WHAT DIFFERENCE WILL THEY MAKE?

Presenters: Syrenthia Anderson – Kelly Cordes – Mark Swanson Developing an Instructional Model to Support the Mathematical Practice Standards Presenters: Syrenthia Anderson – Kelly Cordes – Mark Swanson

NORMS Silence Cell Phones Actively participate Limit “side-bar” conversations Briefly mention…. Respect differing opinions

GOALS Explore conceptual and procedural approaches to mathematical content and lesson structures that engage students in the application of the Standards for Mathematical Practices with the Standards for Mathematical Content. Identify the characteristics of classroom environments that facilitate the development of students as practitioners of mathematics. Developing an Instructional Model to Support the Mathematical Practice Standards Session Description: This session is designed to help participants identify the characteristics of classroom environments that facilitate the development of students as practitioners of mathematics. In this session, participants will explore procedural and conceptual approaches to mathematical content and lesson structures that engage students in the application of the Standards for Mathematical Practice with the Standards for Mathematical Content.

Mathematical Content Standards 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. ***To stay focused on the goals for the session, the conversations about the content standards should be very brief and focused*** Briefly discuss the before/after learning progression as it relates to the upcoming task: In K-2 the focus is understanding partitioning of whole numbers... Note that this concept falls in the Geometry domain

Mathematical Content Standards 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole 4.NF.1 Explain why a fraction a/b is equivalent to a fraction by using visual 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. 4.NF.4 Apply and extend previous understanding of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as multiple of 1/b, and use this understanding to multiply a fraction by a whole number. ***To stay focused on the goals of the session, the conversations about the content standards should be very brief and focused*** Briefly discuss the before/after learning progression: In grades 3 & 4, students apply their understanding of partitioning whole numbers to understanding the part/whole relationship (fractions)… Now we see the overlap of concepts in the Numbers and Geometry domains

Mathematical Content Standards 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way to produce an equivalent sum or difference of fractions with like denominators. 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.2 Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers. ***To stay focused on the goals of the session, the conversations about the content standards should be very brief and focused*** Briefly discuss the before/after learning progression: When students move to grades 5 – 8, they apply their previous understanding of partitioning of whole numbers and part/whole relationships (fractions) to formal algorithms for computing fractions. In the

Opening Draw a diagram of a whole divided into four equal parts. What is each part? What is the whole in terms of parts? Vocabulary review: What is the numerator? What does it tell you? What is the denominator? What does it tell you? The goal of this is to model for participants an opening that activates prior knowlEdge of fractional parts and equivalent fractions to one.

Work Time Task Change 2 ¾ to an improper fraction. Draw a sketch that models your strategy Show how your strategy works for other mixed numbers. Make up a story problem. In the models we want participants to show both the mixed number and the improper fraction. They should be able to do both with one model. Look for participants who use circle, number line and area models to share in the closing. Participants will need to use Accountable Talk to share how their strategy works. Story problem- can participants find a situation that would call for changing a mixed number into an improper fraction. Numbers have meaning in the real world- procedures with connections link to both conceptual understanding and application.

Conceptual vs. Procedural: Debriefing the Problem What does ¼ mean? Draw a model. Prior knowledge needed: Equivalent fractions especially for whole numbers Adding fractions with the same denominator. That 2 ¾ means 2 + ¾

Conceptual vs. Procedural: Debriefing the Problem Work time: Students need to ask themselves how many 1/4ths in 2 3/4? Use your knowledge of fractions to solve this problem. Then:  Show how your strategy works for other mixed numbers.  Draw a sketch that models your strategy  Make up a story problem. Extensions: How many 1/6ths in 8 2/3? How can I write 25/4 as a mixed number? In the models we want participants to show both the mixed number and the improper fraction. They should be able to do both with one model. Look for participants who use circle, number line and area models to share in the closing. Participants will need to use Accountable Talk to share how their strategy works. Story problem- can participants find a situation that would call for changing a mixed number into an improper fraction. Numbers have meaning in the real world- procedures with connections link to both conceptual understanding and application. What is different about the extension problems. Would their strategy still work? A procedure developed with understanding works across situations. If the procedure has conditions, misconceptions will arise.

Conceptual vs. Procedural: Debriefing the Problem Why not just tell students: To convert a mixed number into an improper fraction multiply the denominator by the whole number and add it to the numerator. Is there anything wrong with this procedure? What misconceptions might students have if all they learned was this procedure? The goal here is to have participants understand that there is nothing wrong with this procedure if students understand why they are multiplying the whole number by the denominator. The problem most students have is that they do not understand that 4/4 is one, therefore 8/4 is two. The concept of equivalent fractions is missing as is the link to division. This can present potential problems in algebra where the denominator can symbolize the divisor as a relationship to an abstract whole, or in ratio application, the denominator represents a comparison of parts- I.e. rise to run, with the denominator as run.

Procedures WITHOUT connections to understanding, meaning, or concepts Are algorithmic/procedural Are focused on correct answers rather than developing understanding Require no explanations Require limited cognitive demand Have no connection to concepts

Being a Practitioner of Mathematics Requires complex and non-algorithmic thinking Requires student to explore and understand the math Requires students to access relevant knowledge and apply it Requires considerable cognitive effort

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years. Common Core State Standards for Mathematics, pg. 8 Refer to Common Core Standards (Mathematics) pg. 8 for talking points…

Mathematical Practice Standards Assign each group of participants a “Practice.” Ask each group to highlight the “words or phrases” that they saw evidence of in the mini- lesson that was modeled( “evidence”). Ask each group to share out… ***Keep in mind that K-8 teachers are beginning their focus with Practices 1, 3, & 6. ***

The Role of the Teacher video Ask participants to – watch the video through the lens of the mathematical practices. Highlight the “evidence of the practices” seen in the video Facilitate a conversation around participant observations…

From Standards to the Classroom Environment It is what teachers think, what teachers do, and what teachers are at the level of the classroom that ultimately shapes the kind of learning that young people get. Andy Hargreaves (quoted in Bay-Williams 2010, 37) Classroom Discourse If we believe that there is only one way to learn mathematics, then our students will only do mathematics one way. So, the question becomes, how do we create classroom environments that promote students understanding of the content, therefore, making them practitioners of mathematics? Ask participants - What does this classroom “look like, sound like, feel like?’ Think – Pair – Share Chart participant responses… Address the paradigm shift…. Classroom Norms Classroom Relationships

Making Sense of Mathematics No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students. FACILITATE A CONVERSATION that connects the goals to this slide… As we make this paradigm shift, our charge is to think about how WE create opportunities for students to become mathematicians. * Identify the characteristics of classroom environments that facilitate the development of students as practitioners of mathematics. Explore procedural and conceptual approaches to mathematical content and lesson structures that engage students in the application of the Standards for Mathematical Practices with the Standards for Mathematical Content. (Shifter and Fostnot quoted in Van de Walle, Karp, and Bay-Williams 2010)

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