Joseph G. Curtis and Dr. Robert Speiser, Mathematics Education
When you toss a coin at the start of a football game and call “Heads!” the probability the coin will land “heads-up” is one-half. If the referee does not hear your prediction, he must toss again. The probability the second toss will land heads is again one-half. But what is the probability that both tosses were heads? In the double coin toss, the probability of two heads in a row is one-half times one-half: one-fourth. Although the rule (multiply the probabilities of the two independent events in order to calculate the probability of both events) is simple to carry out, students often have difficulty in recognizing when to apply this problem-solving technique to real-life situations. Because of this difficulty, and because I believe better understanding creates better learners, better teachers, better spiritual beings, and a better society, this research investigated how persons come to understand where and how multiplicative probability accurately describes a given situation.
In order to study how students understand fraction multiplication, I use Marvin Minsky’s cognitive science idea of frames to propose a hypothetical organization of mathematical thinking. Robert B. Davis, whose book inspired me to use frames in this context, gives this example of a frame: when a person hears about a birthday party, he calls to mind paper hats, streamers, cake, singing, etc. This information is contained in the birthday frame. The power of using frames to represent knowledge structures comes from the frames’ capability to organize and gather information for future decision and future action. In this study, I propose some initial frames students may use for fraction multiplication. These proposed frames certainly need further development, and even when fully developed are only hypothesized structures, in other words, tools for teachers, students and learners.
To develop this framework, I videotaped a class of twenty-four future elementary educators in the setting of a Mathematics Education 305 course. This experimental course is the second in a series of two math courses for elementary education majors and focuses on multiplication, division, fractions, and probability. The students are given tasks similar to those they would expect their future students to do. My video record consists of conversations and interactions between students as they solve problems together, usually in groups of six. I served as videographer and later analyzed the data I had recorded.
In this study, a group of six women was given a zip-lock bag containing about one hundred fifty cubic centimeter-sized colored wooden blocks. For each blue block in the bag, there were two white blocks-about fifty blue and one hundred white blocks were in the bag. The women, drawing blocks from the bag one-at-a-time, were to find the probability of drawing two blue blocks on two successive draws. They chose to replace the first block after it was drawn and its color observed; the proportion of blocks in the bag on the second draw would remain exactly the same as on the first draw.
The group of six women began the problem by actually drawing the blocks from the bag. On the first day working on this problem, the group of six women reached a consensus: multiplying the individual probabilities is a valid way to solve this problem although the they could not explain why,. One woman, Joanne, seemed to be particularly frustrated with her initial inability to express her understanding to the other five women in her group. Joanne wrote, “I can see it, I just can’t explain it!”
The next day a student researcher, Kourtney Peters, led the students to connect the block-bag problem with a problem the students had already worked on several weeks earlier. In this previous problem, the students are to consider an auditorium full of people. One-half of all those in the auditorium are female and one third of all those in the auditorium have blue eyes. The question was posed: how many women with blue eyes are in the auditorium? Relating the successive draw question to the auditorium task proved to be useful for the students as they explained their thinking to each other. Several days later, a student from another group in the same classroom explained how her group had connected the probability question with an even earlier problem dealing with combinations of block towers. As her final project, Joanne, a major player in connecting the auditorium problem with the block bag problem, chose to apply the block tower combination problem, which she had not seen before this discussion, to a new situation.
Our research subjects’ and importantly Joanne’s answers to the question “Why do you multiply in this situation?” are important, not only because they shed light on how students understand probability, but also because their answers, and how the student-subjects communicate these answers to others, expose how the subjects organize their thinking on fraction multiplication as well. At this time, the proposed frames that organize students’ thinking for fractional multiplication are still under development but will be developed in a joint paper with Drs. Speiser and Walter. So far we feel strongly that the frame used for fraction multiplication is closely related to the frame used in the successive draw problem. This is significant because it suggests that students may think differently, and at a deeper level, about fraction multiplication than we previously thought.
This present study is important to this work because it assembles data necessary to postulate key frames, and because it sets up an elementary basis for connecting the students’ understanding of fraction multiplication to probability. The data that we have at this time will be further explored as analysis proceeds.