I’ve been introducing the idea of straight line to my third year class recently. This is a topic where I feel that the “just tell them” approach comes crumbling down. After having taught this topic a dozen or more times, I’ve come to be of the opinion that direct instruction, no-matter how clear and how explicit will not suffice to develop procedural and conceptual knowledge we’d like pupils to acquire for this topic. I talked on the recent Craig Barton podcast about pupils (and teachers) requiring the space for sense making. This topic is a prime example of where this is required.
I think it is worth remembering that this is the first time where the relationship between algebra and geometry comes together. Until now these have been discrete topics for our pupils. I’d make the point that Descartes rightly has the coordinate system named after him as it was a big deal for the whole of maths when he came up with it. For instance, imagine calculus without being able to visualise the functions and their stationary points! Given that this was quite a breakthrough in the history of mathematics, then we should be prepared to spend more than a couple of hours on this idea with pupils.
I could talk in length about my treatment of the whole topic of straight line; however, I have chosen to focus on a specific learning episode. To offer some context the class are familiar with the families of lines which are parallel with the axes. I have done very little work on tables of values, and will come back to this later. I previously would always start with this, however, I think that once learners have a feel for gradient and y-intercept we can look at patterns in the coordinates with a fresh perspective and tie the topic together. The class have worked on developing a conceptual understanding of gradient through a slow building up from the idea of vertical over horizontal, counting boxes “for every 1 box along, how many boxes up/down?”, finding gradients by “stepping out” before finally using the standard gradient formula. The idea of coming at this idea from various perspectives is so that pupils appreciate the affect the numerical value of gradient has, as opposed to simply being the result of some calculation.
A task I used to support pupils in developing connections between the formula and the concept is the following from John Mason and Anne Watson. In each case, the pupils have something to relate their answers to, rather than simply churning out meaningless and context less answers. This task appears simple, but, in my experience, for a novice can be quite enlightening.
The approach I’m trying out this year is based upon some excellent work by my colleague @mpcopland. I have not yet talked explicitly about y = mx + c. My experience is that this becomes some abstract notation which bears no correspondence to the notions we are trying to convey. I am happy to delay the introduction of that until the next lesson.
I used Desmos to illustrate each of the following lines. I think using the family of lines with gradient 2 draws more attention to the role of the coefficient of the x.
I asked what was happening. An interesting reply from several pupils was “the lines are moving sideways”. This response is notable as I’ve had pupils come out with this on previous occasions when I’ve been using dynamic software such as Desmos or Autograph. I wonder if I restricted the length of the lines that the vertical movement would be more obvious. It took more probing and specific questioning “what about where the line is crossing the y-axis?” before the idea of vertical movement became apparent.
The demonstration with an invariant gradient drew the attention to the y-intercept. I then explored the lines with gradient of -2 and pupils were able to answer out with successful predictions of how the lines would appear.
By showing a few more examples and cold calling it seemed to be the case that many pupils appreciated that, for instance y = 3x + 2 is a line which has the following properties: A. For every 1 box along it goes up by 3. B. It cuts the y –axis at 3. I was feeling quite confident at this point – this was naïve, as no child will have fully internalised these ideas through simply observing or being told. The opportunity to attend to the ideas for oneself and to play with the mathematics is essential.
The follow up task for learners is shown below, focused on drawing parallel lines with varying y-intercepts. Again this idea of varying just the y-intercept is important, to keep this as the main focus. You’ll notice how this pupil appears to have a limited appreciation of the infinite length of the lines. I’ll probe this more tomorrow.
The next pupil, was one of a few who produced the following. It’s at moments like this where, as a teacher, there are two options: become incredibly frustrated or instead appreciate the mistake and use it as the basis for more conversation.
I shared this with the class and asked them to discuss in their pairs what might be right and what might be wrong about this attempt. Some rich conversation ensued and when I brought the class back together there were a sea of hands eagerly wanting to tell me how to rectify the situation. Interestingly, while this mistake appeared in some pupils work for the first set of lines, nobody repeated it for the second set. I’d said very little, the knowledge, the appreciation of the ideas and the sense of the topic was already in the room. I find it quite important for learners to be able to explain their ideas to each other. I like to take advantage of this to address misconceptions. I conjecture that pupils may be used to hearing my voice and therefore switch off quicker, but are more inclined to listen to a friend and ask demanding questions to clarify their understanding. So rather than me telling the class the mistake and possibly having a positive outcome, a whole host of other positives may have had opportunity to arise in terms of confidence and deepening of understanding, by learners talking to each other.
Exploring the dimensions of the problem
During class discussion a couple pupils posed my favourite type of question. “What would happen if?” They each suggested equations, both of which are shown below. One of them imagined that the line might not be a line at all. I put the equations into Desmos. We looked at their gradient and y-intercept and then I demonstrated some simple rearrangement. This was a nice opportunity to explain that if we have the line in the form y = that we can easily read the gradient and y-intercept. This, I feel, will set us up nicely for the tomorrow’s lesson on formal y = mx + c.
Taking things further
Having spent a few periods on gradient and then today on the y-intercept, I wondered if learners would be able to generate the lines represented by the equations below. Tables of values, as I previously said, have not been emphasised yet. If pupils were to succeed with this task then, I think, it would be from a good understanding of line equations. Richard Feynman has a lovely quote “what I cannot create, I do not understand”. Being able to generate the diagram from the algebra is fundamentally important here. I think that a much higher level of understanding is required to do it without a table of values than is required with one, thus my avoidance for now.
The following is one of many successful attempts.
I then asked pupils to draw a line of their own choice. None of them were trying to vary from the format of y=mx+c now as, I think, they are beginning to see why this representation is useful. A number of interesting lines were drawn correctly, such as y = 0.5x + 0.25.
The last time I wrote about a learning episode like this some people commented along the lines of “why bother?” In this case, I think it is very much worth the effort to use a range of tasks, teaching approaches and representations to develop that fuller understanding. Straight line is a topic that can be taught at a very superficial level. “Here are two points, find the gradient using this formula, come up with the equation of the line using a point and the gradient in this formula.” However, what is the lasting legacy of this? A very limited instrumental understanding. Another critique of a recent post was that “pupils aren’t interested in why”. I disagree entirely. If we offer the material at the appropriate time and in the appropriate manner, ask the right questions and, more importantly, leave space for the learners to make sense of things, ask their own questions and generate their own examples then engagement (whatever that is) can be extremely high.