If you don’t like math…

it might be for a good reason.

Maybe you don’t understand it, or you think it’s difficult, or boring, or not useful.

I’ve flipped through some math textbooks, and I can see why you might feel that way.

My hope is that, with some effort, you start to feel differently.

At its best, math is about solving puzzles, and about surprisingly simple ideas that can solve complex problems.

If you have some time, I’d like to try to share an example of how math can be fun.

(When you’re first learning math, I think it’s more interesting to think of math as fun rather than useful.)

Consider the following question:

What do you get when you add the whole numbers from 1 to 100?

In other words, what is 1+2+3+4+5+…+96+97+98+99+100?

(The “…” means imagine all of those numbers that I didn’t have space to actually write down.)

Let’s try to find the answer the long way:

1+2 = 3

1+2+3 = 6

1+2+3+4 = 10

1+2+3+4+5 = 15

1+2+3+4+5+6 = 21

Argh, this is gonna take too long. There must be an easier way…

When given this same question as a kid, a mathematician named Gauss is said to have answered the question within seconds.

How?

(Take some time to see if you can find a trick to make this question easier.)

Answer:

Gauss made the following observation: The question might be easier if I answer it twice!

1+2+3+4+5+…+96+97+98+99+100=?

1+2+3+4+5+…+96+97+98+99+100=?

He took the second question and reversed the order:

1+2+3+4+5+…+96+97+98+99+100=?

100+99+98+97+96+…+5+4+3+2+1=?

But now look what happens. Let’s find the total of both sequences, one pair at a time: 

1+100=101

2+99=101

3+98=101

4+97=101

5+96=101

All pairs have sum equal to 101! How many pairs are there? 100! So all of the numbers in both sequences sum up to:

101*100=10100

Since we found the sum for two sequences, we need to divide by 2 to answer the original question

10100/2 = 5050

The answer is 5050!

If you are studying math and don’t like it, I hope you find some joy in small tricks like this.

Math has some beautiful ideas, but sometimes it looks so abstract that it’s tough to see how anyone could enjoy learning it.

Even if you know how to do the math, sometimes it’s unclear what is actually going on.

If you ever want to understand more in math class, please ask a friend, a teacher, a parent, or try to learn more online.

I hope that by working hard to understand math, learning math becomes a little more fun for you.

 

How to Make Math Practice A Little More Fun

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Puzzles!

If you struggle with math, you need more practice to master the concepts. 

But drills can get boring.

I’ve tried to design math training exercises that are a little more fun than other exercises.

The math puzzles are in the Puzzles section :)

I hope the puzzles make math practice more fun. You need a lot of practice to master concepts.

My goal is to make math exercises that are fun enough for you to practice on a daily basis.

Practice every day! It’ll help you do better in your math class.

Mental Math as a Stepping Stone to Higher-Level Math

A major stumbling block for math students is fast and accurate recall of basic math facts. For example, although most 5th graders have at one point learned the times table for single-digit numbers, they sometimes struggle to recall these facts quickly.

A mastery of basic math facts is crucial for students attempting to learn more advanced math concepts such as the distributive property:

a*(b+c) = a*b + a*c

Without a mastery of the times table, it is difficult to understand or appreciate the distributive property. However, it is difficult to achieve mastery of basic facts, because students are (understandably) hesitant to endure drill after drill. One way around this reluctance to do drills is to “hide” the drills in a game. But there is a potential downside to “gamifying” drills: Students may start to view math as the boring chore that gets in the way of the game.

One approach that I’ve been considering recently to encourage kids to master basic skills is to introduce them to “advanced” mental math techniques. There are a few useful mental-math techniques that students aren’t typically exposed to:

  • Multiply by 5: Divide by 2, then multiply by 10. (For example, 888*5 = 888/2 * 10 = 444*10 = 4440.)
  • Multiply by 999: Multiply by 1000, then subtract the original number. (For example, 7*999 = 7*1000-7 = 7000-7 = 6993.)
  • Applying the distributive property: To compute a*b + a*c, compute instead a*(b+c). For example, 49*3+49*7 = 49*(3+7) = 49*10 = 490.

It is my belief that there are a large number of students who would greatly benefit from daily practice of math skills. The key challenge is to make the daily practice interesting enough to engage the kids’ attention.

Update: I’ve created an Android app for Mental Math listed on Google Play.

An Intuitive Introduction to Algebra

Algebra students struggle when they attempt to mimic procedures in a mechanical fashion. It would be a better situation if students were simply making errors in their mathematical reasoning. The larger problem is when students stop attempting to reason at all. After years of mimicking math procedures that made no sense in school, students often feel they don’t have the luxury to “understand” what they are doing.

Below I illustrate some common types of “math mistakes”:

3x-x = 3

2/4-1/3 = 1/1

2+x = 2x

Michael Pershan maintains a collection of curated “math mistakes”. These errors seem to be the product of rote learning. When students lack an understanding of math concepts, there are too many “rules” that need to be memorized for them to consistently generate correct answers. 

How can we help students truly learn and *understand* algebra? In the book Vision in Elementary Mathematics, mathematician W.W. Sawyer outlines an intuitive way to introduce students to algebra. The key idea is to motivate algebraic techniques by starting with puzzles. The puzzles are designed to be approachable enough that students can attack the puzzles with intuition, without a teacher telling them how to do the problem.

Here is an example of such a puzzle. A mother has twin daughters, both of whom are the same height. The height of the mother plus the height of one daughter is 9 feet. The height of the mother plus the heights of both daughters is 13 feet. What is the height of the mother, and what is the height of each daughter?

To help the student understand the problem, the teacher can draw pictures illustrating the two clues:

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With enough time (and perhaps a hint or two from a teacher), students can compare the two towers

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to conclude that one daughter is 4 feet tall. Then since the height of the mother and the daughter together is 9 feet

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the height of the mother is 5 feet.

Once students get some more practice with similar problems, they can begin to use equations instead of pictures to reason:

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By introducing algebra using puzzles first, students can develop an intuition for algebra with concrete examples. Students can then begin to develop a true understanding of what algebraic techniques are and why they are useful.

My goal is to create online exercises where students can learn algebra using a “puzzles first” approach. I hope to make an announcement on this blog or at MathPapa. Please email me at rob@mathpapa.com if you would like to be updated or have any comments.

Variable X - Interactive Algebra Lessons

Our first interactive algebra lesson is live! Variable X is a new website where I’ll post interactive lessons and exercises for algebra students.

I’m hoping that the interactive feedback will make learning algebra on Variable X an enjoyable experience. Please send me your feedback and feature requests, and I’ll do my best.

Teaching Negative Numbers

Negative numbers are a major source of frustration for kids preparing for algebra. Below are some tough concepts to explain.

What are negative numbers?

A negative number is a number less than zero.”

Negative numbers are difficult to define, because there are many situations in which negative numbers don’t seem to make sense: You can’t eat -3 cookies, and there aren’t any classrooms with a negative number of students. However, kids are familiar with some examples of negative numbers:

  • Temperature: There are temperatures colder than 0 degrees in Fahrenheit and Celsius.
  • Money: Your total wealth can be -$3 if you owe someone $3.
  • Stocks: The price of a stock can have a negative daily change, which means that the stock had a higher price yesterday than it does today.

These concrete examples of negative numbers can be used to help kids practice arithmetic with negative numbers:

  • If the temperature was 3 degrees Celsius yesterday, and it is 5 degrees colder today, what is the temperature today? (3 - 5 = -2 degrees C)
  • If you owe Bob $3 and you have $2 in cash, what is your total wealth? (2 + (-3) = -$1)

Once students gain more experience using negative numbers, they can start to become comfortable with the idea of negative numbers as numbers in their own right.

Why do I get a positive number when I multiply two negative numbers?

Perhaps the most direct answer to this question is the following: To extend arithmetic from positive numbers to negative numbers in a consistent manner, we have chosen the convention that negative * negative = positive.

But of course, this convention isn’t purely arbitrary. One way to justify this convention is to show that the distributive law for negative numbers implies that -1 * -1 = 1.

Yasha Berchenko-Kogan wrote a very popular answer on Quora to illustrate the multiplication of negative numbers using a setting in which black and red chips represent credits and debts.

Yet another way to suggest why it makes sense to define negative * negative = positive is to show a pattern such as the following:

4*3=12

3*3=9

2*3=6

1*3=3

0*3=0

(-1)*3=?

Since the result decreases by 3 at each step, this pattern suggests that (-1)*3=-3.

None of the justifications above can prove the fact that negative * negative = positive. But taken together, hopefully the student can gain an appreciation for why negative numbers are useful and how they can be used appropriately.

The Computer’s Potential as a Medium for Learning Math

Over the past few months, I’ve been working on MathPapa, a step-by-step calculator to help algebra students learn math interactively. I am excited by the computer’s potential to improve math education.

Of course, the idea to use computers to improve math education isn’t new:

Computers have some natural advantages as a medium for learning math, among them:

  • Instant feedback: For some math questions, computers can instantly tell students whether their answers are right or wrong. Given this feedback, students can avoid wasting time making the same mistakes.
  • Rapid exploration: Some math concepts, such as the multiplication law for exponents, can be understood more easily by observing many examples. Computers can enable the student to generate many examples rapidly.

However, some educators are skeptical of the computer’s potential to improve math education. Dan Meyer, for instance, argues that some important types of math questions are impossible to grade automatically. Furthermore, using computers to teach “math” alters the definition of math to exclude everything that isn’t a natural fit for the computer.

Although I agree that computers can’t replace great teachers, I am hopeful that computers can improve math education for many students. Computers can enable teachers to present math concepts to their students in new ways, and can also enable the students to more effectively practice math on their own. 

Why I Care About Math Education

I’m Robert, a computer science Ph.D. student at Stanford in my final quarter.

Since January, I’ve been tutoring algebra at Jordan Middle School in Palo Alto, CA. I think I’ve had some success as a tutor, but I’ve also felt limited by my inability to provide one-on-one attention to every student.

My goal is to improve middle-school math education by building interactive digital tools. I love math, and I’m met some wonderful students as a tutor. I think many of them would benefit from new ways of presenting math.

In Fall 2011, my Ph.D. advisor, Jennifer Widom, launched a public course on databases. I built the automatic exercises for that course, which had over 60,000 enrolled students and is now hosted by Coursera. Being a part of that course inspired me. It helped me realize that there’s a lot of room for new educational tools on the web.

My goal is to help build the tools that will help students learn math. I’ll be posting updates on this blog, and I’ll be posting the tools at my site: mathpapa.com