You are not stupid!

Teaching math without context is stupid. Historically, the math curriculum has been designed so that we teach skills and techniques without any explanation of why they’re important or how you might use them. It’s harder to learn when you can’t see how topics are related.

Discouraging questions is stupid. Teachers feel pressure to “get through the material” and many don’t know how to cope with unscripted exploration, and 30 brains thinking 35 different thoughts. But when students ask questions, it means they’re ready to learn something. That’s a golden opportunity! It doesn’t happen on schedule, but it’s the whole point of teaching!

Expecting everybody to learn the same way, at the same rate, is stupid. We come in all shapes and sizes, with all different interests, experiences, talents, and struggles. Since our lives are so different, doesn’t it make sense that our thought processes and mental habits are different? When a student doesn’t understand a teacher’s explanation, why do we assume the student is deficient in some way?

Expecting students to sit still and listen while the teacher talks is stupid. Students need to get involved in what they’re learning, not just copy the teacher’s example and try to imitate later. At best, that produces trained monkeys who don’t know what do to when they see a different type of problem. And in the real world, all the problems are different.

Expecting everybody to do the same homework is stupid. Some students need to do lots of examples. Some need to do just a few, but need to really think about the principles. Some need to draw pictures. Some need to prove the theorem for themselves.

Expecting everybody to have the same level of ability is stupid. Not everybody can be good at playing basketball, and that doesn’t mean they’re not trying, they’re bad at sports, they’re a loser, or whatever else discouragers might say. They can still do something for exercise. Not everybody can be good at algebra, but unless they have a severe mental disability, everybody can learn basic math well enough to function, and that’s all some people need.

Teaching new material when students haven’t understood the previous topic is stupid. Math, more than any other subject, builds on previous material. It’s better for a student to learn less, but learn it well, so they can actually use it.

If you’ve struggled academically, maybe you learn differently from the way most teachers teach. Maybe you missed a fundamental concept that everything else depended on. Maybe you can’t remember the information because you don’t see how it relates to anything. Maybe you had a bad experience that left you feeling like a failure.

None of that makes you stupid. You can take the initiative to figure out how you learn best, ask questions until you get satisfactory answers, and study the topics most relevant to you. And I’d love to help point you in the right direction!

A better way to think about solving algebra equations

Most people learn to solve algebra equations in terms of a list of steps for each different type of equation. But really, all those steps are based on a few simple concepts. If you understand those concepts and see an algebra equation within that framework, you can forget about all the different rules and steps that are so overwhelming to beginners.

An algebra equation is like an old-fashioned pan balance used for weighing things. To use a balance, you put an item, say a bag of candy, on one side, and then add weights to the other side until they balance. If the weights are one 5oz and two 1oz, the bag of candy weighs 7oz. In the case of an algebra equation, the equals sign tells you that whatever is on one side weighs the same as whatever is on the other side. Your goal is to figure out what one x “weighs”. You can do whatever you want, like add 5 or multiply by 2, as long as you do it to both sides of the balance so both sides still weigh the same. When you get one x alone on one side of the balance, and only a number on the other side, you’re finished – x “weighs” that many units.

It’s hard to solve algebra equations with an actual balance, because x would have to be a different weight for each equation. However, you can experiment with a virtual balance on the computer. This particular website has some limitations to the size of numbers and what you can do to the equation, but it’s the best program I’ve found for showing a picture of this concept.

So, when solving an equation, you have one goal: get x all by itself. There’s only one rule: whatever you do, you have to do it to both sides. Other than that, you can do whatever you want — whatever you think will help get x by itself. Generally the best strategy is to do the opposite of what’s been done to x. But it’s perfectly acceptable to add, subtract, multiply, or divide by any number, as long as you do it to both sides correctly. You might take the scenic route to a solution, but you’ll get better with practice.

After you practice for awhile and start chafing at the limitations of the virtual balance, there’s another website that will allow you to experiment with steps to solve an equation. There’s no graphical balance, unfortunately, but you can see what happens when you multiply by 3 instead of dividing, or add 5 when 7 would have worked better. You can undo steps and try different ideas, knowing that you won’t be tripped up by an error with negatives when you’re trying to focus on the process.

Of course, with enough practice you’ll be able to solve equations on paper, without any help. And rather than blindly following a list of steps, you’ll be confident that you actually know what you’re doing!

Tips for doing word problems

Word problems (application problems) are a major cause for complaint among students. One reason they’re so difficult is because most of us have learned to do math based on a list of steps – there are steps to add fractions, steps to solve an algebra equation, and so on. But word problems are as varied as the real world, and there’s no list of steps to follow – you have to really think about how the scenario relates to math. Here are some tips for solving  word problems successfully.

  1. Make a list of everything you know – extract the numbers from the written explanation to make it clear what you have and what you need.
  2. Draw a picture or diagram – this helps you see the relationships between the numbers. If some actions have happened in a certain order, draw a timeline.
  3. Remember to convert units as needed before using a formula. Some formulas require particular units, and others just require the same units for each number.
  4. Always write units on every number. As you do the calculations, do the same with the units. If the units come out wrong, you know there’s a mistake in your math.
  5. Use unit analysis (fool-proof method) or proportions whenever possible, to prevent errors. The words “per” or “for each” are a good clue that these methods will work.
  6. Check your answers by putting your result back into the original problem or by using another method to get the same result.

When you’re having trouble, remember…

Nobody can look at a math or science problem and know exactly what to do every time, not even teachers.

Everybody makes mistakes, including your teacher. That’s why we have erasers.

Nobody has every definition, formula, process, and theorem memorized (I have very little memorized.) People who are really good at math focus on understanding, not memorization. If you understand, memorization often comes naturally, or is unnecessary.

There’s more than one way to do anything. If the results are consistently right, the process isn’t wrong (though it may be inefficient). I often learn a new method from my students.

The more you ask why, the more you’ll understand.

In high-level math and science, sometimes you’ll feel like you’re banging your head against a wall. You’re in good company – Einstein did too.

“One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That’s so unlike the true nature of mathematics.” — Leon Henkin

What’s good about mistakes?

People often feel embarrassed or discouraged when they make mistakes. But everybody makes mistakes, even teachers and tutors, and that’s really a blessing in disguise.

  • Mistakes keep us from getting conceited.
  • Mistakes give me an opportunity to show how you can catch them.
  • Mistakes convince you that you need to check your work.
  • The risk of mistakes makes you think about what you’re doing so you learn
  • Mistakes get your attention so you think about why something is wrong and then
    you understand better.