Center for Second Language Teaching and Learning

Kathryn Heinze

Assoc. Professor

Center for Second Language Teaching and Learning

Hamline University A 1790

1536 Hewitt Ave.

St. Paul, MN 55104

Copies of this handout can be downloaded from http://kathrynheinze.efoliomn2.com

The Language of Math

Vocabulary

(1) Mathematics includes words that are specific to mathematics such as divisor, denominator, and quotient. These words are new to most students. (2) But math also includes common everyday vocabulary that takes on a different meaning in math. Examples of these words include: square, rational, column, and table. (3) Math also combines math vocabulary in phrases that can take on a brand new meaning: the least common multiple and a quarter of the apples.

Words for Addition Subtraction

add and subtract from

plus sum take away

combine increased by minus

Words for Multiplication Division

multiplied by divided by

times into

Words for Answers

total

gets you

results in

· It’s not enough for students to learn lists of words. They must learn what they mean in a particular mathematical expression. Examples: (1) 3 multiplied by 10 is vastly different than 3 increased by 10, (2) divided by and divided into will give entirely different results.

4

· When students learn mathematical symbols such as +, ´, ¸, >, <, and they must learn to relate them to (1) mathematical processes or operations, and (2) translate them into everyday concepts.

· If students have had math education in their home countries, they may find differences in symbolic use confusing.

300.000 1,73

Take special note of vocabulary that has both common and technical meanings. This can be particularly challenging for ELLs and needs to be drawn attention to.

column product

quarter place

table

Syntax (Word Order)

The language of mathematics, like all technical or specialized vocabulary, has special syntactic structures and special styles of presentation.

Here are some examples of structures that are frequently used in math and can be difficult to master by ELLs.

greater than/less than as in all numbers greater than 4

n times as much (as) as in Choua earns six times as much as I do. Choua earns

$40,000. How much do I earn?

as . . .as as in The tennis ball is as big as the plastic ball.

-er than as in Fernando is three years older than Frank. Frank is

25. How old is Fernando?

Numbers used as nouns

(Rather than adjectives) as in Twenty is five times a certain number. What is the number?

Prepositions as in eight divided by four and eight divided into four

Passive voice as in when 15 is added to a number, the result is 21. Find

the number.

· One of the principal characteristics of the syntax used in mathematical expressions is the lack of one-to-one correspondence between mathematical symbols and the words that they represent. For example, if the expression eight divided by two is written word for word in the order in which it is written, the resulting mathematical expression could be 8 ¸ 2 =

As a result, ELLs tend to duplicate the surface word order of mathematical statements. Example: “The number a is five less than the number b” is often interpreted by students as:

a = 5 – b instead of a = b - 5

· Logical connectors are another language device used in mathematics: if . . .then, if and only if, because, that is, for example, such that, but, consequently, either . . . or. Students must know what situation is signaled, cause and effect or reason and result, similarity, contradiction, chronological or logical sequence. To use these connectors, they also need to know where logical connectors occur: in the beginning, middle, or end of a clause. Some connectors can only be used in certain positions; others’ meaning changes if the position is changed. Elementary students may, for the most part, be limited to having to puzzle out if. But they do show evidence of having difficulty understanding the hypothetical situations signaled by this connector in problems like those below:

4

If Mohammed can type one page in 20 minutes, how much time will it take him to type two pages?

However, they have much less difficulty when this problem is expressed as:

Mohammed types one page in 20 minutes. How much time does it take him to type two pages? OR How much time does he (Mohammed) need to type two pages?

Ellipsis and Cohesion

Language, all languages, are full of redundancies. But because repetition in a language you know can become annoying, writers of even math problems use several linguistic features to avoid repetition. Two ways that redundancy is avoided include (1) the use of ellipsis (leaving words out), and (2) the use of pronouns and articles.

In ellipsis, words may be left out of a phrase because native speakers can very easily figure out what the missing words are.

All numbers greater than four = All numbers (that are) greater than four

Maria earns six times as much as Peter ( ).

· Check to see if students can supply missing words or phrases. If they can’t, it may indicate that they are having problems inferring the deleted information.

In pronoun reference, pronouns such as she, he, it, her, him, they, them, their, theirs, hers, his, its, etc. refer to nouns that have been mentioned before.

Rachel had 17 toy cars. She gave 11 of them away. How many toy cars does she have now?

· Other pronouns that can be even harder to identify include words such as the one(s), that, these, those, this.

Spread your thumb and first finger as far apart as you can. Do this in the air. Don’t use your other hand to help. Trace them on the board.

· If students can’t figure out what nouns or ideas pronouns refer back to they miss out on understanding how the sentences relate to each other; they don’t understand the cohesion of the text, or how it hangs together. This can be a serious stumbling block to their understanding of a text, whether it is in math or another subject area. Students who claim to know all the words, but can’t understand the message or meaning may have trouble identifying which nouns the pronouns refer back to .

The use of articles and determiners (the, a, an, this, that, these, those) can also be troublesome. Note the following problems:

When 15 is added to a number, the result is 21. Find the number.

Twenty is five times a certain number. What is that number?

One number is ten times another number

·  ELLs need to be explicitly taught the meaning of articles/determiners. Words such as the, this, that, these, those usually indicate that it is a number that has been mentioned before.

4

Other Language Considerations

To be proficient readers, ELLs need to know how sentences, groups of sentences, paragraphs, or whole texts function together to convey a message. Understanding vocabulary alone is not enough. Students have to know what bank of prior knowledge about the world to tap into because the language of math is: (1) very dense and highly packed with concepts, (2) requires up-and-down as well as left-to-right eye movements, (3) requires a slower reading rate and multiple readings, (4) uses numerous symbolic devices such as mathematical symbols, charts, and graphs, and (5) contains a great deal of technical language that conveys precise meanings.

· Students have to apply mathematic concepts, procedures, and applications with language. So students with a weak math background and second language skills are doubly challenged.

· Students need to need to recognize which previously learned math concepts, procedures, and applications must be applied to the text they are reading. They must also know when everyday background knowledge can be applied to what they are learning.

Adapted from Richards-Amato & Snow (1992), Cocking & Mestre (1998), and Carasquillo & Rodriguez (1996).

.

Problems from the MN Basic Standards Test for Math

Kim drove an average of 50 miles per day for the first 4 days of her work week. What would her daily average be for her five-day work week if she drove 35 miles on the fifth day? (if clause)

In a football game, a team gained 4 yards, then another 12 yards, was penalized and lost 15 yards and gained 7 yards. What was the net gain of their three successive plays?

(vocab: net, successive, gained, penalized)

A machine does a job in ¾ of an hour. It takes a person 6 ½ hours to do the same job by hand. How much less time is required to do the job by machine? (articles)

A house plan is drawn to the scale of ¼ inch = 4 feet. What is the length of the house if it measures 3 ½ inches on the floor plan? (synonymous phrases)

Jim spent 1 hour and 30 minutes raking the yard in the morning and 1 hour and 45 minutes in the afternoon. How long did he rake altogether? (ellipsis)

An item was regularly priced at $63. The item went on sale at 1/3 off the regular price. How much was saved?

A pen costs $11.39. Sarah bought 3 pens. Not including tax, how much did Sarah pay?

At 7:00 a.m., Jose’s thermometer registered 5° F. By 5:00 p.m., the temperature had dropped 18°F. What was the temperature at 5:00 p.m.?

The Word Problem Procedure

Word Problem Procedure (WPP)

1.Choose a partner or partners. Write your names above.

2.Choose a problem. Write the problem in the space below.

3.One student read the problem out loud. Discuss the vocabulary and circle words you don't understand. Write the words below.

4.Use a dictionary for help. Ask your partner or teacher for help.

5.What does the problem ask you to find? Write this below:

6.What should you do to solve the problem? Add? Subtract? Multiply? Divide? Write this below.

7.Solve the problem below.

8.Check your answer below.

9.Explain your answer to your partner(s). Write your explanation below.

10.Explain your answer to the class.

11.Write a similar problem on the back of this page.

The various steps of the WPP make it possible for students to practice academic English, apply mathematical rules, and use learning strategies and tactics in the course of the activity. It also provides opportunities for teachers to be explicit and direct in their instruction and to pinpoint specific strategies and tactics that they wish their students to practice. Students use metacognitive strategies such as selective attention in focusing on unknown words (Step 3) and attending to what the problems ask them to find (Step 5), advance preparation in reading the problem to each other (Step 3), and selfmanagement in explaining the problems (Step 9) or writing similar problems (Step 11). They use cognitive strategies such as resourcing in using their dictionaries (Step 4), and elaboration and deduction in solving the problems (Steps 7 and 8). They use socialaffective strategies such as cooperative learning and questioning for clarification in working their way through the various steps. The use of this format thus creates opportunities to introduce and practice learning strategies while solving word problems. In the context of this study, it also makes it possible to analyze student performance in reference to their ability to follow individual steps and sequences of steps, and to learn and apply learning tactics and strategies.

4

4

Discussion of an Example of the WWP

To read a more detailed analysis of how students dealt with each stage of this procedure, go to http://www.ncela.gwu.edu/pubs/symposia/third/spanos.htm. In my presentation, I will only discuss some of the steps in this procedure.

Word problem: Sam's truck weighs 4,725 pounds. The truck can carry 7,500 pounds. What is the total weight of the truck and a full load? (addition of whole numbers)

Step 2 (Choose a problem and write it). Simply copying the word problems from the review sheet was difficult for many of the students. Their work was marked by poor handwriting, spelling errors, sentence fragments, runons, and mechanical errors involving capitalization and punctuation.

Step 3 (Read the problem out loud. Find difficult words). It was difficult to convince students that they should take turns reading the problem to each other. Even after the instructor selected two students to model the practice for the whole class, most pairs still refused to read aloud. However, they did enjoy circling or underlining words they did not understand and writing them in the space provided. Some of the items selected from the word problems were the following:

weighs, carry, load

Step 5 (Write the question). Students were not initially aware that the word problems ended with a question. Once they were, Step 5 became a routine matter of finding the question and copying it. As with Step 2, students made copying errors such as omitting question marks and other punctuation, or writing puzzling statements. For example:

What is carry?

Steps 6 and 7 (Find the operation and solve the problem). There were numerous cases of incorrect selection and application of mathematical operations, incorrect representations of the problems, and computational errors. For example:

4,725 x 7,500 = 31,500

Steps 9 and 10 (Explain orally and in writing). It was difficult for students to give a coherent verbal or written English explanation of their work. A typical response was to simply write the numerical answer from Step 7 along with a statement such as: "This is the answer to the problem." Some of the more interesting written explanations follow: