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1 ELL and Secondary Mathematics Roadblocks and Land Mines

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2 “Traditional” instruction is usually characterized by –teacher lecture, –passive students, –repeated drill and practice, –memorization, and –emphasis on answers rather than explanations.

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3 Roadblocks and Land Mines Many educators experienced “traditional instruction” and tend to teach in the same way as they were taught (Anstrom, 1997; Parker, 1991; Lindquist, 1989; Linn & Herman, 1997; Stigler & Hiebert, 1999). This tendency to teach as we were taught includes the associated language and symbols. Consequently, we are perpetuating some of the language-based problems in mathematics instruction.

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4 Roadblocks and Land Mines These language-based problems in mathematics instruction include I) Use of mathematical and non-mathematical meanings at the same time, II) Use of spatial words when describing arithmetical operations, III) Teachers’ tendency to use “careless” vocabulary, and IV) Confused logic and mismatched symbolism.

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5 Roadblocks and Land Mines I) Use of mathematical and non-mathematical meanings at the same time. Ex: Take a situation where a student is shown several cars and told “there is one red one." 1

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6 Roadblocks and Land Mines II) We make matters more complicated when we use spatial words when describing arithmetical operations. Ex: adding up (where the numbers are arranged vertically and the answer is required at the bottom).

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7 Roadblocks and Land Mines III) Teachers’ tendency to use the same terms that were used when they learned mathematics. Some of it is “careless” vocabulary using terms that have other meanings in standard English.

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8 Roadblocks and Land Mines Discuss with others at your table what is problematic with the following (for both ELL and ALL students!) a) Give your little brother the “bigger half” of the cookie b) “Carry” the one c) “Borrow” a ten

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9 Roadblocks and Land Mines Discuss with others at your table what is problematic with the following: d) Two “goes into” eight e) “Reduce” a fraction f) “Cancel” when simplifying a fraction or rational expression

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10 Roadblocks and Land Mines “Cancel” and “Reduce” 6 = 3 x 2 = 3 8 4 x 2 4 Versus 6 = 3 x 2 = 3 x 2 = 3 x 1 = 3 8 4 x 2 4 2 4 4

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11 Roadblocks and Land Mines How could the use of “cancel” when simplifying a rational expression lead to this common secondary mathematics mistake? 3x + 5y = 3x + 5y= 5y 3x + 7y 3x + 7y 7y 6 = 5 + 1 = 5 + 1 = 1 9 5 + 4 5 + 4 4

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12 Roadblocks and Land Mines IV) Confusion logic and mismatched symbolism A) One objective in mathematics is to determine the reasonableness of a solution. Which form of the following two quantities is easier to estimate?

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13 Roadblocks and Land Mines The first was a “simplified” radical. The second was not in “simplest” form. So which was more “simple” and what should that mean to students in each case?

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14 Roadblocks and Land Mines B) Consider the expression 6 x 7. How is that usually pronounced? Is that problematic? If so, why?

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15 Roadblocks and Land Mines 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?

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16 Roadblocks and Land Mines 6 x 7 How about this as 6 sevens? (More on this to follow later!) What about 7 times 6? Why do we call it the times table?

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17 Related Language-Based Problems There are a multitude of other roadblocks and land mines that are directly tied to language. These are habits and practices that, in many cases, were part of our instruction in K–12 mathematics and are thus perpetuated by teachers, often times unconsciously.

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18 Related Language-Based Problems 1) Problem? Write the following out in words 3 < x < 8 ______________________________

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19 Related Language-Based Problems 2) Problem? Students assert there is no answer for the following: a) 2h = 13 b) If "y" and some other number add up to be 9, find the other number. __________

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20 Related Language-Based Problems 3) Problem? Let y = 3 5y = _______ Student fills in the blank with 53.

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21 Related Language-Based Problems 4) Problem? A) Solve: 24 4 x 2 B) Let y = 2 Solve mentally: 24 4y = _______

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22 Related Language-Based Problems 5) Problem? Solve mentally: 10 = 1/2 For the above, you got an answer. What was the question that you answered? *Explain. (*Do not use the word or any form of divide or goes into.)

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23 Roadblocks and Land Mines Research asserts that we tend to teach how we were taught. Part of how we were taught mathematics includes the language and terminology. Teachers must be mindful of how their use of mathematical terms and non-mathematical vocabulary impact student learning and understanding.

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