# Supporting Rigorous Mathematics Teaching and Learning

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Supporting Rigorous Mathematics Teaching and Learning

Rationale Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it, yet not all tasks afford the same levels and opportunities for student thinking. [They] are central to students’ learning, shaping not only their opportunity to learn but also their view of the subject matter. Adding It Up, National Research Council, 2001, p. 335 By analyzing instructional and assessment tasks that are for the same domain of mathematics, teachers will begin to identify the characteristics of high-level tasks, differentiate between those that require problem solving, and those that assess for specific mathematical reasoning. (SAY) Take a minute and read the rationale for the lesson. The quote comes from the Common Core State Standards. As you can see from the rationale, developing student “understanding” is really important. The Common Core State Standards include grade-level standards that focus on understanding of mathematical concepts and the development of skills. We will engage in the lesson with the goal of deepening our understanding of concepts noted in the standards and we will do this via a set of tasks.

Session Goals Participants will:
make sense of the Number and Operations – Fractions Common Core State Standards (CCSS); determine the cognitive demand of tasks and make connections to the Mathematical Content Standards and the Standards for Mathematical Practice; and differentiate between assessment items and instructional tasks. Directions: Read the goals on the slide.

Overview of Activities
Participants will: analyze a set of tasks as a means of making sense of the Number and Operations – Fractions Common Core State Standards (CCSS); determine the Content Standards and the Mathematical Practice Standards aligned with the tasks; relate the characteristics of high-level tasks to the CCSS for Mathematical Content and Practice; and discuss the difference between assessment items and instructional tasks. Directions: Read the session activities.

The Data About Students’ Understanding of Fractions

Only a small percentage of U.S. students possess the mathematics knowledge needed to pursue careers in science, technology, engineering, and mathematics (STEM) fields. Moreover, large gaps in mathematics knowledge exist among students from different socioeconomic backgrounds and racial and ethnic groups within the U.S. Poor understanding of fractions is a critical aspect of this inadequate mathematics knowledge. In a recent national poll, U.S. algebra teachers ranked poor understanding about fractions as one of the two most important weaknesses in students’ preparation for their course. Directions: Paraphrase the slide. Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, Thompson, & Wray (2010). IES, U.S. Department of Education

The Data about Fractions: Conceptual Understanding
A high percentage of U.S. students lack conceptual understanding of fractions, even after studying fractions for several years; this, in turn, limits students’ ability to solve problems with fractions and to learn and apply computational procedures involving fractions. 50% of 8th graders could not order three fractions from least to greatest; 27% of 8th graders could not correctly shade of a rectangle; 45% of 8th graders could not solve a word problem that required dividing fractions (NAEP, 2004). Fewer than 30% of 17-year-olds correctly translated as (Kloosterman, 2010). (SAY) The challenge is to help children develop understanding of the fractional units with which they are counting. We need to help children understand what is being counted and to place more emphasis on children’s actually partitioning of wholes (Watanabe, 2002; Pothier & Sawada, 1983). Discussion, Turn and Talk: What have you noticed about the understanding of fractions your students demonstrate (now or in the past)? What prior learning do students demonstrate? What misconceptions have you noticed that get in the way of their understanding fractions at the 4th grade level? What are the implications for teaching and the tools that need to be in place so that we can reverse the present United States student statistics about what our students (don’t) know about fractions?

The Data about Fractions: Conceptual Understanding
A lack of conceptual understanding of fractions has several facets, including…students’ focusing on numerators and denominators as separate numbers rather than thinking of the fraction as a single number. Errors such as believing that > arise from comparing the two denominators and ignoring the essential relationship between each fraction’s numerator and denominator. Directions: Paraphrase the slide. Siegler, Carpenter, Fennell, et al; U.S. Dept. of Education, IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten through 8th Grade.

Analyzing Tasks as a Means of Making Sense of the CCSS Number and Operations – Fractions
Directions: (SAY) This data makes a strong case for why we need to spend time on giving students opportunities to make sense of the concept of fractions.

Linking to Research/Literature: The QUASAR Project
The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as they appear in curricular/ instructional materials TASKS as implemented by students Student Learning Directions: (SAY) In this module we will focus on just the written task. Stein, Smith, Henningsen, & Silver, 2000

Linking to Research/Literature: The QUASAR Project

Linking to Research/Literature: The QUASAR Project

Linking to Research/Literature: The QUASAR Project
Low-level tasks Memorization Procedures without Connections High-level tasks Doing Mathematics Procedures with Connections Directions: (SAY) Remember that the QUASAR research identifies two types of high level tasks. They name tasks as “doing mathematics” or “procedures with connections.” Look at the Mathematical Task Analysis Guide in your participant handouts. The characteristics of each of these types of high-level task is listed.

The Cognitive Demand of Tasks (Small Group Discussion)

Directions: (SAY) Take a minute and review the characteristics of tasks on the Mathematical Task Analysis Guide. On the left are the characteristics of low-level tasks and on the right are the characteristics of high-level tasks. The “Doing Mathematics” type of task is at the bottom, right of the chart and the “Procedures with Connections” type of task is in the top right hand corner. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

The Cognitive Demand of Tasks (Whole Group Discussion)

Analyzing Tasks: Aligning with the CCSS (Small Group Discussion)
Determine which Content Standards students would have opportunities to make sense of when working on the task. Determine which Mathematical Practice Standards students would need to make use of when solving the task. Use the recording sheet in the participant handout to keep track of your ideas. Directions: (SAY) We will engage in some of the tasks and then step out and reflect on our engagement in the tasks. We will consider how our learning was supported, and which standards we had opportunities to think about and use when figuring out the solution paths.

Analyzing Tasks: Aligning with the CCSS (Whole Group Discussion)
How do the tasks differ from each other with respect to the content that students will have opportunities to learn? Do some tasks require that students use mathematical practice standards that other tasks don’t require students to use? Directions: (SAY) How does the content in the task differ? Which standards will students have opportunities to make sense of when working on the task? (Chart differences. See notes on the following pages about addressing different standards. Note which representations are provided and which ones students must create.) .

The CCSS for Mathematical Content − Grade 4
Number and Operations – Fractions 4.NF Extend understanding of fraction equivalence and ordering. 4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) 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. 4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Possible Probing Questions and Possible Responses (italics): Which tasks address 4.NF.A.1? Leftover Pizza can require equivalence when students operate with fractions because they need common denominators. Thirds and Sixths requires equivalence. Easting Cereal requires comparison of two fractions and therefore may get at students creating equivalence. Which tasks address 4.NF.A2? Comparing fractions is a skill required by the following tasks: Writing a Rule, Thirds and Sixths, and Eating Cereal. Which tasks address 4.NF.B.3? Students have to add and subtract fractions in Leftover Pizza, and Thirds of Sandwiches. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

The CCSS for Mathematical Content − Grade 4
Number and Operations – Fractions 4.NF Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 4.NF.B.4a Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). 4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? Possible Probing Questions and Possible Responses (italics): What are some distinctions that should be made between these standards? 4a v. 4b are crucial to distinguish because 4a deals with iterations of a unit fraction and 4b deals with the equivalence of two different sets of iterations. Which tasks address 4.NF.B.4? Thirds and Sixths can address this if viewed through the lens of iterating unit fractions. Four-fifths of Homework Three Cakes Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

The CCSS for Mathematical Practice
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Possible Probing Questions and Possible Responses (italics): MP4 Modeling Mathematics: Which mathematical practice standards will students have opportunities to work on when using this set of tasks? The set of tasks give students opportunities to work with visual models and manipulatives. What might a teacher do with students to help them distinguish among different models? Students should be given opportunities to step back and compare the models. Teachers should press students to record fractions, to use the greater than and less than sign, to write explanation. Reason Abstractly and Quantitatively: Which tasks give teachers the opportunities to assess students use of this practice? This can be assessed with any task that has a context. Look For and Make Use of Structure: Not all of these task require students to write about the structure of mathematics. Which tasks ask students to share reasoning related to the structure of mathematics? Writing a Rule for Comparing Fractions uses the structure of same wholes and the concept of the missing piece. Thirds and Sixths uses the structure of the whole and equivalence. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 21

A. Writing a Rule for Comparing
Isabelle is comparing fractions. She says that she can see, without doing any calculations, which one is greater in each of the pairs below: 3 4 and 4 5 and 2 3 and 7 8 Facilitator Notes: This task must get at the piece missing from each fraction; they are each one away from one whole. What rule can be written for comparing the fractions in each pair without finding a common denominator? Does the rule you have written work with all fractions?

B. Leftover Pizza Frankie orders a pizza. He eats of the pizza. His little sister eats of the pizza. How much is left? Facilitator Notes It is important for students to identify that the whole is in fact the same whole and that in order to operate with fractions, there need to be comparable sized pieces.

C. Thirds and Sixths Joel looks at the picture below and says, “I see 2 3 of the picture is shaded.” Sammy says, “No, 4 6 of the picture is shaded.” Who is correct? Write addition and multiplication equations to prove your answer. Facilitator Notes Equivalence can be arrived at visually, symbolically, or by manipulating the model (allow students to cut and move pieces if they want). Possible Probing Questions Which standards do students have opportunities to make sense of? Why? Which math practice standards do students demonstrate and understanding of when solving this task?

D. Four-Fifths of His Homework
Jesse has been working on homework. He looks at the number of problems he has completed and figures out that he has finished of his homework. If he has 20 problems for homework, how many does he still have to complete? Facilitator Notes: If students are struggling with this problem, encourage a diagram that shows 4/5 into which the student can place the 20 dispersed over the 5 pieces in the whole.

E. Thirds of Sandwiches Tara invited friends over to work on homework. She is ordering submarine sandwiches for dinner. They are large sandwiches so she plans on giving of a sandwich to each person. If she wants to feed 7 friends and herself, how many sandwiches does she need to order? Possible Probing Questions and Possible Responses (italics): What makes this number task more difficult for students? (7 is not a compatible number with thirds because 7 is not a multiple of three. Students must consider the context and take the answer that is the nearest whole.)

F. Three Cakes Ashlee brought 3 cakes to school to share with classmates. There are 30 students in the class. How much cake does each student get? Possible Probing Questions and Possible Responses (italics): What are the two ways in which students could consider this problem? (Students have drawn three cakes and created 30ths in each; they have also drawn 3 cakes and cut each into 10th. What are the benefits to each of those?)

G. Decorating Gifts Sarah bought 5 feet of ribbon. She needs to wrap 3 gifts and wants to decorate each gift with an equal amount of ribbon. How many feet of ribbon will be used per gift if she wants to use all 5 feet? Possible Probing Questions and Possible Responses (italics): Why is a linear model a good choice for this problem? (An area model does not necessarily relate to ribbon.) How do the numbers yield an understanding of fractions? (The 5 needs to be divided into three equal sections, thus yielding either an improper fraction or a changing to a mixed number.)

H. Eating Cereal Sam buys a box of his favorite cereal. He eats 1 25 of the box per day. How much of the box has he eaten by the 5th day? Show how you know you are correct. Sam’s sister likes a different cereal. By the 5th day, she has eaten 2 10 of her box. Who has eaten more cereal by day 5 if the boxes are the same size? Facilitator Notes: Students find this task difficult because the task is designed to require careful reading: one fraction refers to daily consumption, one is a 5-day consumption.

Reflecting and Making Connections

Characteristics of Performance-Based Assessments