Meaning, Method and Mastery – a way to understand character development in A Story of Units (and Ratios)

Who we are Brian, John and Mari need no introduction. Dev is a Content Leader at Illustrative Mathematics, has been consulting with Smarter Balanced, and has also been partnering with Eureka Math. Tricia has been running Lane Ignite trainings (UO/ESD/District parternships).

Purpose This brief introduction is a bit of “looking under the hood” to understand the Standards (learning outcomes) on which Eureka Math was built. Not everyone who buys a new car looks under the hood first thing, but it will be helpful to organize our thoughts when we look at the curriculum (and later when you use it!).

A Story? A Story of Units is a story! The main characters are “units”. The main main character is 1. It is the main character of the oldest, most basic story whose characters are 1, 2,3… They can get together, be compared (and contrasted?),…

A Story? Later, new units come on the scene, like 10! As a unit, we think of fellow characters two-tens, three-tens, four-tens,… That is, along with ten we have 10, 20, 30… And there’s some dramatic tension – ten ones can make a ten; ten tens can make a hundred!

A Story? Later, “small” units take the stage: one-half, one- third, etc. New and strange – but they behave in some ways like our previous characters: Just as two tens and four tens make six tens, two eighths and four eights make six eights! And then more characters -.1,.01, etc. And they all live in one place – the number line!

And those Ratios? As appropriate to middle school, ratios are really about two units – pounds and kilograms, miles and hours, etc. – love and reconciliation and understanding where one is relative to the rest of the world…

Plot and character development Those are the characters. How does plot unfold? How do we get to know each of the characters better?

Character development: Meaning, Method and Mastery We all know “conceptual good; procedural bad.” But if “conceptual” is good, shouldn’t we elaborate on what that means? And if “procedural” is bad, does that mean procedure is to be avoided?

What is M 3 ? To elaborate on some aspects of conceptually attentive teaching, we offer the following: Mathematically proficient students can engage both Meaning and Method in situations which require Mastery.

Meaning By Meaning we mean: (internal connections) ability to use and translate between equivalent definitions and models for a concept. (external connections) a situating of a concept in relation to previous concepts, including relevant modeling phenomena.

Meaning Example - the meaning of multiplication encompasses the definitions and models: Repeated addition. Rectangular arrays. Area of rectangles. Scaling on the line. And is situated as building on addition and as useful for modeling quantities accumulated at a constant rate.

Method By Method we mean processes to solve clearly delineated mathematics problems. This includes procedures and algorithms, such as the multidigit multiplication algorithm, but also includes strategies, such as recognizing that = = 415.

Mastery By Mastery we mean the ability to use a mathematical concept: In (age-)appropriate mathematical reasoning. In authentic modeling situations. In the development of more advanced concepts.

Meaning, Method and Mastery. The main Common Core shift: From teaching primarily (only?) for Method to teaching for Meaning, Method and Mastery.

Example: equivalent fractions The flow of equivalent fractions through the CCSSM begins with a standard that identifies the meaning of equivalent fractions: 3.NF.3.a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Example: equivalent fractions Students then use this meaning to develop some intuition around the concept of equivalent fractions: 3.NF.3.b. Recognize and generate simple equivalent fractions (e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

Example: equivalent fractions In Grade 4, students develop some methods that allow them to systematically find equivalent fractions: 4.NF.1. Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. For example 2/3 = (5x2)/(5x3) = 10/15

Example: equivalent fractions In Grade 5, students demonstrate mastery of the skill of finding equivalent fractions by using this skill to add and subtract fractions with different denominators: 5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12.

Common Core Process The Common Core approach to fractions was essentially written by a classroom teacher – Deborah Ball – twenty-five years ago. Other sources for Common Core material include classroom research (equality), high- performing countries (tape diagrams), and mathematicians (transformational geometry).

Single-grade example: 3.MD.5-7 Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 5 Recognize area as an attribute of plane figures and understand concepts of area measurement. a A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 7 Relate area to the operations of multiplication and addition. a Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole- number products as rectangular areas in mathematical reasoning. c Use tiling to show in a concrete case that the area of a rectangle with whole- number side lengths a and bxc is the sum of axb and axc. Use area models to represent the distributive property in mathematical reasoning. d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non- overlapping parts, applying this technique to solve real world problems.

Single-grade example: 3.MD.5-7 Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 5 Recognize area as an attribute of plane figures and understand concepts of area measurement. a A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

Single-grade example: 3.MD Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 7 Relate area to the operations of multiplication and addition. a Find the area of a rectangle with whole- number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

Single-grade example: 3.MD Relate area to the operations of multiplication and addition. b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. c Use tiling to show in a concrete case that the area of a rectangle with whole- number side lengths a and bxc is the sum of axb and axc. Use area models to represent the distributive property in mathematical reasoning. d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non- overlapping parts, applying this technique to solve real world problems.

How Meaning serves Method Retention Addressing misconceptions. Understanding the nature of procedures.

How Meaning serves Mastery Providing language to describe phenomena. Recognizing classes of problems. Awareness of where a method is applicable and where it is not.

Meaning, Method, Mastery in Smarter Balanced Assessment Smarter Balanced is “claim based” – measurements are supposed to support claims about what students can do. Claim 1 is that students know concepts and can perform procedures – that’s meaning and method.

Meaning, Method, Mastery in Smarter Balanced Assessment Claims 2, 3 and 4 pertain to students’ ability to problem solve, provide reasoning and model with mathematics. These are the three aspects of mastery.

Meaning, Method, Mastery in Smarter Balanced Assessment Claims 2, 3 and 4 pertain to students’ ability to problem solve, provide reasoning and model with mathematics. These are the three aspects of mastery.

Meaning, Method, Mastery in Smarter Balanced Assessment Whereas before most of assessment was focused on method, now that will take up about 20% of (summative) assessment. (Claim 1 is 40% with method half of that, Claims 2-4 are 20% each). Instructionally, meaning and method will take up much more than 40% of the time, as they are the foundation for mastery.

For Bethel teachers, M 3 informs What proficiency means at (different stages within) your grade, and thus how to formatively assess readiness to move on. Which classroom activities should be prioritized at different points. What in student’s background might be missing (often meanings), especially in light of transition to CCSS/ Eureka.

For Bethel teachers, M 3 informs There are two different kinds of application problems – Those whose purpose is to promote meaning (Jack’s father gives him $5/week for 7 weeks) – Those whose purpose is to promote/ assess mastery (The above plus: Jill’s mother gives her $6/week for 8 weeks; how much more does she make?)