1. The micro-evolution of mathematical knowledge: the case of randomness
An NCETM research study module

2. This module is based on
Pratt, D. & Noss, R. (2002) Micro-evolution of mathematical knowledge: the case of randomness
Dave Pratt and Richard Noss work at the London Institute of Education. This paper is a draft of one that was eventually published in Journal of the Learning Sciences 11(4),

3. Why this paper is interesting and useful
it describes how complex it is to move from meeting a new idea to recognising when it is useful in a new situation.
it describes how context affects learners’ understanding of a new abstract idea – (the authors call it situated abstraction).
it gives insight into how sequences of tasks might be designed that lead learners to apply a new idea.

4. the paper is about randomness – a difficult idea for students.
it demonstrates how well-designed use of digital technology can provide a window on meaning.
the research method involves a small number of students using a digital resource for an extended period of time, a method that could be used by a teacher to investigate other aspects of learning.

5. Features of this module
This module offers:
Access to a full research paper;
key ideas;*
discussion points;
summaries;
reading excerpts;
critical reflections on research;
suggestions for development work.
* there are more theoretical ideas in the paper than we look at in this module, and if you read the full paper you might want to skip some of them

6. Discussion point: randomness
What is meant by randomness? What understandings of randomness do your students have and how does this develop through experience? (save your ideas for later)

7. Discussion points: what is meant by ‘abstract’ in mathematics?
What do you understand by ‘an abstract idea’?
If students learn by constructing understanding, how do they make new ideas for themselves – especially abstract ones?
How do students learn that a new mathematical idea is useful in other contexts?
(see next slide for two possible models)

8. Idea passes through fuzzy boundary
Context 1
Context 2
Idea passes through fuzzy boundary A
Abstract idea B
Context 1
Context 2

9. Reading excerpt: what do the authors mean by ‘abstract’?

10. Key idea about situated abstraction
Pratt and Noss point out that any context involves people and resources in some activity, with complex connections between external tools and internal knowledge through ‘webbing’. They say that abstraction depends on the webbing.
(you can read about ‘webbing’ if you want to know more)

11. Key idea about situated abstraction
Situated abstractions arise as a way of making sense of situations
‘Situated abstraction’ describes the inherent relations and possible actions in a situation
Situated abstractions are imagined and described in language relevant for that situation
e.g., using a spreadsheet you might say ‘move this to here’ to describe moving a formula into a cell, rather than say ‘substitution’

12. Discussion point: the role of the teacher in abstraction
What is your role in enabling students to shift from situated abstraction to formal mathematical abstract understanding which ‘cuts away from context’? e.g. you might provide formal symbolism, technical language, other situations which use the same mathematics, multi-purpose diagrams.

13. Back to randomness: research method
The authors studied the work of 32 students in all. Students worked in pairs, firstly expressing their ideas of randomness, then working with computer-based tools with the researcher probing their reasoning as they did so (read about the methods used).

14. Summary of students’ understandings of randomness
Unpredictability
students connected randomness with unpredictability, such as when a die is thrown you cannot predict the outcome.
Unsteerability
phenomena for which there was no apparent control, such as how a drawing pin will land when tossed
Irregularity
no regular pattern in the results, such as decimal places for π
Fairness
no bias towards one outcome even when there is steerability, such as operating a spinner; an 'unfair' spinner (one with unequal sectors) might be 'non-random'.
These were not exclusive and the ideas were often associated with each other, for example irregularity and unpredictability were often linked.

15. Discussion point: matching to your experience
How do the categories on the previous slide relate to your earlier thoughts about students’ understandings?

16. Key idea: difficulties with the concept of ‘randomness’
True randomness generates no patterns, so there are no patterns to help students understand randomness. But students often try to make patterns to understand randomness (e.g. “I have thrown ten heads in a row, so there is a higher chance of getting a tail next throw”). Students could only describe randomness in terms observable fairness (deduced from patterns), or the absence of certain characteristics. Students did not talk about aggregating results over a long period of time, which is at the heart of randomness.

17. Summary of a session with one pair of students
The students were given COIN, SPINNER and DICE activities which needed ‘mending’ to give a pie chart with equal sectors (see Chancemaker to do this for yourself)

18. They could tinker with the ‘workings box’ to try and get it to give a fair result.
They could alter the number of trials they could make, up to a large number N.
(Get Chancemaker free)

19. Summary of a session with one pair of students (cont)
They found that the larger the number, the more evenly a pie chart of results would be spread. They could also affect the distribution D by adjusting the ‘workings’ to alter the relative size of sectors of the pie chart. They needed to coordinate these two new ideas, N & D, which had not shown up in their intuitive prior knowledge.

20. Summary continued: what did the students do?
One pair eventually found with the coin that N was a useful tool to explore fairness, but when faced with a similar problem for the spinner they did not use N. With the spinner they tinkered with the appearance of fairness and equality (D) rather than N. After fifteen minutes of tinkering and being prompted by the researcher they reinvented idea N. Finally, with the dice activity, they went very quickly to use N and then to coordinate this with D. It took three similar tasks to shift from ‘behaviour’ ideas of unpredictability, unsteerability, irregularity and fairness to using the abstract ideas of N and D.

21. Reading extract
Read the section about this pair of students in the full text (page 25, paragraph 2)

22. Discussion points: what were they thinking about?
What does the transcript in the paper reveal about their thinking? What does this say about the role of tasks and teachers in enabling learners to adopt new ways of reasoning about an abstract idea?

23. Summary: students’ thinking
Coins, spinners and dice were seen as very different contexts; they did not think of applying an idea that worked for the coin to the spinner and then to the die Ideas N and D were strongly attached to particular situations, but after a few similar experiences learners began to use these ideas together as tools The task design and sequencing allowed them to use their initial ideas, find them inadequate, and then tinker with other options. The researchers prompted students to articulate their reasoning Students did not put aside their intuitive notions just because more sophisticated ideas had proved useful in a particular situation.

24. Key idea: shifting to unfamiliar contexts
new ideas may gain higher priority within the context where they emerged, but low priority in unfamiliar settings
students need situations in which the new idea turns out to be the most powerful
only if students can perceive their importance will new ideas become high priority within the unfamiliar setting.

25. Suggestions for development: researching in your school
How can you find out your students’ understandings of randomness? Where and how do you and your colleagues provide tasks so that learners can develop abstract ideas by finding familiar structures in unfamiliar situations?

26. Reflecting on the project
Revisit the research methods: the original assumptions, the sample used, the task design, the analysis methods, the presentation of findings.
Are these appropriate for the research questions?
Are the links between them clear?
Do the conclusion and recommendations follow from the study?
Is it helpful for your thinking?
This might be a good time to read the whole paper, especially if you are interested in the background theories.

27. Further reading
Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26, Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students’reasoning about probability. Journal for Research in Mathematics Education, 24, Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings: learning cultures and computers. Dordrecht: Kluwer. Pratt, D. (1998b). The co-ordination of meanings for randomness. For the Learning of Mathematics, 18(3), Sfard, A. (1994). Reification as the birth of metaphor. For the Learning of Mathematics, 14(1),

28. Abstraction
A common approach is to regard mathematical abstraction as a strictly hierarchical process, progressing through a series of stages. For example, Dubinsky (1991) forwards the notion of a ―process, an interiorized version of an action in which a repeatable physical or mental transformation of an object or objects takes place. At some point, a process can be transformed by an object, when, according to this theory, the process has been encapsulated to become an ―object.
Sfard (see, for example, 1991) also regards the isolation of mathematical objects as a key achievement of mathematical abstraction. These process/object models have taken steps beyond identifying and labeling stages of mathematical abstraction: For example, Sfard (1994), in a paper in which she adopts a richer and more sociolinguistic stance than that of her 1991 work, has described reification as the birth of metaphor, which renders the knowledge more integrative and manipulable; Gray and Tall (1994) refer to the ambiguous nature of process and object.
Back to main presentation

29. Webbing
Noss & Hoyles (1996) go beyond considering the individual in the process of abstracting knowledge. Building on Wilensky’s theme of connections, they extend the idea to considering a whole network of such links, encompassing not only the individual but also resources external to that person. We will use the word 'resources’ from now on, rather than variations on 'meanings', 'knowledge', 'concepts', etc., to emphasize the complementary roles played by internal (cognitive) and external (physical or virtual) sources of meaning making. Thus the term resource will encompass both external tools and internal knowledge, including both informal knowledge like intuitions and formal conceptual knowledge. To describe this network, Noss and Hoyles use the idea of webbing to evoke the ways that learners come to construct new mathematical knowledge by forging and reforging internal connections through the interaction of internal and external resources during activity and in reflection upon it. The notion of webbing aims, therefore, to recognize the central significance of tools as external resources that shape the nature of the mathematical resources constructed, resources that have been observed to be highly dependent upon the particular attributes of those tools as cognized by students.
Back to main presentation

30. Methods – Students’ ideas of randomness
In these data, children were observed to use four separable resources for articulating randomness, namely unsteerability, irregularity, unpredictability and fairness. In each case, we use a single example to illustrate how the resource was expressed more generally. In this way, we provide a first glimpse of our view of the nature of naïve mathematical knowledge.
(i) Unpredictability
Objects like dice were often described as unpredictable. For example, in response to the question, 'Do you think there is any number which is harder to get than any other number?' one child commented, 'No because it just comes out at random and any number could come out at any time so you don’t really know which one is going to come out or which one is not going to come out.'
The apparent connection between unpredictability and random devices was very commonly expressed. Indeed, for many children, a way of checking the randomness of a device was to see if it was possible to predict outcomes.
(ii) Unsteerability
Phenomena were often seen as random when no known agent was involved in determining the result. For example, when asked to summarize how he would decide on randomness, one child responded, 'Well, you decide by if you’re not controlling it or if you’re not affecting it by doing anything, and if it’s like not bad weather or anything or nothing’s blowing it over or anything, that will be quite random, but if the wind was blowing it or you were putting force or it or something then it’s not that random.'
The primitiveness of unsteerability is also seen in the way it is often used to 'explain' other resources for randomness. Very frequently, children associated unpredictability with unsteerability. On such occasions, unpredictability was usually seen as the outcome of uncontrolled input.
Continued on next slide

31. Methods – Students’ ideas of randomness (cont)
(iii) Irregularity
A third way of making sense of random phenomena was through reference to the lack of a regular pattern in sequences of results. This resource was often linked closely to prediction so that patterns were conjectured on the basis of past results and then used to make predictions, which were tested by further trials. For example, when asked how he would test the fairness of a dice, one child answered, 'Testing it, I’d roll it and if it kept on going on one or another then I might think it’s got like a magnet or something inside it I’d test it about ten, fifteen times.'
(iv) Fairness
Fairness was often a defining characteristic of randomness. For example, one set of questions in the interview was designed to ascertain how the child thought about two spinners, one of which had uniform sectors and another, which had unequal size sectors. On the first uniform spinner, children often expressed concerns that the spinner may not be unsteerable but nevertheless recognized that there was no particular bias towards one number.
In contrast, the same child would often regard the non-uniform spinner as non-random. One child commented, 'No, because whoever made this, made the one and the three bigger so you’ll get the one and the three most of the time.'
The excerpts illustrate the nature of naïve knowledge of randomness as observed throughout our interviews and as frequently observable in early interactions with the software. Below we outline five aspects of this naïve knowledge.
We observed children articulating different naïve resources within moments of each other. For example, a child referred to the unsteerability of a device and moments later referred instead to its unpredictability; the fact that situations that were not controlled were often not predictable apparently encouraged children to express these resources
interchangeably.”
Back to main presentation