0738-7 ReThkAlgebra PW 07.13.11

0738-7 ReThkAlgebra_PW_07.13.11.inddRethinking Algebra Focus on the Content and the Mathematical Practices of the Common Core
Participant Workbook
0738-7 ReThkAlgebra_PW_07.13.11.indd 1 7/14/11 10:07 AM
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ISBN: 97814026-0738-7
Copyright © 2011 Pearson Education, Inc. or its affiliates. All rights reserved. Rethinking Algebra Participant Workbook 3
Contents
Introduction and Overview ................................................................................ 5 Rethinking Algebra: Focus on the Content and the Mathematical Practices of the Common Core ...................................................................................................... 5
Overview ....................................................................................................................... 5
Outcomes ..................................................................................................................... 5
Mathematical Proficiency ................................................................................ 11
Mathematics Goals and Student Profiles .........................................................................14
Kaput’s Five Forms of Algebraic Reasoning ...................................................... 17
Mathematical Practices: Part 1 ........................................................................ 21
Strategic Competence ..................................................................................... 23 Sharing Treats ................................................................................................................ 23
Procedural Fluency .......................................................................................... 25 Pool Border .................................................................................................................... 25
Comparisons and Interventions ...................................................................................... 26
Depth of Knowledge ....................................................................................................... 27
What is it about teaching that influences student learning? ..............................33
Productive Disposition .................................................................................... 35 Video Case Study .............................................................................................................35
4 Rethinking Algebra Participant Workbook Copyright © 2011 Pearson Education, Inc. or its affiliates. All rights reserved.
Contents
Standards for Mathematical Content ............................................................... 39
The Meaning and Uses of Variables ................................................................. 41 Number Trick ...................................................................................................................41
Multiple Scales ............................................................................................................... 42
Doing and Undoing ......................................................................................... 45 Solving Equations and Reversibility ................................................................................ 45
Building Rules to Represent Functions ............................................................. 47 Content Cards ................................................................................................................. 47
Cutting Paper ................................................................................................................. 49
The Number System and Algebraic Thinking ................................................................... 52
Algebra and the Mathematics Standards for High School ................................. 53 A Course Called Algebra ...................................................................................................53
Pizza Pirate .................................................................................................................... 54
Question Stem Cards .......................................................................................................61
Introduction and Overview
Rethinking Algebra: Focus on the Content and the Mathematical Practices of the Common Core
Overview
This comprehensive, two-day workshop examines key algebraic concepts from a mathematical point of view. It addresses the content demand that teachers be prepared to provide students with the opportunity to demonstrate the eight mathematical practices of the Common Core State Standards for Mathematics (CCSSM) in algebra. With a focus on differentiated and active engagement strategies, participants learn how they can successfully pave the way for increased student achievement in algebra by translating the following pedagogical content knowledge into classroom practice:
• Essential Understandings of Algebra
Outcomes
By the end of the workshop, participants will be able to
• identify ways to support the simultaneous development of the mathematical skills and expectations by focusing on these strands of student proficiency:
– Adaptive reasoning
– Strategic competence
– Procedural fluency
– Conceptual understanding
– Productive disposition
• connect instructional strategies for teaching algebra to the Standards for Mathematical Practice;
• identify instructional skills and strategies that facilitate understanding of a student’s capacity for using variables, doing and undoing, building rules to represent functions, and abstracting from computation; and
• outline teaching strategies that make mathematical relationships explicit in an intentional and public way.
Closing the Gap
Generate a list of the greatest challenges that you face, including content, in teaching algebra.
Agree on identifying the strengths and addressing the weaknesses of struggling students.
Pinpoint the strategies that best meet the needs of all students.
SECTION 1
• Conceptual understanding
• Procedural fluency
• Productive disposition
• Adaptive reasoning
• Strategic competence
Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification
Strategic competence: ability to formulate, represent, and solve mathematics problems.
Intertwined strands of proficiency
Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
Figure 2.13 Adding It Up describes five strands of mathematical proficiency (Van de Walle, Karp, and Bay-Williams 2010b, 25).
Mathematical Proficiency
Adaptive Reasoning
Choose the three sets that match. Be prepared to justify your selections.
Situation 1
The number of wolves in Yellowstone National Park from 1872 (when the park was established) to today
Note: For a long period of time, there were no wolves in the park.
Situation 2
A child’s internal body temperature from the time before she contracted the flu, through her illness, to the time when she was completely well
Situation 3
The amount of time your sister spent on her cell phone, beginning with the time before she met her boyfriend, through the time of their relationship, to the time when they broke up
Situation 4
The speed of a taxicab as it makes its way through the traffic in Manhattan
A
Adaptive Reasoning
Goals: What is most important for all students to learn?
Kris is a struggling reader who gets overwhelmed by a large amount of text. She is directionally challenged and embarrassed to ask for help. Kris does much better when her teacher presents directions and instructions visually. As a result, she often draws pictures to solve problems.
K
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D
S
Student’s Weaknesses: Lesson Barriers/Difficulties:
Student’s Strengths: Accessibility Strategies:
Ian is fidgety and distracted during whole-class instruction. Although he is not a good listener, he is able to articulate his own ideas, especially in small groups. In problem solving, Ian tends to make careless errors or omissions because he rushes through the process.
K
I
D
S
Adaptive Reasoning
DeAndre is unorganized, and he loses track of all sorts of things. He often gets confused or loses track in multistep problems and is unable to finish in the allotted amount of time. Despite his lack of organization, he is able to memorize math facts and vocabulary. Because he lacks social graces, his classmates do not like to work with him during small-group activities.
K
I
D
S
Student’s Weaknesses: Lesson Barriers/Difficulties:
Student’s Strengths: Accessibility Strategies:
Shondra is unable to make connections from one lesson to another. Because each problem looks new to her, she works slowly and carefully. She often asks the teacher for assurance or help. She pays attention to detail and is eager to share in discussion, but tends to talk about the steps she used without explaining why.
K
I
D
S
Kaput’s Five Forms of Algebraic Reasoning
1. Generalizations from arithmetic and from patterns in all of mathematics.
“Generalizations begin when students work on early number concepts like addition and place value. If [teachers] see the connection between number and generalizations, then they can better support students’ learning of both number and algebra. Generalizations at the upper elementary and middle [/high] school level include generalizing rules and writing them symbolically” (Van de Walle, Karp, and Bay-Williams 2010a, 97).
2. Meaningful use of symbols.
“Symbols, in particular the equal sign, are often poorly understood. In fact, a primary reason that students struggle in an algebra 1 course is thinking that the equal sign means ‘and the answer is,’ rather than representing equal quantities.
Extending the understanding of symbols is the understanding of variables. [Students] often think the variable is a placeholder for an unknown number, but variables can also stand for numbers that vary. In early grades and in middle school, students need experiences with both variables that represent an unknown and variables that vary” (Van de Walle, Karp, and Bay-Williams 2010a, 97).
Common Misconceptions2
There are 3 feet in a yard. Let f represent the number of feet, and y represent the number of yards.
Which of the following equations correctly shows this relationship?
3f = y or f = 3y?
(Van de Walle and Lovin 2006, 274–275)
3. Study of structure in the number system.
“Rather than memorize properties of numbers (e.g., the commutative property), students should learn properties by exploring structures of numbers within the number system. For example, when [students] note that 4 x 7=7 x 4, students ‘prove’ this relationship is always true” (Van de Walle, Karp, and Bay-Williams 2010a, 97).
4. Study of patterns and functions.
“[Students] need to see the developmental progression of generalizing patterns and functions across the grades. Extending, inventing, and observing patterns begins with repeating patterns in pre-K up through growing patterns that may be quadratic or exponential in middle school. Exploring patterns through the use of contexts and manipulatives is necessary in building an understanding of algebraic concepts. Linear equations and functions are particularly important, and [students] benefit from opportunities to explore linear patterns across [multiple representations]. Proportional relationships are a subset of linear relationships [and] it is important [for students to] see the connection between proportions and algebra” (Van de Walle, Karp, and Bay-Williams 2010a, 97–98).
+ O E
O
E
Complete the addition table. Are the results generated always true? How do you know?
Figure 14.11
Geometric growing patterns using manipulatives (Van de Walle, Karp, and Bay-Williams 2010b, 269).
(b) Color tiles
(a) Pattern blocks
(c) Multilink or centimeter cubes
Step 1 Step 2 Step 3
Step 1 Step 2 Step 3
5. Process of mathematical modeling, integrating the first four list items.
“Kaput defines modeling as the process of beginning with real phenomena and attempting to mathematize them. Mathematical models, or equations, are used to predict other phenomena” (Van de Walle, Karp, and Bay-Williams 2010b, 276).
“Mathematical modeling is often confused with physical models. A physical model is a manipulative, like colored tiles, used to build a concrete representation of a situation. A mathematical model is the equation that represents the situation and can be used to solve real questions, like how much gas will be left after traveling x miles” (Van de Walle, Karp, and Bay-Williams 2010a, 97–98).
Mathematical Modeling of a Taxi Fare
Suppose every day I take the same taxi over the same route from the ______ to ______________. Think about the function notation needed to accurately reflect the following stories which represent the amount paid to the taxi driver. What would x and f(x) represent?
A. I received a door prize at the session so I gave my driver a five-dollar tip.
B. I had a new driver and he got lost. He drove five extra miles and charged me for it.
C. My fare was billed to the hotel for the five-day work week. Today is Friday, and I will pay what I owe.
D. The meter must have been “achy-breaky.” It showed five times the number of miles I actually traveled.
Record the notation below.
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
6. Attend to precision.
8. Look for and express regularity in repeated reasoning.
(Common Core State Standards Initiative 2010, 10)
•
•
•
•
•
Strategic Competence
Sharing Treats
Last Saturday your sister had a birthday party and there were some bags of treats left over. Your mom wants you to pass them out to the kids in the neighborhood so that she can get rid of the extras. You need to visit three houses to drop off the bags. At the first house you stop at, you give them one- half of the bags of treats plus two more. At the second house, you give them one-half of the bags of treats you had left after the first house plus two more. At the third house you stop at, you do the same thing and give them one half of the bags you have left plus two more. When you get home you have two bags left, one for you and one for your sister. How many bags did you start with?
Procedural Fluency
Pool Border
How many different ways can you find to count the border tiles of an 8 x 8 pool without counting them one at a time?
Figure 14.3 (Van de Walle, Karp, and Bay- Williams 2010b, 257)
Procedural Fluency
2 formats
1 caution
Conceptual Understanding
Depth of Knowledge (DOK) is a scale of cognitive demand measuring the knowledge level and complexity required by standards and assessments.
Level One Activities
Recall elements and details of story structure, such as sequence of events, character, plot and setting.
Conduct basic mathematical calculations.
Represent in words or diagrams a scientific concept or relationship.
Perform routine procedures like measuring length or using punctuation marks correctly.
Describe the features of a place or people.
Level Two Activities Identify and summarize the major events in a narrative.
Use context cues to identify the meaning of unfamiliar words.
Solve routine multiple-step problems.
Identify patterns in events or behavior.
Formulate a routine problem given data and conditions.
Organize, represent and interpret data.
Level Three Activities Support ideas with details and examples.
Use voice appropriate to the purpose and audience.
Identify research questions and design investigations for a scientific problem.
Develop a scientific model for a complex situation.
Determine the author’s purpose and describe how it affects the interpretation of a reading selection.
Apply a concept in other contexts.
Level Four Activities Conduct a project that requires specifying a problem, designing and conducting an experiment, analyzing its data, and reporting results/ solutions.
Apply mathematical model to illuminate a problem or situation.
Analyze and synthesize information from multiple sources.
Describe and illustrate how common themes are found across texts from different cultures.
Design a mathematical model to inform and solve a practical or abstract situation.
Level Two (Skill/ Concept)
Describe Explain
Hypothesize
Investigate
Revise
Apply Concepts
Depth of Knowledge (DOK) Levels
Webb, Norman L. and others. “Web Alignment Tool” 24 July 2005. Wisconsin Center of Educational Research. University of Wisconsin-Madison. 2 Feb. 2006. <http://www.wcer.wisc.edu/WAT/index.aspx>.
Depth of Knowledge
• •
• •
Bloom’s Taxonomy
What level of thinking requires one to understand? How does the recent transition of Bloom’s taxonomy to the use of verbs (versus nouns) alter your interpretation of the taxonomy?
Bloom’s Revised Taxonomy
Tile Problem
Your summer job is to repair tiles in the corner of roofs. The tiles you will use are 1 foot by 1 foot squares. Jobs are identified as follows:
Draw jobs #3 and #6 on your graph paper.
a. What do you need to know to draw the figures for job #3 and #6?
#2 #4 #5
(Pearson Education, Inc. 2008, B-25)
After laying the tiles, you need to apply some waterproof caulking around the outside of the pattern. The caulking is sold by the foot. Job #2 requires 8 feet of caulking. Job #5 needs 20 feet of caulking.
b. When the boss assigns you to do job #12, how many tiles and how much caulk will you need to complete the job? Explain how you figured this out.
c. Write a rule that would allow you to detrmine how many tiles and how much caulk you would need for any job. Can you identify any pattrn between the two rules?
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
6. Attend to precision.
8. Look for and express regularity in repeated reasoning.
(Common Core State Standards Initiative 2010, 10)
•
•
•
Read the following article about influences on student learning (quoted in Pearson Education, Inc. 2008, A-10–A-11).
In their literature review about the effects of classroom mathematics teaching on student learning, James Hiebert and Douglas Grouws (2007) pose the question: “What is it, exactly, about teaching that influences students’ learning?” To examine this question, they analyzed research studies of classroom teaching to identify the kinds of teaching that are empirically linked to two critical components of mathematical proficiency: skill efficiency and conceptual understanding.
The authors’ findings point to several key features that facilitate skill efficiency (fast and accurate execution of mathematical procedures). Evidence suggests that skills and procedures are best taught in a fast-paced setting that includes teacher modeling, teacher-directed questions that require a specific correct answer, and substantial time for students to practice.
Pinning down the answer to what influences students’ conceptual development was a more complex task, in part, because of the difficulty of measuring conceptual understanding and disentangling the interactive components of a complex classroom environment. For this reason, Hiebert and Grouws only examined studies where the teaching was described in some detail, enabling them to look carefully at what was really happening during a particular lesson.
In the multiple studies examined by Hiebert and Grouws, evidence revealed that conceptual understanding occurs in a wide range of teaching methodologies and contexts. Some classrooms were characterized as teacher- centered and others student-centered. Some lessons involved group work; others did not. Some relied heavily on teacher demonstrations, while others highlighted class discussions. In other words, the answer to the question, “What is it about teaching that influences student learning?” cannot be answered by pointing to a single best pedagogical approach. However, Hiebert and Grouws found that evidence does suggest a strong link between student conceptual understanding and two key features of teaching—features that can be present in a variety of methodologies and settings. These features are making mathematical relationships explicit and allowing students to struggle with important mathematics.
Making Mathematical Relationships Explicit
What does it mean to make mathematical relationships explicit? Hiebert and Grouws explain that this type of teaching treats mathematical relationships between ideas, facts, and procedures in an explicit and public way. It includes
What is it about teaching that influences student learning?
such things as reminding students of the main point of the lesson and how it connects to previous and future learning, discussing the mathematical meaning that underlies a procedure, questioning students about how solution strategies are similar to or different from each other, or discussing the ways in which mathematical concepts build on each other. The process of making mathematical relationships explicit can occur in a variety of contexts. Conceptual knowledge can be developed through highly structured teacher- centered instruction. It can also occur within looser, more student-centered approaches. In other words, what matters is the regular opportunity for teachers and students to pay attention to—in an intentional and public way— the mathematical relationships between facts, ideas, and procedures.
Allowing Students to Struggle with Important Mathematics
Evidence also suggests a strong link between student understanding and another feature of teaching—the engagement of students in struggling with important mathematical ideas. Hiebert and Grouws use the word struggle to mean that students are required to think hard to make sense of mathematics and to expend effort wrestling with something that is not immediately apparent. This type of struggle occurs when, for example, students are asked to solve problems that are within reach but require them to grapple with key ideas. Struggle does not mean that students are given tasks that are nonsensical or overly difficult. It does not imply a level of challenge that leads to despair and frustration. Rather, struggle, as used in the context of promoting student learning, must be designed as a doable challenge where students are clear about what they need to figure out and can proceed with some confidence that they have the tools to help them succeed.
Struggle allows confusion and doubt to surface, setting the stage for students to restructure their mental processes and make sense of the mathematical ideas in a new way. Teaching that successfully incorporates student struggle by using struggle time—whether it be just 5–10 minutes or a more significant portion of the class period—to give students the opportunity to fit ideas together to make sense of a challenging problem or idea. In addition, struggle connects back to the first feature—making mathematical relationships explicit—by mentally preparing students to engage in meaningful discussions and generate important questions about key mathematical ideas.
Interestingly, Hiebert and Grouws also noted that in addition to developing conceptual understanding, teaching that attended to explicit development of mathematical relationships and allowed students to wrestle with important mathematics also facilitated procedural fluency—skill efficiency plus the ability to use skills flexibly and with meaning.
What is it about teaching that influences student learning?
Productive Disposition
Indicator Example from Video
The task is designed to support the opportunity for students to struggle with important mathematics.
The lesson requires students to see sense in mathematics and to perceive it as both useful and worthwhile.
Classroom interactions support the belief that a steady effort in learning mathematics pays off.
Regular opportunity exists for the teacher and students to pay attention to — in an intentional and public way — the mathematical relationships between facts, ideas and procedures.
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