Teaching Progressions

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I have gathered my research-based teaching ideas into visual and audio Teaching Progressions that show the research-based visual supports that enable children to build math ideas. I discuss how these ideas develop through the year and through the grades. I organized these Teaching Progressions by math topic using the topics in the Common Core State Standards (CCSS) and explain the progression of the standards for each math topic. I show the teaching approaches I developed in classrooms in the Children’s Math Worlds Research Project and then used in Math Expressions. These Teaching Progressions can help people using any program understand more deeply how visual supports support math teaching and learning and can enable viewers to see the progressions of math teaching and learning in action. My development of visual supports was supported by the National Science Foundation and so anyone can use these supports.

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The Math Talk Community overviews what a Math Talk Community is and why is it central to Math Expressions and to the Common Core State Standards. The Solve and Discuss classroom structure is explained, and the four components of the Math Talk Community are described. How the teacher and students assist learning by all is summarized, and how the CCSS 8 Mathematical Practices are used in the Math Talk Community is overviewed. Five short videos of Math Talk are shown near the beginning. This part takes about 22 minutes with the videos.

Getting Started with Math Talk draws on tips from experienced Math Expressions teachers. This part goes more deeply into details of how to build and lead a Math Talk Community. Three short videos of Learning Math Talk are shown near the beginning. This part takes about 14 minutes with the videos.

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Overview Part 1 is a general overview of what the Common Core State Standards are and why they were needed. Some of their central features are identified, and how these Common Core State Standards relate to Math Expressions is summarized. This part could be used with parents or teachers or administrators. Viewers should watch Part 1 first. This part takes 14 minutes.

Overview Part 2 provides a quick overview of each of the CCSS math domains and how Math Expressions supports each domain. Teachers and administrators could watch all of these in succession or focus on particular domains. More in-depth looks at the domains are provided in other Teaching Progressions described below.

Slide 1 is a cover page and slide 2 lists the CCSS math domains. [37 seconds; starts at 0:00]

Slides 3-10: OA: Operations and Algebraic Thinking [9 minutes: starts at 0:37]

Slides 11-18: NBT: Number and Operations in Base Ten [10 minutes; starts at 9:31]

Slides 19-24: NF: Number and Operations– Fractions [6 minutes; starts at 19:10]

Slides 25-31: MD: Measurement and Data and G: Geometry [4 minutes; starts at 25:14]

Slide 32 gives the website where future information will appear. [16 seconds; starts at 29:40]

Part 2 takes 30 minutes.

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OA Part 1 Problem Situations and Problem Solving describes the range of problem situations for different operations and how problem solving varies across these problem situations. The OA Common Core State Standards identify situational meanings of operations for addition/subtraction and for multiplication/division. Three situations are specified for each pair of these operations. Within a given situation, any of the three quantities can be the unknown. Students make math drawings to show a problem situation. They also learn the research-based diagrams used in Math Expressions to support problem solving. These diagrams allow students to generalize situational understandings to larger numbers, fractions, decimals and establish relationships across all of these situations and quantities. Part 1 describes the OA learning path for problem solving from Kindergarten through Grade 5 for single-step problems using any kinds of quantities. All teachers should see this part to understand problem solving in OA and in Math Expressions. This part takes about an hour because of the complexities involved in the problem types.

OA Part 2 The Grades K to 2 Learning Paths for Addition and Subtraction focuses on the Kindergarten, Grade 1, and Grade 2 learning paths for single-digit addition and subtraction calculations. Each operation has levels of increasing abbreviation, abstraction, and internalization reflected in three levels of solution methods specified in the OA standards for Kindergarten, Grade 1, and Grade 2. These levels come from research done all over the world and are used by children everywhere.

Slides 1-6: Overview the levels in the learning path [11 minutes; starts at 0:00]

Slides 7-14: Kindergarten activities and learning supports for the key Kindergarten CC and OA standards [20 minutes; starts at 11:12]

Slides 15-28: Grade 1 activities and learning supports for the key Grade 1 OA standards [25 minutes; starts at 31:17]

Slides 28-29: Grade 2 activities for fluently adding and subtracting within 20 [3 minutes; starts at 55:16]

Slides 30-31: Wrapping up Part 2 [2 minutes; starts at 57:56]

Part 2 takes about an hour. Kindergarten and Grade 1 teachers should watch all of it.

Grade 2 teachers should watch at least from slide 15 on to understand the Level 2 and Level 3 solution methods that will be brought to fluency in Grade 2.

OA Part 3 The Grade 3 Learning Path for Multiplication and Division focuses on the learning path in Grade 3 for single-digit multiplication and division computation. This learning path begins in Grade 3, and students are to come to fluency by the end of Grade 3. The Math Expressions supports for this learning path and for reaching fluency are summarized.

Part 3 takes about 20 minutes.

Grade 3 teachers should watch this Part 3, and it would be helpful also for Grade 4 teachers.

OA Part 4 Properties and Advanced Problem Solving focuses on properties and advanced problem solving in OA. How the commutative, associative, and distributive properties help students with single-digit computations and with problem solving are discussed. Different meanings of the equal sign for different problem situations are described, and the support within Math Expressions for different equations forms is overviewed. Representing and solving two-step and multistep problems from Grades 2 to 4 is overviewed, as is the small CCSS focus on patterns and expressions in Grades 3 to 5. Some aspects of Part 4 are helpful to teachers in each grade. Part 4 takes about 20 minutes.

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NBT Part 1: Overview of the Numbers and Operations in the Base Ten Domain (NBT) overviews the Math Expressions supports for place value and the approaches within the Common Core State Standards and Math Expressions to multidigit addition, subtraction, multiplication, and division. Math drawings that support students to understand and explain written methods are summarized, and acceptable variations in writing the standard algorithm for fluency are overviewed. Teachers at all grade levels should watch this overview. [53 minutes]

Slides 1-6: Introduction [3 minutes; starts at 0:00]

Slides 7-17: Overview of Place Value Concepts and Supports [10 minutes; starts at 2:38]

Slides 18-21: Math Talk Community and Math Practices [3 minutes; starts at 12:33]

Slides 22-28: Meaningful Development of Standard Algorithms in the CCSS [9 minutes; starts at 15:53]

Slides 29-41: Overview of Math Expressions Multidigit Addition and Subtraction Methods

[12 minutes; starts at 24:20]

Slides 42-46: Overview of Math Expressions Multidigit Multiplication and Division Methods

[10 minutes: starts at 36:25]

Slides 47-50: Summary [6 minutes; starts at 46:51]

NBT Part 2: Place Value and Multidigit Addition and Subtraction in Grades K to 4 focuses on place value and methods of multidigit addition and subtraction in Grades K to Grade 4. The whole learning path of visual supports for understanding place value is discussed. Math drawings and accessible methods of adding and subtracting are related and explained. Teachers from K to Grade 4 should watch and discuss certain parts and would benefit from all parts. [75 minutes]

Slides 1-7: Overview of NBT K, G1, G2, G3, G4 teachers watch these [7 minutes; starts at 0:00]

Slides 8 to 32: Kindergarten K and G1 teachers watch these [15 minutes; starts at 6:23]

[8-18: Count to 100 place value, daily routines, quick practice; 19-32: Tens in teen numbers]

Slides 33-59: Grade 1 K, G1, and G2 teachers watch these [21 minutes; starts at 22:51]

[33-40: G1 Count to 120 place value, daily routines, quick practice;

41-59: G1 adding and comparing within 100; starts at 28:33]

Slides 60-80: Grade 2 G1, G2, G3, and G4 teachers watch these [22 minutes; starts at 44:13]

[60-71: G2 place value and adding within 1000; 72-80: G2 subtracting within 1000]

Slides 81-90: Grades 3 and 4 G2, G3, and G4 teachers watch these [8 minutes; starts at 1:05:53]

[81-83: G3; 84-90: G4]

Slides 91 and 92: Point of Use Teaching Background All teachers read this [starts at 1:14:06]

NBT Part 3: Place Value and Multidigit Multiplication and Division in Grades 3 to 6 focuses on place value and methods of multidigit multiplication and division in Grades 3 to 6. Math drawings and accessible methods of multiplying and dividing are related and explained. Teachers from K to Grade 4 should watch and discuss certain parts and would benefit from all parts. [62 minutes]

Slides 1-3: Overview of Part 3 K, G1, G2, G3, G4 teachers watch these [1 minute; starts at 0:00]

Slide 4: Grade 3 G3, G4, and G5 teachers watch this [2 minutes; starts at 1:16]

Slides 5-38: Grade 4 G3, G4, and G5 teachers watch these [42 minutes; starts at 3:03]

Slides 39-54: Grade 5 G4, G5, and G6 teachers watch these [15 minutes; starts at 45:18]

Slide 55: Grade 6 G4, G5, and G6 teachers watch this [1 minute; starts at 59:51]

Slides 56 and 57: Point of Use Teaching Background All teachers read this [1 minute; starts at 1:00:30]

NBT Part 4: Place Value and Decimal Operations in Grade 4 focuses on how Grade 4 students build meanings for decimals as a place value notation for fractions with denominators of 10 and 100. They use concepts of money to understand, read, and write tenths and hundredths as dimes and pennies with the support of secret-code cards that layer to show decimal values. Students relate fraction and decimal notation on number-line diagrams. They use all of these visual supports to reason about the size of decimal values when comparing them, and practice and discuss the decimal values in order to overcome the common error of looking at decimal numbers as whole numbers and not considering their place values (e.g., overcoming the error 0.07 > 0.3 because 7 > 3). The Math Expressions place value poster helps them consider and discuss the symmetry around the ones place instead of around the decimal point and draw conclusions about where they can write a 0 in a number without changing the value (before a whole number and after a decimal number). [26:47 minutes]

NBT Part 5: Place Value and Decimal Meanings, Addition/Subtraction, and Rounding in Grade 5 focuses on how Grade 5 students extend and deepen connections between fraction and decimal concepts and notations including thousandths using money on secret-code cards and metric lengths. They extend their methods for adding and subtracting whole numbers to adding and subtracting decimal numbers by relating money values to a written method and explaining the method using place value and properties of operations. They develop, discuss, and use efficient, accurate, and generalizable methods by generalizing the place value relationships in both directions. They extend rounding to decimal numbers and use a rounding frame of writing the numbers to the nearest tenth or hundredth or thousandth just below and just above the number to see easily which is closer. [23:14 minutes]

NBT Part 6: Place Value and Decimal Multiplication/Division in Grade 5 focuses on how Grade 5 students extend their methods for multiplying and dividing whole numbers to multiplying and dividing decimal numbers by relating money values to a written method and explaining the method using place value and properties of operations. They examine and generalize patterns in shift rules that tell in what direction and how many places a decimal or whole number factor or dividend moves when multiplied or divided by 10, 100, and 1000 and by 0.1, 0.01, and 0.001. They develop, discuss, and use efficient, accurate, and generalizable methods by using the shift rules and creating and justifying patterns about the number of decimal places in the answer. Students express patterns using expanded and exponential notation. They discuss how multiplication as scaling extends from fractions to decimal numbers and explain why multiplying by a decimal less than 1 results in a product smaller than the given number. And they practice identifying word problem situations as multiplying or dividing situations even though the results of multiplying by a decimal number less than 1 is opposite to the result for whole numbers (multiplying results in a smaller number and dividing results in a larger number). [48:33 minutes]

NBT Part 7: Place Value and Decimal Operations in Grade 6 focuses on how Grade 6 students review Grade 5 shift rules and patterns in decimal operations by using the Math Expressions poster showing money for each place value. Students overview all operations with fractions to relate them to the new general division of fractions and become fluent with operations on fractions before extending these operations in Grade 7 (see Fractions NF Part 6 for more details). Students also relate operations with decimals to operations with fractions to understand how the meanings are alike (e.g., you add like units) but the notations create different methods (you find a new equivalent unit in different ways). This enriches and deepens student understanding of number systems and helps students fluently add, subtract, multiply, and divide with decimals. [26:37 minutes]

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NF Part 1: The Grade 3 Learning Path for Fractions overviews Grade 3 concepts of fractions as numbers composed of unit fractions 1/d, finding simple equivalent fractions, and comparing for fractions with like numerators or like denominators. Math drawings that support students to understand and explain are shown. Research has identified errors commonly made by students with fractions. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Grade 3 to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [23:45 minutes]

NF Part 2: The Grade 4 Learning Path for Fractions overviews general fraction equivalence and general comparing fractions, adding and subtracting like denominators, and whole number times a fraction. Math drawings that support students to understand and explain are shown. Research has identified errors commonly made by students with fractions. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Grade 3 to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [32:03 minutes]

NF Part 3: The Grade 5 Learning Path for Fractions Everything Except Division overviews general addition and subtraction (find equivalent fractions if needed), general multiplication of fractions, and multiplication as scaling (resizing). Research has identified errors commonly made by students with fractions. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Grade 3 to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [44:19 minutes]

NF Part 4: The Grade 5 and Grade 6 Learning Path for Fraction Division overviews Grade 5 division of whole numbers results in a fraction, Grade 5 division with unit fractions, and Grade 6 general division of fractions. Research has identified errors commonly made by students with fractions. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Grade 3 to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [33:16 minutes]

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RP Part 1 Unit 1 Grade 6 overviews the approaches within the Common Core State Standards and Math Expressions to understanding and explaining ratio and proportion problem solving. The CCSS specify a relatively new approach extending from multiplication and division and the multiplication table summarized in the Grade 6 Critical Area 1: “Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates.” This approach draws on research of various kinds including research in the Children’s Math Worlds Project that was the basis for Math Expressions. [64 minutes]

RP Part 2 Unit 7 Grade 6 helps students relate and contrast ratios and fractions concepts and notations. Students expand their visual supports and proportion solution methods and use these flexibly to represent and solve more difficult ratio situations including proportion word problems, percentage problems, and converting measurement units. [48 minutes]

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MD Part 1 Geometric Measurement Kindergarten to Grade 2: Length overviews how Math Expressions supports the Common Core State Standards learning path on length from Kindergarten to Grade 2. Seeing and counting length units and comparing lengths are the major emphases. All of these are foundational for all kinds of comparison problems. Understanding length units is necessary for the work on area and volume in the following grades.

The MD standards have three major aspects: geometric measurement (Parts 1 to 5), other measures (Parts 6 and 7), and represent and interpret data (Part 8).

Geometric measurement is a lovely progression from K to G2 length (Part 1) to G3 and G4 length and area (Part 2) to G5 and G6 area (Part 3) to G6 surface area (Part 4) to G5 and G6 volume (Part 5). Length units build up square units and cubic units. The angle measure concepts in Grade 4 are discussed in the Geometry Teaching Progression.

Other measures include the familiar measures money and time (Part 6) and metric and customary liquid volume and mass/weight (Part 7).

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [27:59 minutes]

MD Part 2 Geometric Measurement Grade 3 and Grade 4: Length and Area overviews how Math Expressions supports the Common Core State Standards on length and area in Grade 3 as area is developed as filling with square units and related to multiplication and differentiated from the length units used in perimeter. In Grade 4 these ideas are consolidated into formulas for area and perimeter developed and used by students.

The MD standards have three major aspects: geometric measurement (Parts 1 to 5), other measures (Parts 6 and 7), and represent and interpret data (Part 8).

Geometric measurement is a lovely progression from K to G2 length (Part 1) to G3 and G4 length and area (Part 2) to G5 and G6 area (Part 3) to G6 surface area (Part 4) to G5 and G6 volume (Part 5). Length units build up square units and cubic units. The angle measure concepts in Grade 4 are discussed in the Geometry Teaching Progression.

Other measures include the familiar measures money and time (Part 6) and metric and customary liquid volume and mass/weight (Part 7).

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [31:56 minutes]

MD Part 3 Geometric Measurement Grade 5 and Grade 6: Area overviews how Math Expressions supports the Common Core State Standards on area in Grades 5 and 6. Grade 5 students multiply fractional side lengths to find areas of rectangles. Grade 6 students engage in a fascinating learning path to find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes. They apply these techniques in the context of solving real-world and mathematical problems.

The MD standards have three major aspects: geometric measurement (Parts 1 to 5), other measures (Parts 6 and 7), and represent and interpret data (Part 8).

Geometric measurement is a lovely progression from K to G2 length (Part 1) to G3 and G4 length and area (Part 2) to G5 and G6 area (Part 3) to G6 surface area (Part 4) to G5 and G6 volume (Part 5). Length units build up square units and cubic units. The angle measure concepts in Grade 4 are discussed in the Geometry Teaching Progression.

Other measures include the familiar measures money and time (Part 6) and metric and customary liquid volume and mass/weight (Part 7).

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [48:24 minutes]

MD Part 4 Geometric Measurement Grade 6: Surface Area overviews how Math Expressions supports the Common Core State Standards in Grade 6 to represent 3D figures using 2D nets made up of rectangles and triangles and use the nets to find the 2D surface area of these 3D figures. Students apply these techniques in the context of solving real-world and mathematical problems. Students draw on the knowledge they developed about the area of various shapes to find surface area.

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [22:15 minutes]

MD Part 5 Geometric Measurement Grade 5 and Grade 6: Volume overviews how Math Expressions supports the Common Core State Standards in Grade 5 to find the volume of a right rectangular prism with whole number edge lengths by packing it with unit cubes of the appropriate edge lengths and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Students understand and use the formulas V = l w h and V = b h to find volumes of right rectangular prisms in the context of solving real-world and mathematical problems. Grade 6 students extend these ideas to right rectangular prisms with fractional edge lengths.

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [31:06 minutes]

MD Part 6 Other Measurement Grade 1 to Grade 5: Money and Time overviews how Math Expressions supports the Common Core State Standards on money and time. Money is used to support base-ten concepts in Kindergarten and Grade 1, developed intensively in Grade 2, and used in Grade 4 to support decimal and fraction ideas. Time is developed in Grades 1, 2, and 3, moving from hours and half-hours in Grade 1 to time to 5 minutes in Grade 2 to time to minutes and intervals of time in Grade 3. The many difficulties with concepts of time and the tools used to tell time are discussed and ways to overcome these difficulties are described.

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [50.09 minutes]

MD Part 7 Other Measurement Grade 3 to Grade 5: Liquid Capacity and Mass/Weight overviews how Math Expressions supports the Common Core State Standards on the metric and the customary systems of measure for liquid capacity and mass/weight. In Grade 3 the major units for liquid capacity and mass/weight are introduced. In Grade 4 students focus on all of the metric base-ten relationships, on the meanings of a few important units in the metric measures, and on the main customary measures of liquid capacity and weight. In Grade 5 students can consider the lovely relationships across the systems for which we need the idea of a cube and volume, which are Grade 5 topics.

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [27:02 minutes]

MD Part 8 Represent and Interpret Data Kindergarten to Grade 5 overviews how Math Expressions supports the Common Core State Standards on data. Kindergarten and Grade 1 children classify things into categories and count and compare things. Grade 2 and Grade 3 students focus on picture and bar graphs using length units (G2) and then scale multiples (G3). Students in Grades 2 through Grade 5 discuss line plots, which use length units, first whole number unit lengths and then fractional unit lengths.

Represent and interpret data is the learning path from Kindergarten to Grade 5 in Part 8.

Research has identified many errors commonly made by students. These errors are reviewed, and approaches within Math Expressions to prevent or overcome these errors are discussed. Teachers from Kindergarten to Grade 6 should watch and discuss their grade and any grades earlier than their grade. They would also benefit from parts after their grade to see where their students will go. [43:59 minutes]

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Math Expressions and Geometry (G) in the Common Core State Standards

These will be available at a later date.

Math Talk Community

CCSS and Math Expressions

Math Expressions and OA

Operations and Algebraic Thinking

Math Expressions and NBT Numbers Base Ten, Place Value

Math Expressions and NF

Numbers Fractions

Math Expressions and RP

Ratio & Proportions

Math Expressions and MD Measurement and Data

Math Expressions and G

Geometry

Teaching Progressions