A Learning Episode: Straight Line

Background
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.



Today
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. 

Conclusions

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.

7 comments

  1. Loved your sequencing of explorations. I could imagine young minds productively struggling and feeling a sense of accomplishment when things started to click for them.One minor mathematical error I noticed, which in no way distracted from a thoroughly engaging read:“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.”Suggest, It cuts the y –axis at 2.Thanks again for sharing your classroom adventures and insights.apm Twitter: @autismplusmath

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  2. You have given me much to think about As I read I made notes; this was in order to capture my thoughts so I could offer a detailed response.1. When you wrote \”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\” I found myself thinking about how hard it can be to break away from telling in order to move to causing students to experience a concept based upon an exploratory methodology. Teachers often tell me it is \”quicker to tell them\” and I always think about how many time are students taught to, say, add fractions; and how teacher telling students how to carry out well-worn procedures will not necessarily be quicker in the long term. This clearly begs a question about the usefulness of \”direct instruction\”.2. When you wrote \”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. Spending time on major concepts is something I think 'we' need to see as an important part of co-planning departmental schemes of work, by thinking about the development of these concepts from Y8 (in Scotland) to Y11. This is so this we can check we are not re-teaching concepts from the 'beginning' each time; there can be nothing worse than hearing students say something such as: \”We did this before!\” Progression, therefore, is paramount. There's an important calculation to be done about how many weeks students are taught mathematics in the secondary phase and how long we can afford to spend on major concepts, e.g. Fractions, decimals, percentages, ratio and proportion, (from Y8 to Y11), dealing with Y12 and Y13 separately. Say 40×3 (Ys 8,9,10) + 20×1 in Y11 (130 weeks in total) and 8 major concepts make the numbers relatively easy to work with! The crucial next step is to decide what are the starting points are as a concept progresses and what the extension tasks can be.3. I found your writing about the coming together of geometry and algebra most interesting; partly because I believe the most effective way of teaching algebra is not as a discrete concept, but something which can be used and connected with different concepts; thus exploring rectangles of a constant area of 24 square units and the connection with y = 24/x together with the beautiful graphs which can accrue.4. Learning episodes. I lay great store by writing learning episodes (I call them anecdotes) as a tool for gathering qualitative data for writing at Masters degree level as a practitioner researcher.5. When you wrote: \”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…\” I thought \”Yes\” a slow build up is so important if we want to engage students' \”Relational understandings\” (Skemp); to enable students to find greater depths of cognition.6. I just love the task based upon one point being (4, 3) and others being of the form (x, 12); variation to the fore! I also wondered if extending this so students, say working in pairs, (which you refer to later in your writing) could choose two or three of their own 'other' points, and gather information from the whole class about the gradients of their lines and use this as a stimuli for a whole class discussion pertaining to gradients and/or intersection points on x=0 and y=0.

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  3. 6. I just love the task based upon one point being (4, 3) and others being of the form (x, 12); variation to the fore! I also wondered if extending this so students, say working in pairs, (which you refer to later in your writing) could choose two or three of their own 'other' points, and gather information from the whole class about the gradients of their lines and use this as a stimuli for a whole class discussion pertaining to gradients and/or intersection points on x=0 and y=0.7. I just love what you wrote about the sense of the topic being already in the room: \”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.\”8. When you wrote: \”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.\” I thought this was really powerful because to take this course of action was all about being \”in-the-moment\” and looking for other ways of helping students make sense of something without you, as their teacher, choosing an alternative course of action.9. I just love the Richard Feynman quote.10. I was disturbed by the comments you received: \”why bother\” and “pupils aren’t interested in why”. The first seems to be sadly dismissive; the second was something of a broad generalisation and perhaps the person meant \”my pupils aren’t interested in why”, I won't unpick this further!11. Your reference to a \”lasting legacy\” is a wonderful way of thinking about sense-making and how by taking time and not rushing students too quickly into conceptual understandings is soooooo important.RegardsMike

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  4. Hi ChrisI wonder if you could make more use the powers of imagination that all learners possess? Also, I might want to stress the equation/graph as a *relationship* between x and y. An aim would be for learners to be able to shift backwards/forwards/between equation/words/graph/coordinates for any linear relationship – e.g. x+2y=7 as well as y=mx+c – placing the emphasis on relationship. Take x+2y=7. Describe it in words. Now find a pair of values… and another… not an integer… negative… These are coordinates: imagine the line. Describe it. Imagine shifting all of these points up one. What would the new relationship be? In words? As an equation? Imagine… Your turn: Change *one* word in the word description. Find pairs of values.. imagine, sketch, write the equation… here's someone's sketch, what did they change?Take an example of the form y=mx+c… describe in words, find pairs of values, imagine… what is the effect of changing…? imagine shifting the coordinates… Danny

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  5. Hi Mike, Thanks for taking the time to reply in such depth. I perhaps haven't been thinking of the time aspects of the various pedagogical approaches, but one phrase that did come to mind was \”deep learning stays deep\”. I can't attribute it unfortunately. I'd read your some of your ideas on geometry before, but sadly haven't yet tried this in class. The idea of relationship, which I know Danny has alluded to below, is lacking in some of my teaching prior to straight line. I've been looking at giving pupils increasingly frequent opportunities to generalise. Perhaps looking at this from the perspective of graphing values would be a nice way to move my teaching forward for younger classes. I love the extension idea for the (4,3) task. That's a lovely idea – I will definitely try this approach in the coming lessons! Replying, also to the below… I liked that you mentioned \”in the moment\”. I'm trying to be aware of these pedagogical opportunities in lessons more consciously. I'm currently working on the Discipline of Noticing with Danny and Tom Carson. Thanks, once again, for reading one of my blogs and replying. It means a lot to me. I post these episodes, not in a searching for praise, but to provoke conversation and to learn. Thanks,Chris

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  6. Hi Danny, Thanks again for being so supportive, in taking the time to read and reply. I haven't mentioned the word relationship at all in my blog, and I did mention that I haven't talked about tables of values yet. I will spend significant time working on this. I suppose at the start of this crucial topic I find myself overwhelmed by possible approaches and the available pedagogical choices, which is why – for myself, I fall into the complexity reduction way of thinking. (not the in the conceptual sense, but in a pedagogical sense). A task I am currently writing is a matching activity where pupils will have different graphs, equations and tables of values to group together. Inspired by your comments I think I will add in words, coordinate tuples and alternative rearrangements of the equation. I am intrigued that you mention x + 2y = 7. I think this will be very worthwhile exploring further. I am definitely biased towards y = mx + c. Even at Higher where most people leave it as ax + by + c = 0 I am inclined to to write as y= mx + c, even if it doesn't look \”nice\”, due to fractions etc. I am, perhaps, too much in my own comfort zone mathematically with that format. I love this whole section:\”Describe it in words. Now find a pair of values… and another… not an integer… negative… These are coordinates: imagine the line. Describe it. Imagine shifting all of these points up one. What would the new relationship be? In words? As an equation? Imagine… Your turn: Change *one* word in the word description. Find pairs of values.. imagine, sketch, write the equation… here's someone's sketch, what did they change?Take an example of the form y=mx+c… describe in words, find pairs of values, imagine… what is the effect of changing…? imagine shifting the coordinates…\” There is so much scope here for the start of rich lessons. I find that coming away from worksheets and using these sorts of tasks – that pupils provide comments and insights which result in far richer lessons occurring. I do think that we might differ on our views of the important of well designed tasks, but this is something I'll ask about when you come up in November! Thanks again,Chris

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