Generalising Patterns using Multiple Routes
A core focus of secondary mathematics is for students to develop inductive and deductive reasoning. In particular, this is emphasised in lower secondary mathematics where number patterns are taught. In this chapter, students learn to depict patterns using algebraic expressions and attempt to generalise. This is one route to develop inductive reasoning.
One of the central themes of the book “Developing Thinking in Algebra” (Mason, J, Graham,A, Johnston-Wilder, S, 2005) is the idea of expressing generality. Generality can be expressed in multiple ways which can help students develop algebraic thinking by approaching problems in various ways. Students often only face number pattern problems through a single approach. Here are some examples of problems that can be approached in a variety of ways:
Generalising picture sequences in the reverse manner.
A common number pattern task is to have a series of pictures and then formulate an expression for the nth picture. For instance, take a picture sequence made of sticks, perhaps in the following:
I found this example meaningful as there are many ways of finding number patterns. For instance, what is the number of sticks required to build the roof in the nth house? What is the number of sticks required for perimeter of the nth house? These questions should not be given to the students. Instead, students should come up with their own questions and try to determine the number patterns on their own. This will enable them to develop metacognition through “pattern sense-making”.
Indeed, the reverse process is even more interesting and challenging. For instance, suppose we have the sequence 2+(n-1)+(1+2n+1) and 3(n+1) for the nth house. Find ways to build structures of houses using these expressions. As mentioned by the author, this process helps to support flexibility in algebraic thinking as the student will need to interpret the algebraic expression before visualising the structure.
A cube is painted on all faces. It is then cut up into 27 equal cublets by two plan cuts parallel to each face, (imagine a 1 color Rubik Cube). How many cublets are painted on how many faces? Generalise the problem.
Cube painting is but one way to expose students to inductive reasoning which is not so direct. Students may need to utilise their problem solving skills such as making a list (Creating a table for a two by two cube first, then by a three by three cube) and working backwards in order to attain the answer. Moreover, there are no clear steps to generalise the problem. Students must first think of how to extend this problem. For example, besides cutting the cube into 27 cublets, what is the next smallest number of cublets can be cut using the same operation? What if more than one color is used? If 6 different colors were to be used, like in a Rubik Cube, how many faces would have 3 colors? How many would have two? There are many questions that can be asked to extend this problem.
In summary, number pattern activities listed above require students to apply problem solving skills, metacognition, inductive and deductive reasoning. This will support students’ thinking in algebra.