Mathematics Education Colloquia University of Kentucky December 6, 2005 Linda Jensen Sheffield

1. The New CIA: Curriculum, Instruction and Assessment in Mathematics Education for the 21st Century Mathematics Education Colloquia University of Kentucky December 6, 2005 Linda Jensen Sheffield Regents Professor Northern Kentucky University Sheffield@nku.edu http://www.nku.edu/mathed

2. What we assess defines what we value. What do we value most in mathematics teaching and learning? Linda Sheffield

3. Attitude, Aptitude, and Achievement • Attitude • Efficacy • Usefulness • Enjoyment • Aptitude • ACT • SAT • Achievement • CATS/KCCT • NAEP • TIMSS • PISA Linda Sheffield

4. What is the purpose of assessment?Pre or Post?To prove or to improve? Summative or Formative? Linda Sheffield

5. What is formative assessment? • Assessment to inform instruction • Assessment becomes formative "when the evidence is actually used to adapt the teaching to meet student needs" (Black & Wiliam, 1998) • In fact, research shows that formative assessment has one of the biggest effects on learning, even equal to the effect of parental influence. It is critical that assessment is diverse and divergent and develops through many pathways rather than from a single source. Gilbert Valdez Linda Sheffield

6. For the student Choose Learn Qualify For the teacher Place Monitor Report Uses of Formative Assessment Linda Sheffield

7. 1 Clearly define what you will assess. What do you expect your students to be able to do? 2 Define the purpose of your assessment. Are you intending to conduct placement, formative, diagnostic, or summative assessment? 3 Select or develop assessment procedures that closely match targeted learning goals or abilities. 4 Know the limitations of assessment procedures used. Does the way the test is developed, administered, or interpreted present any limitations? 5 Use a variety of assessment procedures. Is your assessment comprehensive (e.g., do you make use of observations, class work, professional judgment, student and parent input?) 6 Evaluate the assessment or test you develop or use. Are they valid and reliable? 7 Communicate assessment results clearly to all users. Do the students, parents, teachers, and other stakeholders understand the results? 8 Consider and address personal implications. Are your biases influencing your professional judgment? 9 Strengthen the link between assessment and learning. Is assessment helping improve instruction and learning? 10 Assessment should serve a useful purpose and not be an end in itself. Are tests an integral part of your instruction and are they helping you guide instruction? Ten Principles of AssessmentFoertsch (1999) Linda Sheffield

8. Summative assessment is assessment that occurs at the end of a learning unit. The purpose of summative assessment is to measure how well students have learned key content and skills as defined by the unit's learning goals and objectives. Summative assessments can take many forms ranging from traditional or selected response tests like multiple choice, short answer and essay to constructed response or performance-based tests like problem-solving tasks, projects and gathering and analyzing data. What is summative assessment? Linda Sheffield

9. What are state, national and international assessment requirements? • Kentucky Education Reform Act • Commonwealth Accountability Testing System • Kentucky Core Content Test - Test math in grades 5, 8, and 11 • No Child Left Behind - • Test math every year from grade 3 - grade 8 and at least once from grade 10 - 12 • National Assessment of Educational Progress (NAEP) • International Comparisons • Trends in International Mathematics and Science (TIMSS) • Program for International Student Assessment (PISA) Linda Sheffield

10. Standardized testing encourages teachers and administrators to focus instruction on narrow test content as they tend to incorporate the following strategies: • Encouraging drill and practice on rote memory tasks related to content. • Avoiding inquiry-based learning activities in which complex concepts and skills are developed. • Skipping content that may be in the standards but not on the test. (Herman, 1997) • Standards-based assessment is designed to • Improve learning • Monitor and document students’ progress and proficiency • Inform teachers in order to help them make instructional decisions • Help students capture, reflect on, and demonstrate their mathematical growth • Evaluate students’ achievement • From Show-Me Project Assessment Brochure Standardized vs. standards-based

11. Depth of KnowledgeNorman Webb • Level 1 - Recall - Facts, definitions, terms, simple procedures and algorithms or formula applications. This includes one-step, well-defined, and straight algorithmic procedures. • Level 2 - Skills and Concepts - Engagement of mental processing beyond habitual response. This might include classifying, organizing, collecting, displaying, comparing and interpreting data - operations involving more than one step. • Level 3 -Strategic Thinking - Reasoning, planning, using evidence and a higher level of thinking. • Level 4 - Extended thinking - Complex reasoning, planning, developing and thinking most likely over an extended period of time. Linda Sheffield

12. Assessment is not a single event but a continuous cycle.Lynn Steen Linda Sheffield

13. Questions for differentiating the mathematics curricula Who? What? When? Where? Why? and How? Who should learn rich mathematics?–Everyone! What or what if? What patterns do I see? What generalizations might I make from the patterns? What proof do I have? What are the chances? What is the best answer, the best method of solution, the best strategy to begin with … ? What if I change one or more parts of the problem? When? When does this work? When does this not work? Where? Where did that come from? Where should I start? Where might I go for help? Why or why not? Why does that work? If it does not work, why not? How? How is this like other problems or patterns that I have seen? How does it differ? How does this relate to "real-life" situations or models? How many solutions are possible? How many ways might I use to represent, simulate, model, or visualize these ideas? How many ways might I sort, organize, and present this information?

14. Role of a Student Creator ⇛Repeat/rephrase ⇛Agree/disagree...and tell why ⇛Add on to... ⇛Wait, think, and go deeper ⇛Talk to a partner ⇛Record reasoning and questions Role of a Teacher/Mentor: • Ask questions that encourage creativity and reasoning • Elicit, engage and challenge each student’s thinking • Listen carefully to students’ ideas • Ask students to clarify and justify their ideas • Attach notation and language to students’ ideas • Decide when to provide information, clarify, model, lead or let students struggle • Monitor and encourage participation Adapted from Project M3: Mentoring Mathematical Minds Linda Sheffield

15. Linda Sheffield

16. Results from NAEP, TIMSS and PISA TIMSS 2003 showed fourth and eighth graders above the international average in both math and science. This is an increase for eighth grade since 1995. PISA showed 15-year-olds below the international OECD average in mathematical literacy and problem solving. NAEP math scores have gone up for fourth and eighth grade from 1996 - 2003.

17. Because assessment drives instruction, state and national assessment and accountability must include an opportunity for students to demonstrate their abilities to do high-level, in-depth reasoning and problem solving. Linda Sheffield

18. Try a typical fraction question. 6/7 + 7/8 = ________ Estimate: 6/7 + 7/8 is about 1 2 13 15 Write a word problem for 1 3/4 ÷ 1/2. Solve the problem and explain your answer.

19. Try another. What might this represent? Linda Sheffield

20. Question the answers; don’t just answer the questions. Linda Sheffield

21. What if …? this is multiplication? this is division? you use whole numbers? fractions? decimals? algebra?

22. Think deeply about simple things. Linda Sheffield

23. When we think well, we feel good. Understanding is a kind of ecstasy. Carl Sagan (from Broca's Brain, 1979)

24. Assessment Criteria If you wish students to develop deeper understanding of concepts and become creative investigative mathematicians, you should use criteria for assessment that encourage depth and creativity such as: • Depth of understanding - the extent to which core concepts are explored and developed • Fluency - the number of different correct answers, methods of solution, or new questions formulated • Flexibility - the number of different categories of answers, methods, or questions. • Originality - solutions, methods or questions that are unique and show insight • Elaboration or elegance - quality of expression of thinking, including charts, graphs, drawings, models, and words • Generalizations - patterns that are noted, hypothesized, and verified for larger categories • Extensions - related questions that are asked and explored, especially those involving why and what if Linda Sheffield

25. Linda Sheffield

26. Problem from SingaporeSolve the following problem in at least two different ways. Generalize your findings to the nth term.What digit is in the ones place of the sum?1 + 2 + 3 + 4 + 5 + 6 + … + 94 + 95 + 96 + 97 Linda Sheffield

27. Can assessment help us compete on a global level? Consider these recent class averages on end-of-unit test scores: • Class A: 26.67% • Class B: 55.53% • Class C: 47.49% • Class D: 75.44% Linda Sheffield

28. Assessment should support continuous progress for all students. We need to be sure that we are assessing what we value the most. Linda Sheffield

29. References Black, P., & Wiliam, D. (1998b). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 80(2), 139ﾐ148. Retrieved September 6, 2005, December 8, 2003, from http://www.pdkintl.org/kappan/kbla9810.htm Burrill, j., Feijs, E., Meyer, M/, van Reeuwijk, M., Webb, D.; Wijers. M/ The Role of Assessment in Standards-Based Middle School Mathematics Curriculum Materials. Retrieved December 2, 2005, fromhttp://showmecenter.missouri.edu/resources/assessment.pdf#search='mathematics%20assessment%20standards%20based Foertsch, D. J. (1999). Understanding assessment: An introduction to using published tests and developing classroom tests. Unpublished manuscript, North Central Regional Educational Laboratory, Oak Brook, IL. Herman, J. L. (1997, October). Large-scale assessment in support of school reform: Lessons in the search for alternative measures. (CSE Technical Report 446.) Los Angeles: National Center for Research on Evaluation, Standards, and Student Testing. Retrieved September 6, 2005, from http://www.cse.ucla.edu/CRESST/Reports/TECH446.pdf Ma, . (1999). Knowing and teaching elementary mathematics: Teachers’ knowledge of fundamental mathematics in China and the United States. Nahwah, NJ: Erlbaum. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM. North Central Regional Educational Laboratory. Critical Issue: Multiple Dimensions of Assessment That Support Student Progress in Science and Mathematics. Retrieved December 2, 2005, from http://www.ncrel.org/sdrs/areas/issues/content/cntareas/science/sc700.htm Romberg, T. (Project Director). (1995). Assessment standards for school mathematics. Reston, BA: NCTM. Sutton, J. and Krueger, A. (2002). EDThoughts: What we know about mathematics teaching and learning. Arora, CO: McCrel. Linda Sheffield