Eventmath/Quantitative literacy course

About this page
This is an Eventmath course! More precisely, it's a course plan for instructors that provides a complete curriculum for a one-semester, college-level course on quantitative literacy.

Currently, the course has two main components: (1) a list of skills to be learned, and (2) lesson plans that teach those skills. In general, Eventmath lesson plans are each based on a media source, such as a news article, a social media post, or an online video. They are designed for instructors and include activities, assignments, projects, and other resources.

The materials for this course on quantitative literacy are grouped thematically into units inspired by PISA's content categories. A suggested course schedule is provided, and a separate section of this page provides a space for instructors to share customized versions of the course. If any of this is unclear, please don't hesitate to ask questions on this course's Discuss tab!

Notes:
 * Draft status: This page is currently a very early draft.
 * For some skills, we may have lesson plans ready, but they have not yet been listed on this page.
 * For other skills, the community may still need to make lesson plans.
 * Current drafts of lesson plans may be found on the Eventmath Tasks tab.
 * You can help! This is a wiki! You are welcome to improve this draft.
 * You can make edits directly to this page (e.g. to revise the list of skills).
 * You can add to the discussion on the Discuss tab.
 * You can categorize lesson plans by skill (this is similar to typing a hashtag on social media; instructions forthcoming).
 * You can make mistakes: all versions of the page are saved, so any mistakes can be rolled back!
 * Other Eventmath courses: Other potential Eventmath courses may be found in the brand new Eventmath course directory.

Quantitative literacy overview
The terms mathematical literacy and quantitative literacy are sometimes used interchangeably. For the purpose of this course page, it’s useful to distinguish them.

Mathematical literacy is the subject of Eventmath as a whole. The Programme for International Student Assessment defines it as follows : Mathematical literacy is an individual’s capacity to reason mathematically and to formulate, employ, and interpret mathematics to solve problems in a variety of real-world contexts. The Eventmath project interprets this broadly, so that mathematical, statistical, and computational skills are all included; these range from basic skills introduced in primary school to advanced skills applied in research.

Quantitative literacy is the subject of this particular Eventmath page. It’s a subset of mathematical literacy. It consists of the skills that are most fundamental and most widely useful. Each of these skills arises in two contexts: information consumption and information production. Lesson plans in this course emphasize both contexts whenever possible.

Prerequisites

 * Some proficiency with basic arithmetic, including fraction and decimal arithmetic.
 * Curiosity about the world!

Pretest
This section can contain a link to a pretest, along with instructions related to the pretest. Pretests help not only with measuring improvement, but also generating improvement! At least, this seems to be the case, due to the so-called pretest effect. (Here's a Wikipedia page with a little about the pretest effect. There's more literature on this that could be cited here, or possibly on the Eventmath Impact tab, and linked to from here.)

Various standard assessments of quantitative literacy (a.k.a. numeracy) have been developed. A review of the research literature should reveal the available options. We could potentially use those to develop part or all of the pretest.

Notes:
 * It would be best if the pretest is created as a wiki page on this site, so that it can be edited collaboratively.
 * By clicking "Download as PDF" in the left sidebar of the pretest page, it will be possible to print it.
 * If anyone would like to help develop the pretest, please leave a comment on this course's Discuss tab to get help creating the wiki page for it. That way, we can ensure that it gets created in the appropriate place.

Descriptions of the quantitative literacy skills
In this section, you can find descriptions of each skill, along with related ideas (e.g. rough ideas for possible lesson plans).

Design notes:
 * We may need to tinker with the degree of specificity of each skill. For example, we could potentially break up "use percentages" into many smaller skills, such as "convert from a percentage to a decimal." There may be trade-offs involved (e.g. the degree of specificity may affect how easy it is to map skills to lesson plans).
 * The graphic indicators of completion status for individual lesson plans in the course schedule will need to be kept up to date. It's possible that people will discover drafts on the Eventmath Tasks tab and make contributions without updating the graphics on this course page. We may ask a volunteer course coordinator to do periodic updates.

Learn must-know benchmark values
Must-know benchmarks like the ones below will be useful throughout the course. They help us to check if claims are plausible, to make sense of small and large numbers, and to make useful estimates. Typically, rounded values suffice.

Note: A discussion about this list of benchmarks has been started on the Discuss tab.

Population

 * Population of three cities (your own, the largest, and one more)
 * Population of three countries (your own, the largest, and one more)
 * Earth’s total human population

Distance

 * Distance between borders of your country
 * Radius of Earth (polar, mean, and equatorial radii are all within about 1% of 4000 miles or 6400 kilometers)
 * Cruising altitude of commercial passenger planes (or elevation of Mount Everest)

Money

 * Annual governmental budgets (your country’s, the largest governmental budget)
 * Market cap of companies (what's the largest?)
 * Minimum wage in your own region (or alternative statistic if no minimum wage exists)

Use small and large numbers

 * Contextualize extreme numbers using benchmarks
 * Express small and large numbers using scientific notation

Apply basic combinatorial reasoning

 * Multiplication principle
 * Inclusion-exclusion with two sets

Use percentages

 * Convert between percentages, proportions / fractions / ratios, and decimals
 * Compute simple percentages mentally (e.g. 10%, 20%, … of a given value)
 * Distinguish between percentages and percentage points
 * Other percent problems

Apply dimensional analysis
Lesson plan: Dimensional analysis, shipping, and an impossible weight limit

Apply basic geometric formulas

 * Area of a rectangle (e.g. for estimating land area of a state or province)
 * Volume of a box (e.g. for shipping)
 * Circumference of a circle
 * Area of a circle (e.g. for comparing pizzas)

Apply proportional reasoning
See the Wikipedia article on proportional reasoning, for example.

Calculate and employ rates of change

 * absolute rates
 * relative/percent rates
 * average rates (including unit rates)

Develop and interpret linear models

 * analytically
 * graphically
 * numerically
 * verbally

Develop and interpret exponential models

 * analytically
 * graphically
 * numerically
 * verbally

Solve Fermi problems
Fermi problems are estimation problems that may be solved by back-of-the-envelope calculations or mental estimates with little or no data collection, e.g. using known benchmark values.


 * Learn the meaning of "order of magnitude."
 * Estimate order of magnitude based on benchmark values, using scientific notation when convenient
 * Establish lower and upper bounds

Compute and interpret summary statistics

 * Measures of location (including discussion of means vs. medians)
 * Measures of spread (e.g. standard deviation)
 * Measures of dependence / effect size (e.g. correlation coefficient)

See Wikipedia for an overview of summary statistics.

Organize, visualize, and interpret data

 * Arrange data in tables
 * Visualize and interpret data with common chart types
 * Histograms
 * Bar charts
 * Pie charts
 * Line charts
 * Scatterplot
 * Box plot
 * Log plots / logarithmic scale
 * Interpolation vs. extrapolation

Distinguish basic concepts of experimental design

 * Representative samples vs. unrepresentative samples
 * Observational vs. experimental studies

Compute and interpret basic probabilities

 * Empirical probability as relative frequency
 * Simulation
 * Relative vs. absolute risk
 * Theoretical probability with equally likely outcomes

Sketch and interpret normal distribution
Students can collect real data that reveals a normal distribution. See, for example, this classroom activity. The link points to the moment in an hour-long talk when the speaker describes an activity in which students use stop watches to try to measure the duration of a foot race, while they watch a YouTube video. The class then collects the times from all students and plots them, to reveal the distribution.

Interpret margin of error and confidence level from sample surveys
See the Wikipedia article on margin of error.

Identify common statistical paradoxes or errors

 * Cherry picking and selection bias
 * Gambler’s fallacy (compared to law of large numbers)
 * Simpson’s paradox
 * Misleading axes
 * Others worth including?

Employ basic logic

 * Useful propositional logic
 * Inductive vs. deductive reasoning
 * Common logical fallacies (including “correlation implies causation”)

Questions that lead to insights (and lesson plans!)
Part of quantitative literacy is a disposition to ask questions. Some questions, in particular, tend to help with identifying and formulating problems that lead to insight (or a good lesson plan!). It’d be nice to develop a list of such questions, ranked roughly by importance. They ought to be reasonably general, but specific enough to be valuable.

This list may overlap some with the questions suggested in Polya’s How to Solve It, but the focus will be on questions helpful for quantitative literacy, rather than general mathematical problem solving.

Perhaps we could aim to select the top five and incorporate them explicitly into the syllabus. They could then be practiced throughout the various lesson plans. Eventually, we might want to create a longer list of questions on a separate page. Then below the list of top questions, we can link to other questions that are helpful but don't quite make the short list.

Below are just a few questions to get the ball rolling.


 * Does that value/chart seem plausible?
 * Would a different value/chart be helpful?
 * How was that value calculated?
 * What does this axis represent?

Principles for using information and technology
''Like everything on this page, the following principles may be revised by the community! Also, it may be better to put these principles elsewhere on the site, and then incorporate them explicitly into the skills section of the syllabus (e.g. by including specific technology or information literacy skills) or the lesson plan template. Changing the lesson plan template requires some care, since the template is used for all new lesson plans.''

Information literacy

 * It's important that students recognize when information is needed.
 * It’s important that students can find reliable information.
 * It’s important to teach students to evaluate the reliability of sources.

So, lesson plans may often involve brief research components, in which students quickly search for information or data to validate claims they are particularly interested in. This may require explicit instruction (e.g. teaching students to search Google Scholar and quickly assess abstracts).

Computer literacy

 * Using technology such as spreadsheet software is good, when it’s appropriate.
 * Any technology that is used should be available to students outside of class, and after they graduate.
 * If a particular technology will be inconvenient or inaccessible to students in the relevant context, then it’s good to help students work without it.

Overview of the modeling framework for this course
In this course, we list questions, skills, and principles. These are each a part of mathematical modeling, which is the process of using math to answer questions about the world. We can organize this process into four stages, along with tips that tend to be useful:


 * 1) Identify: Identify a question of interest and any necessary data sources.
 * 2) * Tip 1: Is this value/chart plausible?
 * 3) * Tip 2: Would a different value/chart be helpful?
 * 4) Translate: Translate the real-world question into a pure math problem.
 * 5) * Tip 1: Define variables for the quantities we want to know.
 * 6) * Tip 2: State knowns and unknowns in terms of those variables.
 * 7) Solve: Solve the math problem.
 * 8) * Tip 1: Devise a plan.
 * 9) * Tip 2: Implement the plan.
 * 10) Interpret: Interpret the solution in the context of the original question.
 * 11) * Tip 1: Consider the result. (Is it reasonable? Can you communicate it visually?)
 * 12) * Tip 2: Consider the method. (Can you arrive at the same result by another method? Can the method be used in other contexts?)

By choosing a specific framework and explicitly demonstrating the stages throughout the course, we can offer students reliable footholds as they climb difficult terrain. If we refer to each stage by name (identify, translate, solve, interpret) as we demonstrate them, they will be easier to remember.

Sources of inspiration
The stages above encapsulate common themes from various works on problem solving and mathematical modeling. Some examples are below. (Reminder: Turn these into proper references, which are not susceptible to link rot.)
 * In the book How to Solve It (Polya), four stages of the problem-solving process are identified: Understanding the problem, Devising a plan, Carrying out the plan, and Looking back. These stages correspond roughly to our "Translate," "Solve," and "Interpret" stages. The "Looking back" stage includes the helpful advice of considering both the result and the method, which influenced this course's "Interpret" step.
 * A paper published in PRIMUS provides a chart depicting the mathematical modeling process in four stages: Define, Translate, Analyze, Interpret. These stages are similar to the ones used in this course. The "Define" stage is much like our "Identify" stage. The "Analyze" stage includes the instruction "Solve mathematical problem," so it corresponds roughly to our "Solve" step.
 * The PISA 2022 Mathematics Framework identifies three stages: Formulate, Employ, Interpret and evaluate. Based on the descriptions of these steps, they could reasonably be renamed Identify and translate, Solve, and Interpret (these exact terms are actually used in the descriptions), which would essentially match the stages identified in this course.
 * Math in Society (Lippman) has a section on problem solving that identifies five steps. These aren't named, but they could reasonably be called Identify the question, Identify necessary data, Plan a solution, Find data, and Implement the solution. These are reflected in our "Identify" and "Solve" steps.

Notes on each stage of the modeling process
A few notes about each stage are included below. These notes may be refined based on our collective experience helping students learn this process (as well as any relevant research literature).

Identify
''Note: The tips under the "Identify" stage were chosen without much deliberation. It would be good to pick the two most helpful questions, if that's possible. Sticking to two keeps the list structure intact and makes it easier to remember.''

The "Identify" stage is not typically emphasized in traditional math courses. When a student is given a list of questions or problems, identification is done for them: the questions are provided and the necessary data are given. In real-world problem solving, this is not the case.

So, quantitative literacy requires asking questions for oneself (to identify mistakes, to separate information from misinformation, to develop personally meaningful insights, etc.). It also requires us to identify information that will be necessary when solving the problem.

Translate
The "Translate" stage is crucial. It turns an authentic word problem (a.k.a. story problem) into a pure math problem (the translation is the model that represents the real-world situation). If students have particular trouble with word problems but not pure math problems, then the difficulty is in the translation.

To overcome the difficulty, the first step is to recognize that there is a translation stage at all. This often requires explicit, repeated emphasis during instruction. Understandably, there tends to be a strong inclination to find the answer right away, but we can't find the answer if we don't understand the question.

It's also important to emphasize that the goal of this stage is to translate the question, not to answer the question. A rough test of success is whether the translation can be used in place of the question; so, students can be advised to cover up the original question, and if they need to peek, that means their translation needs to be updated.

Sometimes, translation is the most difficult part! Setting up the model makes the mathematical structure more apparent, so the solution can end up being much easier than expected.

Solve
The "Solve" stage is directly influenced by Polya's book How to Solve It (reminder: insert proper reference to this book). With more difficult problems, it's important to explicitly identify the planning and implementation steps in our teaching. Thinking ahead can often save problem solvers from a lot of trouble.

Interpret
Often, we realize that a result is incorrect once we interpret it within the context of the original question. So, this stage includes checking the correctness or accuracy of the solution, to the extent possible.

Quantitative literacy course implementations
This page is intended to serve as a curriculum for a general quantitative literacy course. The goal is for most of the community discussion and effort to be directed here. However, we will invariably want to customize the course for our own needs, in ways such as the following:


 * remove/add skills
 * select different lesson plans
 * adjust timeline
 * modify course principles or policies
 * provide a particular assessment scheme

The easiest way to spin off a version of this course is to do the following (instructions for each task still need to be included below, but in the meantime, anyone can ask for help on this course’s Discussion tab):


 * create a new page (instructions forthcoming, including where to locate the page)
 * paste in the contents of the current page
 * modify the contents to your liking
 * add the category tag “Eventmath QL courses” to your course (instructions forthcoming)

By adding the category tag, each new customized course will be added to the “Eventmath QL courses” category page, automatically. We can link to that category page from here. Instructors can then browse through various implementations on that page, to find one that’s closer to their liking.

Sources of inspiration for this quantitative literacy course
In this section, we can list sources of inspiration for this course. The list below is just a start and may be missing some brilliant resources. If you've taught a quantitative literacy course before, you're welcome to include your syllabus here! If you've created other relevant materials, you're welcome to include them here as well! Consulting these resources may help us to improve the current draft.

To make sure this list stays valuable and easy to use, we’ll use the following two criteria for inclusion.


 * 1) No lists of resources. We don’t want to have to click multiple levels deep to find something good.
 * 2) Web-based materials, including journal articles hosted on the web, should be openly accessible if possible.

''We may want to reformat the list below. Hopefully this will work well enough for now.''

Free Textbooks

 * [F1] Math in Society (David Lippman)
 * [F2] Liberal Arts Mathematics (College of the Canyons)
 * [F3] College Mathematics for Everyday Life (Inigo et al.)

Web Resources

 * [W1] Teaching Quantitative Reasoning with the News
 * [W2] Numeracy in the News
 * [W3] National Numeracy
 * [W4] SIGMAA on Quantitative Literacy
 * [W5] Mathematics Educators Stack Exchange Post “Examples of Innumeracy”
 * [W6] Innumeracy.net
 * [W7] News Literacy Model Curriculum in Math
 * [W8] Skew the Script
 * [W9] Calling Bullshit syllabus
 * [W10] How to Spot Fake News
 * [W11] thinkingquantitatively blog
 * [W12] PISA 2022 Mathematics Framework

Videos

 * [V1] Math for Informed Citizens (David Kung @ TedxGreatMills)
 * [V2] 5 tips to improve your critical thinking (Samantha Agoos @ Ted-Ed)
 * [V3] The Vaccine for Fake News (Cambridge University)
 * [V4] Can you outsmart a troll (by thinking like one)? (Claire Wardle @ Ted-Ed)
 * [V5] Reflections on a 119 Year Old Curriculum (Nils Ahbel)

Print Books

 * [P1] Common Sense Mathematics (Bolker and Mast)
 * [P2] Innumeracy: Mathematical Illiteracy and Its Consequences (Paulos)
 * [P3] Calling Bullshit: The Art of Skepticism in a Data-Driven World (Bergstrom and West)
 * [P4] How Not to Be Wrong: The Power of Mathematical Thinking (Ellenberg)
 * [P5] Why Numbers Count: Quantitative Literacy for Tomorrow's America (Steen)
 * [P6] The Numbers Game: The Commonsense Guide to Understanding Numbers in the News, in Politics, and in Life (Blastland and Dilnot)
 * [P7] Calculated Risks: How to Know When Numbers Deceive You (Gigerenzer)
 * [P8] How Charts Lie: Getting Smarter about Visual Information (Cairo)
 * [P9] Math with Bad Drawings (Orlin)
 * [P10] Radical Equations: Civil Rights from Mississippi to the Algebra Project (Moses & Cobb)
 * [P11] The Art of Statistics: How to Learn from Data (Spiegelhalter)
 * [P12] Case Studies for Quantitative Reasoning: A Casebook of Media Articles (Madison et al.)
 * [P13] Making Numbers Count: The Art and Science of Communicating Numbers(Heath & Starr)

Research

 * [R1] Numeracy (open-access, peer-reviewed, electronic journal of the National Numeracy Network)
 * [R2] Misinformation and Fake News in Education (Current Perspectives on Cognition, Learning and Instruction) (Kendeou, P., Robinson, D.H., & McCrudden, M.T. (Eds.))
 * [R3] Improving Comprehension of Numbers in the News (Barrio, P.J., Goldstein, D.G., & Hofman, J.M.)
 * [R4] Designing and Assessing Numeracy Training for Journalists: Toward Improving Quantitative Reasoning Among Media Consumers (Ranney, M., Rinne, L., Yarnall, L., Munnich, E., Miratrix, L., & Schank, P.)
 * [R5] A Study Into the Value Placed on Numeracy As Symbolic Capital Within the Journalistic Field (Harrison, S.)
 * [R6] How Does One Design or Evaluate a Course in Quantitative Reasoning? (Madison, B.L.)
 * [R7] Quantitative Literacy in the Media: An Arena for Problem Solving (Watson, J.M.)
 * [R8] Teaching Statistics Using the News Media (Ceesay, T.P.)
 * [R9] Misinformation in and about science (West, J.D., Bergstrom, C.T.)

Notes:
 * [R3] The researchers provide templates for turning quantitative sentences in news articles into sentences that are easier to understand. The research finds that the completed templates "substantially improve people’s ability to recall measurements they have read, estimate ones they have not, and detect errors in manipulated measurements." Maybe Eventmath lesson plans could provide exercises asking students to fill in these templates. This could be useful for any student, and in particular, it may help journalism students learn to communicate quantitative information more effectively. Overall, this research is definitely worth a closer look.
 * [R5] This is a gold mine. It's a PHD dissertation on numeracy among journalists. It includes a “numeracy audit” of two newspapers, with a taxonomy of errors. Perhaps the most common errors could be explicitly incorporated into the syllabus in some way. It also includes a guide to writing with numbers for journalists; this may be useful for creating exercises where students are asked to improve the writing in an existing source, to help them become better information producers. That's just a sample of what this resource contains.
 * [R9] Here's a relevant quote: "As society increasingly relies upon quantitative data, data reasoning skills become paramount. In 2017, we began developing a curriculum to address these issues of quantitative literacy... We stress concepts such as selection bias, correlation vs. causation, relative vs. absolute risk, and plausibility checking via Fermi estimation."