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The Impact of a Simulation Based Inference Introductory Statistics Course on Pre-Service Teachers’ Perception of Elementary Students’ Probabilistic Reasoning

Dr. Laura Kyser Callis & Dr. Jennifer McNally, Curry College, Milton, MA

On a final exam, students were asked the following question:

Mr. Barnes is teaching a unit on probability. They are using a computer app to flip a coin. He overhears students talking as he circulates.

Evan: I think it's broken. We got 4 out of 10 heads. Aren't we supposed to get 5?

Kyle: Maybe we set it up wrong.

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Comment on the student's thinking. Be sure to comment on:

• The difference between theoretical and empirical probability and the connections between them.
• How likely a result like 4 out of 10 heads is.

You may want to use apps to explain your answer.

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There are five conceptions of random events and probability that were identified in student responses:

1. No evidence of understanding, or incorrect claims
2. A focus on randomness, “anything can happen,” sometimes with 1-4 trials of their own
3. An indication that some results are more likely than others, probability as an estimate
4. Quantifying the variation of events other than the expected value, either numerically or graphically
5. In addition to quantifying events, connecting theoretical probability to the center of the distribution, distribution-thinking

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• Fewer PSTs who experienced the children’s visit made incorrect claims.
• PSTs who experienced the children’s visit were more likely to use technology to simulate the event to show how different outcomes could occur.
• PSTs who experienced the SBI course were more likely to use many trials and use distribution-thinking

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In a mathematics content course for prospective preK-8 educators (n=10), PSTs conducted simulations with virtual manipulatives to represent different outcomes with games of chance in order to address hypothetical student thinking. They then played and discussed a probability game from the book Alice in Randomland with five children, ages 3 to 8, (Guiñez et al., 2021). Three of the PSTs who had taken or were currently taking a general education introductory statistics course that used SBI curriculum were interviewed. A final exam question asked about hypothetical children’s probabilistic thinking.

Methods

In this and previous iterations of the math-for-teachers course, simulations had been used to demonstrate different potential outcomes in games of chance as a way to respond to students’ thinking. Typically, simulations were a topic for a single class session. In the simulation-based introductory statistics course, simulations were used throughout the course, with different real-world contexts. Formal probability rules were not addressed during class time in either course, though supplemental reading and videos were available. We wondered if PSTs who had taken the SBI course would be able to transfer what they learned in the SBI course to making sense of student thinking. Transfer is a significant challenge in mathematics education, and the context of the SBI course was more directed at studies of psychology, animal behavior, and consumer issues rather than games of chance and children’s thinking. We found that not only were PSTs able to make the connection, but they were able to use their learning in the SBI course to better address children’s thinking. We conjecture that the recurring nature of the simulations in the SBI class, rather than a single lesson, supported the PSTs to internalize the dual nature of probability: local uncertainty but global patterns (Ingram, 2022). For other mathematics teacher educators, we might suggest introducing and revisiting simulations throughout the semester or encouraging PSTs to take an SBI introductory statistics course.

Simulation based inference curriculum can support PST in making sense of students’ thinking in productive ways. In particular, they position children as knowledgeable and having reasonable ideas while still identifying mathematical goals, rather than positioning children as having mathematical deficits. PSTs report that SBI helps them to appreciate patterns within random events, particularly how frequently outcomes other than the expected value occur. Interviews with all three PSTs who experienced the SBI curriculum spoke to this, whereas the written response only elicited these ideas for 2 of 3 PSTs. Interviews may be a better tool for understanding PSTs conceptions of random events in regard to children’s thinking.

Conclusions

Conceptions of Random Events

Introduction

The Guidelines for Assessment and Instruction in Statistics Education (GAISE College Report ASA Revision Committee, 2016) call for probability to be taught in the service of inference. As such, instruction should focus less on probability rules and games of chance and more on using probability as a tool for modeling variation in random events, such as sampling. Simulation based inference (SBI) curricula follow these recommendations, minimizing attention to probability rules in favor of supporting students to develop chance models, such as flipping coins or spinning spinners, for concepts like the null hypothesis. Students then conduct many iterations of these simulations and graph the outcomes. Through this process, they are able to visualize if results as extreme as those that occur in an actual study would be likely to happen by chance, if the null hypothesis were true. SBI curricula have been shown to improve students’ conceptual understanding of challenging concepts and decrease inequities among marginalized groups (Chance et al., 2016; Tintle et al., 2011, 2012, 2018).

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However, probability concepts at the elementary school level often focus on games of chance and a classical, theoretical probability perspective (Ingram, 2022). Teachers, therefore, are still responsible for teaching such concepts. Moreover, teachers have a complex challenge: they must have deep understanding of the many facets of probability themselves, as well as an understanding of children’s thinking, including learning progressions and common problematic conceptions (Batanero et al., 2004; Ingram, 2022). In particular, understanding variation in random events is a particularly challenging concept (Fife et al., 2020). This study sought to determine whether the SBI curricula supported elementary pre-service teachers in making sense of children’s probabilistic thinking and whether the PSTs saw connections between the SBI curriculum and children’s thinking.

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Research Questions:

• How do elementary PSTs who have taken an Simulation Based Inference (SBI) course compare in their responses about children’s probabilistic thinking, compared to those who have not taken the course? �
• Do PSTs see connections between SBI and children’s probabilistic thinking?

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The Game

In the interviews, pre-service teachers who had previously taken the SBI course report productive dispositions toward children’s thinking and awareness of connections between the SBI curriculum and children’s thinking.

Pre-service teacher C,

• attention to variability, but also that some events are more likely than others
• attention to students’ using data to choose a strategy
• productive orientation to student thinking

Pre-service teachers A and H

• Indicated that the SBI curriculum helped them appreciate how often events other than the expected value happened - and they gave examples from the game and connected to student thinking and how they would address student thinking
• Held productive views of student thinking
• Noticed addition strategies
• Noticed that counting mistakes may be results of the manipulative features
• Indicated that “misconceptions” are reasonable, coming from children’s lived experience of other games and quantitative situations
• Were eager to help children connect to the theoretical probability - how more combinations would lead to a sum of 7 compared to 12
• Planned to use data from the game to demonstrate the combinations to connect to theoretical probability

Perception of Children’s Thinking,

Perceived Connections to SBI Curriculum

Batanero, C., Godino, J. D., & Roa, R. (2004). Training teachers to teach probability. Journal of Statistics Education, 12(1), 2. https://doi.org/10.1080/10691898.2004.11910715�Chance, B., Wong, J., & Tintle, N. (2016). Student performance in curricula centered on simulation-based inference: A preliminary report. Journal of Statistics Education, 24(3), 114–126. https://doi.org/10.1080/10691898.2016.1223529�Fife, J. H., James, K., & Peters, S. (2020). A learning progression for variability. ETS Research Report Series, 2020(1), 1–22. https://doi.org/10.1002/ets2.12286�GAISE College Report ASA Revision Committee. (2016). Guidelines for Assessment and Instruction in Statistics Education (GAISE) College Report 2016 (p. 141). http://www.amstat.org/education/gaise�Guiñez, F., Vásquez, C., Brito, C., & Martínez, S. (2021). Alice in Randomland: A resource for improving attitudes towards probability and its teaching. Statistics Education Research Journal, 20(2), 14. https://doi.org/10.52041/serj.v20i2.410 Ingram, J. (2022). Randomness and probability: Exploring student teachers’ conceptions. Mathematical Thinking and Learning, 1–19. https://doi.org/10.1080/10986065.2021.2016029�Tintle, N., Clark, J., Fischer, K., Chance, B., Cobb, G., Roy, S., Swanson, T., & VanderStoep, J. (2018). Assessing the association between precourse metrics of student preparation and student performance in introductory statistics: Results from early data on simulationbased inference vs. nonsimulation-based inference. Journal of Statistics Education, 26(2), 103–109. https://doi.org/10.1080/10691898.2018.1473061�Tintle, N., Topliff, K., Vanderstoep, J. L., Holmes, V., & Swanson, T. (2012). Retention of statistical concepts in a preliminary randomization-based introductory statistics curriculum. Statistics Education Research Journal, 11(1), 21–40.�Tintle, N., VanderStoep, J., Holmes, V., Quisenberry, B., & Swanson, T. (2011). Development and assessment of a preliminary randomization-based introductory statistics curriculum. Journal of Statistics Education, 19(1).

References

Discussion

1. Toss 2 dice
2. Find the sum
3. If the sum is your horse’s number, you get to move 1 space