Mathematics Done in English
Next year my school is creating an International Studies program, which is a study track for high school students who plan on spending their junior year of high school abroad. One of the new courses we're offering is called Applied English. In that course, we will teach four months of math, four months of science, and two months of English. The math text is one that I'm writing now, Mathematics Done in English.
This textbook and related materials are under a Creative Commons Attribution 4.0 license. You can do most anything with it, as long as you cite the author (me) and source (i.e., this site). I think education materials should be free wherever possible. This makes it easy for teachers to share ideas and teach as effectively as possible. If this textbook is of interest or use to you, please talk, blog, or write about it.
- Part I: Numbers
- Part II: Statistics
The files here are all connected with the ESL textbook, Mathematics Done in English, written by Douglas Perkins. The textbook and associated files that I've produced are all under the license as noted below. In a few cases, snippets of externally-copyrighted work are included. These are labeled as such. This information is brief, limited in scope, non-commercial, and included for nonprofit educational purposes. As such, its inclusion is fair use.
This work is licensed under a Creative Commons Attribution 4.0 International License.
It is common for students in Japan to have difficulty saying the numbers eleven and twelve. Also, pair such as thirteen/thirty and fourteen/forty can be a challenge. Both listening and speaking practice are needed. A lot of mathematics terminology in English is decipherable to Japanese students because the words appear in katakana in Japanese. Showing lots of examples is valuable.
One good way to practice arithmetic and numbers is to sum up the cards in a deck, saying the running total aloud. It takes about five minutes to do, is challenging enough to be interesting, and is good pronunciation practice.
Coin tossing and dice rolling are great ways to learn about probability. If you have the coins and the dice, that is ideal. It's fun to play with four-sided, eight-sided, twenty-sided, and otherwise unusual dice, and they're available for a few dollars at any decent gaming shop.
One topic in this chapter is the game of roulette. Your students probably have not seen the game before, so you'll want to show a video explaining (part of) the game. There are many available, and here is one I like. To demonstrate playing the game, use a Flash website. Students are too young to gamble, and you might think teaching them how is bad. On the other hand, the odds of winning at roulette are bad. Calculate the expected return, and students will realize that playing this game for money is a losing proposition.
In any language there are often multiple ways to say the same thing. When we're studying a foreign language, it's important to be able to express things in at least one way and understand things stated in many ways. So, for example, you could say five over three or five divided by three, and both are perfectly acceptable. Some numbers are easier to say if you forget about the special cases. For example, one and a half is a little tricky, whereas three over two and one point five are relatively simple. The teacher should be sure to use multiple formulations for these numbers, so that students are familiar with them.
In the previous chapter we began doing word problems, and the obvious extension of that is to do word problems with variables — that is, equations. Conceptually, this is not a great leap. One goal is for students to get used to doing math with both numbers and letters. Another is for students to get used to switching between written form (i.e., several sentences) and mathematical form (i.e., an equation).
Although included here as a branch of mathematics, statistics rears its head in a myriad of subjects. This part of the textbook starts off with averages and ends with graph making and graph reading.
The most obvious place to start calculating averages is in the classroom. Find out the mean shoe size, the median height, and the mode of the times when students wake up. In this chapter, students learn three notions of the average and how to calculate them.
The median of a list is the number that occurs in the middle. If you have an odd number of items in your list, like
[ 1 , 2 , 5 ], the middle is obvious:
2. But if you have an even number of items, like
[ 1 , 2 , 4 , 9], it's unclear what to do. In some cases people prefer you write the middle two numbers:
[ 2, 4 ], and in other cases they prefer the mean of those two numbers:
3. Which of these is better depends on the circumstance, and both are acceptable here.
A simple analysis of birthdays suggests that there ought to be more January birthdays than February birthdays because there are 31 days in January but only 28 in February. Similarly, if a class has 14 members, we expect that on average two of them were born on a Friday, because there are seven days in a week. In this chapter, we first ask students to think about and solve these kinds of problems.
You might think that birthdays are distributed randomly. For example, you might expect that one in seven children in the country is born on a Sunday. Surprisingly, that's not true. There are obvious trends in when children are born. Some days of the week are more and less common, and some months are more and less common. It's interesting to think about why that might occur.
If you have a math savvy class, you can calculate the mean day of the month for birthdays. Let's ignore which month and focus on the day of the month. For example, if we have three students, one born on the 12th, one born on the 15th, and one born on the 18th, the mean is the 15th. Assuming that on average around the world birthdays are distributed randomly throughout the year, what is the mean birthday?
As an opening to world statistics, we look at the foreign resident population of Japan. It is a surprise to many, but the country with the largest number of foreign residents in Japan is China. Following that are several other Asian countries. The United States, the top non-Asian country, ranks at sixth.
Another good way of playing with statistics is to restate things in the negative. For example, consider the following two statements.
55% of students in this class are girls.
45% of students in this class are boys.
Those two statements are stating the same essential idea. Also consider the following three statements.
On average, cats are smaller than dogs.
On average dogs are larger than cats.
On average, cats are not larger than dogs.
These three are also quite similar. One might object that the final sentence is not logically equivalent to the two above it, but then again, the odds of average cat and dog sizes being precisely equal are quite low, so it should be considered reasonable enough. On a pedagogical level, one of the goals is to convert English statements about statistics into other English statements about statistics. This is a great way to learn about and practice both subjects simultaneously.
Some countries use the metric system, but many only use the metric system for some things. Knowing a little about Fahrenheit and feet and inches is a good thing for internationally-minded students. It is not of critical importance to memorize exactly how to convert from miles to kilometers, or from centimeters to feet and inches, particularly when our students end up studying in several different countries that don't share the same units. That being said, the ability to do a conversion given the formula is of great value.
Many of the activities in the textbook benefit greatly from the teacher bringing a computer to class. That lets us show videos, listen to listening tracks, and use websites. If you have a Windows machine, the calculator SpeedCrunch is free, open source, and lets you show your calculations to the class clearly.
For playing video and audio, probably your computer's default media player can do the job. If you'd like to try a new one, VLC is a good choice. It's free, open source, and runs on all major operating systems. If you like, you can play audio and video files at reduced speed, which can be a nice way to spice up a listening activity.
The textbook, tests, homework, and most other materials were produced in LibreOffice. It's an office suite that's better than Microsoft Office in most ways. For editing
SVG files, I use Inkscape. For editing photos and other pictures, I use the GIMP. These programs are all free and open source.