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Fast horses, slow cows: Context and mathematical tasks

Merilyn Taylor and Bronwen Cowie
Abstract: 

This article reports on the analysis of a videotape of four children participating in a group mathematical task. The authors discuss the ways that the context, social organisation, and resources of the task shaped the children's approach to the task, and the implications of this for teachers.

Journal issue: 

Fast horses, slow cows: Context and mathematical tasks

Merilyn Taylor and Bronwen Cowie

In this article we describe and discuss how four children went about solving the Farmyard Race, a National Education Monitoring Project (NEMP) assessment task on co-operative logic (Flockton & Crooks, 1998, p. 56). As part of a NEMP probe study, we analysed the video of the children’s solution processes from a sociocultural perspective (Boaler & Greeno, 2000). This led us to take particular note of the ways four children worked together to use speech and equipment to help solve the task. The insights we gained from this close analysis have implications for how teachers might present and enact mathematical tasks with groups of children.

Considering contextualised mathematical tasks

Contexts and group work are recommended in Mathematics in the New Zealand Curriculum (Ministry of Education, 1992) and in the draft curriculum (Ministry of Education, 2006). Group work and handson activities were identified by New Zealand teachers of mathematics in a recent study as being particularly effective in the teaching and learning of mathematics for children of all ages (McGee et al., 2001). Contextualised group tasks are usually presented via written text and accompanied by diagrams, verbal instructions, and /or supporting equipment. Contextualised tasks, whilst they can be engaging, place a number of demands on children. Children have to be able to discern the mathematical ideas embedded within a contextualised story. They need to appreciate if and how their everyday knowledge and experiences can help with their understanding and resolution of the mathematical intent of a task (Taylor & Biddulph, 1994). Yet another challenge is that more than one meaning can sometimes realistically be taken from a word or phrase (Barwell, 2005). Added to this, the equipment supplied as part of a contextual task can both support and divert children from the mathematical intent embedded within a context. On the one hand, it can act as a tool that helps children explore, describe, represent, and communicate mathematical ideas to others (Gravemeijer, 1994). On the other hand, children can use equipment in ways that lead to idiosyncratic interpretations and resolution of a contextually based task (Kanes, 1998). The issues involved in interpreting a contextual task are all the more salient when children are expected to work together to make sense of, and solve, it. Children working towards a joint solution need to know and be able to employ social interaction and decision-making skills in conjunction with mathematical thinking. Research suggests their expectations about how to go about this are shaped by their past experiences of solving mathematical problems (Cobb, Gravemeijer, Yackel, McClain, & Whitenack, 1997), adding yet another level of complexity. It was for these reasons that we were interested in studying in depth a group’s responses to a contextualised mathematical group task.

Our research agenda

We are interested in children’s mathematical thinking from a sociocultural perspective. This perspective places attention on the interactions between people, the tasks they seek to accomplish, and the setting they are in (Cobb, 2000). Knowledge and understanding of mathematics are not seen as attributes that an individual possesses but as social practices in which they engage. The focus is on how people use language, materials, and representations as they work together (Boaler, 1999). This frame therefore allows for a focus on the ways in which children go about solving a task—their talk and interactions as well as how they use material resources. To conduct such an in-depth analysis, we accessed videotapes of groups of students working on NEMP mathematics tasks via a NEMP probe study. Here we report on the interactions of one of the groups we studied as they solved the Farmyard Race problem (see Cowie & Taylor, 2005, for the full report of the study).

Children working on the Farmyard Race task

In this section we begin by describing the problem. We then describe and discuss the way the context of the task intersected with the children’s everyday knowledge and experience, the issues associated with perceived ambiguities in the language of the task, the role played by the equipment within the task, and the social decision-making practices the children employed.

Introducing the task

The Farmyard Race task required the children to appreciate and aggregate ideas to do with order represented by ordinal number words such as first and third, and spatial markers such as before, after, between, and followed, in order to determine the finishing order of eight common farmyard animals (see appendix). The video we analysed for this paper began with the NEMP administrator placing a recording strip on to a table and introducing the problem to the children by stating that the task was about an animal race. She placed eight plastic animals, one by one (two horses, a cow, a sheep, a pig, a piglet, a goat, and a dog) on the table beside the strip, naming each in turn. The children responded with associated animal noises as each one was put down! Next, the administrator gave each child one of four clue cards. She instructed the children that their task was to find the finishing order of the animals. The implication was that the children needed to aggregate information on the clue cards to decide the order in which each of the eight animals finished the race, and, indeed, the four children (Cam, Katy, Andrew, and James) showed a commitment to work together to solve this problem. Information and ideas were pooled. The children reiterated ideas, re-voiced statements made by others, and made links to earlier propositions.

The intersection of the children’s ideas and the context

Observing how the children went about solving the Farmyard Race co-operative logic word problem highlighted the complexity of the interaction and connections between a contrived context and children’s own knowledge and experiences. The willingness with which the children participated in the task underlined that a context can be motivating. The context served as a source of opportunities for mathematical reasoning and thinking in a way that co-ordinated the children’s work. However, it also set up a tension between the children’s knowledge and experience and the mathematical ideas embedded in the task; a tension that required the children to negotiate the boundary between school mathematics and everyday concerns (Cooper, 1996). Katy and James struggled to reconcile their everyday knowledge of horses with the finishing order that was emerging from consideration of the clues. They argued that, in Katy’s words, “Horses are one of the fastest things in the world.” Late in the solution process, James said: “I don’t think a horse could be last.” The finishing place of the cow also posed a dilemma for James and Katy. None of the eight clues specified the relationship of the cow to other animals in the race; rather the placement of the cow required the children to synthesise information on the clues. James was first to speculate about the cow’s position.



James:

This one [the cow] is really slow, so it must go there. [James put the cow near the back of the recording strip.]

Katy:

No, they’re not [slow] … we have bulls next door.

Andrew:

It might of [come last— Andrew pointed to the last place on the recording strip], but we don’t know that, do we James?

Here James used his everyday experience to assert that cows were slow and hence this one would have to end up towards the back of the field. Katy offered an alternative view based on her experience of “two bulls next door that could run really fast”. In contrast, Andrew argued against the use of everyday knowledge when he asserted it was not certain that the cow would come last. Of the four children, only Andrew seemed to be aware that the task was not grounded in reality but rather it provided a particular setting for engaging with mathematical ideas. Andrew often questioned whether it was appropriate to draw upon everyday experiences to solve the task and at one point he proposed that the story was a fake. At another time, Andrew held up the cow and asserted “It’s not as if this guy’s going to start running is it? It’s plastic.”

Perceived ambiguities in the language of the task

The children made a number of reasonable interpretations of the clues but some of the language proved to be ambiguous to them in the context of the task. For instance, Katy pointed out that, in relation to the clue “The white horse saw four legs beat him home”, all the animals in the race had four legs. She questioned whether the clue contributed any worthwhile information:



James:

Read your clue again Andrew, read your clue again.

Andrew:

Goat wanted to bite mother pig’s tail as he followed her across the finish line. The white horse saw four legs beat him.

Cam:

What’s your first clue again?

James and Andrew (in unison): The white horse saw four legs beat him home.



Katy:

They all have four legs though.

Cam:

All of them have four legs, that’s stupid.

Katy:

Yeah, that’s true.

The children did not discern this clue implied that only four legs crossed the finish line before the white horse and so therefore just one animal could be in front of the white horse. The clue “The brown horse led the pack until he stopped to eat” was also a source of confusion. The children recognised the word “until” signalled the brown horse had led the pack up to the time it stopped to eat. They explored different possibilities for its final placement. Katy, implying from her perspective that the brown horse had to win the race, suggested that “Maybe he stopped to eat after he had finished?” These examples illustrate some of the ways in which children can rationalise or misread the intended meaning of language in mathematical tasks.

The role of the equipment

For the task, the equipment served as a tool for representing the children’s taken-as-shared reasoning. The recording strip and the plastic animals allowed the children to explore and manage the possibilities arising from their interpretation and aggregation of the clues in a way that was accessible to them all. The movement of the equipment supported group discussion of alternative placements. Possible finishing orders were made visible and could be readily discussed and changed. However, the children’s affinity with the brown horse created tension. When the administrator first produced the brown horse, James, then Katy, reached for it. James stated: “I bet the brown horse wins.” Katy and James were committed to the idea of the brown horse winning the race throughout the problem-solving process. After the task had been concluded, Katy lamented: “I can’t believe my favourite animal [the brown horse] came last. This is really hard.” Her comment highlighted the affective element involved in reconciling everyday knowledge and beliefs with the mathematical demands of the task.

Social decision making or mathematical logic

The challenges involved in collaborating on contextual tasks are important when children are expected to work together. The question of where the cow would be placed recurred several times. When James could not persuade the others that cows were slow and so one could not win the race, he suggested taking a vote to resolve the question about its position. The vote was three to one for the cow being first:



James:

Hey everybody, we’ll have a vote, have a vote then.

Andrew:

No, that doesn’t solve anything then, James.

James:

Yeah, but there’s no point arguing then. Right, who votes for the horse to be last and the cow to be first?

Here, James called on the practice of voting to invoke a process of social decision making. James accepted the consequences of a majority gracefully. None of children queried the use of a popular vote to resolve a difference of opinion in a logic problem, which suggests they confused the social and mathematical aspects of the task.

Discussion and implications

Contextualised mathematical tasks provide a powerful forum for helping children to think and talk about mathematical ideas. In the case of the Farmyard Race, the clue cards, the plastic animals, and the recording strip provided a focus for talk and a way for the children individually and collectively to explore possibilities for the finishing order in the race. However, these children’s responses to the Farmyard Race task suggest it cannot be assumed that children will always appreciate the mathematical intent of a contextualised task as they can draw on their own experiences in varied and idiosyncratic ways. The nuances of language and the subject-specific ways in which it is used in mathematical tasks, particularly those that are contextually based, can pose a challenge for children. A significant part of this challenge arises because, typically, contextualised tasks include a blend of everyday and mathematical meanings for words. While the ambiguities this produces can lead to fruitful discussion as children share their thinking about possible meanings, children can also be diverted from thinking mathematically. Equipment can also help to focus and coordinate shared endeavour, or it can serve as a distraction from the mathematical intent of a task. In this case, two children formed an attachment to the finishing order of individual animals independent of the logic of the task. As we have shown, groups of children may use social norms rather than mathematical logic to resolve any differences of opinion if they do not appreciate the mathematical intent of a problem. The implication of these issues for teachers is that children may need support to understand the contrived nature of some school mathematical tasks. The successful interpretation of, and solution to, many school contextualised tasks as mathematical activities, depends on children understanding the nuances of contextualised tasks.

Acknowledgement

This work was conducted with and made possible by Probe Study.

References

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Appendix: The clues for the Farmyard Race task

Dog was third to cross the finish line.

Piglet finished second to last.

Two animals came between mother pig and her piglet.

Goat wanted to bite mother pig’s tail as he followed her across the finish line.

Sheep and piglet ran together until the end when piglet tripped.

The white horse finished before the goat.

The white horse saw four legs beat him home.

The brown horse led the pack until he stopped to eat.

Merilyn Taylor is a lecturer in mathematics education at the University of Waikato. She is interested in children’s perpectives on learning mathematics.

Email: meta@waikato.ac.nz

Bronwen Cowie teaches in science education and educational research methods at the Centre for Science and Technology Education Research, the University of Waikato. Her research interests include assessment of learning, teacher–student classroom interactions around science and technology ideas, and the use of ICT for teaching and learning.

Email: b.cowie@waikato.ac.nz