In the last blog entry I described some concrete strategies for differentiating instruction in mathematics class. That entry described how a teacher can differentiate instruction when teaching the whole class, by placing responsibility on the whole class to work collaboratively to share mathematical ideas and strategies.
A second way to promote differentiation in the maths classroom is through the use of suitable maths problems. Some problems can be solved at different levels by children who are have different levels of mathematical achievement. Furthermore, the challenge contained in some problems can be extended or eased to suit children who need more or less of a mathematical challenge.
Let me make this point more concrete with reference to the problems (a) and (b) below. I begin with problem (a).
(a) Mary went into the shop and bought three sweets. One sweet cost 10 cent, one sweet cost 5 cent and the third sweet cost 1 cent. How much did Mary spend altogether?
Problem (a) is typical of many word problems that you’ll find in Irish maths textbooks. In order to answer it, children have to decide which operation to use and then they add the three amounts of money. In a typical first class some children will find this difficult, some will find it easy and most will manage to solve it within a short time. Those who find it difficult will need more support before they can solve it, and those who find it easy will finish quickly. If early finishers are not given additional work, they will become bored or disruptive. But more of the same work will be of little benefit to early finishers if they can already do this type of problem.
Closed problems like (a) make differentiation of instruction difficult. When faced with such problems in a textbook, a teacher can take steps to adapt the problems to make them more accessible and mathematically challenging for a wide range of learners. The teacher could, for example, begin with the answer and make the problem open-ended by saying that “Mary bought three sweets and paid 16c. What might she have paid for each sweet?” The teacher could adapt the problem by changing the prices of the sweets, or by asking how much change the child would receive if €1 was given to the shopkeeper in payment for the sweets, or by asking how much more one sweet cost than another. The teacher could ask children to write down the calculation they used to solve the problem, or to explain why they know their answer is correct. Such strategies can be applied by a teacher to make closed problems more accessible and more suitably challenging to more learners. Now let’s look at problem (b).
(b) I have 10 cent, 5 cent and 1 cent coins in my money box. If I open the box and take out three coins, how much money could I have?
Problem (b) has multiple correct answers, without needing any adaptation. The answer 16c is correct. But there are other answers. Some children in a class might get just one answer. Some children might get several of the answers and others might get them all. A big challenge would be to prove that you knew you had all the possible solutions. This open-ended problem is different from the previous one because children at different levels of mathematical achievement can be challenged by the same problem. And after the children spend a while working on the problem independently, all of them can participate in a follow-up class discussion contributing what they have discovered and listening to what others have done. The problem requires the children to add, but it also requires them to decide which numbers to add and to be systematic in how they decide what coin values need to be added. Like problem (a), the problem could be extended. This could be done by taking out four coins, five coins etc. Or the denomination of the coins in the box could be changed.
These examples show that the problems available to a teacher have the potential to make differentiation of instruction easier or more difficult. Adapting problem (a) takes time and effort. Problems like (b) may need little, if any, adaptation; but problems like that can be difficult to source. They need to be gathered over time and shared among teachers. I have posted some open-ended mathematics problems in the resource section of this website to get you started. If you use or come across any others, please post them in the “Comment” section.