The last discussion on teaching the quadratic function was fun, but it left me with some questions, and these clarify that there is a long road ahead.
The recipe for re-engineering didactics of mathematics is:
When you see a student struggling with a topic, ask two (introspective, sympathetic) questions:
– What causes that a (and/or this particular) student experiences this struggle ?
– What causes the mathematics community to create this didactic situation ?
Subsequently put your hunches to the test, and recycle till you have a satisfactory answer.
Then proceed with experimental design and hypothesis testing.
Conventional studies in didactics consider only the first question. They assume that the math is sound and that the struggle is caused by external factors like the use of the calculator or perhaps the order of topics in the curriculum. I agree that much can be improved on these external factors, but my diagnosis is that we better first solve the internal problems within “mathematics” itself. A student who struggles with a math question is often quite capable in dealing with complex issues (check what kids can learn) and a struggle indicates that the cards are rigged. The large number of examples of crummy “mathematics” that has already been identified in my book Elegance with Substance (pdf online) shows that “mathematics” education and its conventional research have lost their innocence and even the benefit of the doubt, and are actually the prime suspects.
Didactic divide from counting to addition
I am not qualified for elementary school and the following is only a hypothetical idea, to provide us with a useful example for below.
In elementary school some kids don’t quickly master addition, and get stuck in counting only. For sums like 7 + 2 = 9 they rely on their fingers, which eventually becomes awkward for a sum like 2 + 7 = 9. The didactic question is rather not why these kids get stuck but why others take the leap across this divide from counting to addition. Apparently, a majority of kids have a way of mastering both commutation that 2 + 7 = 7 + 2 and also the tables of addition, so that the outcomes are automated, and are available directly from permanent memory without the need to find the result by actually counting (and check and double check). Didactics helps to identify the points for a natural learning process: (1) awareness of the didactic divide, (2) diagnosis of what is the cause for slow progress (not math itself), (3) treatment, namely create the tables of addition, create awareness of these and the properties, and practice makes perfect. In this manner the teaching environment can help the slow kids to find the learning route they were unfortunate not to find themselves – and later they may be fast themselves again on another subject. (See Henk Boonstra on elementary school reform w.r.t. structural slow and fast.) Learning goals are, even at this age: (a) knowing how, (b) knowing why, (c) knowing about knowing. A child who has mastered addition has every right to be proud of it, and can be invited to explain what it has learned.
PM 1. For numbers above 10 see A child wants nice and not mean numbers.
PM 2. This also clarifies why the tables of multiplication must be known by heart, to cross the divide between addition and multiplication.
Quadratic functions revisited, conventionally
The last discussion on teaching the quadratic function caused a focused search on what others wrote about its didactics. We find a similar issue of exploration versus automation (counting versus addition). Perhaps we may speak about a didactic divide between quadratic equation solving (factoring or completing the square) and automation (quadratic formula). The situation is a bit more complex than with counting and addition (perhaps also because it happens in highschool and not in elementary school). The quadratic formula can be automated only for the standard polynomial form. Handling the quadratic function requires a certain level of algebra overall.
Let us first follow conventional didactics that assumes that the “mathematics” is sound. In conventional didactics, “completing the square” is a solution technique, and not the basic form. The quadratic formula is called an algorithm, a fancy word for recipe. However, the other solution methods follow recipes too, and the key notion is automation.
Colin Foster (2014), ‘Can’t you just tell us the rule?’Teaching procedures relational, distinguishes procedural knowledge (knowing how) from relational knowledge (knowing why), and finds two uses for the quadratic formula: (1) to learn more about how and why, perhaps prove it, (2) to use it automatically with a focus on a different target. In itself this is sound, but this doesn’t generate a specific didactic strategy yet, except perhaps the order first (1) then (2). Foster laments the current “political” focus on teaching to the test, and thus (perhaps) emphasis on the quadratic formula. Foster usefully indicates that tradition has been strong and research has been weak:
“Vaiyavutjamai and Clements (2006) comment on the lack of research into students’ difficulties with quadratic equations, and since then a number of studies have explored this area (…).”
David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), Evolving a three-world framework for solving algebraic equations in the light of what a student has met before, suffer from Tall’s idiosyncratic verbosity on “embodied” versus “crystaline” notions (see related critique), but it is helpful to see some of the literature and experiments and student confusions. De Lima (2011), summarizes a study in Portuguese:
“In this paper, we present an analysis of 77 14-15 year-old students’ work with a non-familiar situation: the solution of a quadratic equation, written in a factorized form. Data is analysed in the light of a theoretical framework that considers three different worlds of Mathematics and the influence of ‘met-befores’ derived from learning experiences related to them. We show that having the quadratic formula as the only met-before to solve quadratic equations may not help the students to face all kinds of situations involving such equations. In addition, we claim that it is necessary to present to students learning situations that involve at least two worlds, the embodied and the symbolic, but in ways which also allow consideration of characteristics of the formal world, without which students may create their own inappropriate techniques.”
Alwyn Olivier (1989), Handling pupil’s misconceptions, is informative. Students have minds of their own and they may induce themselves to create a recipe like:
(x – u)(x – w) = c solves as x – u = c or x – w = c, which works for c = 0 and hence for any c
One way to approach such frequent misconceptions is to explicitly discuss them, and for example let students prove a theorem that the above is true only for c = 0. Mathematicians create theorems for what they regard as relevant steps. At the level of students, such issues can be considered to be relevant steps that eliminate their confusions. PM. For this particular equation, it is a step further to analyse it more. We can observe that the vertex lies at h = (u + w) / 2 and that solutions will be x1,2 = h ± d for some d. Substitution of h + d generates d = √(c + (u – w)2 / 4). Perhaps this must be mentioned too, perhaps this distracts, and the traditional approach has chosen the latter.
The traditional approach causes the following didactic issue.
- When a student knows only the quadratic formula, and doesn’t know about factoring or the basic (vertex) form, then teachers will tend to see this as problematic. An equation (x – u)(x – w) = 0 must first be expanded to fit the standard polynomial form, with b = -(u + w) and c = u w, to subsequently apply the quadratic formula, and then generate the solution that already could be seen at the start, namely x = u or x = w. Teachers may still opt for the latter level of competence, since the quadratic formula generates a solution for all cases.
- Teachers may reason that mastery of the quadratic formula still involves some algebraic competence. The latter may also be a delusion, since not-seeing the direct solution to the factored form isn’t quite competent. Lack of algebraic competence can also result in other errors, like when expanding the factored form. A student may however still earn some points for “knowing the quadratic formula” and “making proper substitutions (even from the wrong derivation)”.
- When a student is mathematically proficient and uses the quadratic formula, few teachers will object. However, such a student would tend to recognise the basic (vertex) and factored forms, and directly state their solution. It would only be an oversight (inattention) when such a student would still adopt the quadratic formula.
Quadratic functions revisited, re-engineered
In the re-engineered approach, teaching starts with the basic form. This is not mistaken as a solution method (“completing the square”). It is only a solution method when one starts from the standard polynomial form, but such is a wrong place to start from.
- The three forms – basic, factored, standard polynomial – arise in structured order.
- The relationships between parameters and solutions directly follow from this.
- The relevant questions concern the turning point (vertex) and zeros. Above discussion concerns the quadratic formula and focuses on the zeros, but there is also the turning point.
- Each form has its own solution method for turning point and zeros.
- A 3 by 3 table can show how one can transform one form into the other. It can be emphasized that transformation need not be remembered since the questions in the upcoming test concern vertex and zeros. Special attention would be given to special forms like a x^2 = – b x. The latter can be solved directly for x = 0 or assuming x ≠ 0 as x = – b / a. It can be clarifying to also write it as a (x – 0)(x + b / a) = 0.
- The quadratic formula is derived directly from the basic form, and is usefully remembered for automated application for the standard polynomial form only.
Essential for the handling of the quadratic function are the recognition of the graph and the three algebraic forms and the associated solution approaches. Essential is the attitude to check the solution and correct errors. Essential are the knowing how, knowing why, and knowing about knowing.
After the above has been clarified and assessed in examination, subsequent lessons can be spent on derived questions. The above has a specific learning goal, but the overall goal of competence in algebra and analytic geometry remains. For example, given two points and a particular parabola, state the formula in this form and / or that form. For example, check reactions to solving forms like x^2 – h^2 + v = 0 or a x^4 + b x^2 + c = 0. It is fair that students have a stage during which the names of parameters help support memory, but the next step is to recognise forms and infer the role of parameters.
As said, this re-engineered form must be tested in experiments before it can replace current convention. Didactics remains an empirical science, and the students themselves must show what works for them.
There is one issue of design that needs close scrutiny. Consider students who didn’t get adequate arithmetic in elementary school, who didn’t quite learn the tables of addition and multiplication, and thus fail to directly recognise the factors of say x^2 – 10 x + 24 = 0. I always feel sorry for students who have diligently learned the quadratic formula and who lose time on tedious arithmetic, only because of this background, so that they are severely punished on their weak spot.
- My null hypothesis is: It is only optional to factor a standard form by using the rules on adding and multiplying zeros. This is a different kind of competence. Focusing on this competence is derivative, inessential, and distracting from the real learning goals on the quadratic function. It is proper to show the method of direct factoring and train on this, since it helps algebraic insight and skill, but in another chapter. It should not be a prime element in the discussion of the quadratic function. Having test questions with functions in polynomial form that are easy to factor by students with that level of algebra, puts a bonus on a competence that is irrelevant here, and puts a malus on students that follow the longer route of the algebraic formula.
- The alternative hypothesis: It might empirically still be possible that students are served by this repeat training on the tables of addition and multiplication and this factoring, perhaps since they better learn what factors are, and perhaps since this is adequate skill in arithmetic so that the quadratic formula can be put on the graphic calculator. (See also Filloy et al..)
There is a curious key remark by David Tall, Rosana Nogueira de Lima & Lulu Healy (2014), p18-19, on the relevance of the didactic divide. It is correct that the form ‘quadratic expression = number’ doesn’t have a simple recipe, see Olivier, but this does not warrant a conclusion that students need not cross a divide. Also, Lima and Healy (2010) mention the basic (vertex) form of the quadratic function and are close to finding the re-engineered approach. Unfortunately, their frame of reference is that “mathematics” is sound and does not need to be questioned itself.
PM 1. Relevant is also Jan Block (2016), Flexible algebraic action on quadratic equations, who advises “tasks which focus not on finding solutions of equations by on classifying different types of equations” (p397). Block uses dashed lines and particular shapes to indicate kinds of relations or operations. This might be useful for teachers. For students, it is indeed a reminder that recognition of shapes can by supported by icons. The basic (vertex) form can be identified with an icon of a vertex (U), the factored form by an eye since the zeros can be seen directly (ʘ), and the polynomial can have this (√) icon to remind of the quadratic formula.
PM 2. Here is a long discussion on how to prove the quadratic formula, which doesn’t quite use the easy one of the re-engineered approach (but of course the methods are the same implicitly).
The quadratic function is an example of the usefulness of the recipe for re-engineering. Do not take “mathematics” for granted but accept it as the prime suspect itself for blocking student learning.
Empirical testing of this issue requires the design of integrated lesson plans, with detailed learning goals and exam questions, and worked-out presentations and materials for practice.
Once the re-engineered method has been developed and field tested and shown successful, then publishers can step in and reap the profits from the hard work of the re-engineers.