Development of the Algebra II Units

The Teaching PrincipleThe Teaching Principle

Effective teaching requires Effective teaching requires understanding what understanding what ALL ALL students students

know and need to learn and know and need to learn and challenging and supporting them to challenging and supporting them to

learn it welllearn it well.. (NCTM, 2000)(NCTM, 2000)

Slide developed by Gail Burrill, MSU 2009

TIMSS Video Study - TasksTIMSS Video Study - Tasks

Stigler & Hiebert, 2004Slide developed by Gail Burrill, MSU 2009

TIMSS Video Study - TIMSS Video Study - ImplementationImplementation

Stigler & Hiebert, 2004Slide developed by Gail Burrill, MSU 2009

Students learn ifStudents learn if

they are actively involved in choosing they are actively involved in choosing and evaluating strategies, considering and evaluating strategies, considering assumptions, and receiving feedback. assumptions, and receiving feedback.

they encounter contrasting cases- they encounter contrasting cases- notice new features and identify notice new features and identify

important ones.important ones. National Research Council, 1999

Slide developed by Gail Burrill, MSU 2009

Which shape will hold the same amount Which shape will hold the same amount of spaghetti and be the most of spaghetti and be the most

economical?economical?

Area of Area of basebase

Surface Surface areaarea

Volume Volume Ratio of Ratio of surface area surface area to volumeto volume

CylinderCylinder

Rectangular Rectangular prismprism

Shape 3Shape 3MDE, 2003 Slide developed by Gail Burrill, MSU 2009

Technology is often usedTechnology is often used

For For PerformancePerformance::- To carry out difficult computations or To carry out difficult computations or

manipulationmanipulation- To make the graph or find the equation of a line To make the graph or find the equation of a line

- After the content has been taughtAfter the content has been taught

- Review, consolidate concepts- Review, consolidate concepts

NOT as a Tool for NOT as a Tool for LearningLearning

Slide developed by Gail Burrill, MSU 2009

Technology canTechnology can

explicitly engage teachers in exploration, inquiry, and conjecture with their students

around specific mathematical concepts.

create new opportunities for students to reason and make sense out of the mathematics world

Slide developed by Gail Burrill, MSU 2009

Ways technology materials have Ways technology materials have gone astraygone astray

Emphasis on mastering the technology - mathematics is Emphasis on mastering the technology - mathematics is of secondary importanceof secondary importance

Demonstration of concept - students are spectatorsDemonstration of concept - students are spectators

Revisiting a topic in a simple way - studentsRevisiting a topic in a simple way - students ’’ role is role is verificationverification

Replicating activities from the point of view of current Replicating activities from the point of view of current instructional materials, underutilizing the technologyinstructional materials, underutilizing the technology ’’s s

potentialpotential

Fragmented ideasFragmented ideas - - obtaining a formula as an objectiveobtaining a formula as an objective(adapted from Belfort & Guimaraes, 2004)(adapted from Belfort & Guimaraes, 2004)

Slide developed by Gail Burrill, MSU 2009

A Possible SolutionA Possible Solution

Use technology to provide different learning Use technology to provide different learning experiences that allow students to interact with experiences that allow students to interact with

the mathematics in new ways. the mathematics in new ways. Structure lessons that take account of well Structure lessons that take account of well

documented trouble spots.documented trouble spots.Ask questions that reveal student thinking - Ask questions that reveal student thinking - questions that promote reasoning and sense questions that promote reasoning and sense

making.making.

Slide developed by Gail Burrill, MSU 2009

In the context of:In the context of:

The tasks in which students engageThe tasks in which students engageDeveloping conceptual understanding as Developing conceptual understanding as

well as supporting fluencywell as supporting fluencyWhat teachers do with a taskWhat teachers do with a task

What questions teachers ask and howWhat questions teachers ask and howWhat questions teachers answer and howWhat questions teachers answer and how

What research tells us about learningWhat research tells us about learning

Slide developed by Gail Burrill, MSU 2009

Reasoning/Sense Making QuestionsReasoning/Sense Making QuestionsCompare and contrastCompare and contrast:: How are they alike? How How are they alike? How

different?different?

PPredict forwardredict forward:: “What would happen if . . ?” “What would happen if . . ?”

Predict backwardPredict backward:: “How can I make . . happen?” “Is it “How can I make . . happen?” “Is it possible to ... ?”possible to ... ?”

Analyze a connection/relationshipAnalyze a connection/relationship: “When will . . . be : “When will . . . be (larger,equal to, exactly twice, …) compared to . . .?” “When (larger,equal to, exactly twice, …) compared to . . .?” “When

will . . be as big as possible?”will . . be as big as possible?”

Generalize/make conjecturesGeneralize/make conjectures: “When does . . . work?” : “When does . . . work?” “Under what conditions does … behave this way?” “Describe “Under what conditions does … behave this way?” “Describe

how to find . . .?” “Is this always true?”how to find . . .?” “Is this always true?”

Justify/proveJustify/prove mathematically mathematically: “Why does . . . work?”: “Why does . . . work?”

Consider assumptionsConsider assumptions inherent in the problem and what inherent in the problem and what would happen if they were changedwould happen if they were changed

Interpret information, make/ justify conclusionsInterpret information, make/ justify conclusions: : “The data support… ;“The data support… ; “ “This… will make ….happen because…”This… will make ….happen because…”

Burrill & Dick, 2008Slide developed by Gail Burrill, MSU 2009

“…“…when teachers learn to see and hear when teachers learn to see and hear students’ work during a lesson and to use students’ work during a lesson and to use

that information to shape their instruction, that information to shape their instruction, instruction becomes clearer, more focused, instruction becomes clearer, more focused,

and more effective.” and more effective.” (NRC, 2001, p.350 )(NRC, 2001, p.350 )

Managing DiscussionsManaging Discussions

Slide developed by Gail Burrill, MSU 2009

How Students Learn Mathematics in How Students Learn Mathematics in the Classroomthe Classroom

Principle #1:Principle #1: Teachers Must Engage Students’ Teachers Must Engage Students’ PreconceptionsPreconceptions

Principle #2:Principle #2: Understanding Requires Factual Understanding Requires Factual Knowledge and Conceptual FrameworksKnowledge and Conceptual Frameworks

Principle #3:Principle #3: A Metacognitive Approach A Metacognitive Approach Enables Student Self-MonitoringEnables Student Self-Monitoring

NCR 2004NCR 2004

Implications for teaching:Implications for teaching:Recognize that many students are Recognize that many students are

learning for the moment not for learning for the moment not for understanding and retention.understanding and retention.

Take advantage of technology that can Take advantage of technology that can allow students into math in new ways.allow students into math in new ways.

Focus on the questions, plan for them, Focus on the questions, plan for them, and use them to create classrooms and use them to create classrooms where the focus is on developing where the focus is on developing

understanding of the math understanding of the math Slide developed by Gail Burrill, MSU 2009

ReferencesReferencesBelfort, E. & Guimaraes, C. (2004). Teachers' practices and dynamic Belfort, E. & Guimaraes, C. (2004). Teachers' practices and dynamic geometry. In Hoines, M. & Fuglestad, A. (Eds.), geometry. In Hoines, M. & Fuglestad, A. (Eds.), Proceedings of the 28Proceedings of the 28thth Conference of the International Group for the Psychology of Mathematics Conference of the International Group for the Psychology of Mathematics EducationEducation. (2, pp. 503-510).. (2, pp. 503-510).Burrill, G. & Dick, T. (2008). What state assessments tell us about Burrill, G. & Dick, T. (2008). What state assessments tell us about student achievement in algebrastudent achievement in algebra. . Paper presented at NCTM 2008 Research Paper presented at NCTM 2008 Research PresessionPresessionFlorida Department of Education (2006). FCAT Mathematics Released Florida Department of Education (2006). FCAT Mathematics Released Items, Grade 9.Items, Grade 9.Maine Department of Education. Maine Educational Assessment (MEA) Maine Department of Education. Maine Educational Assessment (MEA) Released Items 2004, 2005. Mathematics Augmentation Sample Items.Released Items 2004, 2005. Mathematics Augmentation Sample Items.Michigan Department of Education. (2003). MMLA Lesson Study Michigan Department of Education. (2003). MMLA Lesson Study Project. Burrill, G., Ferry, D., & Verhey R. (Eds). Lansing, MIProject. Burrill, G., Ferry, D., & Verhey R. (Eds). Lansing, MIMichigan Department of Education. (2007). Released items mathematics Michigan Department of Education. (2007). Released items mathematics grades 7, 8 fall. grades 7, 8 fall. www.michigan.gov/mde/0,1607,7-140-22709_31168_31355-95470--,00.html National Assessment for Educational Progress (2005). Released Item. National Assessment for Educational Progress (2005). Released Item. National Center for Educational Statistics. (2005). National Center for Educational Statistics. (2005).

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National Council of Teachers of Mathematics. (2000). National Council of Teachers of Mathematics. (2000). Principles and Principles and Standards for School MathematicsStandards for School Mathematics. Reston VA: The Council. Reston VA: The Council

National Research Council (2001). National Research Council (2001). Adding It UpAdding It Up. Kilpatrick, J., . Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web at Press. Also available on the web at www.nap.edu..

New Hampshire Department of Education. (2006). End of Grade 10 New Hampshire Department of Education. (2006). End of Grade 10 Common Items Released ItemsCommon Items Released Items

Ohio Department of Education. (2006). Released end of course tests Ohio Department of Education. (2006). Released end of course tests algebra & geometry, Ohio Graduation Test. Spring. Columbus OH.algebra & geometry, Ohio Graduation Test. Spring. Columbus OH.

Ohio Department of Education. (2006). Ohio graduation tests item Ohio Department of Education. (2006). Ohio graduation tests item analysis. March. Columbus OH.analysis. March. Columbus OH.

Schwartz, & Branford, J. (2000) A time for telling. Schwartz, & Branford, J. (2000) A time for telling. Cognition and Cognition and Instruction. Instruction.

Stigler, J. & Hiebert, J. (2004). Improving Mathematics Teaching, Stigler, J. & Hiebert, J. (2004). Improving Mathematics Teaching,

Improving Achievement in Math and Science, 61(5), p. 12 – 17Improving Achievement in Math and Science, 61(5), p. 12 – 17

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Texas Education Agency (2004). TAKS Texas assessment of knowledge Texas Education Agency (2004). TAKS Texas assessment of knowledge and skills. Spring. Exit level mathematics, pp. 37-74. Austin TX.and skills. Spring. Exit level mathematics, pp. 37-74. Austin TX.

Texas Education Agency (2004). TAKS. Texas assessment of knowledge Texas Education Agency (2004). TAKS. Texas assessment of knowledge and skills item analysis summary report, all students. Spring. Austin TX.and skills item analysis summary report, all students. Spring. Austin TX.

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