Laying foundations in chance and data

Jenni Way, University of Western Sydney

Chance in our lives

We all deal with situations involving the element of chance every day - Will it rain today? What will I do if I miss the train? Is this a fair game? Much of our decision making involves trying to predict events and considering possible results of our actions. Whenever chance is involved, the unpredictability of the situation means that our normal logical thinking processes may not be entirely appropriate. For example, when you fill out Lotto entries, do you deliberately spread the numbers out? Why? Each number has the same chance of winning. The Lotto machine knows nothing of numerical order! (Actually, if you choose a string of consecutive numbers and win, you’d probably have the prize pool to yourself!). Our Australian culture includes some classic expressions that refer to our encounters with chance: ‘Buckley’s chance’; ‘Murphy’s law’; ‘Against all odds’; ‘Fat chance!’; ‘In your dreams!’, and of course some less polite ones.

Learning to reason ‘correctly’ using probability concepts, such as randomness and chance, is a skill that very few people have mastered, mainly because we have had very little assistance to do so. A fairly large body of research, for example Piaget and Inhelder (1965), Fischbein (1975), Green (1983), Jones et al. (1996), Way (1996), has shown that young children develop intuitive understandings of basic probability concepts without instruction. There is also evidence that many of these intuitions are misleading or incorrect, and that by adulthood, they develop into misconceptions that are extremely difficult to correct. However, studies such as Fischbein’s and Jones’s have established that children are responsive to appropriate instruction on probability concepts.

Some researchers, such as Shaughnessy (1992) and Fischbein (1975), believe that intuitions and misconceptions need to be challenged through experimental activities involving concrete materials. This is a difficult task for many teachers because they hold misconceptions or limited understanding about chance concepts themselves. Most published teacher resource books, mathematics programs, and pupil workbooks currently being used in NSW classrooms contain activities based on chance concepts. Unfortunately some of these publications present mathematically incorrect or confusing ideas about basic probability. The development of chance concepts, and the associated language, are not catered for in the current K—6 NSW Syllabus, even though all other state syllabuses (except the Northern Territory) have included work on chance for several years. The new draft chance and data strand proposes a carefully prepared sequence of activities and information that seeks to support both teachers and children in learning and clarifying chance concepts.

Data in our lives

Society has placed an ever-increasing demand on people’s ability to deal with, understand and produce data. Informed decision making is dependent on reliable information. Which product should I buy? Who should I vote for? Which health cover is best for me? Without sufficient understanding of data handling techniques, it is not possible to make critical judgements about the reliability of data. There is research evidence that suggests that children need assistance in developing the basic concepts and skills over a long period (for example, Pereira-Mendoza & Swift, 1981; Watson, 1995).

In our schools, children engage in inquiry processes across all the key learning areas (KLA). Some of the syllabuses, for example science and technology, specify data concepts and skills in outcomes statements. Obviously the children need assistance in learning the required concepts and skills - hence the data component of the new draft chance and data strand. The NSW K—6 Syllabus already contains some units on graphs, but this represents only one aspect of working with data, that is, displaying data. There is a need for coherent development of all the processes and skills associated with working with data.

Probability concepts

The basic concepts of probability include chance, randomness, sample space (all available possibilities), independence of events, and order of likelihood. Development of the associated language is an important aspect in acquiring and communicating probability ideas. Development of these concepts takes place in two main contexts: social and experimental. The social contexts use normal day-to-day experiences of the children to focus on language, beliefs, and decisions involving chance. The experimental contexts use games and concrete activities involving chance to expose the underlying mathematics. Experimental activities often rely on the gathering of data to make or evaluate predictions.

Data skills

Although concepts and understandings are also important, work on data has a strong skills base. These skills include: posing suitable questions, gathering and organizing data, displaying and summarizing, interpreting data, drawing conclusions, and designing investigations and surveys. By the end of primary school many children are capable of quite sophisticated statistical thinking, particularly if firm foundations have been built in the early years.

Some chance and data activities for K—3

One of the strengths of working with chance and data is that the activities overlap with many other areas of the curriculum, both from mathematics and other KLAs. The following activities are examples of the chance and data work that can be carried out in the early school years, Kindergarten to Year 3. Consider the chance and/or data ideas and skills being developed in each activity. Also consider how the activities draw on mathematical understandings, develop language skills, and support topics in other KLAs.

Shoes (Years K—1)

Invite at least twenty children to remove one shoe (or use hats, lunch boxes, bags, pencils, etc.) and place it on a mat. Pose the question: ‘Do more of us wear rough-soled shoes than smooth?’. The children can then sort the shoes into the two groups, and compare them to discover the answer. Encourage the children to pose other questions about the shoes, then reclassify the shoes appropriately. Year 1 children could explore ways to represent what they have done and found out through drawings or symbols, such as squares of paper forming a simple graph.

In this activity the children are learning to pose questions that can be answered through data collection and organization. They are sorting and classifying real objects and comparing groups using one-to-one correspondence or counting. There is opportunity to develop observation skills, descriptive language, and comparative language in a meaningful context. Simple graphing procedures can be either introduced or consolidated. The concept of representing real data through drawing or symbols is an important step into the abstract world of graphing.

Possible/impossible (Years 1—2)

The children individually draw and/or write something they consider to be either possible or impossible to occur at lunchtime. As a class, the children share their ideas, discuss differences of opinion, and form a display under the headings ‘Impossible’ and ‘Possible’.

This activity does more than clarify the meaning of ‘possible’ and ‘impossible’. It also provides opportunity for children to confront the concept of chance in a social context. The children should be encouraged to challenge each other’s thinking. For example, one child might say that it would be ‘impossible to see Grandma in the playground’, while another child could argue that Grandma just might turn up for a surprise visit, etc. There is a strong tendency for people to use past experiences and routines as a basis for making decisions about chance events, rather than considering the element of chance itself. It is easy for us to make the mistake of equating ‘very unlikely’ with ‘impossible’.

Heads and tails game (Years 2—3)

The children stand up and decide whether to place a hand on their head or their hip. Flip a coin. If it comes up heads, those children with their hand on their head remain standing while the others sit down. Ask the children still ‘in’ to make their choice again, then flip the coin. This continues until only one child remains standing: the winner. After playing the game several times, discuss the likelihood of getting out for each choice. Is there an equal chance of getting heads or tails? Play the game several more times, keeping a record of each flip of the coin.

An extension for older children (and adults) is to flip two coins and have the children use both hands to indicate their choices. (2 heads, 2 tails, or 1 head & 1 tail). Keep a record of all the results on the board. After playing the game several times, ask if there are any strategies that increase your chances of winning. Play the game several more times so that the children can try out different strategies. Eventually the children should realize that the outcomes are not equally likely (there are two ways to get a head and a tail).

This game provides the opportunity for the children to encounter the concepts of sample space (the two possible outcomes of flipping a coin), equal likelihood, and randomness. The data collected might confirm expectations, or it might contradict expectations. Logic might tell us that there should be an equal number of heads and tails occurring, but the element of chance means that something unlikely (like 90% heads) could happen. One of the more sophisticated understandings in probability is the ‘law of large numbers’, that is, if you flip the coin enough times the results will come out even ‘in the long run’. One misconception that children (and adults) hold is that somehow the coin should ‘know’ what it is supposed to do, and produce alternating, or at least balanced, results of heads and tails. In reality, each flip is a separate, independent event.

The extension game challenges the assumption that the sample space (possible outcomes) remains the same when two coins are used.

Lucky dip (Years 2—3)

Organize the children into five teams, with a scorer for each team. Allocate each team a colour, then place a corresponding block for each team into a paper bag. Make 20 to 30 draws from the bag, while the scorers keep a count of how many times their team’s colour comes out. These scores could be written up on the board in a table. Ask: ‘Is this a fair game?’ Do you think the same team will win next time?’ Play the game a couple of times. Without showing the class, change the composition of the colours in the bag, deliberately leaving out a colour and making one colour clearly dominant (such as 3 of one colour). Play the game again and discuss the results in comparison to the previous games. After a few more games with the biased bag, invite the children to guess the contents of the bag and explain their reasons. Show the contents of the bag, and invite predictions for the winner of the next game.

This type of activity can be a powerful way to develop basic probability concepts. Through this game, the relationship between the sample space (the coloured blocks) and the likelihood of specific outcomes (the block drawn from the bag) can be explored. In other words, if each team is represented by one block, the game would be fair, because each team has the same chance of winning. However, because the draws are random, there is no guarantee that all the different colour blocks will be drawn out with equal frequency. When the biased sample space is created, there is likely to be a noticeable change in the outcomes. It also becomes impossible for one of the teams to win. Obviously this could lead to some quite abstract discussion, but most young children would be able to deal with these ideas within the concrete context of the game. This activity also illustrates the overlap between chance and data. The game generates data which are then used to assist decisions about the sample space, likelihood and fairness.

Conclusion

Activities in chance and data are of limited value if the teacher does not understand the underlying concepts and skills, and perceive the potential for challenging children’s thinking and introducing new skills. Data activities are easily integrated with other areas of mathematics and other KLAs. Chance and data activities are rich in mathematics, and link closely with real world or ‘social’ mathematics.

References