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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
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!
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.
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.
You can explore alone. But playgrounds are more fun with friends.
With graph theory we explore math without numbers.
Which kinds of measurements make the best pattern for foot size and height?
Which of ten famous ideas would you pick to win a beauty contest?
Challenge classmates to a math duel in the spirit of the dangerous world of 16th century algebra.
Watch tradgedy unfold as error propagation wrecks estimation!
The start of term reflection helps you start the term with mindfulness and purpose.
The CLO reflection affirms that core learning happens in every class.
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.
Practice tests can be playgrounds too.
Do not take them too seriously. Seldom do an entire practice test.
Instead, do the first few problems. Try these again and again on different versions the same practice test. 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 practice test. 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 tests 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.
Topic tests happen during class time. They teach you how to take a math test.
The first half of the class time focuses on solving problems using the characteristics of well-written step-by-step problems. Pretend this is a traditional math test where you work by yourself.
The second half of each topic test focuses on group collaboration to finish and fix answers. There will be a lot of valuable learning involving explanations out loud and pointing at each other's papers—two resources that will forever vanish! Capture this learning by making fixes to your scratch work, and by finishing any test problems you did not finish by yourself.
Each group member should write his or her own fixes. Write your fixes and finishings distinctly. Use another color pencil, pencil versus pen, separate pieces of paper, clouds around the fixes, etc. The ability to clearly see which topics were difficult or promoted careless mistakes will help you be properly wary when studying for future tests.
Before the end of the day, text or email your instructor a one sentence study plan.
This study plan should include using helpful study skills and habits and be relevant to the problems you fixed or the problems your group answer inaccurately. It should describe how you will study before the next test. Optionally, it may also include reminders about improving test-taking technique.
The end result is a written record of which problems you need to study and your plan about how to study them. It also has every problem correctly solved in a step-by-step manner to be optimally helpful when studying for the final exam.
The end of term tests are simply larger collections of topic test 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.