1. Story telling tasks: The case of Mpho Nicky Roberts University of Johannesburg

2. ‚Full service‘ public school in South Africa
National Context
‚Full service‘ public school in South Africa
Full Service schools are designated by South African policy (Department of Education, 2001) to be the frontrunners of inclusive education.
They are to develop the human resource and infrastructural capacity to include learners with low and moderate support needs in natural proportion to the incidence of these learners in the communities served by the schools.

3. Design experiment (6 cycles)
Assess
First 3 cycles: England and SA (Year 1 and Grade R) resourced context focusing on parity (odd and evenness)
Analyse
Teach
Reflect
Next 3 cycles: South Africa, full service school, resource constrained context focusing on additive relations
Plan

4. All the learners were from poor socio economic families.
Mpho’s class context
All the learners were from poor socio economic families.
All the learners in the school were black children. Only two learners had English as a home language.
Mpho was 9-year old boy in Grade 2 class of 30 learners (19 boys – 11 girls).
Mpho’s class was considered to be particularly challenging:
A high number (10 out of 30) of learners who had previous repeated a Grade, and
A high number (8 out of 30) of learners who had been, or were waiting to be, formally assessed by an educational psychologist to diagnose specific learning difficulties.

5. Community and school context

6. Mpho
Mpho had Sesotho as his home language and learnt mathematics in English.
Repeated Grade 1.
Although his Grade 2 teacher was concerned that he was still not meeting the grade level requirements, he was not on the list of learners waiting to be formally assessed, as other learners’ needs were considered greater than his.
Written pre-test (5 additive relations word problems and 2 bare additive relation calculations tasks)
Pre-test
0 / 7
Post-test
3 / 7
Delayed post-test
5 / 7

7. Theoretical framing
Drawing on Variation Theory (Marton and Booth) and the Watson and Mason concepts of
Example spaces (Watson and Mason)
Learner-generated examples
Eg 1
Eg 2
Eg 5
Eg 3
Eg 4
Dimensions of possible variation

8. Additive relation word problems
Numbers: Type and magnitude
Contexts: Discrete / continuous objects
Problem types: Actions/relationships

9. Collection = subset A + subset B 4. Compare
ACTION - Change
STATIC - Relationship
1. Change increase Start + change = result 2. Change decrease Start – change = result
3. Collection
Collection = subset A + subset B
4. Compare
Whole = referent +/- difference

10. Additive relations problem types: CHANGE
TYPE 1: Change increase problems An action of joining that increases the number in a set.
‘I have 5 apples. I get 3 more apples. How many apples do I have now?’
TYPE 2: Change decrease problems An action of separating which decreases the number in a set.
‘I have 8 apples. I eat 3 of them. How many are left?’

11. Additive relations problem types: STATIC
TYPE 3: Collections problems Two parts make a whole but there is no action. The situation is static.
‘I have 8 apples. 3 are red. The rest are green. How many are green?’
Type 4: Compare problems The numbers of objects in two disjoined sets are compared.
‘I have 8 apples. You have 3 apples. How many more apples do I have than you?’

12. Collection = subset A + subset B 4. Compare
Varying the position of the unknown
1. Change increase Start + change = result 2. Change decrease Start – change = result
3. Collection
Collection = subset A + subset B
4. Compare
Whole = referent +/- difference
Common whole-part-part structure

13. Task 1: Assessment item (individual interview)
Can you tell me a story/ make up a word problem for 10 – 7 = ... Can you keep the same numbers but give me a harder problem? And another one?

14. Mpho’s pre-test story activity
Collection?
Two states: some broken (and some not broken)
T: Let me give you an example of a story and then maybe you can also tell one. … [I had 10 balls. I kicked 7 balls were over the fence. How many balls are left?]
T: Okay now can you tell me a story? I told you a story about balls. Mpho: Mmmm. There was 10 cars. 7 was broken. How much is left?
Change?
Something happens, then some are left
T: Can you tell me another story? Maybe a bit harder one. Mpho: There was 20 cars 10 broke-ted, there was 10 left.
Harder:
Numbers get bigger:
Evidence of change decrease (result unknown) and collection?

15. Mpho’s pre-test stories
[Tell me another one, but keep the 10 and 7] Mpho: There was 10 hearts, 7…10 hearts, 10 heart, 4 choc… 10 chocolates, 7 was eaten. Is 3.’ T: That’s a nice one. So we’ve eaten the hearts, they’re chocolate hearts? Mpho:[nods] N: So we’ve got 10 chocolate hearts…10 chocolate hearts. 7 were eaten. How many were left?. Mpho: [smiles and nods]
Change decrease (result unknown)
Start: 10
Change: eat 7
Result: Is 3

16. Mpho’s post-test stories
Can you tell me a story/ make up a word problem for
10 – 7 = ...
Mpho: Mpho have 10 sweets. My friend take aways 7 sweets. How many Mpho have?’
Can you keep the same numbers but give me a harder problem?
And another one?
Mpho: My friend have 70 sweets

17. Mpho’s post-test stories
Use 10 and 7, but make the problem harder
Mpho: My friend have 14…13 sweets, he take away 7 sweets, 15, 16 sweets [Mpho reaches for structured number line, counting back in ones from 16] Mpho: Is 1, 2, 3, 4, 5, 6. My friend have 17 sweets, he take away 7, [Mpho touches a structured number line, counting back in ones from 17] Mpho: 1, 2, 3, 4, 5, 6, 7 T: And you get to? Mpho: 10
Change decrease (start unknown)
Start: ?
Change: Take away 7
Result: Is 10
... – 7 = 10

18. Mpho’ s numbers:
6-4-2
Task: Whole class task
Varied characters:
Mom, dad, friend
Varied objects:
cats, cars, eggs
Varied actions:
bring, bring, breaks
Change decrease (change unknown)
Change decrease (result unknown)

19. As diagnostic assessment task (individual interview)
Concluding remarks
Story-telling tasks (problem-posing / learner generated examples) can be used to reveal learning gains or absences which could be analyzed to inform structured and targeted support to LSEN (and all learners):
As diagnostic assessment task (individual interview)
As differentiated learning activity (whole class)
Working with example spaces and dimensions of possible variation be of value to other teachers engaged in inclusive education practices.

20. nickyroberts@icon.co.za Thank you
Roberts, N. (in press). Learners exemplifying for themselves: Grade 2s telling additive relations stories. In H. Venkat & M. Graven (Eds.), Improving primary mathematics education, teaching and learning: Research for development in resource constrained contexts: Palgrave.
Roberts, N., & Stylianides, A. J. (2013). Telling and illustrating stories of parity: a classroom-based design experiment on young children’s use of narrative in mathematics. ZDM Mathematics Education,