characteristics of the graph.
- The shape of the graph is known as a parabola:
- the parabola “bottoms out”, i.e. it has a minimum turning point;
- The coordinates of the turning point are (0, 0) (since 0 is its own square);
- the parabola is symmetrical about the y-axis, i.e. the line x = 0 is an axis of symmetry (this is, of course, a direct consequence of the fact that the squares of any number and its negative have the same value);
- the graph meets the x-axis at x = 0 (a root);
- the graph meets the y-axis at y = 0 (the y-intercept).
The whole (enactive) process was repeated for the graph of
y = –x2
which, of course, produced an inverted but otherwise identical parabola. This illustrates the idea that the curve can “max out”, ie have a maximum turning point. I introduced the term nature to highlight the distinction between the two types of turning point.
Employing Desmos now to pick up the pace of learning, I displayed a range of quadratic functions allowing the pupils, in each case, to both notice and describe, or locate, the five key features of the resultant parabola, namely:
- the nature (N) and coordinates of the turning point (T)
- the equation of the axis of symmetry (A)
- the location of the roots (R) …
- … and of the y-intercept (Y)
(Note that the terms in bold are those used by the Scottish Qualifications Authority, which administers the exam system here; I’m aware that other jurisdictions may use different language.)
Pupils, working in pairs with around 30 seconds thinking and discussion time, appeared to quickly become conversant with the terms and adept at providing correct answers, including cases with no roots, a repeated root and distinct roots. As a homework task, each pupil was asked to take the five letters denoting the key variable features of the parabolic graph of a quadratic function and come up with a mnemonic, i.e. a memorable phrase to encapsulate this learning. You may, by now, have realised the significance of the title of this article!
Here are some examples of their efforts.
I felt this activity gave pupils a sense of ownership of the intended learning; as a bonus, it also lightened up the topic – we all had a few chuckles as these were shared around the room! In fact, in subsequent lessons, I have used a different mnemonic each time to acknowledge the individual efforts and to foster that sense of constructing a shared understanding, another underpinning principle of Bruner’s pedagogical work.
On the topic of using mnemonics, this was discussed during a wonderful CPD event (Multiple Representations in Mathematics) I attended last week hosted by La Salle Education and delivered by Chris McGrane. We all agreed they do what they say on the tin (i.e. they aid remembering), but that they should never be a replacement for deep conceptual understanding. I must confess I have fallen into this trap in the past, not only by giving my own mnemonic phrase (“Never Read Your Auntie’s Texts”) but, more significantly, by neglecting to plan learning experiences that allowed pupils to make sense of these concepts, fundamental as they are to Cartesian analytic geometry and crucial in accessing the study of maths at a higher level.
I believe I have done a better job with this group of learners, building on the more careful and better-informed planning and delivery that comes with increasing experience, critical reflection and invaluable professional dialogue, for which I sense there is a real and growing appetite at present, particularly among the online mathematics-teaching community. Long may it last.
Post-script: learning, teaching, doing … and blogging maths
As noted above, engaging with the online maths-teaching community has helped me begin to transform my teaching. My decision to start this blog-site was inspired by those who have been brave enough to hold up their principles and practice for public scrutiny in order to generate productive discussion among others who are determined to move mathematics education forward at grass-roots level. Furthermore, I’m finding that the blogging process allows me to stand back and critically examine my own practice in ways I have been unable to do before. In fact, while re-constructing this particular lesson here I have identified an opportunity to improve my future practice. When displaying graphs generated by Desmos to prompt discussion among my learners and to assess their understanding of the intended learning, I used the standard form,
y = ax2 + bx + c,
to input each function, in full view of the learners. With hindsight, and with the ability to read this narrative as an observer of my own practice, I can now see that I could have better begun to address my earlier conjecture that
“the different algebraic representations of a quadratic function mean that pupils have to learn the ways in which each form links the algebraic and graphical features: this can overwhelm some learners”
by expressing the functions in question in a mixture of standard, factorised and completed square forms, whilst verbally emphasising these different but equivalent algebraic representations. Another small step towards becoming the teacher I want to be.
I hope, like me, you have found this reflection on a learning episode useful. Thanks, once again, for your time.