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<div class=Section1>
 
<p class=MsoNormal align=center style='text-align:center'><span
style='font-size:14.0pt;font-family:"Times New Roman"'><b>Mathematics Learning
Activity Types</b></span><a style='mso-footnote-id:ftn1' href="#_ftn1"
name="_ftnref1" title=""><span class=MsoFootnoteReference><span
style='font-size:14.0pt;font-family:"Times New Roman"'><span style='mso-special-character:
footnote'><![if !supportFootnotes]>[1]<![endif]></span></span></span></a><span
style='font-size:14.0pt;font-family:"Times New Roman"'><sup>, 2</sup><b><o:p></o:p></b></span></p>
 
<p class=MsoNormal align=center style='text-align:center'><span
style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>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. <o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>Essentially,
these mathematics activity types are designed to be <i>catalysts</i></span><span
style='font-family:"Times New Roman"'> to thoughtful and creative instruction
by teachers. We have conceptualized seven genres of activity types for
mathematics that are derived from the NCTM's process standards. To encourage
active engagement by all students, these activity types are expressed using
active words (verbs) to represent the pursuit of a dynamic and student-centered
learning environment.<span style="mso-spacerun: yes">&nbsp; </span>Many of these
words are drawn directly from the NCTM standards.<span style="mso-spacerun:
yes">&nbsp; </span>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. <o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<h1>The &quot;Consider&quot; Activity Types</h1>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>When learning
mathematics, students are often asked to thoughtfully consider new concepts or
information.<span style="mso-spacerun: yes">&nbsp; </span>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 &quot;Consider&quot; 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. <o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 1: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Consider&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
 
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example
  Technologies<o:p></o:p></b></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Attend to a
  Demonstration<o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students gain
  information from a presentation, videoclip, animation, interactive whiteboard
  or other display media <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Powerpoint,
  iMovie, YouTube, podcasts, videoconferencing, or other display media <o:p></o:p></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Read Text<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  extract information from textbooks, or other written materials, in either
  print or digital form <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Electronic
  textbooks, websites (i.e. the Math Forum), informational .pdfs <o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:22.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Discuss<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  discuss a concept or process with a teacher, other students, or an external
  expert<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Ask-an-expert
  sites (e.g., Ask Dr. Math), online discussion groups, videoconferencing<span
  style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:44.0pt'>
 
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:44.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Recognize a
  Pattern<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:44.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  examine a pattern presented to them and attempt to understand the pattern
  better<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:44.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Graphing
  calculators, virtual manipulative sites (e.g., the National Library of
  Virtual Manipulatives), spreadsheets<o:p></o:p></span></p>
 
  </td>
 </tr>
 <tr style='height:22.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Investigate a
  Concept<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  explore or investigate a concept (such as fractals), perhaps by use of the
  Internet or other research-related resources<o:p></o:p></span></p>
 
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Web searching,
  informational databases (Wikipedia), virtual worlds (Second Life),
  simulations<span style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:22.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Understand or
  Define a Problem<o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  strive to understand the context of a stated problem or to define the
  mathematical characteristics of a problem<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Web searching,
  concept mapping software, ill-structured problem media (i.e. Jasper Woodbury)<o:p></o:p></span></p>
  </td>
 </tr>
 
</table>
 
<p class=MsoNormal style='margin-left:5.4pt'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>The
&quot;Practice&quot; Activity Types<o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>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 application. Some
educational technologies can provide valuable assistance in helping students to
practice and internalize important skills and techniques.<span
style="mso-spacerun: yes">&nbsp; </span>This table provides some examples of
how technology can assist in these important student practice efforts.<o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 2: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Practice&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
 
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example Technologies<o:p></o:p></b></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><span
  style="mso-spacerun: yes">&nbsp;</span>Do Computation<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  undertake computation-based strategies using numeric or symbolic processing <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Scientific
  calculators, graphing calculators, spreadsheets, Mathematica<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><span
  style="mso-spacerun: yes">&nbsp;</span>Do Drill and Practice<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Students
  rehearse a mathematical strategy or technique, and perhaps uses
  computer-aided repetition and feedback in the practice process <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Mathblaster
  drill and practice software, online textbook supplements, online homework
  help websites (WebMath). <o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:22.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><span
  style="mso-spacerun: yes">&nbsp;</span>Solve a Puzzle<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>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 <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Virtual
  manipulatives, Web-based puzzles (magic squares), brainteaser Web sites
  (CoolMath)<o:p></o:p></span></p>
 
  </td>
 </tr>
</table>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><span
style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>The
&quot;Interpret&quot; Activity Types<o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>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.<span
style="mso-spacerun: yes">&nbsp; </span>Often students need to spend some time
deducing and explaining these relationships to internalize them.<span
style="mso-spacerun: yes">&nbsp; </span>Educational technologies can be used to
help students investigate concepts and relationships more actively, and assist
them in interpreting what they observe.<span style="mso-spacerun: yes">&nbsp;
</span>This table displays activity types that can support this thoughtful
interpretation process, and gives some examples of the available technologies
that can be used to support forming the interpretations.<span
style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 3: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Interpret&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example
  Technologies<o:p></o:p></b></span></p>
 
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Pose a
  Conjecture<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  poses a conjecture, perhaps using dynamic software to display relationships<o:p></o:p></span></p>
 
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Dynamic
  geometry software (Geometer’s Sketchpad), widgets (Explore Learning), e-mail<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Develop an
  Argument<o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  develops a mathematical argument related to why they think that something is
  true.<span style="mso-spacerun: yes">&nbsp; </span>Technology may help to
  form and to display that argument.<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Concept
  mapping software (Inspiration), presentation software, blogs, specialized
  word processing software (Theorist), e-mail<o:p></o:p></span></p>
  </td>
 
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Categorize<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  attempts to examine a concept or relationship in order to categorize it into
  a set of known categories<o:p></o:p></span></p>
  </td>
 
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Database
  software (Microsoft Access), online databases, concept mapping software,
  drawing software<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:22.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Interpret a
  Representation<o:p></o:p></span></p>
  </td>
 
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  explains the relationships apparent from a mathematical representation
  (table, formula, chart, diagram, graph, picture, model, animation, etc.) <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Data
  visualization software (Inspire Data), 2D and 3D animations, video (iMovie),
  Global Positioning Devices (GPS), engineering visualization software
  (MathCad) <o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:22.0pt'>
 
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Estimate<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  attempts to approximate some mathematical value, by further examining
  relationships using supportive technologies <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Scientific
  calculator, graphing calculator, spreadsheets, student response systems
  (Clickers)<o:p></o:p></span></p>
 
  </td>
 </tr>
 <tr style='height:22.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Interpret a
  Phenomenon Mathematically<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Assisted by
  technology as needed, the student examines a mathematics related phenomenon
  (such as velocity, acceleration, the Golden Ratio, gravity, etc.) <o:p></o:p></span></p>
 
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:22.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Digital
  cameras, video, computer-aided laboratory equipment, engineering
  visualization software, specialized word processing, robotics, electronics
  kits<o:p></o:p></span></p>
  </td>
 </tr>
</table>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>The
&quot;Produce&quot; Activity Types<o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>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.<span style="mso-spacerun: yes">&nbsp; </span>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. <span
style="mso-spacerun: yes">&nbsp;</span>The activity types listed below suggest
technology-assisted efforts in which students become “producers” of
mathematics-related products. <o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 4: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Produce&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
 
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example
  Technologies<o:p></o:p></b></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Do a
  Demonstration<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  makes a demonstration on some topic to show their understanding of a
  mathematical idea or process.<span style="mso-spacerun: yes">&nbsp;
  </span>Technology may assist in the development or presentation of the
  product. <o:p></o:p></span></p>
 
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Interactive
  whiteboard, video (YouTube), document camera, presentation software, podcasts<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Generate Text<o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  produces a report, annotation, explanation, journal entry or document, to
  illustrate their understanding. <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Specialized
  word processing (Math Type), collaborative documents (Google docs), blogs,
  online discussion groups <o:p></o:p></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Describe an
  Object or Concept Mathematically<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Assisted by
  the technology in the description or documentation process, the student
  produces a mathematical explanation of an object or concept<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Logo graphics,
  engineering visualization software, concept mapping software, specialized
  word processing, Mathematica<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Produce a
  Representation <o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Using
  technology for production assistance if appropriate, the student develops a
  mathematical representation (table, formula, chart, diagram, graph, picture,
  model, animation, etc.)<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Spreadsheet,
  virtual manipulatives (digital geoboard), spreadsheets, Inspire Data, concept
  mapping software, graphing calculator <o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Develop a
  Problem<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  poses a mathematical problem that is illustrative of some mathematical
  concept, relationship, or investigative question<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Word
  processing, online discussion groups, Wikipedia, Web searching, e-mail <o:p></o:p></span></p>
 
  </td>
 </tr>
</table>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<span style='font-size:12.0pt;font-family:"Times New Roman";mso-ansi-language:
EN-US;mso-fareast-language:EN-US'><b><br clear=ALL style='page-break-before:
always'>
</b></span>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>The
&quot;Apply&quot; Activity Types<o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>The utility of
mathematics in the world can be found in its authentic application.<span
style="mso-spacerun: yes">&nbsp; </span>Educational technologies can be used to
help students to apply their mathematics in the real world, and to link
mathematical concepts to real world phenomena.<span style="mso-spacerun:
yes">&nbsp; </span>The technologies essentially become students’ assistants in
their mathematical work, helping them to link mathematical concepts to the
reality in which they live.<span style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 5: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Apply&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
 
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example
  Technologies<o:p></o:p></b></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Choose a
  Strategy <o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  reviews or selects a mathematics related strategy for a particular context or
  application.<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Online help
  sites (WebMath, Math Forum), Inspire Data, dynamic geometry/algebra software
  (Geometry Expressions), Mathematica, MathCAD<o:p></o:p></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Take a Test<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  applies their mathematical knowledge within the context of a testing
  environment, such as with computer-assisted testing software.<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Test-taking
  software, Blackboard, survey software, student response systems<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Apply a
  Representation<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  applies a mathematical representation to a real life situation (table,
  formula, chart, diagram, graph, picture, model, animation, etc.).<span
  style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Spreadsheet,
  robotics, graphing calculator, computer-aided laboratories, virtual
  manipulatives (algebra tiles) <o:p></o:p></span></p>
  </td>
 </tr>
</table>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>The
&quot;Evaluate&quot; Activity Types<o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>When students
evaluate the mathematical work of others, or self-evaluate their own
mathematical work, they utilize a relatively sophisticated understanding of
mathematical concepts and processes.<span style="mso-spacerun: yes">&nbsp;
</span>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.<span style="mso-spacerun:
yes">&nbsp; </span>The following table lists some of these evaluation-related
activities.<span style="mso-spacerun: yes">&nbsp; </span><o:p></o:p></span></p>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 6: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Evaluate&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example
  Technologies<o:p></o:p></b></span></p>
 
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Compare and
  Contrast<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  compares and contrasts different mathematical strategies or concepts, to see
  which is more appropriate for a particular situation.<o:p></o:p></span></p>
 
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Inspiration,
  Web searches, Mathematica, MathCad<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Test a
  Solution<o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  systematically tests a solution, and examines whether it makes sense based
  upon systematic feedback, which might be assisted by technology.<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Scientific
  calculator, graphing calculator, spreadsheet, Mathematica, Geometry Expressions<o:p></o:p></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Test a
  Conjecture<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  poses a specific conjecture and then examines the feedback of any interactive
  results to potentially refine the conjecture. <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Geometer
  Sketchpad, statistical packages (e.g/, SPSS, Fathom), online calculators,
  robotics <o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Evaluate
  Mathematical Work<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  evaluates a body of mathematical work, through the use of peer or technology-
  aided feedback. <o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Online
  discussion groups, blogs, Mathematica, MathCad, Inspire Data<o:p></o:p></span></p>
  </td>
 </tr>
</table>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>The
&quot;Create&quot; Activity Types<o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'>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.”<span style="mso-spacerun: yes">&nbsp; </span>It is said that this quote
represents his strong belief that mathematics is a very inventive, inspired,
and imaginative endeavor.<span style="mso-spacerun: yes">&nbsp;
</span>Educational technologies can be used to help students to be creative in
their mathematical work, and even to help other students to learn the mathematics
that they already know.<span style="mso-spacerun: yes">&nbsp; </span>The
activity types below represent these creative elements and processes in
students’ mathematical learning and interaction. <o:p></o:p></span></p>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Table 7: </b></span><span
style='font-family:"Times New Roman"'>The &quot;Create&quot; Activity Types<b><o:p></o:p></b></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 bgcolor=white style='margin-left:
 5.4pt;background:white;border-collapse:collapse;mso-table-layout-alt:fixed;
 mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr style='height:11.75pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;background:#CCCCCC;
  padding:0in 0in 0in 0in;height:11.75pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Activity
  Type<o:p></o:p></b></span></p>
  </td>
  <td width=203 style='width:202.5pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Brief
  Description<o:p></o:p></b></span></p>
  </td>
  <td width=148 style='width:148.1pt;border:solid black .25pt;border-left:none;
  mso-border-left-alt:solid black .25pt;background:#CCCCCC;padding:0in 0in 0in 0in;
  height:11.75pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'><b>Example
  Technologies<o:p></o:p></b></span></p>
 
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Teach a Lesson<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  develops and delivers a lesson on a particular mathematics concept, strategy,
  or problem.<o:p></o:p></span></p>
 
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Presentation
  software, interactive video, video, podcasts<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Create a Plan<o:p></o:p></span></p>
 
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  develops a systematic plan to address some mathematical problem or task.<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Concept
  mapping software, collaborative writing software, MathCad, Mathematica<o:p></o:p></span></p>
  </td>
 </tr>
 
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Create a
  Product<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  imaginatively engages in the development of a student project, invention, or
  artifact, such as a new fractal, tessellation, or other creative product.<o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Word
  processor, animation tools, MathCad, Mathematica, Geometer Sketchpad<o:p></o:p></span></p>
  </td>
 </tr>
 <tr style='height:33.0pt'>
  <td width=112 style='width:111.75pt;border:solid black .25pt;border-top:none;
  mso-border-top-alt:solid black .25pt;padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Create a
  Process<o:p></o:p></span></p>
  </td>
  <td width=203 valign=top style='width:202.5pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
 
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>The student
  creates a mathematical process that others might use, test or replicate,
  essentially engaging in mathematical creativity.<span style="mso-spacerun:
  yes">&nbsp; </span><o:p></o:p></span></p>
  </td>
  <td width=148 valign=top style='width:148.1pt;border-top:none;border-left:
  none;border-bottom:solid black .25pt;border-right:solid black .25pt;
  mso-border-top-alt:solid black .25pt;mso-border-left-alt:solid black .25pt;
  padding:0in 0in 0in 0in;height:33.0pt'>
  <p class=MsoNormal><span style='font-family:"Times New Roman"'>Computer programming,
  robotics, Mathematica, MathCad, Inspire Data, iMovie <o:p></o:p></span></p>
  </td>
 </tr>
</table>
 
<p class=MsoNormal style='text-indent:.5in'><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=MsoNormal><span style='font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
</div>
 
<div style='mso-element:footnote-list'><![if !supportFootnotes]><br clear=all>
 
<hr align=left size=1 width="33%">
 
<![endif]>
 
<div style='mso-element:footnote' id=ftn1>
 
<p class=Body style='margin-left:13.5pt;text-indent:-13.5pt;tab-stops:13.5pt'><a
style='mso-footnote-id:ftn1' href="#_ftnref1" name="_ftn1" title=""><span
class=MsoFootnoteReference><span style='font-size:10.0pt;font-family:"Times New Roman"'><span
style='mso-special-character:footnote'><![if !supportFootnotes]>[1]<![endif]></span></span></span></a><span
style='font-size:10.0pt;font-family:"Times New Roman"'><span
style="mso-spacerun: yes">&nbsp;&nbsp;&nbsp; </span></span><span
style='font-size:9.0pt;font-family:"Times New Roman"'>Suggested citation (APA
format, 6<sup>th</sup> ed.):<o:p></o:p></span></p>
 
<p class=Body style='margin-left:13.5pt;text-indent:-13.5pt;tab-stops:13.5pt'><span
style='font-size:9.0pt;font-family:"Times New Roman"'>Grandgenett, N., Harris,
J., &amp; Hofer, M. (2009, February). <i>Mathematics learning activity types</i></span><span
style='font-size:9.0pt;font-family:"Times New Roman"'>. Retrieved from College
of William and Mary, School of Education, Learning Activity Types Wiki: </span><span
style='font-size:9.0pt;font-family:"Times New Roman";color:blue'>http://activitytypes.wmwikis.net/file/view/MathLearningATs-Feb09.pdf</span><span
style='font-size:9.0pt;font-family:"Times New Roman"'> <o:p></o:p></span></p>
 
<table border=0 cellspacing=0 cellpadding=0 width=473 style='width:473.4pt;
 margin-left:5.4pt;border-collapse:collapse;mso-padding-alt:0in 5.4pt 0in 5.4pt'>
 <tr>
  <td width=13 valign=top style='width:13.3pt;padding:0in 5.4pt 0in 5.4pt'>
  <p class=Body style='margin-top:0in;margin-right:-.9pt;margin-bottom:0in;
  margin-left:-5.4pt;margin-bottom:.0001pt'><span style='font-size:9.0pt;
  font-family:"Times New Roman"'><sup>2 </sup></span><span style='font-size:
  9.0pt'><o:p></o:p></span></p>
 
  </td>
  <td width=386 valign=top style='width:386.25pt;padding:0in 5.4pt 0in 5.4pt'>
  <p class=MsoNormal style='margin-left:-5.4pt'><span style='font-size:9.0pt;
  font-family:"Times New Roman"'>“Mathematics Learning Activity Types” by Neal
  Grandgenett, Judi Harris and Mark Hofer is licensed under a <a
  href="http://creativecommons.org/licenses/by-nc-nd/3.0/us/">Creative Commons
  Attribution-Noncommercial-No Derivative Works 3.0 United States License</a>.
  Based on a work at <a href="http://activitytypes.wmwikis.net/">activitytypes.wmwikis.net</a>.</span><span
  style='font-size:9.0pt'><o:p></o:p></span></p>
  </td>
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  style='font-size:9.0pt;font-family:"Times New Roman"'><o:p></o:p></span></p>
 
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 </tr>
</table>
 
<p class=Body><span style='font-size:10.0pt;font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
<p class=Body><span style='font-size:10.0pt;font-family:"Times New Roman"'><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>
 
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