Algebraic Readiness Standards Topic 4 Operations on Rational Numbers N.S. 1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, terminating decimals) and take positive rational numbers to whole-number powers. N.S. 1.3 Convert fractions to decimals, and percents and use these representations in estimations, computatins, and applications.
Algebraic Reasoning and Algebraic Thinking What is Algebraic Thinking? How would you define it?
Algebraic Reasoning Shelley Kriegler (2001) Prof of Mathematics at UCLA separates algebraic thinking into two components: tools and ideas. Mathematical Thinking Tools: 1. Problem solving skills 2. Representation skills: express relationships in a variety of ways such as tables, graphs, and equations 3. Inductive and deductive reasoning skills: use valid reasoning to reach and justify conclusions Fundamental Algebraic ideas: 1. Algebra as generalized arithmetic 2. Algebra as a language 3. Functions and mathematical modeling
5 NCTM on Algebra… Many adults equate school algebra with symbol manipulation--solving complicated equations and simplifying algebraic expressions. …But algebra is more than moving symbols around. Students need to understand the concepts of algebra, the structures and principles that govern the manipulation of symbols and how the symbols themselves can be used for recording ideas and gaining insights into situations. (NCTM, 2000, p. 37)
6 What does this mean to a teacher of early grades? Elementary children are capable of this thinking but are not often given the opportunity to do so. Children can learn arithmetic in a way that provides a basis for learning algebra.
Discussing a + b – b = a Following is a discussion with a second grader as she attempts to justify the conjecture a + b - b = a. She has been in a class that spends quite a lot of time discussing conjectures and how you know whether a conjecture is true for all numbers. What elements of algebraic reasoning can you recognize? 7
Describing Susie’s Algebraic Reasoning The letters a and b have meaning. The way she carries out calculation suggests insight (b-b)–she computes “out of order.” She understands basic properties: additive inverses and additive identities. She recognizes the limitations of checking different numbers, but at the same time recognizes the value of checking different numbers in looking for numbers for which it may not be true.
Middle school students’ strategies for word problems Dana (7th grader) asked to solve this problem: A carpenter has a board 200 inches long and 12 inches wide. He makes 4 identical shelves and has a piece of board 36 inches long left over. How long is each shelf? She added 36 and 4 wrote 200x12 tried 2400-36 4 x 36 and subtracted that from 200, then subtracted 12 and got 44.
Middle school students’ strategies for word problems 1. Just add 2. Guess at the operation to be used 3. Look at the number sizes and use those to tell you which operation to use. 4. Try all operations and choose the most reasonable answer. 5. Look for “key words.” 6. Decide whether the answer should be larger or smaller than the given numbers, then decide on the operation. 7. Choose the operations with the meaning that fits the story.
How can we help students develop knowledge and ways of thinking in elementary school and middle school that support the learning of algebra?
Support for these tools and Ideas Mathematical Thinking Tools: 1. Problem solving skills 2. Representation skills: express relationships in a variety of ways such as tables, graphs, and equations 3. Inductive and deductive reasoning skills: use valid reasoning to reach and justify conclusions Fundamental Algebraic Ideas: 1. Algebra as generalized arithmetic 2. Algebra as a language 3. Functions and mathematical modeling
Quantitative Reasoning Requires a rich understanding of how quantities are related and how those relationships can be used to draw inferences, and how numerical values can be inferred from those given. Builds an understanding of the situation that will support and justify arithmetical reasoning. With such an understanding, it’s easy to decide (and justify) what calculations are needed.
Quantitative Reasoning 1) Draws heavily on everyday experience 2) State general relationships and make inferences from them 3) Quantitative relationships –especially complicated multiplicative ones–can be expressed in many ways 4) Requires and supports development of a classroom orientation towards making sense of situations (Thompson, Philip, Boyd, & Thompson, 1994)
Quantitative Reasoning Questions Can I imagine the situation as though acting it out? What do I know about the quantities involved? Do some change over time? Are some constant? What is the quantity of interest? What quantity am I asked to find or describe? How could making a diagram help reveal relationships? What quantities and values do I know that help me find the quantity of interest? What quantities and values can I derive?
Problem Solving My brother and I walk the same route to school every day. My brother takes 40 minutes to get to school and I take 30 minutes. Today, my brother left 8 minutes before I did. a. How long will it take me to catch up with him? b. Part of someone’s work on this problem included 1/30 and 1/40. What quantities do the two fractions represent? c. Suppose my brother’s head start is 5 minutes instead of 8 minutes. Now how long would it take me to catch up with him?