Emily Pearce Understand the meaning of mathematical concepts and symbols and how to communicate

mathematical ideas in writing.

Often times, students, especially English Language Learners (ELL), have difficulties with translating, interpreting, and comprehending mathematical language (Adoniou & Qing 3, and Manqian & Lapuk 288). Although it is important to teach students the definitions of mathematical terminology, it is just as important to develop the skills needed to derive and conceptually understand the complex language of mathematics. Strategies for comprehending mathematical language is especially helpful for ELLs, but can be utilized by any and all students that struggle with mathematics.

I. Translating among algebraic, graphic, numeric, and written modes of presenting mathematical ideas.

Presenting the same material in multiple representations is useful to students when translating among mathematical ideas. It is reasonable to ask students to compare and contrast various modes of representations. Consider the following problem that asks the same question with three different presentations of the material:

Q: How much does 1150 minutes cost? How much does 1500 minutes cost? How many

minutes can you get for $220?

Mode 1: The U-Stationary cell phone plan allows 1000 whenever minutes for $59.99 per month. Additional minutes cost $0.45 each. Mode 2: Under the BU&U cell phone plan, the total cost for a month’s usage is expressed in the table below.

Minutes Total cost

0 $49.99

300 $49.99

600 $49.99

900 $49.99

1000 $99.99

1100 $149.99

1200 $199.99

Mode 3: Under the Scamper cell phone plan, the total cost C( m ) for a month’s usage ( m minutes) is expressed by the piecewise-defined function below:

By comparing the three different representations, students will gain conceptual understanding of the mathematical idea and thus, be able to translate these ideas whether they be graphical, algebraic, numeric, or written. For more examples of how to visually display information after it is collected, or to find interesting questions to pose, go online to USA TODAY at http://www.usatoday. com/snapshot/news/snapndex.htm. News-related graphs and questions are categorized and displayed. These visual displays of information are easy to read and understand which could be initially helpful first introducing multi-modal problems (Edwards 3).

II. Converting between mathematical language, notation, and symbols, and standard

English language. Word problems often include language, mathematical notation or terminology, names, or phrases that are difficult to comprehend. There are three main components that can have an impact on understanding mathematics: (1) mathematical register, (2) language used in the classroom, and (3) technical communication (Galligan 20). A couple strategies to assist reading comprehension include substituting the first letter of the proper name, their own names, or the names of friends or family members and ignoring or replacing unfamiliar words with familiar words (Edwards 1-2). More importantly, it is essential that students are familiar with mathematical vocabulary. According to Edwards, “Math terminology in word problems is a comprehension as well as a decoding challenge...many math language terms have a counterintuitive, conversational meaning...teaching math vocabulary has been shown to improve student performance on math tests” (3). This can be accomplished by incorporating journal writing, mathematics portfolios, or cooperative learning activities. Journal writing was chosen for its most obvious benefit, of encouraging student to write and express their ideas more often in a mathematical frame. Cooperative Laming was chosen to encourage students to communicate verbally while problem solving with their peers. The portfolio was chosen to help students see growth in both their writing abilities as well as in their mathematical abilities (Hackett 32). Moreover, there are a number of processes that describe how mathematical terminology came about. It is helpful to know these processes because we can use them to cultivate understanding of mathematical discourse. Galligan describes some of the following processes: (1) Derivation

New words produced by affixes (prefixes, suffixes, and infixes) (e.g. cycloid, circumference, divisor, etc.). However, some derivations are not as intuitive as others. For instance, derivative is classified as a rate of change coming from the word “derive,” yet the calculation implies “differentiation.” The reason it appears in derivative is to look at the difference in quantities. (2) Coinage

The invention of a new word (e.g. googol). (3) Acronyms

The first initials of each word used to create a new word (e.g. PIN, QED, DNE, radar, sonar, etc.) (4) Back-formation

When new words develop from other words by reducing them (e.g. evaluate (evaluation), quantitate (quantitative), approximately (approximation), etc.). (5) Clipping

The abbreviation of words (e.g. math (mathematics), trig (trigonometry), sin (sine), graph (graphic formula), etc.). (6) Conversion

When the function of a word changes, without reduction. Often times from a noun to a verb and vice versa. (e.g. “graph the function” and “draw the graph of a function,” or “round the number” and “round number,” etc.) (7) Borrowing

Words that are borrowed from other languages. For example, algebra is borrowed from the Arabic word “al-jabr” meaning the reunion of borrowed parts. This process of introducing new words demonstrates the importance that some cultures place on certain words. (8) Compounding

The joining of two separate words to form a new word. For example, seventeen is a combination of the words seven and ten. Many words that are of Greek or Latin origin are also compounded in the English language (e.g. parabola comes from Greek para ‘alongside, nearby, right up to’ and -bola from the verb ballien ‘to cast, to throw’). This process is similar to derivation. Furthermore, some mathematical symbols and notation are frequently used to create statements from open sentences. This is often done by using a quantifier. The phrase “for every” (or its equivalents) is called a universal quantifier. The phrase “there exists” (or its equivalents) is called an existential quantifier. The symbol is used to denote a universal quantifier, and the∀ symbol is used to denote an existential quantifier (Sundrom 63). For example, consider the∃ following phrase: We would translate this into standard English as: “For all x∀x ε R)(x ).( 2 > 0 in the real numbers, is greater than zero” such that ‘ ’ translates to ‘in,’ ‘ R’ refers to the realx2 ε numbers, and ‘>’ means ‘greater than.’ To negate the statement using mathematical notation we can write: These are the first of many symbols used to(∀x ε R)(x ) ∃x ε R)(x ).¬ 2 > 0 ≡ ( 2 ≤ 0

consolidate standard English into mathematical notation. Likewise, compound statements are also commonly used in formal mathematical writing. A compound statement is a statement that contains one or more operators. Because some operators are used so frequently in logic and mathematics, we give them names and use special symbols to represent them (Sundrom 33). Here are just a few of the basic compound statements discussed on page 33 of Sundrom’s text:

1. The conjunction of the statements P and Q is the statement “ P and Q ” and its denoted by . The statement is true only when both P and Q are true.P ⋀ Q P ⋀ Q

2. The disjunction of the statements P and Q is the statement “ P or Q ” and its denoted by . The statement is true only when at least one of P or Q is true.P ⋁ Q P ⋁ Q

3. The negation (of a statement) of the statement P is the statement “not P” and is denoted by . The negation of P is true only when P is false, and is false only when P isP¬ P¬ true.

4. The implication or conditional is the statement “If P then Q ” and is denoted by P → Q . The statement is often read as “P implies Q, and is false only when PP → Q P → Q is true and Q is false.

III. Deducing the assumptions inherent in a given mathematical statement, expression or

definition. Questions written in compositional, not conversational English, i.e. difficult to read and comprehend. Mathematician George Polya’s (1973) classic problem solving framework offers a model for teaching students how to understand word problems:

Figure 1: Problem Solving Steps

Step 1 What kind of question is this?

• Identify the question type • Connect to already learned approaches

Step 2 What is the question asking for?

• Find the keywords

Step 3 What information am I given to solve the problem?

• Focus on relevant information • Organize information in a table or drawing

Step 4 How can I solve this problem?

• Use the information already given • See if the problem can be broken down into smaller steps • Eliminate obvious wrong answers

Step 5 Did I solve the problem?

• Decide if you solved what is being asked

(Edwards 2). According to Edwards, “In Polya’s framework, a problem solver first understands what type of problem is being posed, then clarifies what is being asked for, investigates the problem to see what information is already given, formulates a plan for solving the problem, and checks the computational work for any missteps or errors before finalizing an answer” (2). IV. Evaluating the precision or accuracy of a mathematical statement.

There are many ways in which people should be aware of their level of precision and accuracy when making mathematical statements. For instance, when writing mathematical arguments or statements, one should never refer to mathematics as “this,” “that thing,” “it,” etc. It is unclear to the reader what one could be referring to when they are not explicit in their statements. It’s also a good idea to read over one’s proofs or arguments in making sure that not only is one’s mathematics correct, but also one’s mathematical discourse. In order to do so, it’s necessary to know how the language of mathematics differs from ‘everyday’ English language and the language of other discipline areas: text level, sentence level and word level, as we have briefly discussed in the previous exemplars (Adoniou & Qing 5). Notice that there are multiple expressions in mathematics to express similar algorithmic functions e.g. subtract, take away, minus, take the difference of, etc. It is important to use these expressions appropriately and in everyday conversation to bridge the gap between mathematical vocabulary and colloquial vocabulary (Adoniou & Qing 9). For example, it is appropriate and worthwhile to say, “Wow, the probability of that outcome happening is very low and it still happened,” then to say, “It’s unbelievable that that happened! The laws of probability have been shifted just for me.” In a given situation, the probability of an event occurring is not altered by some “higher power” and so one must be careful with what they say especially to non-native speakers of English. This kind of colloquialism is confusing and contradictory. Probability is often an area of mathematics that people have trouble translating back and forth between mathematical notation and standard English due to it’s ambiguous discourse. Nonetheless, we must hold ourselves accountable for what we say and continue to say or write precisely what we are thinking so that we do not contradict ourselves or the mathematics.

Questions: note: Roman numerals (I, II, III, IV) indicate which exemplar is being demonstrated.

1. A student, Hannah, sees the fraction and is asked to reduce it to lowest terms. Shea−ba −b2 2

says, “Cancel the a on the bottom with the on the top, to get the a on the top, anda2 cancel the b on the bottom with the on the top to get the b on the top and finally, noteb2 that when you divide two negatives (the one on the top between and and the one ona2 b2 the bottom between a and b ), we get a positive. So, the answer is .” Now, we knowa + b the answer is , but what was done is nonsense. Respond to the following questions,a + b and then describe how asking Hannah these questions would help her see the flaws in her procedures.

a. (III) Would the same thing work for to get ? How can you check it?a+ba +b3 3

a2 + b2 b. (II, IV) When can you “cancel” a term in the numerator and a term in the

denominator? What are you really doing when you are “cancelling”? c. (III, IV) Can you “cancel” zeros? That is, ? Explain.0

0 = 1

2. Let P be the statement “It is raining,” Q be the statement “Matthew is reading a book,” and R be the statement “Janice is sleeping.”

a. (II) Translate the following mathematical notation into standard english language: i. Q )P → ( ⋁ R

ii. Q )P ⋁ ( ⋀ R iii. P )( ↔ R iv. ¬P )( v. (Q ) ¬P¬ ⋁ R →

b. (III) Assume Janice is not sleeping. What conclusions can we make about the above statements?

3. (I) The graph of the function is given below (January 2018, Algebra(x) ax x f = 2 + b + c

I Regents Exam).

Could the factors of be and ? Based on the graph, explain why or(x)f x )( + 2 x )( − 3 why not.

4. (II, IV) In 2005, the Pythagorean Theorem was a deciding factor in a case before the New York State Court of Appeals. A man named Robbins was convicted of selling drugs within 1,000 feet of a school. In the appeal, his lawyers argued that the man wasn’t actually within the required distance when caught and so should not get the stiffer penalty that the school proximity calls for. “The arrest occurred on the corner of Eighth Avenue and 40th Street in Manhattan. The nearest school, Holy Cross, is on 43rd Street between Eighth and Ninth Avenues. Law enforcement officials applied the Pythagorean Theorem to calculate the straight-line distance between the two points. They measured the distance up Eighth Avenue (764 feet) and the distance to the school along 43rd Street (490 feet). Using the data to find the length of the hypotenuse, (x) feet. Robbins’ lawyers contended that the school is more than 1,000 feet away from the arrest site because the shortest distance should be measured as a person would walk the route. However, the seven-member Court of Appeals unanimously upheld the conviction, asserting that the distance in such cases should be measured ‘as the crow flies.’” Explain why the lawyers argued the way they did.

5. (III) Consider the following statement: “If the thorbean is floopyfloop at a, then thorbean is dingydip at a, such that a is a real number.” Using only this true statement, is it possible to make a conclusion about thorbean in each of the following cases? Explain.

a. It is known that thorbean is floppyfloop at 0. b. It is known that thorbean is not floppyfloop at 35. c. It is known that thorbean is dingydip at .1717 d. It is known that thorbean is not dingydip at .π

6. (I, II, III) A machinist created a solid steel part for a wind turbine engine. The part has a

volume of 1015 cubic centimeters. Steel can be purchased for $0.29 per kilogram, and has a density of 7.95 . If the machinist makes 500 of these parts, what is the cost ofg

cm3 the steel, to the nearest dollar . Describe the steps you would have to do in order to calculate the cost of steel of 500 parts.

7. (II) (1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. (2) Write the negation of the statement in a symbolic form that does not use the negation symbol.

(3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

a. ∀a ε R)(a )( + 0 = 0 b. ∃x ε Q)(x x )( 2 − 3 − 7 = 0 c. ∃x ε R)(x )( 2 + 1 > 0 d. ∀n ε Z)(∃m ε Z)(m )( 2 > n

8. (II) An integer m is said to have the divides property provided that for all integers a and

b, if m divides ab, then m divides a or m divides b. a. Using the symbols for quantifiers, write what it means to say that the integer m

has the divides property. b. Using the symbols for quantifiers, write what it means to say that the integer m

does not have the divides property. c. Write an English sentence stating what it means to say that the integer m does not

have the divides property.

References

Adoniou, Misty, and Yi Qing. “Language, Mathematics and English Language Learners.”

Australian Mathematics Teacher , vol. 70, no. 3, Jan. 2014, pp. 3–13.

Edwards, Sharon A., et al. “Reading Coaching for Math Word Problems.” Literacy Coaching

Clearinghouse, Literacy Coaching Clearinghouse, 30 June 2009.

Galligan, Linda. “Creating Words in Mathematics.” Australian Mathematics Teacher , vol. 72,

no. 1, Jan. 2016, pp. 20–29.

Hackett, Kimberly, and Theresa Wilson. Improving Writing and Speaking Skills Using

Mathematical Language. 1 May 1995.

Sultan, Alan, and Alice F. Artzt. The Mathematics That Every Secondary School Math Teacher

Needs to Know . Routledge, 2018.

Sundrom, Ted. Mathematical Reasoning: Writing and Proof. Mar. 2019

Zhao, Manqian, and Karen Lapuk. “Supporting English Learners in the Math Classroom: Five

Useful Tools.” Mathematics Teacher , vol. 112, no. 4, Jan. 2019, pp. 288–293.