One of the ways to help students to make connections among concepts is to give them problem solving tasks that have many correct solutions or answers. Another way is to make sure that the solutions to the problems involve many previously learned concepts. This is what makes a piece of knowledge powerful. Most important of all, the tasks must give the groundwork for future and more complex concepts and problems the students will be learning. These kinds of task need not be difficult. And may I add before I give an example that equally important to the kind of learning tasks are the ways the teacher facilitates or processes various students’ solutions during the discussion.
I would like to share the problem solving task I made to get the students have a feel of the existence negative numbers.
We tried these tasks to a public school class of 50 Grade 6 pupils of average ability and it was perfect in the sense that I achieved my goals and the pupils enjoyed the lessons. This lesson was given after the lesson on representing situations with numbers using the sorting task which I describe in my post on introducing positive and negative numbers.
Sorting is a simple skill when you already know the basis for sorting which is not case in the task presented here.
Just like all the tasks I share in this blog, it can have many correct answer. The aim of the task is to make the students notice similarities and differences and describe them, analyse the relationship among the numbers involved, be conscious of the structure of the number expressions, and to get them to think about the number expression as an entity or an object in itself and not as a process, that is speaking of 5+3 as a sum and not the process of three added to five. The last two are very important in algebra. Many students in algebra have difficulty applying what they learned in another algebraic expression or equation for failing to recognize similarity in structure.
Here are some of the ways the pupils sorted the numbers:
1. According to operation: + and –
2. According to the number of digits: expressions involving one digit only vs those involving more than 2 digits
3. According to how the first number compared with the second number: first number > second number vs first number < second number.
4. According to whether or not the operation can be performed: “can be” vs “cannot be”.
5. This did not come out but the pupils can also group them according to whether the first/second term is odd or not, prime or not. It is not that difficult to get the students to group them according to this criteria.
Solution #4 is the key to the lesson:
During the processing of the lesson I asked the class to give examples that would belong to each group and how they could easily determine if a number expression involving plus and minus operation belongs to “can be” or “cannot be” group. From this they were able to make the following generalizations: (1) Addition of two numbers can always be done. (2) Subtraction of two numbers can be done if the second number is smaller than the first number otherwise you can’t. You can imagine their delight when they discovered the following day that taking away a bigger number from a smaller number is possible.
One pupil proposed a solution using the result of the operations but calculated for example 3-10 as 10-3. This drew protests from the class. They maintained that 3-10 and similar expressions does not yield a result. Note that class have yet to learn operations on integers. And obviously they could not yet make the connection between the negative numbers they used to represent situation from the lesson they learned the day before to the result of subtracting a bigger number from a smaller. To scaffold this understanding I ask them to arrange the number expressions from the smallest to the biggest value. This turned out to be a challenging task for many of the students. Only a number of them can arrange the expressions for smallest to the biggest value. My next post will show how the task I gave to enable the class to make the connection between the negative number and the subtraction expressions.