## Rich Questions and Re-Defining Mathematical Concepts

Over the last few weeks, Patrick Honner’s post on ‘Which triangle is more equilateral?’, has promoted some amazingly rich discussion (I won’t restate the problem here – click the link to take a look). I love it when people share rich questions. Generally because I find that I can’t keep track of all the resources which come my way but I do seem to remember great questions. This seems to apply to many maths educators nowadays which is one of the reasons why I agree with Dylan Wiliam and think that sharing rich questions is the way forward.

When I started this blog, I envisaged that many of the posts would be about great questions but alas, Geogebra beckoned and has kept me busy for the last 4 months. Patrick Honner has inspired me to write more posts on this so here’s hopefully the first of many (unless Geogebra distracts me in which case I can’t promise anything).

So firstly, why is Patrick’s question so great?

To mention a few reasons…

1) Low Floor – Anyone can at least take a guess at which triangle they think is more equilateral. The mere act of making that guess feeds our curiosity to find that answer. Indeed in this case you may believe that both triangles have the same degree of ‘equilateralness’.

2) Re-defining Concepts – The problem forces us to look at a common mathematical concept from a completely new angle (excuse the pun) and re-define our notion of what an equilateral triangle is. It helps in this case that there’s a high number of possible ways to re-define what it means to be equilateral and then explore whether our re-definition sheds light on the given problem. Some ways of re-definement are more advanced than others which allows access at different levels.

Is this the only reason why Patrick’s post is so popular?

Take a look at Dan Meyer’s tweet below.

I hadn’t noticed this until Dan pointed it out. Patrick’s responses to the comments on his blog are outstanding. If every teacher replied to students’ comments in this way then Maths classrooms around the World would be invigorating to be a part of. I may even go as far as showing Patrick’s feedback to some of my students to help them gain a higher understanding of how to effectively feedback to their peers.

A few randomly picked questions which may induce a re-definement

I really like the idea of a student re-defining a concept given a new scenario or interesting problem. Often we talk about this as addressing misconceptions but sometimes it may be that a students’ only ‘misconception’ is that they haven’t been exposed to the right type of question before.

As a simple example take a question on solving linear equations.

3x + 2 = 3x – 5

This to me is a quintessential re-definement question. The only way I have to understand this is to graph both sides (y=3x + 2 and y=3x – 5, or indeed any balanced combination of these e.g. y=3x and y=3x – 7) to show that they have the same gradient and hence don’t intersect – Would anyone do it differently? Most students don’t think of solving equations in this way so how would you introduce the re-definition of this concept? Would you set this question during a unit of work on solving linear equations or would you set it during a graphing linear functions unit of work? Would it influence your decision as to which unit of work you would teach first so that students have a higher liklihood of understanding it or discovering it for themselves? Would it even influence a decision to teach both topics at the same time?

Here’s another example of a question posed in Dylan Wiliam’s book (Embedded Formative Assessment) to help re-define a fractions concept/address misconceptions.

In which of the following diagrams is one-fourth of the area shaded?

How awesome a question is this?!? Just let this question loose in the classroom and enjoy facilitating the re-definements and addressing the misconceptions.

These questions, in a similar way to Patrick’s triangle question, force students to analyse the current definitions they have about concepts and engages them in re-defining those that are incorrect or incomplete. As Patrick has shown, the way in which we respond during this phase of re-definement is crucial.

Whether they be questions for exploration like Dan Meyer’s 3 ACT questions or media which promote questions, questions which address misconceptions or questions which force us to think differently about mathematics, they’re all brilliant to hear about and I’m hugely grateful to anyone who shares them. For me, having these 3 questions in my mental bank is much more beneficial than gaining three worksheets on equilateral triangles, solving linear equations and shading fractions of shapes.