How to spark off a Maths topic ? – Part 1: concept processors

This post is about sparking off the core topic of a Maths lesson. (NB: This is different from starting a lesson with a Do Now or Starter, which are often used (and quite rightly so) as retrieval activities or stimulating/settling activities, but are not necessarily related to the core topic of the day’s lesson). Therefore, the spark will often be the second item in the lesson, but it is the first point of contact with the main topic.

Choosing the right spark to introduce a topic is important for 3 reasons:

  • First of all, the spark should totally serve the core topic, either through raising a question that will inevitably lead to the new concept or skill being introduced, or by catching attention and reinforcing how the lesson will be remembered.
  • Secondly, the spark should be in direct correlation with the way the main point of the lesson will be wrapped up (more about that in a future post).
  • Thirdly, the spark should introduce the first increment on prior knowledge, i.e. building up on what students already know.

But beyond these three criteria, the choice of the ideal spark is very personal. There isn’t one right answer. It depends on your teaching style, it depends on the students, it depends on how long you have taught the class and how far you feel you can go with surprising ways to reinforce material.

The potential diversity of sparks is vast, but it is useful to have 2 categories in mind: concept processors and memory reinforcers. I will start with concept processors because, due to their virtuous in-built logic, they are often most favoured by Maths teachers.

 

  • Concept processors:

Concept processors are sparks that make the new concept or skill appear totally inescapable, necessary and logical.

The most common concept processor is to present students with a problem that can’t be solved without introducing the new concept/skill of the day. Because the new concept/skill appears so inescapably necessary, this way of sparking off the topic is often regarded as virtuous, and it actually is, although that does not always make it the most efficient solution.

Let us look at 6 examples of concept processors.

Example n°1: a simple word problem

A great classic is the bow and arrow question, which is often used to introduce simultaneous equations.

A bow and an arrow cost £11 in total. The bow costs £10 more than the arrow. What does the arrow cost ?

More challenging and much more rewarding in terms of teaching and learning: UKMT questions. UK Maths teachers are very lucky to have UKMT, because many questions from Junior and Intermediate challenges are great stimulators of Maths thinking processes.

Ideally, if it wasn’t for monotony, every lesson could be sparked off with a UKMT question, as they cover a broad range of Maths notions.

 

Example n°2: exploration and deduction

To introduce straiStraight linesght line equations, students are asked to cluster the following straight lines.

In a second step, students pick up clues that lead to writing the equation in each of the identified situations.

 

Example n°3: error games

Using another kind of deduction, error games are also an interesting possibility:

  • Are these answers sensible ?
  • There are 3 errors on this page, find them.

This can be used for a whole range of techniques, particularly where numbers are involved (multiplication/division by powers of 10, etc.)

 

Example n°4: a picture

‘Conceptual’ does not necessarily mean ‘verbal’. Sometimes, a couple of pictures are enough to make the concept self-evident, like in the following example on compound units (speed).

161-drivinghomeforxmas simplified

This example, by the way, comes from the 101qs website, which I strongly recommend. It is a key resource in the world of open-ended Maths.

 

Example n°5: collective processing

Another highly efficient concept processor is to organise a collective process in order to learn skills that require sequential thinking, like Bidmas or solving equations.

Let’s take Bidmas as an example. When this is learned for the first time, it is possible to work with groups of 6 students, each student representing one of the Bidmas steps (Brackets, Indices, Division, Multiplication, Addition, Subtraction). Each student is only allowed to perform his own part of the process. Then roles are switched. Then each student combines 2 roles, then 3, etc., until each student is able to perform all operations in the right order. This is quite thrilling because students gradually discover they can do very complex operations, with the support of others to begin with, then on their own.

This works for any skill that requires sequential thinking, like solving linear equations. There again, one student can be a Collector (collecting like terms), another can be an Expander (expanding brackets), another will be the Adder (adding the same number to both sides of the equation), and another will be the Divider (dividing both sides of the equation by the same number). By proceeding in this way, students will be surprised to find they can easily solve equations which would normally be regarded as complex, like algebraic fractions.

 

Example n°6: conceptual discussion

Although this may sound frighteningly intellectual, it isn’t: this is actually a concept processor where students are fully taken on board, and in a way that they immediately buy into. This processor can only be used from time to time, but its results are likely to trigger other processors.

The idea is simply to show students 3 or 4 videos explaining the same concept/skill. Here are 4 videos explaining how to expand brackets:

This is just a selection from many possible videos available on YouTube on the topic of expanding brackets. For the longer videos, you would only use the relevant section.

The point is to have students discuss these videos. Which they will, because they love to compare, argue, vote, etc. It is widely hoped that students will criticize these videos. You, as their teacher, will learn a lot from what they like and dislike. There are even secret hopes: for example, that students may come to realize how excessive scaffolding can make things more complicated for them. In the end, teacher and students will agree on which rules and examples should be copied in their exercise books.

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