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Mathematics
Learning
Activity
Types1,2
The purpose of
presenting an activity types taxonomy for mathematics is to introduce
the full
range of student learning activities for teachers to consider when
building
lessons that strive to effectively integrate technology, pedagogy, and
content.
In doing so, we attempt to scaffold teachers’ thinking about how to
best
structure their learning activities, best support those activities with
educational technologies, and to spark their creativity during
instructional
planning.
Essentially,
these
mathematics
activity types are designed to be catalysts
to thoughtful and creative instruction by teachers. We
have conceptualized seven genres of activity types for mathematics that
are
derived from the National Council of Teachers of Mathematics’ (NCTM’s)
process
standards. To encourage active engagement by all students, these
activity types
are expressed using active words (verbs) to focus instructional
planning on
student rather than teacher actions. Many
of these words are drawn directly from the NCTM
standards. Each of the seven genres is
presented in
a separate table that names the activity types for that genre, briefly
defines
them, and then provides some example technologies that might be
selected by a
teacher while undertaking each activity. Please note that the specific
software
titles referenced in the Possible Technologies columns are meant to be
illustrative. The taxonomy’s
authors do not specifically endorse any of the listed products.
The "Consider" Activity Types
When learning
mathematics, students are often asked to thoughtfully consider new
concepts or
information. This request is a
familiar one for the mathematics student, and is just as familiar to
the
teacher. Yet, although such learning activities can be very important
contributors to student understanding, the "Consider" activity types
also often represent some of the lower levels of student engagement,
and
typically are manifested using a relatively direct presentation of
foundational
knowledge.
Table 1: The
"Consider" Activity Types
|
Activity Type |
Brief Description |
Possible Technologies |
|
Attend to a Demonstration |
Students gain information from a
presentation, videoclip, animation, interactive whiteboard or other
display media |
Document camera, Content-Specific interactive tool (e.g., ExploreMath)
presentation or video creation software, video clips, videoconferencing |
|
Read Text |
Students extract information
from textbooks or other written materials, in either print or digital
form |
Electronic textbook, websites
(i.e. the Math Forum), informational electronic documents (e.g. .pdfs) |
|
Discuss |
Students discuss a concept or
process with a teacher, other students, or an external expert |
Ask-an-expert site (e.g., Ask
Dr. Math), online discussion group, videoconferencing |
|
Recognize a Pattern |
Students examine a pattern
presented to them and attempt to understand the pattern better |
Graphing calculators, virtual
manipulative site (e.g., the National Library of Virtual
Manipulatives), content-specific interactive tool (e.g., ExploreMath),
spreadsheet |
|
Investigate a Concept Used in:
Calculating a Monthly Car Payment (grades 11-12)
Blogging: Part to Whole (elementary)
Math All Around Us (elementary) |
Students explore or investigate
a concept (such as fractals), perhaps by use of the Internet or other
research-related resources |
Content-Specific interactive tool (e.g., ExploreMath), Web searching, informational databases (e.g.,
Wikipedia), virtual worlds (e.g., Second Life), simulations |
|
Understand or Define a Problem Used in:
Geometry at Work: Architects (high school)
Arizona Road Trip (grade 4+) |
Students strive to understand
the context of a stated problem or to define the mathematical
characteristics of a problem |
Web searching, concept mapping
software, ill-structured problem media (e.g., CIESE Projects) |
The "Practice" Activity
Types
In the learning
of mathematics, it is often very important for a student to be able to
practice
computational techniques or other algorithm-based strategies, in order
to
automate these skills for later and higher-level mathematical
applications.
Some educational technologies can provide valuable assistance in
helping
students to practice and internalize important skills and techniques. This table provides some examples of how
technology can assist in these important student practice efforts.
Table 2: The
"Practice" Activity Types
|
Activity Type |
Brief Description |
Possible Technologies |
|
Do
Computation Used in:
Arizona Road Trip (grade 4+) |
Students undertake
computation-based strategies using numeric or symbolic processing |
Scientific calculators, graphing
calculators, spreadsheet, Mathematica |
|
Do
Drill and Practice |
Students rehearse a mathematical
strategy or technique, and perhaps uses computer-aided repetition and
feedback in the practice process |
Drill and practice software,
online textbook supplement, online homework help websites (e.g.,
WebMath). |
|
Solve
a
Puzzle |
Students carry out a
mathematical strategy or technique within the context of solving an
engaging puzzle, which may be facilitated or posed by the technology |
Virtual manipulative, Web-based
puzzle (e.g., magic squares), mathematical brainteaser Web site (e.g., CoolMath) |
The "Interpret" Activity
Types
In the
discipline of mathematics, individual concepts and relationships can be
quite
abstract, and at times can even represent a bit of a mystery to
students. Often students need to spend
some time
deducing and explaining these relationships to internalize them. Educational technologies can be used to
help students investigate concepts and relationships more actively, and
assist
them in interpreting what they observe. This
table displays activity types that can support
this thoughtful
interpretation process, and provides some examples of the available
technologies that can be used to support forming the interpretations.
Table 3: The
"Interpret" Activity Types
|
Activity Type |
Brief Description |
Possible Technologies |
|
Pose a Conjecture |
The student poses a conjecture,
perhaps using dynamic software to display relationships |
Dynamic geometry software (e.g.,
Geometer’s Sketchpad), Content-specific interactive tool (e.g.,
ExploreMath), e-mail |
|
Develop an Argument |
The student develops a
mathematical argument related to why they think that something is true. Technology may help to form and to display
that argument. |
Concept mapping software,
presentation software, blog, specialized word processing software (e.g., Theorist) |
|
Categorize |
The student attempts to examine
a concept or relationship in order to categorize it into a set of known
categories |
Database software, online
database, concept mapping software, drawing software |
|
Interpret a Representation |
The student explains the
relationships apparent from a mathematical representation (table,
formula, chart, diagram, graph, picture, model, animation, etc.) |
Data visualization software
(e.g., Inspire Data), 2D and 3D animation, video clip, Global
Positioning Devices (GPS), engineering-related visualization software (e.g., MathCad) |
|
Estimate |
The student attempts to
approximate some mathematical value by further examining relationships
using supportive technologies |
Scientific calculator, graphing
calculator, spreadsheet, student response system (e.g. “clickers”) |
|
Interpret a Phenomenon
Mathematically |
Assisted by technology as
needed, the student examines a mathematics-related phenomenon (such as
velocity, acceleration, the Golden Ratio, gravity, etc.) |
Digital camera, video,
computer-aided laboratory equipment, engineering-related visualization
software, specialized word processing software (e.g., Theorist),
robotics, electronics kit |
The "Produce" Activity
Types
When students
are actively engaged in the study of mathematics, they can become
motivated
producers of mathematical works, rather than just passive consumers of
prepared
materials. Educational technologies
can serve as excellent “partners” in this production process, aiding in
the
refinement and formalization of a student product, as well as helping
the
student to share the fruits of their mathematical labors.
The activity types listed below suggest
technology-assisted efforts in which students become “producers” of
mathematics-related products.
Table 4: The
"Produce" Activity Types
|
Activity Type |
Brief Description |
Possible Technologies |
|
Do a Demonstration
Used in:
Blogging: Part to Whole (elementary) |
The student makes a
demonstration on some topic to show their understanding of a
mathematical idea or process. Technology
may assist in the development or presentation of the product. |
Interactive whiteboard, video
creation software, document
camera, presentation software, podcast, video sharing site |
|
Generate Text |
The student produces a report,
annotation, explanation, journal entry or document, to illustrate their
understanding. |
Specialized word processing
software (e.g., Math Type), collaborative word processing software,
blog, online discussion group |
|
Describe an Object or Concept
Mathematically
Used in:
Math All Around Us (elementary) |
Assisted by the technology in
the description or documentation process, the student produces a
mathematical explanation of an object or concept |
Logo graphics, engineering
visualization software, concept mapping software, specialized word
processing software, Mathematica |
|
Produce a Representation
Used in:
Geometry at Work: Architects (high school)
Build Your Own Business (grades 5-8) |
Using technology for production
assistance if appropriate, the student develops a mathematical
representation (table, formula, chart, diagram, graph, picture, model,
animation, etc.) |
Spreadsheet, virtual
manipulatives (e.g., digital geoboard), document camera, concept
mapping software, graphing calculator |
|
Develop a Problem |
The student poses a mathematical
problem that is illustrative of some mathematical concept,
relationship, or investigative question |
Word processing software, online
discussion group, Wikipedia, Web searching, e-mail |
The "Apply" Activity Types
The utility of
mathematics in the world can be found in its authentic application. Educational technologies can be used to
help students to apply their mathematical knowledge in the real world,
and to
link specific mathematical concepts to real world phenomena. The technologies essentially become
students’ assistants in their mathematical work, helping them to link
the
mathematical concepts being studied to the reality in which they live.
Table 5: The "Apply"
Activity Types
|
Activity Type |
Brief Description |
Possible Technologies |
|
Choose a Strategy |
The student reviews or selects a
mathematics-related strategy for a particular context or application. |
Online help sites (e.g.,
WebMath, Math Forum), Inspire Data, dynamic geometry/algebra software
(e.g., Geometry Expressions), Mathematica, MathCAD |
|
Take a Test |
The student demonstrates their
mathematical knowledge within the context of a testing environment,
such as with computer-assisted testing software. |
Test-taking software,
Blackboard, online survey software, student response system (e.g.
“clickers”) |
|
Apply a Representation |
The student applies a
mathematical representation to a real life situation (table, formula,
chart, diagram, graph, picture, model, animation, etc.).
|
Spreadsheet, robotics, graphing
calculator, computer-aided laboratories, virtual manipulatives (e.g.,
electronic algebra tiles) |
The "Evaluate" Activity
Types
When students
evaluate the mathematical work of others, or self-evaluate their own
mathematical work, they engage in a relatively sophisticated effort to
try to understand
mathematical concepts and processes. Educational
technologies can become valuable allies
in this effort,
assisting students in the evaluation process by helping them to
undertake
concept comparisons, test solutions or conjectures, and/or integrate
feedback
from other individuals into revisions of their work.
The following table lists some of these
evaluation-related activities.
Table 6: The "Evaluate" Activity Types
|
Activity Type |
Brief Description |
Possible Technologies |
|
Compare and Contrast
Used in:
Calculating a Monthly Car Payment (grades 11-12) |
The student compares and
contrasts different mathematical strategies or concepts, to see which
is more appropriate for a particular situation. |
Concept mapping software (e.g.,
Inspiration), Web searches, Mathematica, MathCad |
|
Test a Solution
Used in:
Build Your Own Business (grades 5-8) |
The student systematically tests
a solution, and examines whether it makes sense based upon systematic
feedback, which might be assisted by technology. |
Scientific calculator, graphing
calculator, spreadsheet, Mathematica, Geometry Expressions |
|
Test a Conjecture |
The student poses a specific
conjecture and then examines the feedback of any interactive results to
potentially refine the conjecture. |
Geometer Sketchpad,
content-specific interactive tool (e.g., ExploreMath), statistical
packages (e.g., SPSS, Fathom), online calculator, robotics |
|
Evaluate Mathematical Work |
The student evaluates a body of
mathematical work, through the use of peer or technology-aided
feedback. |
Online discussion group, blog,
Mathematica, MathCad, Inspire Data |
The "Create" Activity Types
When students
are involved in some of the highest levels of mathematics learning
activities,
they are often engaged in very creative and imaginative thinking
processes.
Albert Einstein once suggested that
“imagination is
more important than knowledge.” It
is said that this quote represents his strong belief that mathematics
is a very
inventive, inspired, and imaginative endeavor. Educational
technologies
can be used to
help students to be creative in their mathematical work, and even to
help other
students to deepen their learning of the mathematics that they already
understand. The activity types
below represent these creative elements and processes in students’
mathematical
learning and interaction.
Table 7: The "Create"
Activity Types
|
Activity Type |
Brief Description |
Example Technologies |
|
Teach a Lesson |
The student develops and
delivers a lesson on a particular mathematics concept, strategy, or
problem. |
Document camera, presentation
software, videoconferencing, video creation software, podcast |
|
Create a Plan Used in:
Build Your Own Business (grades 5-8) |
The student develops a
systematic plan to address some mathematical problem or task. |
Concept mapping software,
collaborative word processing software, MathCad, Mathematica |
|
Create a Product Used in:
Arizona Road Trip (grade 4+)
Blogging: Part to Whole (elementary) |
The student imaginatively
engages in the development of a student project, invention, or
artifact, such as a new fractal, a tessellation, or another creative
product. |
Word processing software,
videocamera, animation tools, MathCad, Mathematica, Geometer Sketchpad |
|
Create a Process |
The student creates a
mathematical process that others might use, test or replicate,
essentially engaging in mathematical creativity. |
Computer programming, robotics,
Mathematica, MathCad, Inspire Data, video creation software |
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