For my daily commute, I take one route to get to work in the morning and a different route to get home in the afternoon. The shortest route is quicker in the morning but takes twice as long to get home because of traffic in the afternoon. I’ve taken these journeys so many times now that I can navigate my way with very little thought!
One day, I was driving to work later than usual and so decided to use my ‘afternoon route’ instead, to avoid the traffic. I’d only ever driven home this was before, so it meant I was navigating this route in the reverse direction to what I was used to. I thought it would be a doddle but was surprised at how tricky I found it! Despite driving down these roads so many times, I had to pay more attention to where I was going than I’d done for years! “I know I usually turn right onto this main road from a side street, but which side street is it?” It also made me realize that I didn’t know this part of town as well as I thought I did; I was just used to following the same sequence of turns each day. All this from the simple act of driving in the reverse direction to what I was used to!
How is this relevant to learning mathematics?
It’s a slightly tenuous analogy, but I also find that the same can be said for many aspects of learning mathematics. There are many topics where I often find myself asking students to work through problems in one direction far more often than the other. With enough practice, students would get to the point where they could carry out the correct processes in the correct order routinely with little thought or effort. However, the simple act of turning these problem around can encourage students to re-concentrate their efforts, deconstruct processes that they are familiar with or look at things in a different way. While this sometimes serves to increase challenge in a task, other times it can help students assimilate a more complete image of the concept they are studying.
For example, take the two questions below which require students to manipulate surds. I reckon I’ve given my classes far more practice on questions where they simplify surds (like the one on the left) than questions where they ‘unsimplify‘ surds (like the one on the right).
Even though ‘unsimplifying’ a surd might seem like a strange thing to do, it can still be useful from a concept learning point of view. There are two key things that I usually want my students to think about when simplifying surds, like in the question on the left: factor pairs of 45 that contain square numbers and remembering to square root the 9 when they take it out of the radicand. The question on the right also shines a light on these points but from a different angle: they have to square the 3 before they can bring it into the root, which means they multiply the 5 by a square number.
Working both ‘forwards’ and ‘backwards’ through a concept can facilitate a fuller inspection of it. This could even be done with the tiniest of tasks. For example, in a previous blog post I looked at some activities that students can do during lessons about corresponding angles. In one activity students were given two angles and asked to determine if they were corresponding; in another activity students were given one angle and asked to mark the angle that corresponds with it.
There are a lots of opportunities for students to work ‘forwards’ and ‘backwards’ through concepts. In some cases, it is because a process has an inverse that is either definable, easy to describe or taught explicitly as a separate learning objective. Examples of these include expanding brackets and factorising, converting numbers into and out of standard form, or drawing straight lines for equations and writing equations for straight lines.
When students learn a formula, it is typical for textbooks, worksheets and assessments to ask students to work with that formula in different ways by varying the information that is provided. Area tends to be a common one for this.
There are some topics where students need to know that certain conditions are met before they can perform a calculation (i.e. “if A is true, then B will work”). For example, they need to know a triangle has a right angle before they can use Pythagoras’ formula. These situations can also be turned around to ask students to perform calculations in order to evaluate the conditions (i.e. “if B works, then A is true”). E.g. students can substitute the three lengths on a triangle into Pythagoras’ formula to determine if it contains any right angles.
This idea can also be applied to similar shapes. A student can either use the fact that two shapes are similar to work out a missing length, or use the lengths on both shape to decide if they are similar.
It also works well for getting students to think about angle rules in different ways too…
With both Pythagoras and similar shapes, I usually teach students how to calculate missing lengths before teaching them how to check for right angles or similarity. However, this year I tried it the other way around for the first time. In other words, students first used Pythagoras’ Theorem to determine if a shape contained right angles before they used it to calculate missing sides. With similar shapes, they practised testing for similarity before they practised calculating missing sides. It seemed to work quite well but I think I need to try it again with another class before I decide which way I prefer.
Other opportunities to ‘work backwards’ also present themselves in situations where students learn to construct things, such as graphs. I’ve found that asking students to deconstruct these things helps them to think more carefully about their features and properties. This sometimes helps later on when students move on to interpreting graphs and making conclusions because they have already practised pulling raw data out of images.
Students can also be encouraged to work in the reverse direction to what they are used to by providing students with solutions to think about rather than (what would conventionally be seen as) the question. Sometimes this gets phrased as, “Here is the answer. What was the question?” But they can also be problems in their own right.
The quadratic equation example also benefits from becoming more open. This means that once a student has found the most straightforward equation (x2 + x – 6 = 0) they can then look for ways to manipulate it to find less obvious ones (e.g. x2 + x – 4 = 2; 2x2 + 2x – 12 = 0; -x2 – x + 6 = 0).
A slightly more contrived way to achieve a similar effect to the trigonometry example could be to provide both a question and its solution, but to conceal aspects of the question. This can encourage students to think about how aspects of the solution relate to aspects of the problem in a different way to what they might be used to.
The examples above don’t necessarily provide extra challenge but encourage students to think about concepts in multiple ways. However, sometimes working backwards can give additional opportunities to increase the difficulty of a task or include extra stages of working.
The cumulative frequency and scatter graph images were created using Desmos.
If you would like to use or share any of the images from this blog, feel free to use this Power Point to help you: This Way, That Way, Forwards and Backwards
Previous blog posts:
‘Planning Topics’ Series:
- Filling in the Gaps With Histograms
- Twists and Turns With Straight Line Graphs
- Thinking About Corresponding Angles
- Thinking About Areas of Circles
- Thinking About Areas of Parallelograms
- Thinking About Ratio and Algebra
‘What I Learned From Shanghai’ Series:
‘Making Connections’ Series