Students Can Understand
Concepts Using Mathematical
Software
December 8, 2012
Gail Burrill, Michigan State University
burrill@msu.edu
Thomas Dick, Oregon State University
tpdick@math.oregonstate.edu
Wade Ellis, West Valley College (retired)
wellis@ti.com
using and choosing technology
in the mathematics classroom
Driving question: Does the technology afford
real advantages to the teacher for…?
using and choosing technology
in the mathematics classroom
Driving question: Does the technology afford
real advantages to the teacher for…?
1) illustrating mathematical ideas,
using and choosing technology
in the mathematics classroom
Driving question: Does the technology afford
real advantages to the teacher for…?
1) illustrating mathematical ideas,
2) posing mathematical problems,
using and choosing technology
in 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,
using and choosing technology
in 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, or
4) eliciting evidence of students’ mathematical
thinking.
using and choosing technology
in the mathematics classroom
Bottom line: What leverage does technology
provide to help teachers ask questions that
“push” or “probe”?
using and choosing technology
in the mathematics classroom
Bottom line: What leverage does technology
provide to help teachers ask questions that
“push” or “probe”?
PUSH
students’ mathematical thinking forward
using and choosing technology
in the mathematics classroom
Bottom line: What leverage does technology
provide to help teachers ask questions that
“push” or “probe”?
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 students
and 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 overdirect 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 Understanding
Key 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
Action-Consequence Principle
Technology-based learning scenarios should
• allow students to take deliberate, purposeful
and mathematically meaningful actions
Action-Consequence Principle
Technology-based learning scenarios should
• allow students to take deliberate, purposeful
and mathematically meaningful actions
• provide immediate, visual and mathematically
meaningful consequences
EXAMPLES
from fractions
to calculus
Fractions:
Building Understanding
Research on learning algebra:
Making links to the classroom
1988 NCTM Yearbook on Algebra:
Common Mistakes in Algebra (Marquis, 1988)
2. 5
a
x+y
x+z
b = (ab)
= y
z
7
2
2
(x+4) = x + 16
x
+ r = x+r
y
s
y+s
3a-1=
10 of 22 were related to
fractions
1
3a
* 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
What does fraction as a point on a
number line buy us?
• 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 1/3
Larger 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
Research on learning algebra:
Making links to the classroom
1988 NCTM Yearbook on Algebra:
Common Mistakes in Algebra (Marquis, 1988)
2. 5
a
x+y
x+z
b = (ab)
=y
z
7
2
2
(x+4) = x + 16
x + r
y
s
= x+r
y+s
3a-1= 1
3a
10 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/