Formative Assessment in the Mathematics Classroom Coding for the Core

Assessment Card Sort Sort the descriptors into two categories –Summative –Formative

Diagnostic: To identify preconceptions, errors, types of reasoning, and learning difficulties Summative: To measure and document the extent to which students have achieved a learning target Formative: To inform and provide feedback to students on their learning

FACTS Purposeful Planned Often-spontaneous Teacher-to-student Student-to-teacher Student-to-student Oral Written Formative Assessment Classroom Techniques

FACT Layout –Description –How the FACT Promotes Student Learning –How the FACT Informs Instruction –Design and Administration –General Implementation Attributes Ease of Use Time Demand Cognitive Demand –Modifications –Caveats –Use with Other Disciplines

Uncovering Student Thinking About Mathematics in the Common Core

Math Starters! 2 nd Edition 5 to 10 Minute Activities Aligned with the Common Core Math Standards

#4 Card Sorts Students collaboratively sort a set of cards with pictures, numbers, symbols, or words according to a specific characteristic or category. –Based on preexisting knowledge about the concept or procedure –Discuss reasons for placing each card into a designated group.

#4 Card Sorts Proportional Reasoning Sort 7.RP.A.2 1.Sort cards into 2 piles: Proportional and Not Proportional 2.Create two additional examples of each category 3.Choose 3 cards from Proportional pile – explain how you know they belong. 4.Choose 3 cards from the Not Proportional pile - explain how you know they belong. Teaching Idea: ask ss to develop a list of characteristics that could be used to decide if a relationship is proportional. As ss share and come to agreement, the teacher records and draws an example and nonexample to further illustrate. Provide opportunity to regroup cards using listed characteristics – add to list as needed.

#4 Card Sorts Number Card Sort – 8.NS.A.1 1.Sort cards into 2 piles: Rational and Irrational. 2.Create two additional examples of each category. 3.Explain choice for 3 cards in Rational pile 4.Explain choice for 3 cards in Irrational pile

Compare Card Sorts to…. #8 Concept Attainment Cards #9 Concept Card Mapping #30 Matching Cards #49 Sequencing Cards #75 Word Sort Select one of these strategies and briefly outline a topic you could use it with at your grade level

#11 Create the Problem Students are given the solution and are asked to figure out what the real-world problem might be. Another way to look at this… An equation is like a newspaper headline. Ask students to write stories to go along with math headlines (equations).

#11 Create the Problem Solve the equation and write a word problem that goes with it. 5 x 1.5 = n

#36 Opposing Views Probes These probes present two or more characters with conflicting mathematical ideas. Students are asked to pick the person whom they most agree with and explain why.

#36 Opposing Views Probes Which friend do you agree with? Explain why you agree. Provide evidence. I think π is an irrational number. I think 1/4 is an irrational number.

#10 Concept Cartoon Cartoon characters’ comments include an idea that is more mathematically acceptable than the others as well as alternative ideas based on common misconceptions and errors. Ss decide which character in the cartoon they agree with most and why.

#10 Concept Cartoon

Why is the circle not invited to the Polygon Party? What other shapes would be invited to the Polygon Party? How do you know they would be invited? Are there any other shapes who wouldn’t be invited? Who are they and why?

#36 Opposing Views & #10 Concept Cartoons Compare with #19 Friendly Talk Probes Brainstorm concepts you and think of at your grade level for use with these FACTs? How can you see these FACTs as part of your math instruction?

#2 Agreement Circles Kinesthetic way to activate thinking & engage ss in discussing and defending their mathematical ideas. –Ss stand in circle –Teacher reads statement –Ss who agree step to center of circle –Ss who disagree remain standing on outside of circle. –Small groups engage in discussion to defend their thinking.

#2 Agreement Circles y = -3/5 x + 2 is a linear function If you agree with this statement step to center of circle. If you disagree with or are unsure of this statement remain on the outside of the circle.

#7 Commit & Toss An anonymous elicitation technique used to make students’ thinking visible to the class.

#7 Commit & Toss 1.Students are given an assessment probe. 2.After completing the probe, students crumple their papers into a ball and, upon a signal from the teacher, toss the paper balls around the room until the teacher instructs them to pick up or hold on to one paper. 3.Students take the paper that they end up with and share the answer and explanation that is described on the paper that they are holding.

#7 Commit & Toss Percents and Tips – Multistep Problem 7.RP.3 Intro: A tip is a gratuity, or a payment, for good service. People commonly tip waiters and waitresses, hairdressers, bellhops, and cab drivers. In most cases, a tip of 15% is acceptable.

#51 Strategy Harvest Ss complete a problem solving task and then circulate among their peers to find ss who used a strategy different from theirs to solve the problem. Ss record the other strategies and describe how the strategy differs from the one they used. Ss give feedback to each other on their strategy.

Example Strategy Harvest Sheet My Strategy__________’s Strategy

#51 Strategy Harvest Choosing Methods and Solving Systems of Linear Equations There are several ways to solve a system of linear equations: graphing, substitution, addition-or-subtraction, and multiplication with addition-or-subtraction. Solve each system of linear equations. Use the method you prefer.

#22 Human Scatter Graph Ss position themselves around the room according to their response to a question and their confidence level. Creates a visual graph of results. Choose selected response questions with at least3 and no more than four choices. Label the way (Y-axis) on one side of the room with the choices, A, B, C, D. Label the adjacent wall (X-axis) with a range of low confidence to high confidence.

#22 Human Scatter Graph Example See page 106 in the book. Let’s practice… Value of the inequality…If m>0 and n<0, decide if each inequality is Always True, Sometimes, True, Never True, or Can’t be Determined. m+n<0

#39 Pass the Problem Begin by working in pairs to respond to a problem, partially completing a solution to a problem. When time is up, exchange partially completed solution with another pair to finish – modify, add to, or change it as deemed necessary.

#39 Pass the Problem Drawing Comparative Inferences about Two Populations 7.SP.4 Problem: According to the prices in the tables, which place, the store at the mall or the outlet, has less expensive jeans? Explain your answer.

#50 Sticky Bars Low-tech version of personal response systems (clickers). Present ss with a selected response question. The answer is anonymously recorded on a Post-It note and passed to the teacher. The teacher or a student arranges the sticky notes on the wall or board as a bar graph representing the different responses.

#50 Sticky Bars There are 45 more students than teachers on the field trip. a.45 = s + t b.s + 45 = t c.t + 45 = s

#35 Odd One Out Odd One Out provides an opportunity for students to access mathematical knowledge to analyze relationships between items in a group.

#35 Odd One Out Which expression does not belong because it is not equivalent to the others? a.9a 2 b.-9a 2 c.-(3a) 2 d.-3a 2  3

#17 Four Corners Four Corners provides an opportunity for students to make their ideas public. It is used with selected-response questions to identify and group students who have similar responses to the question asked.

#17 Four Corners Division of Fractions Model Page 33 Uncovering Student Thinking About Math in the Common Core

FACTs focused on Writing…Reflection #33 Muddiest Point #43 POMS: Points of Most Significance #56 Ten-Two #57 Thinking Log #62 Three-Two-One #67 Two-Minute Paper #73 What are You Doing and Why (or use as discussion)

#21 Hot Seat Questioning Used when all students are expected to be ready to respond to math questions at any time. Students are selected to sit in the “hot seat” to respond to questions asked by the teacher while the class provides feedback on the students’ responses. Think – what are two FACTs that you think you will use and why….

#21 Hot Seat Questioning Variation Check under your seat for an index card with a “hot seat” label.

Using Classroom Assessment to Improve Student Learning

Make students thinking visible Asking students to explain what they were thinking when they solved a problem may reveal… –Appropriate procedural knowledge –No conceptual understanding –Can’t explain the mathematical ideas underlying why or how the procedure worked When asked why? Because it’s the rule!

Middle school teachers Need both conceptual and procedural math content knowledge that goes beyond grades they teach. Need PCK – that enables them to teach the conceptual and procedural content to EVERY student…. Differentiation.

Teacher proficiencies needed Math basis of formulas and procedures and explain why they work Should use many different representations of concepts and procedures Recognize and encourage deep thinking in classroom discussions

Providing Feedback Information about performance in light of some goal. Descriptive vs. Evaluative Feedback Effective descriptive feedback gives students an idea of what they are doing well and offers specific input on what they must do next to advance learning.

Students’ Role in Formative Assessment By the middle grades, ss should be able to articulate areas in which they are proficient and those in which they need more support. They need opportunities to be engaged in: –Criteria and goal setting –Self-reflection –Self-evaluation –Sharing their work –Defending their ideas publicly

Students’ Role in Formative Assessment Teachers need to: –Provide exemplar work samples that illustrate expectations –Provide assistance in prompts to help ss self- reflect such as: Describe the strategies you used to solve this problem. What other strategies might you use the next time to solve a similar problem? Did you collaborate with peers when you were solving the problem? What is helpful? If so, how? If not, why not?

Formative Assessment Tools Jigsaw Group 1: Range Questions & Observation protocols p Group 2: Gallery Walks p Group 3 – Round-Robin Activities p Group 4 - Focused Questions and Hinge Questions p Group 5 - Mathematical Discourse – accountable talk & math congress p Group 6 - Neutral feedback – p Group 7 - Exit cards – p.33-38