ADVANCING RESEARCH, IMPROVING EDUCATION |

## Transcript of Presentation Slides:
The presentation slides for The Trouble with Math is English are available online in PDF format.
## Title Slide: The Trouble With Math is English by Concepcion Molina, EdD, Program Associate ## Slide 2: Objectives- Examine how mathematics language and symbolism impact students’ understanding.
- Examine the merger of mathematics language and symbolism with difficulties in mathematics instruction.
## Slide 3: Connecting Mathematics and LanguageIn one word, state what comes to mind for most people when you say “mathematics.” ## Slide 4: Connecting Mathematics and LanguageImagine you are a sophomore taking the TAKS and you come upon problem number 14, below. Solve it and discuss with others at your table.
## 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: ## Slide 7:
Solve problem 2: ## 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 LanguageIt is not an ELL problem. (ELL students) It is an ALL problem. (ALL students) And ALL students are MLLs. ## Slide 11: Mathematics and LanguageThere 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: AmbiguityReduce: 6/8 Answer: 6/8 (displayed in a reduced font size) ## Slide 18: AmbiguityExpand: (x + 3) (x – 4)
Answer: ( x + 3 ) ( x - 4 ) ## Slide 19: AmbiguityFind 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: Ambiguity1) Which is larger? The numbers 35 and 5 are displayed, but the 5 is displayed in a much larger font. ## Slide 21: Ambiguity1) Which number is larger? The numbers 35 and 5 are displayed, but the 5 is displayed in a much larger font. ## Slide 22: Ambiguity1) Which numeral is larger? The numbers 35 and 5 are displayed, but the 5 is displayed in a much larger font. ## Slide 23: Ambiguity- Which is larger?
- Which number is larger?
- 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 IAdditional 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 IIConfusion — Pronunciation A) What are some ways to say 2632?
B) How can these two addresses be confused? ## Slide 26: Mas (more) Ambiguity IIIConfusion — Pronunciation
C) 2560 and $25.60
D) 25.6 ## Slide 27: Mas (more) Ambiguity IVA 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: MeaningsPart 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 ## Slide 30: Even More Ambiguity INumber - 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. . .
Example: -(-x) = x The Op-Op property ## Slide 32: Even More Ambiguity II
2) elementary or middle school definitions with new meanings in higher level mathematics.
3) more than one definition or meaning for the same math terms. ## Slide 33: Traditional Instruction and Language“Traditional” instruction is characterized by - teacher lecture,
- passive students,
- repeated drill and practice,
- memorization, and
- an emphasis on answers rather than explanations.
## Slide 34: Traditional Instruction and LanguageMany 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, 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 LanguageThis slide displays a graphic with the word "tradition" with a red line through it. ## Slide 37: Traditional Instruction and LanguageLanguage-based problems in mathematics instruction include the following: - Use of mathematical and non-mathematical meanings at the same time
- Use of spatial words when describing arithmetical operations
- Confused logic and mismatched symbolism
## Slide 38: Traditional Instruction and Language(Language-based problems, continued): - The use of “shortcuts”
- Teachers’ tendency to use “careless” vocabulary
- Dominant use of “naked” numbers
## Slide 39: Careless VocabularyTeachers 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 VocabularyConsider 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 VocabularyDiscuss with others at your table what is problematic with the following (for both ELL and ALL students!) - Give your little brother the “bigger half” of the cookie
- “Carry” the one
- “Borrow” a ten
## Slide 45: Careless VocabularyDiscuss with others at your table what is problematic with the following (for both ELL and ALL students!) - “Cancel” when simplifying a fraction or rational expression
- “Reduce” a fraction
- Two “goes into” eight
## Slide 46: Careless VocabularySolve 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” NumbersAbout 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 What will most students do to answer the question and why is that? ## Slide 49: Use of “Naked” NumbersConnecting 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 ## 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 ApplicationWhat 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 ApplicationWhat 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: Would that confuse you? Consider: 1 inch (on a map) = 20 miles (real life) ## Slide 54: Emphasize Both Symbolism and LanguageStudents’ Interpretation of Symbols
1) Do students see ⁵⁄₄ as 5 x ¼?
Do students see ⁵⁄₄ as 5
2) Do students see 1 and one-half as 1 + one-half? ## Slide 55: Ambiguity as an AssetHow can we use ambiguity as an asset in our classroom instruction rather than it being a liability? Examples: - One pig grew from 5 pounds to 10 pounds. Another grew from 100 to 108 pounds. Which pig grew more?
- Are a square and a rectangle similar?
End of presentation slides. Copyright ©2010 by SEDL. All rights reserved. No part of this presentation may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from SEDL (4700 Mueller Blvd., Austin, TX 78723), or by submitting an online copyright request form at http://www.sedl.org/about/copyright_request.html. Users may need to secure additional permissions from copyright holders whose work SEDL included after obtaining permission as noted to reproduce or adapt for this presentation. |