INFORMative Assessment of Student Understanding; Ratio and Rate of Change

1. INFORMative Assessment of Student Understanding;Ratio and Rate of Change August, 2012

2. Thinking About Your Practice • Why do you assess students? • What purposes do your assessments serve? Turn to your neighbor and share your list

3. INFORMative Assessment • To determine what students know and understand • To determine potential misunderstandings • To facilitate student recognition of what the student knows or needs to know • To make instructional decisions for/with a student • To make instructional decisions for/with a class Not for evaluation and grading

4. Teaching-Learning Cycle

5. Standards for Math Practice • In your table groups highlight the verbs that you see within the Standards of Mathematical Practice • Talk in your groups and be ready to share... • Who will be doing these actions? • What actions are already present in your classroom? • Which actions will you be working to include?

6. Ratio, Rate, Proportional Reasoning • It is estimated that less than half of the adult population can be viewed as proportional thinkers Teaching Fractions and Ratios for Understanding,Lamon (1999) • This implies that proportional reasoning is not a natural development, but rather requires some special experience or learning- perhaps from the classroom

7. Ratio and Rate for Proportional Reasoning In order to reason proportionally students need to be able to- • Select equivalent ratios • Compare ratios using a variety of strategies • Reason multiplicatively • Reason multiplicatively as the quantities change, but the relationship remains the same

8. Rates and Rates of Change To reason multiplicatively as the quantities change, but the relationship remains the same • A rate represents a single relationship that is shared by an infinite number of ratios. • The rate shows how one quantity in the pair will change with respect to how the other quantity changes.

9. Seeing the Forest... • Each table group will be creating a poster that shows their solution strategy, explaining how that strategy works, and employs the strategy to count the trees • With your table groups read through the “Counting Trees Task” • When your group is ready have 1 person come get the poster paper and markers your group will need

10. Seeing the Forest and the Trees • Be ready to share your solution strategy with the group • If your group finishes before the whole group is done, discuss how your students might approach this problem and then list the criteria that an “exceptional student response” would have to include

11. Teaching-Learning Cycle What Standards for Mathematical Practices could be potential Learning Targets corresponding to a lesson involving this task? Or in other words, what Practices could this task be used to assess?

12. Seeing the Forest in Action The following video is from MARS- Mathematics Assessment Resource Service Before watching this video let’s assume • There is a lot we don’t know • That the teacher has good intent and teaching expertise

13. Seeing the Forest in Action Be ready to share and discuss... • What do you notice most about this video and • The task in which students are working • What the students are doing • What the teachers are doing • The understandings about ratios that the students utilize • What aspects of the Teaching-Learning Cycle are shown

14. Providing Actionable Feedback • When working with your groups you listed the criteria that an “exceptional student response” would have to include • Use your list and work with your group to provide feedback to each of the students in the student work samples • Ideally, your feedback will provide the student with an action they can perform to improve their thinking.

15. Impede or Empower “Involving students in their learning is a key characteristic of formative assessment.” (Black and Wiliam, 1998) “ When children continually participate in the assessment process, they learn to recognize their own expertise. As active assessors, they necessarily exercise a more autonomous and decision-making role in their learning. Consequently, instead of being used to gain power over a child, assessment empowers the child.” (Anderson, 1993)

16. Impede or Empower • Read the classroom scenarios given in your handouts • With a partner determine whether each scenario impedes or empowers students in developing their responsibility for themselves as learners of mathematics Are there other examples you could add to this set of scenarios? What practices do you employ that empower your students to have an active role in their learning?

17. Teaching-Learning Cycle

18. Summing It Up… • We can look to BOTH the CCSS content standards and the mathematical practices to help us define clear learning targets. • By including our students in the assessment process, we share the assessment responsibility with them and empower them to distinguish “right” from “wrong”. • By implementing INFORMativeassessment in a more deliberate manner, we could help students make important connections between “big” mathematical ideas.