Teaching and Learning Mathematics
Question 1:
1.1 Instrumental understanding: doing something without real understanding
1.2 Less to remember: a benefit of rational understanding
1.3 An integrated network of concepts can also be called: cognitive schemes
1.4 Constructivist theory: The idea that learners are creators of their own learning, that integrated
networks, or cognitive schemas, are both a product of constructing knowledge and tools with which
additional new knowledge can be constructed and that the idea of reflective thought being used to
modify our existing schemas to incorporate new ideas.
1.5 Understanding is a measure of the quality and quantity of connections that an idea has with existing
ideas.
Question 2:
2.1 A Grade 6 learner added the fractions to obtain the answer shown below
+ + = =
2.1.1 Is the calculation correct?
No, it isn’t. + + = + + =
2.1.2 Explain the misconception that they displayed in their calculation:
The misconception is that the learner added the fractions as you would if they were ‘like
fractions’. That is that they added all the numerators together and then added all the
denominators together. However, these fractions are ‘unlike fractions’ and when the
denominators are different, we have to find a whole that 5, 10 and 20 can divide into equally.
, 2.1.3 What teaching strategies could be used to assist the learner to do the given calculations
correctly?
The learner could use the area model or the length model to illustrate the addition. This could
help in their understanding of why we use lowest common multiples and lead them to the
correct answer.
2.2
2.2.1 Arrange the numbers from small to large and then plot them on a number line 10cm long and
drawn to scale.
0.01 0.09 0.1 0.5 0.9 0.99
2.2.2 0.9-0.5=0.4
2.2.3 0.9+0.1=1
2.2.4 I believe it could be useful to learners as they can visualise the numbers on the timeline and by
counting spaces up or down, they are able to solve the sums on their own.
2.3 Instrumental understanding is doing something without understanding, for example, doing a simple
equation based on a formula. Relational understanding is knowing what to do and why.
2.4 Rational understanding allows learners to give examples, find equivalencies and build connections
among ideas wheras instrumental understanding only allows learners to complete one idea without
understanding why.
2.5 Problem solving is when one finds the solution to a particular situation by a means which was not
immediately obvious.
2.6 Three ideas to make lessons more effective:
1. Begin where the learners are, not where you as teachers are.
2. Teaching should begin with the ideas that learners already have – the ideas they will use to create
new ones.
3. Engage learners in tasks or activities that are problem based and that require thought.
Question 1:
1.1 Instrumental understanding: doing something without real understanding
1.2 Less to remember: a benefit of rational understanding
1.3 An integrated network of concepts can also be called: cognitive schemes
1.4 Constructivist theory: The idea that learners are creators of their own learning, that integrated
networks, or cognitive schemas, are both a product of constructing knowledge and tools with which
additional new knowledge can be constructed and that the idea of reflective thought being used to
modify our existing schemas to incorporate new ideas.
1.5 Understanding is a measure of the quality and quantity of connections that an idea has with existing
ideas.
Question 2:
2.1 A Grade 6 learner added the fractions to obtain the answer shown below
+ + = =
2.1.1 Is the calculation correct?
No, it isn’t. + + = + + =
2.1.2 Explain the misconception that they displayed in their calculation:
The misconception is that the learner added the fractions as you would if they were ‘like
fractions’. That is that they added all the numerators together and then added all the
denominators together. However, these fractions are ‘unlike fractions’ and when the
denominators are different, we have to find a whole that 5, 10 and 20 can divide into equally.
, 2.1.3 What teaching strategies could be used to assist the learner to do the given calculations
correctly?
The learner could use the area model or the length model to illustrate the addition. This could
help in their understanding of why we use lowest common multiples and lead them to the
correct answer.
2.2
2.2.1 Arrange the numbers from small to large and then plot them on a number line 10cm long and
drawn to scale.
0.01 0.09 0.1 0.5 0.9 0.99
2.2.2 0.9-0.5=0.4
2.2.3 0.9+0.1=1
2.2.4 I believe it could be useful to learners as they can visualise the numbers on the timeline and by
counting spaces up or down, they are able to solve the sums on their own.
2.3 Instrumental understanding is doing something without understanding, for example, doing a simple
equation based on a formula. Relational understanding is knowing what to do and why.
2.4 Rational understanding allows learners to give examples, find equivalencies and build connections
among ideas wheras instrumental understanding only allows learners to complete one idea without
understanding why.
2.5 Problem solving is when one finds the solution to a particular situation by a means which was not
immediately obvious.
2.6 Three ideas to make lessons more effective:
1. Begin where the learners are, not where you as teachers are.
2. Teaching should begin with the ideas that learners already have – the ideas they will use to create
new ones.
3. Engage learners in tasks or activities that are problem based and that require thought.
