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Kids at a library know they are readers. They know how to read. They discuss the books they read with their friends. They devour books for pleasure. Other people tell them they are readers.
They feel done with "learning to read" and have moved on to "reading to learn". They use books to better understand the world.
They use text to answer questions that arise naturally in their life and thoughts. If they forget a fact, they can read to refresh their memory. If they need to make a tricky decision, they can read something to clarify the issues.
When do people become mathers?
Let's compare learning math to learning language.
To ask the right question is harder than to answer it.
- Georg Cantor
Everyone knows that reading and writing are very different.
Everyone can read at a higher level than they can write. Of course this is normal!
Imagine if elementary schools tied creating and devouring language together. "Sorry, little Suzie. You cannot check out any Harry Potter books from the library until you can write as well as J. K. Rowling."
Absurd! But inexplicably familiar for math, where too often the only math books shown to students are textbooks that tie together creating and devouring math at the same level. Too often students have never heard of the fun math books that people enjoy devouring, such as the books in our library.
There are certainly aspects of literacy and studying the English language that go way beyond what Suzie and her friend at the library call being a reader. These are considered hobbies, eccentricities, and academic specialties. People can enjoy that others know those things without feeling incomplete as a reader without them.
For example, not everyone can diagram sentences.
And not everyone can read The Canterbury Tales.
On the other hand, society believes math is like a ladder or a flowchart. Students are taught that they are incomplete until they reach a place on the ladder far beyond the level of math useful in most people's daily life.
But people who think about math as a ladder or flowchart are believing three lies.
In truth, most branches of mathematics are almost independent. People can enjoy geometry, algebra, number theory, topology, and business math with minimal skill in the other branches.
In truth, people can understand and devour these branches of mathematics in both a pure form (thinking about shapes, groups, and patterns) and in applied settings (thinking about business, construction, science, finance, codes, health formulas, etc.) at a much higher level than they can personally create. Any student is truly at several places among the clouds, not at one place on a flowchart.
Most people looking at this website can use math decently to balance a checkbook, compare prices at the store, do household carpentry, and lots of other things. Sadly, using math to answer problems that naturally arise in your life and thoughts is something that happens almost exclusively outside of the classroom. These activities have never been appreciated, assessed, or celebrated in the classroom.
Imagine a class about art where you learn about many famous paintings, and painters, and styles of painting, and historical influences for painters, etc. But you never picked up a brush and actually painted. That would be an "art appreciation" class, right?
And imagine a class about music where you listen to many famous songs, and learn about composers and orchestras, and styles of music, etc. But you never wrote notes and actually composed, or used an instrument to play a song. That would be a "music appreciation" class, right?
Now imagine a class about math problems where you learn about famous, old math problems that millions of other people have already solved, but you never actually create any new and original and personal math. That would be a "math appreciation" class, right?
Too many students have never taken a real math class! They have only had "math appreciation" classes, and were duped into believing these were real math classes. Only at college—maybe—do they finally see issues without right answers: personal decisions about budgeting priorities, retirement plans, renting versus home-ownership, dieting and exercise, business decisions, etc.
If I was an "painting appreciator" who had never picked up a paintbrush, and someone called me an "painter", then I would sense something was wrong, even if all my life people had mistakenly called painting appreciation "painting". If a new class finally guided me through using a paintbrush, I would so clearly see it as the most genuine painting class I ever had.
Too many students are in a equivalent predicament. They feel in a deep yet vague way that they are only "math appreciators" and so cannot call themselves mathers.
The personal math they have done to make real life more successful has been undervalued, exiled from the classroom, and uncelebrated. Their unhappy efforts at math appreciation have been valued, central in the classroom, and celebrated with smiley faces, check marks, and letter grades.
Here the math that has been a part of your real life since childhood is what becomes central and celebrated: money and budgets, patterns and building, shopping and borrowing, health and recipes, prioritizing and scheduling.
So, what is a mather?
1. Can you participate in a math discussion? Can you follow what the instructor does on the board in class? Can you talk about planning a road trip? Do you pause the conversation when you need to ask for clarification? When it gets boring can you ask a question about a tangent that you find more interesting?
We all use language to talk. We are better at talking than reading or writing. Math shares some conversational skills with language, as well as having some specific math-conversation skills.
2. Can you enjoy devouring a math book? Not a textbook that teaches math appreciation, but a practical math book about real-life issues and problems and decisions that other people have struggled with and used math to help understand. Can you use external resources when you seek clarifications or purse tangents?
We all use language to read. We are better at reading for fun than writing. Math shares some pleasure reading skills with language, as well as having some specific math-reading skills.
3. Can you use math to explore a branch of mathematics without a teacher? Perhaps topology, behavioral economics, business math, or number theory?
We all use language to learn. We are better at reading to learn than writing. Math shares some comprehension skills with language, as well as having some specific math-comprehension skills.
4. Can you use math to analyze your own or other people's or business' issues and problems and decisions, and communicate about the results?
We all use language to write. We are best at writing about our own interests and hobbies. Math shares some composition and communication skills with language, as well as having some specific math-related skills.
5. Can you use people, videos, books, or other resources to refresh your rusty or forgotten math skills?
We all use language to write, and might become rusty with or even entirely forget a certain writing skill that we have not used in months or years. What is the rule for a possessive apostrophe on a plural noun that already ends with the letter s? What is the differences between the two major types of sonnets? Math shares most review and refresh skills with language, but there are some specific math-review skills.
6. Can you use math to do problems on a test?
We all use language to write. The most difficult writing is about language-specific tasks, such as composing a sonnet or writing an essay about the Great Vowel Shift. Math shares some advanced or eccentric skills with language, as well as having some specific math-related skills.
7. Can you plan how much math is in your life?
We all use language to write. But some people are more than writers: their job or hobby requires them to plan how much they write. We call them poets, novelists, essayists, and authors. Language classes ask students to become those for a while. Similarly, people who use math in their job or hobby are called mathematicians, and your math class asks you to become a mathematician for a time. This involves time management. You should structure your schedule, your day, and your mind to do the math you should get done even if you are not in the mood, and also to cut short an enjoyable math exploration because of other real life responsibilities.
Can you do these seven things? Some better than others? Then you are a mather!
Get ready to become a better mather.
Thanks to many people for the ideas and discussions that helped form these ideas, especially Scott Kim, Paul Lockhart, Gary Mort, Cathy Miner, and Karen Louise White.
We all agree that your theory is crazy, but is it crazy enough?
- Niels Bohr
Besides the topics to the left, you will learn math study skills. Math work has its own best ways to take notes, do homework, cooperate in groups, and prepare for tests.
If you have never been taught these study skills then of course you felt overwhelmed by previous math experiences. You were actually trying to learn two classes at once: the math topics and these study skills. Then there was the extra problem that the second class is barely acknowledged.
We emphasize these study skills. Once you learn them, other people will look at you and say, "You are such a good math student!" What they mean is, "I am still doing a double curriculum but you're not. You already know the study skills and are doing half the work."
This is one-third of becoming "good at math".
Our first three topics on the left discuss foundational tools: dealing with numbers, expressions, and formulas. For the sake of simplicity, we first learn these tools with minimal real-life application. But there is much more to learn about context.
As an analogy, a young child knows what a "pliers" is. They understand how to use one. But in real life there are many kinds of pliers, each appropriate in different situations, and each used slightly differently.
Our latter three topics on the left discuss that context. We use those foundational tools to make decisions for health, finance, and business.
Adding context to skills is another one-third of becoming "good at math".
This is how real math works:
First, a problem appears. It is not an classic problem in a textbook that generations of students have used to build fluency in a skill. Instead, someone is making a decision. Should they rent or buy a house? How much food should they buy when planning a party? Do they have enough money saved for a certain vacation? Is their higher BMR a big deal? Is it worthwhile to open a new credit card for the sign-up bonus?
Second, the complexity of real life is shrunk down to a few numbers on paper. Which math concepts describe the situation? Which facts and formulas are used to deal with those concepts? Which numbers are measured or looked up? How will estimating these numbers affect the trustworthiness of the results?
Third, numbers and symbols are moved around on paper, using both algorithms and guessing, to discover results. This step does indeed include the broadest and deepest set of skills. But that does not excuse how too often it is the only part of real math done in a classroom.
Fourth, the results are evaulated. The numbers and symbols are translated back into concepts, and then back into the real-life context. Are these results actually helpful for making that real-life deicision? After all, any time we move numbers around on paper it poops out some kind of answer. What have we learned? How firm is our certainty? Is our margin of error acceptable? Need we start over with more or different concepts or numbers to improve our results or to come up with a backup plan that has redundancy or failsafes?
Finally, either the entire process or just a summary of the results is communicated to family, friends, or colleagues.
All five steps involve skills and strategies specific to math. We will discuss all five steps.
(Using math in science or computer programming often involves a similar process. But the specific skills and strategies will be different when the application is physics, chemistry, or programming: different from each other, and different from pure real life math.)
Math is where natural laws are clearest, where patterns are most beautiful, where systems are most pure, where truth is most intuitive, and where big problems are most solvable.
Real math offers guidance, not right answers. The isssues are too big and involve depth, estimation, and uncertainty. They issues ask what does this kind of math mean for you? You and I will make different health decisions, personal finance decisions, and business decisions. Yet the same math can guide us. Our answers will mature as we gather more data and clarify the context.
Real math problems introduce you to seeing life with a math worldview, which helps your quality of life. This is the final one-third of becoming "good at math".
Allow our math time to challenge you. Achieve your potential. Be pleasantly surprised by the height of your accomplishments.