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Math OER

Explorations  •  Gymnasia

What playgrounds do is provide kids with a relatively safe way to learn about using their bodies to navigate the world—how to balance, how to get from here to there, what to do when you get stuck. In other words, how to solve problems in the physical world.

As I was watching my daughter, I realized that math too is a playground. But it’s not a playground for our bodies, it's a playground for our minds.

- Jason Marshal

If you are a LCC student, please share during our class meetings what you did with two explorations.


Here are places you can use math to explore, analyze, and estimate without being expected to accomplish any specific goals or reach any specific answers. Just play!

playground Image

Kids first engage with a physical playground when they see it. They start thinking about the slide, or the tire swing, or whether they could step from one place to another. Math playgrounds are less visual. So these explorations provide a little bit of descriptive guidance to help you "see" the math clearly as you approach.

another playground Image

There are suggestions: questions to think about, and challenges to attempt. But please do not treat these suggestions like homework problems. Approach the exploration in your own way.

Be mindful as you play. What goes quick? What goes slow? What gives a sence of progress? What makes you feel off-balance? How do you recover from getting stuck?

You can explore alone. But playgrounds are more fun with friends.

graph theory

With graph theory we explore math without numbers.

comparing expensive and budget event

Make a comparison between an expensive and budget version of either a vacation or a wedding.

sample budget

With a pretend and real budget explore the 50-30-20 Rule.

rent or buy a home

Explore the issues about renting versus buying a home.

grocery shopping strategizing

Try some grocery shopping strategizing by recording what you buy and comparing prices by category.

foot size and height

Which kinds of measurements make the best pattern for foot size and height?

beauty contest

Which of ten famous ideas would you pick to win a beauty contest?


Watch tradgedy unfold as error propagation wrecks estimation!

famous algebraists

Challenge classmates to a math duel in the spirit of the dangerous world of 16th century algebra.



The ancient Greeks called their training facilities gymnasia. These buildings were places for athletic practice and be instruction, and also gatherings for socializing about intellectual pursuits. Both the athletic activities and the socializing respected that mental foritude and flexibility went hand-in-hand with physical training. The building usually bordered outdoor lawns or courtyards, where most of the physical activity happened.

The Greek word gymnasia literally means "naked school". Historically this was literal. The ancient Greeks practiced and competed nude for two reasons. First, they realized that training happened best when athletes were honest about their abilities and shortcomings. An important part of training is identifying and monitoring our strengths and weaknesses without feeling needless embarassment or shame. Second, the ancient Greeks believed a well-deveoped body and mind became a tribute to their dieties worthy of display.

Our math training should share these values. Taking a test is both a physical and mental activity. Effective training requires awareness and openness about strengths and weaknesses. Feeling embarassment or shame about today's performance is inappropriately counterproductive. (But please wear clothes to class.)

3-minute multiplication speed quiz

Pass a speed quiz to feel confident with your multiplications tables.

representational fluency speed quiz

Test your representational fluency with 47 hundredths.

Crystal's No Calculator Test

A student named Crystal took a No Calculator Test. Can you find any errors?

Roper's Gymnasium in Philadelphia

Try a midterm as practice or self-assessment.

Math 20 uses Shapeshifting, Mad Science, or Justice midterms.

Math 25 uses Health Decisions, Personal Finance Decisions, or Business Decisions midterms.

Olympic torch ceremony

As end of the term tests you can assess yourself with a No Calculator Test, Math 20 Overall Test, or Math 25 Overall Test.

Notice that all the midterms and the end of term tests are merely collections of random homework problems.

A high school math class is a bit like becoming a concert pianist. No one really sees or cares about your hours of practice. They watch you as you get up on stage at the end. You demonstrate your ability, by yourself, under pressure.

In graduate school, more assessment is oral exams. Instead of creating written answers privately, you stand at the chalkboard while your instructors ask you to do work similar to the practice tests. You do fewer problems, but are expected to be smoother. Sometimes this is a social gathering in which a group of students take turns demonstrating their mastery and celebrate afterwards.

In our class, as an undergraduate class, the end of term tests can use either format. The choice of format might be yours to make, or the instructor might assign you a format.

Continuing with our history commentary, the similar buildings in Asian countries are called a dojo (Japan), dojang (Korea), wugan (China), or Akhara (India).

Imperial Civil Service Examination in Song Dynasty The buildings also blended physical and mental training. They were often larger complexes, with dormitories and guest rooms. Compared to our math work, these Asian terms might place too little emphasis on vulnerability, and too much emphasis on formality, cleanliness, hospitality, honor, and competitive ranking.

Yet the Asian training mindset can remind us to not take these gymnasia activities too seriously. They are also playgrounds. Seldom start with a goal of finishing completely.

tai chi in a park

Instead, do the first few parts. Try these again and again on different versions. Work on good form. Enjoy affirming your mastery!

Then add another one or two problems. Again, practice your form with a shorter to-do list.

Eventually you will have mastery over the entire activity. But do not rush it.

Practice mindfulness. You are working on good form by repeating a task with attentiveness to detail. Acknowledge the frustrations and the joys while not focusing on them.

Your actual assessment will look just like these practice tests—almost. For some problems, instead of creating answers you will be required to analyze already complete work to explain steps or find errors.

Big Issues

These big issues without right answers do not have any step-by-step guidance.

You are on your own, released from the rigor and expectations of the classroom to do real math.

Learn the habit of using math to estimate and explore. Math can help make decisions even if there is no right answer. Issues like these are the "icing on the cake" that sweetens all the math skills and algorthms you have learned.