MAKING STATISTICS FUN - How left-handed are you? (grades 2-4)
When you look at the mathematics assessments for the EQAO in grade 3, the problem
solving utilizing data involves very little thinking. Students might have to add missing
data to a tally chart of favourite numbers from 1 - 6 making sure the mode is 4. They
could just as easily be asked to make a bar graph from data in a pictograph. None of
this requires much thought; none of this fosters thinking about data to predict; none
of this helps us to tell a story about ourselves or others. In other words, none of this is
surprising, pleasing, or . . . fun.
We wondered if there might be a way to involve the students in investigating an
interesting and novel question, and to let them use data to predict and tell a story about
ourselves. This we did with the very simple question: How left-handed are you?
- What is your favourite fruit?
- Is your favourite colour red, blue or yellow?
- How many pets does your family have?
These are all typical polling questions that students ask when required to poll their classmates so they can make and interpret graphs in the primary (and unfortunately, at times, junior and intermediate) grades. As far as the Ontario mathematics curriculum is concerned this is all fine. Leveled exemplars from the grade 3 EQAO mathematics test reward making sure that all graph elements are present and accounted for, not deep or complex thinking about issues or ideas. If we are to foster mathematically literate children, the answers to the polling questions above might better be:
- Why do you want to know?
- How are you going to use your data?
- Does your data matter?
With this as a mathematical backdrop, we wondered if our students might not be able to tackle something a little bigger and more complex than a simple polling question. Would it be possible for the students to engage with the big idea that data can tell us something about ourselves and that it can be used to predict?
We started out our investigation with a very simple activity. We asked our students how left-handed they were. As they looked at their hands, they gave us a variety of responses that ranged from “very” to “not at all.” We gave them a paper with two simple grids on them. One was marked right, the other left. The task was to write as many X’s as they could with each hand within 30 seconds.
In order to compare, we showed the students how to make their scores with each hand into an ordered pair and to plot them on an x-y graph creating a scatter plot (x = left hand score, y = right hand score).This proved to be quite easy as almost every student had played Battleship before, and they were quite familiar with coordinates. The resulting scatter plot was very interesting. We had large clusters of data near the x-axis and small clusters near the y-axis. There were a few scattered data points in the middle.
The students quite easily identified the bottom cluster as being right-handed, and the top as left-handed. We drew a straight line (x = y) and asked them what they thought it might represent. The few students whose data points were near the line identified themselves as being equally good with their right and left hands. We gave them the word ambidextrous. The students went back to their graph and started looking at how close they were to the ambidextrous line. One young student figured that, “The farther away from the ambidextrous line I am the more right-handed I am.”
With this experience under their belts, we proposed to the students that they make up their own math experiments to find out how left-handed they were doing other activities. Different types of tests were created by the students: bouncing a ball, picking up jacks, making balls of plasticene, or cutting strips of paper to name but a few. Throughout, the students were able to tell how right- or left-handed they were by comparing their results to the original scatter plot.
The students were positioned as investigators for these data activities. As investigators, they were predisposed to asking questions and making meaning of the results of their math experiments with respect to the baseline results we plotted as a class. This is the type of inquisitive attitude that we wished to engender in our students. Once they had conducted two different math experiments, the students seemed to wane in interest in conducting further experiments. They had found what they needed and made sense of their results. It was time to move on to something new.
For the first math experiment, the teachers were the primary investigators. They lead the students through the first experiment, demonstrated how to make ordered pairs from the data, and they ensured that the students knew how to plot the coordinates. Once this had been demonstrated once, providing a model of how to conduct an investigation, the teachers were able to step back, and take on a more supportive role. Always circulating and asking questions to check on how the students were making meaning of their results, the teachers acted as facilitators rather than providers of knowledge.
Implications for Practice
If we are to truly teach our students to understand and use data in a meaningful way, we must reach beyond the curriculum for bigger ideas than completing bar graphs and finding the mode of a set of data. Creating a scatter plot was easy and allowed the students to see the relationship between the performance of their right and left hands. Letting the students use this type of graph before grade 8 where it is introduced, allowed them to move beyond working on data to just create graphs to becoming young mathematicians working with statistics. In other words, they are well on their way to becoming mathematically literate.
A few weeks after we had wrapped up the left-hand right-hand experiments, one of the students shared that they had heard that feet grow faster outwards than you height grows up when you are getting close to being a teenager. After much measuring and plotting of their foot length versus their height, we did indeed find that some children seem to have faster growing feet by looking at the scatter plot.
Another activity they tried that was related was to plot their height versus their arm spans. The teacher drew a straight, diagonal line along x = y. The students were asked what to call this line and they responded with, “Squares!” They could clearly see who was a tall rectangle, who was a wide rectangle, and who was a perfect square. Young students are much more able to think mathematically, statistically than the mathematics curriculum gives them credit for. We now knew that if we design interesting, puzzling, novel mathematical experiences involving data, the students are completely capable of thinking, and finding relationships so that they can tell a story about themselves. “We aren’t too left-handed. How left-handed are you?”
The Math Performance Festival is funded by the Imperial Oil Foundation, the Fields Institute, Research Western, the Faculty of Education at UWO, and the Canadian Mathematical Society. A project by George Gadanidis (UWO), Marcelo Borba (UNESP, Brazil), Susan Gerofsky (UBC), and Rick Jardine (UWO).