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Build Fluency through Mathematical Reasoning and Problem-solving while Breaking Down Language Barriers
for All Students – Especially English Learners
Dean Ballard Director of Mathematics
Presented by
www.Corelearn.com
The Ground We Will Cover
• The specific challenges EL students face in math
• Proven research-based strategies for addressing those challenges
• Models for applying specific techniques effectively in the classroom
EL Student Population Data – USA 2013-14: 9.3% (4.5 million) 2011-12: 9.2% (4.4 million) 2003-04: 8.8% (4.2 million) Calif: 22.7% (now at 22.1%)
From NCES (National Center for Educational Statistics)
Challenges for ELs to Overcome • Limited background knowledge
– Language, mathematics?
• Cultural differences – Pair/group work – Symbols, measurements
• Linguistics – Math vocabulary – Words-Symbols-concepts – Structures – passive voice – Complex phrases – "measure of central tendency"
—California Math Framework, Universal Access, 2015
Challenges for ELs to Overcome • Polysemous words
– Put dinner on the table. – Put data in the table.
• Homonyms – Sense, cents, since; left vs. right, left over
• Syntax and Semantics – "A square has volume of 20 cm."
• Translating words into symbols – "x is 5 less than the number y" à x = 5 – y;
x = y – 5 California Math Framework, Universal Access, 2015
Challenging Syntactic Features • Long dense noun phrases
– The volume of a rectangular prism with sides 8, 10, and 12 cm
• Classifying adjectives that precede the noun – Prime number, rectangular prism
• Qualifiers that come after the noun – A number which can be divided by one and itself
• Conjunctions: – If, when, therefore, given, assume
—Schleppegrell, 2007
Identify Challenging Words Double Meanings: table fraction even base rational tangent side irrational variable point operation volume mean expression etc.
Homophones: cent à sent or scent plane à plain two à to or too sum à some sine à sign
Multiple Terms for Same Idea: • altitude, height or length • add, sum • solve, answer, compute • justify, explain, prove
Small Words or Phrases: or fewer less than many then increase and of decrease any all left
Unique Terms: hypotenuse, parallelogram, coefficient, quadratic
Similar Sounding Words: tens vs. tenths then vs. than sixty vs. sixteen
Recommendations from Research 1. Focus on students’ mathematical reasoning, not accuracy in
using language. (important semantics not syntax)
2. Focus on mathematical discourse practices, not language as words, or grammar.
3. Recognize the complexity of language in math classrooms.
4. Treat everyday language as a resource, not as an obstacle.
5. Uncover the mathematics in what students say and do.
—Moschkovich, 2012
Specific Strategies & Techniques
• Make vocabulary and concepts explicit and visual.
• Make good use of repetition.
• Use scaffolds such as sentence frames, partnering, talking in native language first, etc.
• Use engaging activities.
• Create, manage, and process opportunities for students to talk about the math with each other.
Create Equations with the Digits 1–9 from Spend Some Time with 1 to 9 (2014)
Create as many equations as you can with the digits 1 – 9: • Use some or all of the digits in each equation.
8 ÷ 4 = 5 – 3
• Do not use any digit more than once within any equation.
non-example: 8 ÷ 4 + 3 = 7 – 3 +1
• Do not use the digit zero. non-example: 16 ÷ 2 = 40 ÷ (8 – 3)
• You may use any math operation. Add, subtract, multiply, divide, exponents, etc.
6 × 7 = 42 23 = 8 7 × 5 + 8 – (6 + 1) = 29 + 3 + 4
Specific Strategies & Techniques
• Make vocabulary and concepts explicit and visual.
• Make good use of repetition.
• Use scaffolds such as sentence frames, partnering, talking in native language first, etc.
• Use engaging activities.
• Create, manage, and process opportunities for students to talk about the math with each other.
Create Equations with the Digits 1–9 from Spend Some Time with 1 to 9 (2014)
Create as many equations as you can with the digits 1 – 9: • Use some or all of the digits in each equation. • Do not use any digit more than once within any equation. • Do not use the digit zero. • You may use any math operation.
Dean's workspace: 56-43=12+(9-8) 65-43-8=12+(9-7)
Spend Some Radical Time with 1 to 9 {2, 4, 6, 8}
Place any of the digits from the set above into the blank boxes in each inequality shown to the right to make the statement true.
For example, below we have used 2, 6, and 8 to make a true statement:
• Do not use a digit more than once in a statement.
• Do not use a calculator.
2 5 3 8 6 7< <
a. b. c.
5 3 7< <
3 9 7< <
7 3 5< <4 5 3 2 8 7< <2 5 3 6 6 7< <2 5 3 8 6 7< <
38 67 38 67, ,therefore< <
36 6 38 6; therefore= >
4 5 9 2 5 3; therefore< < < <
2 2 2 5 2 3, ( ) ( )So < <
4 2 5 6< <
2 5 38<
Place any of the digits from the set above into the blank boxes in each inequality shown to the right to make the statement true.
Spend Some Radical Time with 1 to 9 from Spend Some Time with 1 to 9 (2014)
{3, 5, 7, 9} Place any of the digits from the set above into the blank boxes in each inequality shown to the right to make the statement true.
For example, below we have used 3, 5, and 7 to make a true statement:
• Do not use a digit more than once in a statement.
• Do not use a calculator.
3 2 4 5 7 8< <
a. b. c.
2 4 8< <
4 2 8< <
8 4 2< <
Best Deal
• Individually: Guess which is the better deal. Record.
• Pair-share: Share guess and reasoning.
• Individually: Solve the problem. Show your work.
• Pair-share: – Explain what you would have to pay in each sale. – Compare similarities/differences between the two sales. – Share your solution and justification.
One store is having a 50% off sale. Another store has a 40% discount, with an additional 15% off of the sale price. Which sale should you take advantage of if you want the best reduction on a sweater that costs $68.79?
• I think the 50% discount is _________ than the 40% plus an added 15% discount off of the sale price because _________.
• Turn to a partner and share your statement.
Turn and Talk with Sentence Frame
Sentence Stems and Frames
• Specific sentence frames: A. The store with the best reduction is _____ because _____.
B. The sales at the two stores are similar because they both ______.
C. The sales at the two stores are different because ______.
D. Comparing the sales at the two stores is tricky because ______.
Frayer Model (Frayer, Frederick, & Klausmeier, 1969)
Definition (in own words) Part of a whole. One number divided by another number and written as one number over the other number.
Facts/Characteristics
The denominator tells you how many parts the whole is split into. The numerator tells you how many of the parts you are taking.
Examples
2 7 5 4
Non-examples
5.12 50%
0
Fraction
Agree/Disagree – Why?
Michael says the graph of the function y = 2x2 intersects the y-axis but not the x-axis.
Janice says the graph of the function y = 2x2 intersects both the y-axis and the x-axis.
• Who do you agree with? Why?
• Turn to a partner and discuss for 2 minutes. Be ready to explain your reasoning.
Sorting & Matching Activities
• Sort cards or objects into groups based on common properties or characteristics.
• Students identify the rationale for how cards/concepts are sorted or matched.
• Each student keeps his/her own record.
Symbol and Cue Card Matching Match Symbol cards (yellow) with Cue cards (salmon)
8 – n n less than 8 The difference between 8 and n
8 < n
8 is less than n The difference
between n and 8 n – 8
n less than 8
n is less than 8
n is greater than 8
8 is greater than n
Guess My Polygon Activity
Questions • Yes/No questions only • Must be related to shape characteristics
Density of Text
Mathematics is the most difficult content area material to read because there are more concepts per word, per sentence, and per paragraph than in any other subject; the mixture of words, numerals, letters, symbols, and graphics requires the reader to shift from one type of vocabulary to another.
—Braselton & Decker, 1994
Math Text Reading
• Preview vocabulary
• Preview examples
• Connect examples – describe what changes
• Anticipation/Prediction Guides
Anticipation Guide for Unit Rates Directions: Under Me write T or F next to each statement based on your current thinking. After reading the text, under Text write T or F for your final opinion based on what you have learned from the reading. Be ready to explain how the text proves your final answer is correct.
Me Text Anticipation Statements 1. A rate is a ratio. 2. All rates are unit rates. 3. All unit rates are rates. 4. Unit rates can be written as fractions or decimals. 5. Equal rates are like equal fractions.
6. Unit rates are a good way to compare prices.
Anticipation/Prediction Guide
Word Problem Strategies
Plan “Read to Understand the Word Problem”
1. Pick a word problem to be assigned to students.
2. Make a second version of this word problem that is exactly the same in every respect except the numbers. Make sure your numbers work.
• Read the last sentence first. • Read to understand rather than solve.
Wrap Up - What techniques were used to minimize or address language challenges? • Provide clear directions and instruction.
• Use and connect visual, oral, and written instruction and directions.
• Be very clear and repetitious with key vocabulary.
• Focus some instruction on vocabulary – Frayer Models and other charts.
• Use scaffolds such as sentence frames, partnering, allowing students to talk and translate for each other in their native language, etc.
Wrap Up - What techniques were used to minimize or address language challenges? • Focus on the mathematical meaning students are
communicating about not the syntax
• Utilize cultural differences as opportunities for discussing and comparing meanings and interpretations
• Recognize the complexity of language in math classrooms
• Use engaging activities
• Require writing along with oral communication.
• Create, manage, and process opportunities for students talking about the math with each other.
Let’s Connect! Dean Ballard
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