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Engage in the Mathematical Practices: Strategies to Build Numeracy and Literacy With K-5 Learners

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Increase student learning with engaging lesson plans and high-level tasks. In this user-friendly guide, mathematics teachers will discover more than 40 strategies for ensuring students learn critical reasoning skills and retain understanding. Each chapter is devoted to a different Standard for Mathematical Practice and offers an in-depth look at why the standard is important for students’ understanding of mathematics.

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17 Chapters

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Chapter 1 Standard for Mathematical Practice 1: Make Sense of Problems and Persevere in Solving Them

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It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. . . . Every new discovery in mathematics results from an attempt to solve some problem.

—HOWARD EVES

When Sarah’s oldest son was in third grade and working on homework, he asked, “Mom, can you please help me with this one?” while pointing to a word problem. Naturally, she requested he read the problem aloud. He quickly responded, “Mom, I don’t need to read it. These are all multiplication word problems. Every problem has two numbers; I just don’t know how to multiply these two numbers together.” Sure enough, after reading the problems posed, Sarah saw that he was right.

It struck her that her son’s teacher thought students were reading the word problems to make sense of the problem, but students knew every question had two numbers, always factors, and they simply needed to grab the two numbers (without reading) to multiply them.

 

Chapter 2 Standard for Mathematical Practice 2: Reason Abstractly and Quantitatively

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Creating classroom opportunities for developing higher-order thinking is essential for helping students become critical thinkers, problem solvers, innovators, and change makers upon which our society thrives.

—PÉRSIDA HIMMELE & WILLIAM HIMMELE

Kit’s brother Mike shared the following incident when his fourth-grade daughter came home from school with questions on her division homework. Specifically, she was working on 456 divided by 6. Kit’s brother’s conversation with his daughter went something like the following.

Mike: Tell me what you know.

Daughter: Well, my teacher said that I could break apart 456.

Mike: What do you mean, “break apart”?

Daughter: I can break it into smaller numbers that make 456.

Mike: This problem is division. Are you dividing when you break numbers apart?

Daughter: I think so, but I can’t remember.

 

Chapter 3 Standard for Mathematical Practice 3: Construct Viable Arguments and Critique the Reasoning of Others

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Higher-order thinking thrives on interaction.

—PÉRSIDA HIMMELE & WILLIAM HIMMELE

Subitizing, the ability to quickly determine a small number of objects without having to count, enables young students to connect number names with quantities and begin to develop their sense of number, which leads to an understanding of addition and subtraction. Students practice subitizing when they quickly state the number shown using different configurations of ten frames, dots on a number cube, tallies, or base-ten blocks, to name a few. The following scenario illustrates the power of subitizing.

While working in a first-grade classroom, we used two ten frames to help students practice finding sums within 20. The frames appeared on the SMART Board as quick images. Students looked at the images (figure 3.1) and then shared with a partner how many dots they saw and how they saw them. The images remained on the screen for only two or three seconds so students would use strategies other than counting.

 

Chapter 4 Standard for Mathematical Practice 4: Model With Mathematics

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Mathematics is a performance, a living act, a way of interpreting the world.

—JO BOALER

How many students understand that mathematics happens outside of the designated mathematics block of time each day? How do students use mathematics to make sense of their world by quantifying? For example, when are students telling time or determining how much time is left until lunch? How do they figure out how many carrot sticks they need for a party if each student gets two? How can they find the difference between the number of students who walk to school and the number who ride the bus? In other words, how do students experience the relevance of mathematics?

When working in a second-grade classroom, we overheard a student say, “I hate math. I’m never going to use this.” Not twenty minutes later, the student asked a friend if he would share his apple slices with her. The friend replied, “That means I would only get three slices instead of eight.” She quickly corrected him and let him know they would each get four slices to eat. This seemed acceptable to both students, and the apples were shared. Unknowingly, they had used mathematics to solve a real-world problem.

 

Chapter 5 Standard for Mathematical Practice 5: Use Appropriate Tools Strategically

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The ability to strategically select and use tools means a student is able to decide when and how to use it and, most of all, when not to.

—CATHY SEELEY

While entering an elementary school for parent conferences, a couple of bins near the door struck Sarah’s eye. A sign over the enormous tubs read “Free Resources—Please Take.” No further directions were provided, and parents were left to wonder why or how the resources would benefit their children’s learning.

When peering into one bin, Sarah saw many three-dimensional geometric models, some fraction bars, ten frames with plastic discs, and spinners. Many of the objects, covered in a thin layer of dust, looked relatively new and unused. Why were these mathematical tools sitting in the hallway? Why were they offered free to the community? Were these extras?

It turns out the tools in the bins were simply discovered in the back of closets and classrooms that were recently renovated and cleaned. Though teachers did not have similar tools in their classrooms, they were willing to let these go. This led Sarah to wonder:

 

Chapter 6 Standard for Mathematical Practice 6: Attend to Precision

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Language is as important to learning mathematics as
it is to learning to read
.

—NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

“How many minutes old are you?” a fifth-grade teacher asks her students. The question seems easy enough until students begin discussing what they need to know in order to answer this question. Working in small groups, students begin identifying some facts that they need, such as the number of days in the year, the number of hours in a day, the number of minutes in each hour, and their exact date and time of birth.

In the fourth-grade classroom down the hall, students are given a different question: “Thirteen candy bars are to be shared with seven friends. What is the size of each friend’s share of the candy bars?” Two students in the class have the following dialogue.

Alfredo: Let’s use a calculator. We just divide . . . 7 ÷ 13.

Alicia: Don’t you mean 13 ÷ 7? We need 7 shares.

 

Chapter 7 Standard for Mathematical Practice 7: Look for and Make Use of Structure

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Getting it happens when students engage their minds in mathematical activity and connect what they are now learning to what they already know or to emerging structures in their minds.

—CATHY SEELEY

While working in an elementary school, we happened to observe two lessons related to structure that showcased the need for Mathematical Practice 7, “Look for and make use of structure.” In a first-grade classroom, we observed a teacher working with students to understand the commutative property of addition. Through investigation and repeated focused problems, students began to notice a pattern and articulate that the sum will be the same regardless of which number they write first in the expression. For example, 1 + 3 will be equal to 3 + 1. As this unfolded and students built on their own understanding of addition to make sense of the commutative property, the teacher acknowledged the structure and then proceeded to call it the flip-flop property because the numbers can be flip-flopped.

 

Chapter 8 Standard for Mathematical Practice 8: Look for and Express Regularity in Repeated Reasoning

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To grow mathematically, children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.

—LYNN ARTHUR STEEN

Recently, Kit had an opportunity to work with Courtney, a second-grade student. Courtney was asked to find the sum of 43 + 29. Figure 8.1 illustrates how her teacher presented the problem emphasizing the use of the traditional algorithm.

Figure 8.1: Place value example.

Courtney was stuck in the “counting on” strategy for each place value. She used a hundreds chart and started at 9 and then counted 3 more to find the sum of the numbers in the ones column. She had not considered any patterns that might make finding totals easier. Kit began by asking her to find the sum of 10 + 2. She looked at her hundreds chart and counted two more from ten. Kit asked her what she noticed about the sum 10 + 2 = 12 and recorded the following number sentences as Courtney found other sums.

 

Epilogue: What Do I Do With These Strategies?

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In an excellent mathematics program, educators hold themselves and their colleagues accountable for the mathematical success of every student and for personal and collective professional growth toward effective teaching and learning of mathematics.

— NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

Throughout this book, you have explored strategies that promote student learning of both the content standards and the Standards for Mathematical Practice. With the emphasis on the Mathematical Practices, students learn to make sense of problems and persevere in solving them (using a variety of strategies purposefully), reason abstractly and quantitatively, and construct and critique viable arguments. Additionally, students learn to look for and use patterns, tools, and models to explain relevant situations, all while being precise with language and notation. With these critical reasoning habits of mind, students become true problem solvers and effective users of mathematics.

 

Appendix A: Alternate Table of Contents—Literacy Connections

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Appendix B: Alternate Table of Contents—Strategies Building Numeracy and Literacy

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Appendix C: Standards for Mathematical Practice—Background

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The Standards for Mathematical Practice were based on two transformative resources, Principles and Standards for School Mathematics (NCTM, 2000) and Adding It Up: Helping Children Learn Mathematics (Kilpatrick, Swafford, & Findell, 2001). Principles and Standards for School Mathematics contributed significantly to the teaching-and-learning-mathematics landscape. NCTM expanded and extended the original 1989 standards to not only include content standards but also process standards. These process standards answered how students were to learn, whereas the content standards described what K–12 students should learn. The process standards included five key elements.

1.Problem solving

2.Reasoning and proof

3.Communications

4.Connections

5.Representations

These process standards actively engage students in solving problems that enable them to build new knowledge. Through the problem-solving process, students reason, verify their solutions, communicate their ideas, make connections, and use multiple representations of their solutions.

 

Appendix D: Standards for Mathematical Practice

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Source: NGA & CCSSO, 2010, pp. 6–8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

 

Appendix E: CCSS for Mathematics Grades K–5

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The following table shows the domains and clusters of the content students in grades K–5 must learn in the Common Core State Standards for Mathematics. The Mathematical Practices cannot be taught without content standards. This table shows the content addressed throughout this text when giving strategies and examples for teaching the Mathematical Practices as habits of mind for students to develop.

Table E.1: Grades K–5 Domains and Clusters

Source: NGA & CCSSO, 2010.

 

Appendix F: The Task-Analysis Guide

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The following table delineates tasks featured throughout this book as lower- or higher-level-cognitive-demand mathematical tasks.

Table F.1: Cognitive Demand Levels of Mathematical Tasks

Lower-Level Cognitive Demand

Higher-Level Cognitive Demand

Memorization Tasks

•These tasks involve reproducing previously learned facts, rules, formulae, or definitions to memory.

•They cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use the procedure.

•They are not ambiguous; such tasks involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated.

•They have no connection to the concepts or meaning that underlie the facts, rules, formulae, or definitions being learned or reproduced.

Procedures With Connections Tasks

 

Appendix G: Sources for Higher-Level-Cognitive-Demand Tasks

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Source: Kanold, 2015a. Used with permission.

Common Core Conversation

www.commoncoreconversation.com/math-resources.html

Common Core Conversation is a collection of more than fifty free website resources for the Common Core State Standards in mathematics and ELA.

EngageNY Mathematics

www.engageny.org/mathematics

The site features curriculum modules from the state of New York that include sample assessment tasks, deep resources, and exemplars for grades preK–12.

Howard County Public School System Secondary Mathematics Common Core

https://secondarymathcommoncore.wikispaces.hcpss.org

This site is a sample wiki for a district K–12 mathematics curriculum.

Illustrative Mathematics

www.illustrativemathematics.org

The main goal of this project is to provide guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience upon implementation of the Common Core State Standards for mathematics.

 

Appendix H: Mathematics Vocabulary and Notation Precision

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Table H.1 shows common imprecise classroom language with suggestions for precise language (Mathematical Practice 6).

Table H.1: Common Imprecise Language

Imprecise Terms

Precise Terms

Cancel and canceling

Remove common factors or divide if multiplication

Communative or flip-flop property

Commutative (commute means move)

Plug in

Substitute

Using amount and number interchangeably

Number is countable; amount is not countable, only measurable.

Using fewer and less interchangeably

Fewer is number; less is amount.

Using singulars and plurals interchangeably

Radius is singular and radii is plural; die is singular and dice is plural; parenthesis is singular and parentheses is plural, and rhombus is singular while rhombi or rhombuses are plural.

 

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