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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)


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)


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)


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


Grandgenett, N., Harris, J., & Hofer, M. (2011, February). Mathematics learning activity types. Retrieved from College of William and Mary, School of Education, Learning Activity Types: http://activitytypes.wm.edu/Math.html

   “Mathematics Learning Activity Types” by Neal Grandgenett, Judi Harris and Mark Hofer is licensed under a
Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License
. Based on a work at http://activitytypes.wm.edu.