Thinking Probabilistically in the Classroom

Education and research magazine Elements Ed published an article written by the Alliance in Issue 02 from 2024. Read the article by Jinsol Lee and Hannah Diamond below or view the full issue, “Making Kids Smarter: Are We Up to the Task?” on their website. Read more about Thinking Probabilistically, one of the learning domains of Decision Education.

Thinking Probabilistically in the Classroom

Equipping children with strong decision-making skills prepares them to successfully approach and solve complex problems in our modern world. Decision Education—or, the teaching and learning of skillful judgment formation—promotes lasting competencies that children can benefit from well into the future. Thinking probabilistically—a pillar of Decision Education—involves making predictions about future outcomes and then considering the risks, rewards, and consequences of all options in a given decision.¹

The ability to think probabilistically is critical given children’s unprecedented access to an “overload of information.”² Children require support in learning how to evaluate the reliability of information and detect when information is incomplete or misleading.³ Furthermore, because many everyday decisions involve uncertainty, it is important for children to be able to understand and utilize probabilistic thinking to make optimal decisions. Research has shown that an inability to do so “can be extremely costly, not only at the individual level but also for society in general.”⁴ Developmental researchers have also found that children exhibit biases in probabilistic thinking;⁵ while the ability to think probabilistically improves with age, adolescents still performed poorly on probability tasks.⁶

Because poor performance on probability tasks is often the product of a lack of relevant skills and knowledge,⁷ education is a promising means of improving children’s probabilistic abilities. Children who learn to think probabilistically in the classroom can engage comfortably with numbers, identify and address areas of uncertainty in their own knowledge, maximize their chances at achieving desired outcomes, and form well-calibrated judgments based on available information. Promoting the use of probabilistic thinking in classroom conversations supports a culture of intellectual humility and truth-seeking, equipping children with the skills needed to successfully navigate a rapidly changing world.

Using Numbers To Think

Numeracy—or, the ability to understand and work with numbers—is foundational to skillful decision-making. When numbers are involved in a decision situation, “highly numerate decision makers make better decisions than the less numerate.”⁸ Children who develop comfort with numbers in their youth will benefit for years to come as they encounter more consequential personal, academic, professional, financial, medical, and other life decisions as adults. More numerate individuals are less susceptible to misinformation and emotions that can impair their judgment⁹ and are thus better able to engage in clearer decision-making. They consider relevant numbers when making decisions and have a richer, more complex understanding of information.

However, despite the importance of numeracy for skillful decision-making, “even highly educated individuals do not always comprehend numbers when making decisions.”¹⁰ Including numeracy as part of the standard educational experience can have lasting effects on improved decision-making.¹¹ Numeracy is an answer tostudents who protest, “why are we learning this?” in their math classes. Of course, not all students will pursue advanced education or a career in mathematics, but being numerate will increase the likelihood that they continue to apply and benefit from mathematical principles throughout their lives.¹²

Teachers can promote numeracy in students by encouraging increased use of numbers in everyday situations. While individuals tend to be more comfortable with and inclined to use words like “possibly” and “maybe” to convey likelihoods, there is tremendous variability in how people interpret these terms. In a survey capturing the extent of this variability, respondents understood “real possibility” to mean anywhere from 20 to 80 percent.¹³ Use of imprecise language can lead to misunderstandings and misinformed decision-making.

In order to make this broad range of interpretations salient for students, teachers can provide a list of common terms (such as “frequently,” “maybe,” “rarely,” and so on) and instruct students to assign percentages to them. Displaying the students’ responses visually—on a line plot or bar graph, for instance—enables students to reflect on and appreciate the extent of agreement or disagreement in their answers. Teachers can also provide students with specific, relevant examples to contextualize the use of these words. For example, teachers could pose the following scenario to a class: “Your friend says they can probably come to your house after school today. How likely is this to happen?” To scaffold or differentiate this exercise, teachers can start with percentages that might feel intuitive to students—like 0, 25, 50, 75, and 100 percent—and then gradually incorporate more granular percentages as students’ comfort level builds.

Establishing a norm of number use in the  classroom can increase children’s understanding  of numbers and their meaning. This practice can also be adopted at home. When children use words  to convey likelihoods, parents and caregivers can  encourage them to include a percentage. Adults  can model this practice for children in their own  communication as well, and students can and  should be encouraged and empowered to request  clarification from the adults in their lives when they use ambiguous words.  

Degrees of Confidence

Another variant of the above practice involves  providing degrees of confidence—that is, numerical  representations of one’s confidence in a belief or  prediction. This practice encourages individuals  to reflect on the information they know and  the information they are lacking and then  communicate their level of certainty to others.  Degrees of confidence are well-calibrated when  they accurately reflect the extent of one’s  knowledge or correspond with the actual  frequency of a predicted event.  

Research have long stated that well calibrated confidence assessments are “critical to effective decision making.”¹⁴ Overconfident decision-makers may attempt more than they are capable of, may refrain from asking for help, may make riskier decisions, and may “neglect signs that decisions are going awry.”¹⁵ Likewise, underconfident decision makers are more likely to rely unnecessarily on others and to feel paralyzed by hesitation and self-doubt.¹⁶ As children become more  independent decision-makers, their ability to  assess their confidence accurately is particularly  important as it increasingly informs their  decision-making process.¹⁷ 

Behavioral decision research has identified a tendency for individuals of all ages to  make “poorly calibrated confidence judgments.”¹⁸  However, confidence assessments are most poorly  calibrated in childhood; by adolescence, accuracy  improves, and by adulthood, it stabilizes.¹⁹ This  developmental trajectory is reflected in decision making behavior: adolescents evaluate the advice  they receive more thoroughly than children do  before choosing whether to follow it.²⁰ 

To incorporate increased number use, teachers  can prompt children to express their degrees  of confidence as percentages when making estimates and predictions or expressing their  beliefs. Students can assess their confidence in  predictions about course material (for example: what will happen to the main character in a story)  or their mastery of content and readiness for an  assignment (for example: how well-prepared  they are for an upcoming science exam). The goal is not to express 100 percent confidence,  but to reflect honestly about the state of one’s  knowledge and belief. To encourage the use  of degrees of confidence, teachers along with  parents and caregivers can prompt students to  provide rationales for their stated confidence levels and to identify what information might shift their confidence in either direction. As with other numeracy practices, sharing degrees of confidence can become a norm in the classroom and at home. 

Children can also track changes in their confidence about a particular matter as new information presents itself, as circumstances shift, or as they uncover new insights  based on their existing knowledge. For example, if a student is 75 percent confident that their favorite football team will win the Super Bowl, they might  consider how this confidence would change if the quarterback becomes injured. A biology teacher in the Decision Education Teacher Fellowship program, run by the Alliance for Decision Education, found  that her ninth-grade students embraced this lesson, regularly providing confidence levels on their own accord.²¹ 

This metacognitive practice directs students to the gaps in their knowledge, cultivating focused, eager learners who are “hungry for information” to increase their confidence levels.²² When an  individual specifies a degree of confidence, they are implicitly acknowledging that beliefs are nuanced and are rarely merely “right” or “wrong.” Expressing confidence helps people embrace, rather than resist, information that challenges their beliefs, “since it  feels better to make small adjustments in degrees of certainty instead of having to grossly downgrade from ‘right’ to ‘wrong.’”²³ 

From Intuition To Calculation

Skillful decision-making requires individuals to think deliberately, accounting for probabilistic considerations that intuitive preferences often  overlook.²⁴ People have instinctive, emotional responses to the possibility that a desired or dreaded outcome will occur—for example, excitement after purchasing a lottery ticket or fear of an accident when boarding a plane. But these intuitive responses do not account for the (slim) probability of these outcomes.²⁵ 

When weighing decision options, it is natural to favor options that may lead to desirable outcomes and to avoid options that may lead to undesirable outcomes, regardless of the probability of these outcomes actually occurring. This tendency often “leads to inferior outcomes,” or decisions that are unnecessarily risky or not risky enough.²⁶ Decision makers must consider both how much each choice is worth to them and how likely each choice is to occur; this encourages selection of the option that  maximizes overall expected utility.  

The expected utility calculation is a concrete way of orienting students to probabilistic considerations. When children think in terms of expected utility, they ask themselves questions like: “what outcomes do I want (or not want)?”; “how much satisfaction (or dissatisfaction) will I experience from each possible outcome?”; and “how likely are each of these outcomes?” Then, they assign numbers to their answers to such questions. To calculate expected utility, the decision maker multiplies how much each option is worth to them by the probability it will happen. In certain decision situations, each option has an objective, measurable value and probability. These situations call for a  more straightforward, introductory calculation of expected value. For example, if a lottery ticket offers a 20 percent chance of winning $100, the expected  value is 0.20 x $100, or $20. More abstract decisions may require the decision-maker to evaluate and  identify the subjective desirability of each potential  outcome and to use subjective probabilities derived  from their estimates and personal judgment.  Consider the following scenario: a child is deciding whether to audition for the school play. They rate  the anticipated experience of being in the play an  80 out of 100. But many of their classmates are also  planning to try out, so they estimate that there is a  65 percent chance they will get a part. The expected  utility in this case is 80 x 0.65, or 52. The child would repeat this process for other extracurricular  activities they are considering, pursuing the one  with the largest expected utility.  

More advanced calculations of this nature hone students’ ability to “quantify and compare the  probabilities of each possible event.”²⁷ After making  a decision, children can then reevaluate their initial  probability estimates, gradually improving the  accuracy of these estimates.²⁸ With practice, this way of thinking can become habitual: children can  become more mindful of probabilities when deciding  among options to maximize their expected utility.²⁹ 

Consider, too, the following anecdote from  the Alliance for Decision Education: a high school  teacher in the Alliance’s Decision Education Teacher Fellowship program encouraged student  government of receiving administrative approval for each  initiative as a percentage. Then, they calculated  and discussed the expected utility of each initiative,  distinguishing exciting ideas that were feasible from  those that were not. The teacher reported that it  was “the most helpful tool” they used to support these students.  

Base Rates

The tendency to neglect probability also applies to base rates—or, numbers that represent the naturally occurring frequency of something in a general population. Base rates provide decision-makers with information about the likelihood that certain events will occur and should inform judgments and decisions in uncertain situations.³⁰ However, people tend to overweigh the importance of characteristics specific to their situation and tend to disregard objective statistical information; this phenomenon is called base rate neglect.³¹ For example: imagine a child who is deciding where to sit at tonight’s baseball game in order to maximize their chances of catching a fly ball. According to base rates, fans catch fly balls most often in section 50. But a friend shares anecdotally that they caught a ball in section 10.³² The child may feel more inclined to sit in section 10, neglecting the base rate. This tendency is “one of the most significant departures of intuition” from rational decision-making.³³ Decision-makers often dismiss base rates as irrelevant, so it is especially important to demonstrate their relevance to decision-making for students.³⁴

Teachers can help students conceptualize how to incorporate base rates into their decision-making by introducing the “outside view” and “inside view.”³⁵ Taking the outside view means considering general trends and objective evidence from similar situations, including base rates. Students might ask themselves: “How do situations like this usually go? How have they gone in the past? How likely is this in the general population?” For example, if students are using the outside view to predict the likelihood that the school debate team will beat their rivals, they might refer to how the team has performed in similarly high-stakes situations. Children can work through a variety of predictions about a range of topics, identifying which base rates they would refer to when taking the outside view and perhaps even researching them. By contrast, taking the inside view means relying on evidence from the current situation. Students might ask themselves: “What makes this situation different or unique?” To continue the example: maybe the best student on the debate team has been out sick with the flu, thus impacting the team’s chance of winning.

Teachers can guide students as they make predictions that are relevant to course material or, for that matter, their lives beyond the classroom. To yield the most accurate predictions, students should take the outside view first, generating predictions based on relevant base rates. Then they can take the inside view, identifying details specific to the example and discussing how these details modify their initial predictions, if at all.

Why Teach Probabilistic Thinking?

The benefits of promoting probabilistic thinking skills in the classroom—as well as in students’ homes and communities—are both significant and widespread. The aforementioned practices invite students to engage actively in classroom learning. Probabilistic thinking skills empower students to be authorities on their own knowledge and agents of their own learning. By using their probabilistic thinking skills, students can learn to tolerate and navigate a world riddled with uncertainty, forming judgments and making decisions with the best information available to them. Probabilistic thinking cultivates students who are open to changing their minds, acknowledging multiple possibilities and perspectives, pursuing an accurate worldview, and making informed and improved decisions. This leads to better outcomes not only in their own lives and in the lives of those around them, but also in society as a whole.

Endnotes

1. “The Decision Education K-12 Learning Standards: An Overview,” Alliance for Decision Education (March 27, 2023),https://alliancefordecisioneducation.org/blog/decision-education-k-12-learning-standards/.

2. Julie Hooft Graafland, “New Technologies and 21st Century Children: Recent Trends and Outcomes,” Organisation for Economic Co-operation and Development, Education Working Papers, no. 179 (2018): 14, https://doi.org/10.1787/e071a505-en.

3. Graafland, “New Technologies”; Ida K.R. Hatlevik and Ove E. Hatlevik, “Students’ Evaluation of Digital Information: The Role Teachers Play and Factors That Influence Variability in Teacher Behaviour,’’ Computers in Human Behavior 83 (2018): 56–63.

4. Caterina Primi et al., “Measuring Probabilistic Reasoning: The Construction of a New Scale Applying Item Response Theory,” Journal of Behavioral Decision Making 30, no. 4 (2017): 933, doi: 10.1002/bdm.2011.

5. Francesca Chiesi, Caterina Primi, and Kinga Morsanyi, “Developmental Changes In Probabilistic Reasoning: The Role Of Cognitive Capacity, Instructions, Thinking Styles, and Relevant Knowledge,” Thinking & Reasoning 17, no. 3 (2011): 315–50, doi: 10.1080/13546783.2011.598401; Paul A. Klaczynski, “Analytic and Heuristic Processing Influences on Adolescent Reasoning and Decision-Making,” Child Development 72, no. 3 (2001): 844–61.

6. Klaczynski, “Analytic and Heuristic Processing.”

7. Chiesi, Primi, and Morsanyi, “Developmental Changes”; Klaczynski, “Analytic and Heuristic Processing.”

8. “Beyond Comprehension: The Role of Numeracy in Judgments and Decisions,” Current Directions in Psychological Science 21, no. 1 (2012): 31–35, doi:10.1177/0963721411429960.

9. Ellen Peters et al., “Numeracy and Decision Making,” Psychological Science 17, no. 5 (2006): 407–13; Peters, “Beyond Comprehension.”

10. Peters, “Beyond Comprehension,” 31.

11. Peters, “Beyond Comprehension.”

12. Peters et al., “Numeracy and Decision Making.”

13. Andrew Mauboussin and Michael J. Mauboussin, “If You Say Something Is ‘Likely,’ How Likely Do People Think It Is?” Harvard Business Review (July 3, 2018), https://hbr.org/2018/07/if-you-say-something-is-likely-how-likely-do-people-think-it-is; Annie Duke, How to Decide: Simple Tools for Making Better Choices (New York: Penguin Random House, 2020).

14. Andrew M. Parker and Baruch Fischhoff, “Decision-Making Competence: External Validation through an Individual-Differences Approach,” Journal of Behavioral Decision Making 18 (2005): 14, doi: 10.1002/bdm.481.

15. Parker and Fischhoff, “Decision-Making Competence,” 6; Daniel Kahneman, Thinking, Fast and Slow (New York: Farrar, Straus and Giroux, 2011).

16. Parker and Fischhoff, “Decision-Making Competence.”

17. Madeleine E. Moses-Payne et al., “I Know Better! Emerging Metacognition Allows Adolescents to Ignore False Advice,” Developmental Science 24, no. 5 (2021): 1-13, doi: 10.1111/desc.13101.

18. Klaczynski, “Analytic and Heuristic Processing,” 852; Parker and Fischhoff, “Decision-Making Competence,” 2.

19. Moses-Payne et al., “I Know Better”; Leonora G. Weil et al., “The Development of Metacognitive Ability in Adolescence,” Consciousness and Cognition 22, no. 1 (2013): 264–71, https://doi.org/10.1016/j.concog.2013.01.004.

20. Moses-Payne et al., “I Know Better.”

21. https://alliancefordecisioneducation.org/teach/fellowships/

22. Duke, How to Decide, 70; Weil et al., “Metacognitive Ability.”

23. Duke, How to Decide, 70.

24. Chiesi, Primi and Morsanyi, “Developmental Changes”; Kahneman, Fast and Slow; Valerie Thompson and Kinga Morsanyi, “Analytic Thinking: Do You Feel Like It?” Mind and Society 11 (2012): 93–105, https://doi.org/10.1007/s11299-012-0100-6.

25. Paul Slovic, “Rational Actors and Rational Fools: The Influence of Affect on Judgment and Decision-Making, “Roger Williams University Law Review 6, no. 1 (2000): 163–212.

26. Kahneman, 321.

27. Kinga Morsanyi and Dénes Szücs, “Intuition in Mathematical and Probabilistic Reasoning,” in The Oxford Handbook of Numerical Cognition Online, eds. Roi Cohen Kadosh and Ann Dowker (Oxford: Oxford University Press, 2015): 8, https://doi.org/10.1093/oxfordhb/9780199642342.013.016.

28. Duke, How to Decide.

29. Morsanyi and Szücs, “Intuition.”

30. “Why Do We Rely on Specific Information over Statistics?” The Decision Lab, accessed August 9, 2023, https://thedecisionlab.com/biases/base-rate-fallacy; Amos Tversky and Daniel Kahneman, “Judgment under Uncertainty: Heuristics and Biases,” Science 185, no. 4157 (1974): 1124–31.

31. Daniel Kahneman and Amos Tversky, ”On the Psychology of Prediction,” Psychological Review 80, no. 4 (1973): 237–51, https://doi.org/10.1037/h0034747.

32. Judite V. Kokis et al., “Heuristic and Analytic Processing: Age Trends and Associations with Cognitive Ability and Cognitive Styles,” Journal of Experimental Child Psychology 83, no. 1 (2002): 26–52.

33. Kahneman and Tversky, “Psychology of Prediction,” 57.

34. Maya Bar-Hillel, “The Base-Rate Fallacy in Probability Judgments,” Acta Psychologica 44, no. 3 (1980): 211–33.

35. Kahneman, Fast and Slow, 247.