Maria Hernandez, NCSSM/Amplify learning. Angela Teachey, NCSSM. and

Transcription

1 Maria Hernandez, NCSSM/Amplify learning. Angela Teachey, NCSSM and

2

3

4 Instructions: Let f x = x + 1 2x. Calculate f 4. Student s Question: For the fraction part, do I plug in 4 and leave it in the denominator, or do I multiply 4 by 1 2? My response: That should be apparent from the notation. I can t answer your question. Student: It depends on who you ask.

5

6

7

8 CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. CCSS.Math.Content.5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 ( ) is three times as large as , without having to calculate the indicated sum or product.

9 CCSS.Math.Content.6.EE.A.2c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s 3 and A = 6 s 2 to find the volume and surface area of a cube with sides of length s = 1/2.

10 1. More complex grouping symbols? 2. Students struggle to recognize and understand structure in algebraic expressions? 3. Students must construct expressions with correct order of operations rather than just interpreting given expressions? 4. Distributive property a confounding factor in the transition to algebraic structures and functions?

11 Algebra: Seeing Structure in Expressions CCSS.Math.Content.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context. CCSS.Math.Content.HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. CCSS.Math.Content.HSA-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. CCSS.Math.Content.HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

12 Look for and make use of structure Mathematically proficient students look closely to discern a pattern or structure. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

13 1. More complex grouping symbols? New types of notation: radical signs exponentiation notation fraction bars absolute value symbols arguments of trigonometric and logarithmic expressions Memory aids like PEMDAS and Please Excuse My Dear Aunt Sally not easily transferable to all algebraic grouping symbols

14 unfortunately, the rules have serious drawbacks the criterion of addition before subtraction still does not allow us to solve correctly. Finally, use of a rule as complicated and arbitrary as these rules tend to become, involves far too much of the flavor of rote learning which has made arithmetic so unsavory in the past Source: Bender, M.. (1962). Order of Operations in Elementary Arithmetic.The Arithmetic Teacher, 9(5), pp

15 2. Students struggle to recognize and understand structure? Is there a cognitive gap when students transition from operating on numbers to operating on unknowns (variables)?

16 it seems that many teachers and textbook authors are unaware of the serious cognitive difficulties involved in the learning of algebra. As a result, many students do not have the time to construct a good intuitive basis for the ideas of algebra or to connect these with the prealgebraic ideas they have developed in [earlier coursework]; they fail to construct meaning for the new symbolism and are reduced to performing meaningless operations on symbols they do not understand. Herscovics, N., & Linchevski, L. (1994). A Cognitive Gap between Arithmetic and Algebra. Educational Studies in Mathematics, 27(1), pp

17 3. Students must construct expressions with correct order of operations rather than just interpreting given expressions? At the algebra, precalculus, and calculus levels, students must often construct expressions that convey the correct order of operations, requiring them to reverse their thinking. Particularly relevant when entering functions properly into graphing utilities and other computing software. Perhaps it is not wise to assume that this is always a straightforward transition.

18

19 An algebra student is using a calculator to investigate a horizontal compression, y = 1, on 2x the toolkit function y = 1 and is confused to see x that the transformed reciprocal function is looks linear.

20

21 4. Distributive property a confounding factor in the transition to algebraic structures and functions? Distributive property receives heavy emphasis in middle grades. CCSS.Math.Content.6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y) Students may transfer this knowledge inappropriately leading them to believe that other operations distribute over addition? x + y = x + y x + y = x + y log x + y = log x + log y sin x + y = sin x + sin (y)

22 many other teachers and researchers identify a culprit as responsible for student difficulties with expression transformation, namely a strong tendency for students to overgeneralize. Schwartzman (1986) provides examples of overgeneralization. In an article entitled The A of a B is the B of an A, he writes about how students have difficulty restraining themselves from overgeneralizing the notion of the distributivity of one operation over another. He provides a list of twenty non-equivalencies that students are prone to assume. These include a + b n a n + b n, , and n n n a+b a b a + b a + b. All of these examples are variations on true distributive statements, such as the power of a product is the product of the powers... However, they are overgeneralizations in that the student who accepts them is assuming fewer constraints on which operations distribute over which others than actually exist. Merlin, E. M. (2008). Beyond PEMDAS: Teaching Students to Perceive Algebraic Structure (Masters thesis). Retrieved from Dissertations and Theses database. (UMI No. 5636)

23

24 Structure and Grouping Symbols: How do we prioritize our work? Work from the inside out. Examples When computing an expression such as , the fraction bar has the same function as parentheses. Any operations that appear above or below a fraction bar should be completed first. Compute (6) 2

25 2+ 4 x 5 Students tend to solve expressions based on how the items are listed, in a left-to-right fashion, consistent with their cultural tradition of reading and writing English. Therefore, the rules underlying operation order actually contradict students natural way of thinking. Welder, R. (2006). Prerequisite Knowledge for the Learning of Algebra. Hawaii International Conference on Statistics, Mathematics, and Related Field.

26 However, Kieran suggests that if an equation such as 3 x 5 =15 were 3 x 3+2 =15, students would realize that bracket/parentheses usage is necessary to keep the equation balanced (Kieran, 1979). Welder, R. (2006). Prerequisite Knowledge for the Learning of Algebra. Hawaii International Conference on Statistics, Mathematics, and Related Field.

27 They found that students tended to over generalise the order, usually giving addition priority over subtraction; or using operations in left to right order; they can show lack of awareness of possible internal cancellations; they can see brackets as merely another way to write expressions rather than an instruction to act first, for example: and 926 ( ) yielded different answers (Nickson, 2000 p. 120); they also did not understand that signs were somehow attached to the following number review of research literature on how children learn mathematics, commissioned by the Nuffield Foundation

28 From Nephew s Homework:

29

30 Equality Equivalence Calculate Which of these do we mean?

31 Beginning algebra students tend to see the equal sign as a procedural marking that tells them to do something, or as a symbol that separates a problem from its answer, rather than a symbol of equivalence (Behr, Erlwanger, & Nichols, 1976, 1980). Even college calculus students have misconceptions about the true meaning of the equal sign (Clement, Narode, & Rosnick, 1981). Welder, R. (2006). Prerequisite Knowledge for the Learning of Algebra. Hawaii International Conference on Statistics, Mathematics, and Related Field.

32

33

34

35

36

37

38 Algebra Touch Mathination SymCalc HD (CAS) Other ideas for how to use CAS?

39 From Steve Leinwand s Accessible Mathematics many students need different modes of access to mathematical concepts to be successful. It should be clear that we will never attain the important and ambitious goals of algebra for all specifically, and mathematical power for all generally, until and unless we change how the material is taught.

40 Frequent imbedding of them mathematics skills in real-world situations and contexts Given F = 4(S-65)+10, find F when S = 81. OR The speeding fine in Vermont is $4 for every mile per hour over the 65 mph speed limit, plus a $10 handling fee. Then we can build algebraic understanding such as: What is the fine if you are caught going 81 mph? How fast must have you been going if the fine was $102? Create a graph that shows the relationship between the speed and the fine.

41 Learn and reflect Can help us to understand each other s perspective and help us to meet students where they are. Share ideas How do students learn Order of Operations in middle grades? How do middle grades teachers use alternatives representations geometric models, others?

42 How can HS teachers remind students to make sense of the algebraic notions based on their experience with operations of numbers? How can we help them see when they are overgeneralizing? How can we help students see the algebraic structure in their work?

43