Blogging from The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics

Prof Maxine Pfannkuch from The University of Auckland.

**The first Judy Paterson Speaker**

Visualising chance in introductory probability

**The problem**

- Random events and chance phenomena permeate our lives and environments: volcanic eruptions, epidemics, crashes – need probabilistic reasoning to understand these.
- Current teaching approach in intro prob is mathematical – obscures nature and role of randomness in processes and the nature of chance phenomena
- Many probability misconceptions are prevalent in people’s thinking (Kahneman, 2011)
- Appear resilient to teaching – how do we get students to confront these misconceptions?
- After 30 years of research a re-think is needed

**Possible solutions**

- Use a modelling approach to probability
- Construct a model (Konol and Kazak)
- Explore model behavior (Pratt, 2005)
- Use dynamic visualizations to give students an opportunity to:
- Experience random behavior through simulations
- Visualize chance through creating new representation infrastructure
- gain access to previously inaccessible concepts

**Study: Part 1**

Interviewed seven practitioners of probability (teachers,etc.) (at least 2 hours per interview)

- What probability concepts need to be promoted?
- What are perceived problem areas for students when learning probability
- What learning strategies might mitigate these problems

**Findings**

To understand what concepts need to be promoted:

- Seeing structure and applying structure are important aspects of probability modeling

What needs to be promoted?

- Viewing a situation in context – having an intuition about seeing the underlying structure – Markov, Poisson processes etc.
- Breaking the problem up into parts – seeing the processes in each part.

**Building blocks:**

- Randomness
- Distributions
- Conditioning
- Mathematics

Given the traditional word problem, how do you get students **engaged**? Engaged means digging deep into the problem and the concepts of probability working in the problem. What is the representation of the problem, going from the words, to the structure and model?

Randomness, Distribution, Mathematics (in many different ways – from number sense to mathematical reasoning) were all problems.

**Possible strategies:**

**Incentivize**: Get the students to make a conjecture and then test that conjecture (Garfield, deMas Zieffler, 2012)

- Statisticians have a model in their head about what data should look like. If the data turns out to be different, they have to change their model.

**Use Visual imagery**:

- Make previously inaccessible concepts accessible and visible
- Playing around with chance-generating mechanisms
- Stimulate students’ intuitions about randomness, variation, distribution
- change inputs and observe resultant outputs

**Link representations**

- Flexible use of representations essential for conceptual understanding:
- Students should be able to seamlessly switch between them

**Relatable contexts**

- Integration of contextual and statistical/probability knowledge fundamental
- Initiate concept formation through engaging students…

**Second part of study:**

- Development of four tools and tasks trial on introductory probability students
- To ascertain conceptual understanding and probabilistic reasoning being enhanced
- Trial each tool on three pairs of students
- Capture student thinking and interaction with tools and tasks

**Software tools:**

**Eikosogram****Pachinkogram**– question about diabetes – chances that someone who tests positive actually has diabetes?- Seeing fallacies of thinking of students: Example of a Pachinkogram, but not from this particular study. Taken from here.

**Poisson processes tool**- Aim was to show the link between the exponential distribution and the Poisson process. Watching a Poisson process and asking the students to imagine the distribution (eg. of waiting times of buses).
- Gave an intuition into the factor.

**Markov processes tools**- Research question: Given that the students have learnt about Markov processes from a mathematical perspective (due to ethics issues), what new understanding seem to emerge when exposed to a dynamic, visual environment
- Tasks
- Two office car rental agency
- Four office car rental agency
- Snakes and ladders

**Task 2 from the Markov processes tool:**

- Given four rental agencies (A,B,C and D), probabilities of picking up and dropping off at different agencies, write down a Markov state diagram for the system. See here for an example, but not from this study. Example, but not taken from study, taken from here:
- Rank the equilibrium distributions (of the cars at each of the four rental agencies) from highest to lowest.
- Students had never understood about equilibrium distribution stabilising over time.
- Students do simulation and test conjecture.
- Students analyse why conjecture may be right – big arguments “don’t know it it is true. Might be. Because these are the probabilities coming in (to each state) and the probabilities coming in should be directly related to the equilibrium distribution if there is one” – not correct, but leading in the right direction.
- If a car is picked up at agency A – how many times would you expect it to be hired out before it arrived at B, C or D or was returned to A?
- Intuitively rank and estimate the expected times of these scenarios from highest to lowest
- Sketch the distributions
- What range do you expect?

- It’s very clear that the students are learning as they play, and they discuss what’s happening with the models.
- Students amazed at expected times and the variation and range of the dsitrubitions
- Analysed why by explaining the variation by noting the self-looping, and by connecting the probabilities with the…

**In summary:**

- Make a conjecture
- Analyse underlying structure
- Make a test
- Analyse why they were right or wrong
- Linking and switching between representations
- Developing some intuition for seeing structure
- The tool had visual imagery
- Students could see how probabilities operate in Markov processes
- Never previously seen equilibrium distribution: Image, awareness of stabilization in long run

**What new understandings seem to emerge?**

- They students were developing intuition
- They were engaging in the task
- They were linking representations: Enabling them to see the underlying structure
- Seeing what randomness looked like
- They were engaging with the mathematics of the problem

**Student reflections:**

- Makes you think a lot more about what you are doing
- Graphs make it really clear
- Students appreciated seeing the equilibrium and hitting times distribution
- Having fun…
- Like going through the process of going through your own intuition: Good to make predictions. I kind of think that it is a really good way to introduce the idea

**Concluding remarks**

- All our tasks are dynamic visual imagery tools
- Seem to have the potential to deepen and enhance the students’ learning.

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