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Transcript of Presentation Slides:
The Trouble With Math is English
by Concepcion Molina

Screenshot of the presentation title slide.
The presentation slides for The Trouble with Math is English are available online in PDF format.

The Trouble with Math is English was presented by Concepcion Molina at the Conference for the Advancement of Mathematics Teaching, San Antonio, TX, in July, 2010.

Title Slide: The Trouble With Math is English

by Concepcion Molina, EdD, Program Associate
como.molina@sedl.org
July 16, 2010
©2010 SEDL

Slide 2: Objectives

  1. Examine how mathematics language and symbolism impact students’ understanding.
  2. Examine the merger of mathematics language and symbolism with difficulties in mathematics instruction.

Slide 3: Connecting Mathematics and Language

In one word, state what comes to mind for most people when you say “mathematics.”

Slide 4: Connecting Mathematics and Language

Imagine you are a sophomore taking the TAKS and you come upon problem number 14, below. Solve it and discuss with others at your table.

14) Find the ugloft of a bipkad if the rexnuza is 20.

Slide 5: Find the ugloft of a bipkad if the rexnuza is 20.

You look on the information sheet of the TAKS test and see the following:

  • ugloft = area
  • bipkad = circle
  • rexnuza = diameter

Now solve the problem. What made the difference?

Slide 6: Connecting Mathematics and Language

Solve problem 1:
4
∑ 2n-1
n=1

Slide 7:

Solve problem 2:
X = 1 + 3 + 5 + 7

Slide 8:

What is your Reaction/Interpretation to the following?

“I kill cats and eat them.”

Slide 9:

“I kill cats and eat them.”

A photo shows a fisherman with a number of catfish that were caught for food.

Slide 10: Mathematics and Language

It is not an ELL problem. (ELL students)

It is an ALL problem. (ALL students)

And ALL students are MLLs.

Slide 11: Mathematics and Language

There are many factors that contribute to language being problematic in mathematics:

  • Total isolation of mathematics and language arts
  • Learning the vocabulary is part of the learning
  • Difficulty of the English language itself

Slide 12: Mathematics and Language (Factors, continued)

  • Mathematical terms are rarely used outside of the mathematics classroom
  • Abstract nature of many mathematical terms
  • Ambiguity

Slide 13: Difficulty of the English Language

  • Extra
  • Ordinary
  • Extraordinary

Slide 14: Difficulty of the English Language

  • Logarithm
  • Algorithm
  • Paradigm

Slide 15: Difficulty of the English Language

  • Hot water heater
  • Water heater
  • 30 – second exercise
  • 32nd exercise

Slide 16: Difficulty of the English Language

(Whose idea was this?)

Two images are displayed: a pair of pants and a pair of socks.

Slide 17: Ambiguity

Reduce: 6/8

Answer: 6/8 (displayed in a reduced font size)

Slide 18: Ambiguity

Expand: (x + 3) (x – 4)

Answer: (  x      +     3  )    (  x      -      4  )
The text on the answer has been expanded by adding more space between the numbers and operators.

Slide 19: Ambiguity

Find x.

A picture of a right triangle is displayed, showing the numbers 6 and 8 along the two short sides of the triangle and the letter X along the longest side of the triangle. An arrow points to the x with the text, "Here it is!"

Slide 20: Ambiguity

1) Which is larger?

The numbers 35 and 5 are displayed, but the 5 is displayed in a much larger font.

Slide 21: Ambiguity

1) Which number is larger?

The numbers 35 and 5 are displayed, but the 5 is displayed in a much larger font.

Slide 22: Ambiguity

1) Which numeral is larger?

The numbers 35 and 5 are displayed, but the 5 is displayed in a much larger font.

Slide 23: Ambiguity

  1. Which is larger?
  2. Which number is larger?
  3. Which numeral is larger?

This is an example of how language details are important and how ambiguity exists within mathematics.

Slide 24: Mas (more) Ambiguity I

Additional Confusion

“Number” can still be about mathematics, but not refer to a quantity (e.g., number as a name, code, or location).

For instance, in a phone number such as 512-476-6861 or in a street address such as 30218 Main Street.

Slide 25: Mas (more) Ambiguity II

Confusion — Pronunciation

A) What are some ways to say 2632?

B) How can these two addresses be confused?
3218 Main Street and 30218 Main Street

Slide 26: Mas (more) Ambiguity III

Confusion — Pronunciation

C) 2560 and $25.60
How are these often pronounced and what is problematic about it?

D) 25.6
How is this often pronounced and what is problematic about it?

Slide 27: Mas (more) Ambiguity IV

A big idea in mathematics can have several terms associated with it that must also be learned. There may be added complexity if there is ambiguity or confusion with the true meaning of those additional terms.

Example: Number

Related terms: numeral, amount, place value, digit, and quantity.

Slide 28: Meanings

Part of the difficulties occur because of polysemous words whose multiple meanings can cause confusion.

Polysemous words can have multiple meanings within mathematics or have one meaning in mathematics and another in standard English.

Slide 29: Even More Ambiguity!

There is also the problem of mathematical terms having other meanings outside of mathematics. Students (and teachers) need to decipher the meanings of words in mathematics from the meaning in standard English.

Examples: Degree
Translation

Slide 30: Even More Ambiguity I

Number

  • The opening number was the highlight of the show.
  • His last girlfriend really did a number on him.
  • My finger is number than it was five minutes ago

Slide 31: Even More Ambiguity II

If that wasn’t enough, we make mathematics language more difficult because of. . .
1) “fluid” mathematical terms — invent “new” names or change meanings in mathematics terminology.

Example: -(-x) = x The Op-Op property
Commutative property = “Order property”
“FOIL” Method
New definition of “average”
Trapezoid

Slide 32: Even More Ambiguity II

2) elementary or middle school definitions with new meanings in higher level mathematics.
Example: Exponent ( 7½)

3) more than one definition or meaning for the same math terms.
Examples: ounce (weight or volume?)
inverse (function, operation, multiplicative and additive, matrix, variation, proportion, trig. function, etc.)

Slide 33: Traditional Instruction and Language

“Traditional” instruction is characterized by

  1. teacher lecture,
  2. passive students,
  3. repeated drill and practice,
  4. memorization, and
  5. an emphasis on answers rather than explanations.

Slide 34: Traditional Instruction and Language

Many educators experienced “traditional instruction” and studies indicate that teachers tend to teach in the same way as they were taught.

What would this tendency to teach as we were taught include?

Slide 35: Traditional Instruction and Language

This tendency to teach as we were taught includes the associated language and symbols. In addition, we not only tend to teach HOW we were taught, but also WHAT we were taught.

Traditional instruction pays little attention to mathematics language and symbolism and their impact on learning. In many ways, traditional instruction not only perpetuates, but also adds to the language-based problems in mathematics instruction.

Slide 36: Traditional Instruction and Language

This slide displays a graphic with the word "tradition" with a red line through it.

Slide 37: Traditional Instruction and Language

Language-based problems in mathematics instruction include the following:

  1. Use of mathematical and non-mathematical meanings at the same time
  2. Use of spatial words when describing arithmetical operations
  3. Confused logic and mismatched symbolism

Slide 38: Traditional Instruction and Language

(Language-based problems, continued):

  1. The use of “shortcuts”
  2. Teachers’ tendency to use “careless” vocabulary
  3. Dominant use of “naked” numbers

Slide 39: Careless Vocabulary

Teachers have a tendency to use the same terms that were used when they learned mathematics. Some of it is “careless” vocabulary. This often involves using terms that have other meanings in standard English.

Slide 40: Careless Vocabulary

Consider this expression: 6 x 7

How is that usually pronounced? Is that problematic? If so, why?

Slide 41: Careless Vocabulary

  • 6 x 7 is pronounced as 6 times 7.
  • In most contexts this represents 6 groups of 7, which is actually 7 + 7 + 7 + 7 + 7 + 7
  • So shouldn’t this really be described as 7 six times, as opposed to 6 times 7?

Slide 42: Careless Vocabulary

  • How about this as 6 sevens?
  • Would it make more sense for students to see 6y as six “y’s” instead of 6 times y?
  • (Why do we even call it the times table?)

Slide 43: Careless Vocabulary

  • Makes sense for 6 x 7 to mean 6 sets of 7.
  • Students can make the algebraic connection to an expression: 3(y + 5).
  • 3(y + 5) is 3 sets of (y + 5), or
    (y + 5) + (y + 5) + (y + 5)
  • Eliminates the common mistake that 3(y + 5) = 3y + 5

Slide 44: Careless Vocabulary

Discuss with others at your table what is problematic with the following (for both ELL and ALL students!)

  1. Give your little brother the “bigger half” of the cookie
  2. “Carry” the one
  3. “Borrow” a ten

Slide 45: Careless Vocabulary

Discuss with others at your table what is problematic with the following (for both ELL and ALL students!)

  1. “Cancel” when simplifying a fraction or rational expression
  2. “Reduce” a fraction
  3. Two “goes into” eight

Slide 46: Careless Vocabulary

Solve mentally: Ten divided by one-half = ?

How much sense does this question make: “How many times does ½ go into 10?”

Slide 47: Use of “Naked” Numbers

About this problem……

Solve mentally: Ten divided by one-half = ?

Did we leave the answer as JUST 20, or did we clarify that the answer was 20 halves?

Slide 48: Use of “Naked” Numbers

Given: 25 x 28 and 24 x 28
How much more is the first product?

What will most students do to answer the question and why is that?

Slide 49: Use of “Naked” Numbers

Connecting back to 6 x 7 … taking this beyond “how to” multiply—

Students are not usually taught to see numerals also as words, which would enable them to interpret the above as 6 sevens.

25 cows is 1 more cow than 24 cows.
25 twenty-eights is 1 more twenty-eight than 24 twenty-eights.

Slide 50: Language, Symbols, and Instruction

“The difference between the right word and the almost-right word is like the difference between lightning and lightning bug.” A quote by Mark Twain.

Part of teachers’ content knowledge is an awareness of how their use of mathematical terms, symbolism, and non-mathematical vocabulary impact student learning and understanding.

Slide 51: Classroom Application

What can we do to teach MATHEMATICS (not arithmetic and efficiency) to ALL students?

  • To the extent possible, use concrete examples or manipulatives
  • Emphasize both symbolism and academic language.
  • Organize thinking with graphic organizers
  • Use ambiguity to your advantage, not a disadvantage.

Slide 52: Classroom Application

What can we do to teach MATHEMATICS (not arithmetic and efficiency) to ALL students?

  • Simplify, yet deepen.
  • Use the deep knowledge in one topic to make connections and to leverage the learning of related topics.
  • Teach mathematics as relationships.

Slide 53: Emphasize Both Symbolism and Language

A map indicates the following scale:
1 inch = 20 miles

Would that confuse you?

Consider: 1 inch (on a map) = 20 miles (real life)

Slide 54: Emphasize Both Symbolism and Language

Students’ Interpretation of Symbols

1) Do students see ⁵⁄₄ as 5 x ¼? Do students see ⁵⁄₄ as 5 one-fourths? Why or why not?

2) Do students see 1 and one-half as 1 + one-half?
Do students see that 9 = 1 and one-half x 6 means that 9 is 1 and one-half sixes (1 six and half of another six)?

Slide 55: Ambiguity as an Asset

How can we use ambiguity as an asset in our classroom instruction rather than it being a liability?

Examples:

  1. One pig grew from 5 pounds to 10 pounds. Another grew from 100 to 108 pounds. Which pig grew more?
  2. Are a square and a rectangle similar?

End of presentation slides.

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