students can understand concepts using mathematical software december 8, 2012 gail burrill, michigan...

Students Can Understand Concepts Using Mathematical Software

December 8, 2012Gail Burrill, Michigan State University

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Thomas Dick, Oregon State University

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Wade Ellis, West Valley College (retired)

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mailto:[email protected]

mailto:[email protected]

using and choosing technologyin the mathematics classroom Driving question: Does the technology afford

real advantages to the teacher for…?



1) illustrating mathematical ideas,




2) posing mathematical problems,





3) opening opportunities for students to

engage in mathematical sense making

and reasoning,





3) opening opportunities for students to

engage in mathematical sense making

and reasoning, or

4) eliciting evidence of students’ mathematical thinking.

using and choosing technologyin the mathematics classroom Bottom line: What leverage does technology

provide to help teachers ask questions that “push” or “probe”?



PUSH

students’ mathematical thinking forward



PUSH

students’ mathematical thinking forward

PROBE

how students are thinking mathematically

categories of technology

Conveyance technologies transmit and/or receive information

Conveyance technologies are not specific to mathematics.

types of conveyance technology• Presentation interactive whiteboards, slide presentation

(e.g., powerpoint), document cameras, projectors/monitors

• Communication intranet (within classroom/school) internet (allowing extended “classroom”)

• Sharing/collaboration shared documents or workspaces• Assessment/monitoring/distribution

clickers, monitoring software for networks

categories of technology

Mathematical action technologies perform mathematical tasks and/or respond to the user’s actions in mathematically defined ways

Mathematical action technologies can play the role of another actor along with the studentsand teacher.

types of math action technology• computational/representational toolkits

graphing calculators, CAS, spreadsheets• dynamic geometry environments Geometer’s Sketchpad, Cabri• Microworlds constrained environments with mathematically

defined “rules of engagement”• Computer simulations parameter driven virtual enactments of

physical phenomena

Tools for Doing Math(technology as computational or construction task servant)vs.Tools for Developing Understanding(technology to create scenarios for insight)

Technology as Tool for Doing

Key issue for teachers: Helping students to become good managers of the technology –

*making decisions of which tool to use and when

*monitoring/interpreting results

Danger: technology-based activities that over-direct step-by-step solutions to problems

Bottom line: Need for rich problems as well as good questions that demand reasoning and sense making around solutions or strategies

Technology as Tool for Developing UnderstandingKey issue for teachers: asking good questions • Predict consequence in advance of action

(what would happen if…?)• Consider action that would produce a given

consequence (what would make … happen?)• Conjecturing/Testing/Generalization (When…?)• Justification (Why…?)

Danger: activities that simply prescribe actions and ask for recording of observations

Bottom line: need good questions that demand reflection, sense making and reasoning

Action-Consequence Principle

Technology-based learning scenarios should


Technology-based learning scenarios should• allow students to take deliberate, purposeful

and mathematically meaningful actions


Technology-based learning scenarios should• allow students to take deliberate, purposeful

and mathematically meaningful actions• provide immediate, visual and mathematically

meaningful consequences

EXAMPLESfrom fractionsto calculus

Fractions: Building Understanding

Research on learning algebra: Making links to the classroom

1988 NCTM Yearbook on Algebra: NCTM Yearbook on Algebra: Common Mistakes in AlgebraCommon Mistakes in Algebra (Marquis, 1988)(Marquis, 1988)

aa22. . bb

55 = (ab) = (ab)

77(x+4)(x+4)

22 = x = x

22+ 16 + 16

x+y yx+y y x r x+rx r x+r

x+z x+z ==

z z y y + +

s s = =

y+s y+s

3a3a-1-1= 1= 1

3a3a10 of 22 were related to fractions

* Conceptual Knowledge:– Makes connections visible, – enables reasoning about the mathematics, – less susceptible to common errors, – less prone to forgetting.

* Procedural Knowledge: – strengthens and develops understanding– allows students to concentrate on relationships

rather than just on working out results

NRC, 1999; 2001

A fraction

Is typically thought of as:• Quotient,• Part to Whole, or• Ratio

Rethinking Fractions: Based on Part 2, Fractions by H. Wu

Department of Mathematics #3840

University of California, Berkeley

Berkeley, CA 94720-3840

http://www.math.berkeley.edu/~wu/

Common Core State Standards

A fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts; a fraction a/b is the quantity formed by a parts of size 1/b.

• a fraction is a number on the number line• 2. a. Represent a fraction 1/b on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.• b. Represent a fraction a/b on a number line by marking off

a lengths of 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

CCSS, 2010

CCSS: Fractions

• Two fractions are equivalent (equal) if they are the same size, or the same point on a number line.

• Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

• Build fractions from unit fractions by applying /extending previous understandings of whole number operations.

CCSS, 2010

• A constant way to think: k/p is k copies of 1/p - the length of the concatenation of k segments each of which has length 1/p .

• Behavior similar to whole numbers: k/3 is a multiple of 1/3Larger fraction is to the right on the number line

• Connection of whole number to fractions.

• One number has many names and none more important than another.

• No difference between proper and improper fractions

What does fraction as a point on a number line buy us?

Research on learning algebra: Making links to the classroom

1988 NCTM Yearbook on Algebra: NCTM Yearbook on Algebra: Common Mistakes in AlgebraCommon Mistakes in Algebra (Marquis, 1988)(Marquis, 1988)

aa22. . bb

55 = (ab) = (ab)

77(x+4)(x+4)

22 = x = x

22+ 16 + 16

x+y yx+y y x r x+r x r x+r

x+z x+z ==

z z y y + +

s s = =

y+s y+s

3a3a-1-1= 1= 1

3a3a10 of 22 were related to fractions

References Black, P. Harrison, C., Lee, C., Marshall, E., & Wiliam, D. (2004).

“Working Inside the Black Box: Assessment for Learning in the Classroom,” Phi Delta Kappan, 86 (1), 9-21.

Common Core State Standards Mathematics. (2010). Council of Chief State School Officers & National Governors Association. www.corestandards.org/

Marquis, J. (1988). Common mistakes in algebra.The Ideas of Algebra K-12. 1988 Yearbook. Coxford, A. (Ed). Reston, VA. National Council of Teachers of Mathematics.

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

National Research Council. (1999). How People Learn: Brain, mind, experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press.

Wu, H. (2006). Fractions, Part 2. In Understanding Numbers in Elementary School Mathematics, Amer. Math. Soc., 2011.] //www.math.berkeley.edu/_wu/

students can understand concepts using mathematical software december 8, 2012 gail burrill, michigan...

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