The New CIA: Curriculum, Instruction and

Assessment in Mathematics Education for the 21st

Century

The New CIA: Curriculum, Instruction and

Assessment in Mathematics Education for the 21st

CenturyMathematics Education Colloquia

University of Kentucky

December 6, 2005

Linda Jensen Sheffield

Regents Professor

Northern Kentucky University

Sheffield@nku.edu

http://www.nku.edu/mathed

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

Linda Sheffield 2

What we assess defines what we value.

What we assess defines what we value.

What do we value most in mathematics teaching and

learning?

What do we value most in mathematics teaching and

learning?

Linda Sheffield 3

Attitude, Aptitude, and Achievement

Attitude, Aptitude, and Achievement Attitude

Efficacy Usefulness Enjoyment

Aptitude ACT SAT

Achievement CATS/KCCT NAEP TIMSS PISA

Attitude Efficacy Usefulness Enjoyment

Aptitude ACT SAT

Achievement CATS/KCCT NAEP TIMSS PISA

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What is the purpose of assessment?

Pre or Post?

To prove or to improve?

Summative or Formative?

What is the purpose of assessment?

Pre or Post?

To prove or to improve?

Summative or Formative?

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What is formative assessment?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

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

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Uses of Formative AssessmentUses of Formative Assessment

For the studentChooseLearnQualify

For the studentChooseLearnQualify

For the teacher Place Monitor Report

For the teacher Place Monitor Report

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Ten Principles of Assessment Foertsch (1999)

Ten Principles of Assessment Foertsch (1999)

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?

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?

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What is summative assessment?What is summative assessment?

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.

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.

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What are state, national and international assessment requirements?

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)

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)

Standardized vs. standards-basedStandardized vs. standards-based 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

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Depth of KnowledgeNorman Webb

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.

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.

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Assessment is not a single event but a continuous cycle.

Lynn Steen

Questions for differentiating the mathematics curriculaWho? 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?

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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 reasoningElicit, engage and challenge each student’s thinking Listen carefully to students’ ideasAsk students to clarify and justify their ideasAttach 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

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Results from NAEP, TIMSS and PISA

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.

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.

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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.

Try a typical fraction question.Try a typical fraction question. 6/7 + 7/8 = ________

Estimate: 6/7 + 7/8 is about

a. 1

b. 2

c. 13

d. 15

Write a word problem for 1 3/4 ÷ 1/2. Solve the problem and explain your answer.

6/7 + 7/8 = ________

Estimate: 6/7 + 7/8 is about

a. 1

b. 2

c. 13

d. 15

Write a word problem for 1 3/4 ÷ 1/2. Solve the problem and explain your answer.

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Try another.Try another.

What might this represent?What might this represent?

Linda Sheffield 20

Question the answers; don’t just answer the questions.

What if …?What if …?

this is multiplication?

this is division?

you use whole numbers?

fractions?

decimals?

algebra?

this is multiplication?

this is division?

you use whole numbers?

fractions?

decimals?

algebra?

Linda Sheffield 22

Think deeply about simple things.

When we think well, we feel good. Understanding is a kind

of ecstasy.Carl Sagan (from Broca's Brain, 1979)

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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

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AssessmentCriteria

1 Novice

2 Apprentice

3 Proficient

4 Distinguished

Depth ofUnderstanding

Little or nounderstanding

Partialunderstanding;minor mathematicalerrors

Goodunderstanding;mathematicallycorrect

In-depthunderstanding;well-developedideas

Fluency One incompleteor unworkablestrategy ortechnique

At least oneappropriatesolution withstrategy ortechniqueshown.

At least twoappropriatesolutions, may usethe same strategyor technique

Several appropriatesolutions, may usesame strategy ortechnique

Flexibility No methodapparent

At least onemethod(e. g., all graphs, allalgebraic equationsand so on)

At least twomethodsof solution (e. g.,geometric,graphical, algebraic,physical modeling)

Three or moremethods ofsolution (e. g.,geometric,graphical, algebraic,physical modeling)

Originality Method maybe differentbut does notlead to asolution

Method will lead toa solution but isfairly common

Unusual, workablemethod used byonly a few students

Unique, insightfulmethod used onlyby one or twostudents

Elaboration orElegance

Little or noappropriateexplanationgiven

Explanation isunderstandable butis unclear insome places

Clear explanationusing correctmathematical terms

Clear, concise,precise explanationsmaking good use ofgraphs, charts,models, or equations

GeneralizationsandReasoning

Nogeneralizationsmade, or they areincorrect andreasoning isunclear

At least one correctgeneralizationsmade; but notwell-supported withclear reasoning

At least onewell-made, supportedgeneralization, ormore than onecorrect butunsupportedgeneralization

Severalwell-supportedgeneralizations;clearreasoning

Extensions No relatedmathematicalquestion explored

At least one relatedmathematicalquestionappropriatelyexplored

One related questionexplored in-depth,or more than onequestionappropriatelyexplored

More than onerelatedquestion exploredin-depth

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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

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?

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%

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%

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Assessment should support continuous progress for all

students.

Assessment should support continuous progress for all

students.

We need to be sure that we are assessing what we

value the most.

We need to be sure that we are assessing what we

value the most.

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ReferencesBlack, 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, from http://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.