# How to Have Students Make and Test Conjectures

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It is through the process of having students make and test conjectures that higher levels of reasoning and more complex learning will occur.

Suggestions for using the Make and Test Conjecture Method

- Engage the students by having them make a prediction(s) about possible outcomes to this question and explain and share their reasoning.
- Have students collect, access, or simulate data to answer the research question.
- Have students analyze the data to see possible data-based answers to the research question.
- Create disequilibrium by having students compare their prediction(s) with actual outcomes.
- Promote discussions so that encourage students to come up with explanations for the predicted and actual outcomes in order to strengthen associations between concepts and develop the students reasoning abilities.

### Engaging Students

Adding the "make and test conjectures" method to an activity can be as simple as asking students to first think about a question before examining the data. In this way students become engaged with reasoning about the data and develop an interest in seeing the resulting data. It is important to have the students explain their reasons for the conjectures (often predictions) and then later try to verbally explain why they turned out to be correct or incorrect.

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Allan Rossman and Beth Chance (1998)* utilized the method of making and testing conjectures in their book

*"Workshop Statistics: Discovery with Data and Minitab"*.

For

**example:**In one activity they ask the students to guess the number of different states a typical student at their school may have visited. The students are also to guess which states would be visited least and which states would be visited most. They are then to guess the proportion of students at their school who have been to Europe. The students record their own personal data on these questions. Finally the students collect the actual data for these questions from their classmates. The students are then asked to compare their estimates with the actual class data and write a sentence or two comparing the two distributions.

*In this example,*students are engaged in reasoning about data, and have a reason to be interested in examining the graphs of data produced by the class.

In another **example** *Rossman and Chance (1998)* ask students to guess the number of people per television set in the United States, China and Haiti for 1990. The students then have to make a prediction about which countries will have few people per television, which will tend to have longer life expectancies, shorter life expectancies, or if there will be no relationship between televisions and life expectancy. Then students are given the data to analyze and use to test their conjectures.

### Confronting Misconceptions

*Chance, delMas and Garfield (2004)*use this method to develop student reasoning about sampling distributions. For

**example,**they found it beneficial to have students confront the limitations of their knowledge so they could correct their misconceptions and construct strong, correct connections about sampling variability and distributions. They used technology to generate simulations to test students predictions about how samples and sampling distributions behave; confronting some of the strong misconceptions students have about sampling and helping them construct an understanding of the Central Limit Theorem. They used the predict/test/evaluate method to create disequilibrium in the students and then used discussions to make sure the students fully integrated the information about the concept into their schemes.

Another **example** of using this method to confront misconceptions involves learning about confidence intervals. Many students do not understand what the 95% means (in a 95% confidence interval), and there is a frequent misconception that a 95% confidence interval means that 95% of the data are in the interval. Students can be asked to predict what percentage of the data from a sample are in the confidence interval, and what percentage of the data in a population are within a particular interval. They can then run simulations using a web applet *(e.g, Sampling Words at RossmanChance.com)* to test these conjectures.

### Developing Reasoning

*Cob and McClain (2004)*also used this method for developing the statistical reasoning abilities in elementary school children. For

**example,**they had students make predictions about the effectiveness of a new AIDS treatment. The students then examined two different sets of data related to the treatment, compared their intuitions with the real world data, and then discussed and wrote up their analyses. Their goal was to make sure the students would view data not merely as numbers but as measures of an aspect of a situation that was relevant to the question under investigation.

**Chance, B., delMas, R., & Garfield, J. (2004)**. Reasoning About Sampling Distributions. In D. Ben-Zavi & J. Garfield (Eds.), *The Challenge of Developing Statistical Literacy, Reasoning, and Thinking*. Kluwer Academic Publishers; Dordrecht, The Netherlands.

**Cobb, P., & McClain, K. (2004)**. Principles of Instructional Design for Supporting the Development of Student Statistical Reasoning. In D. Ben-Zavi & J. Garfield (Eds.), *The Challenge of Developing Statistical Literacy, Reasoning, and Thinking.* Kluwer Academic Publishers; Dordrecht, The Netherlands.

This chapter proposes design principles for developing statistical reasoning in students. To learn more: Developing Statistical Reasoning

**Rossman, A. L., & Chance, B. L. (1998)**. The Workshop Mathematics Project –*Workshop Statistics: Discovery with Data and Minitab.*Springer-Verlag: New York.