Writing in the Mathematics Classroom

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=56 BGCOLOR="#003333" TEXT="#F7F7FF" bgproperties="fixed" background="011a0a03.jpg"> <div> Writing in the Mathematics Classroom</div> <div> </div> </td> </tr><tr> <td valign="top" width=135 BGCOLOR="#003333" TEXT="#F7F7FF" bgproperties="fixed" background="21fa0c01dduajq.jpg"> <div> <A HREF="index.htm" TARGET="_top" TITLE="Educators Science and Math Institute Ser"><U><B>Educators' Science and Math Institute Series</B></U></A></div> <bR> <div> <A HREF="id2.htm" TARGET="_top" TITLE="About the Series"><U><B>About the Series</B></U></A></div> <div> <A HREF="id117.htm" TARGET="_top" TITLE="Goals"><U><B>Goals</B></U></A></div> <div> <A HREF="id3.htm" TARGET="_top" TITLE="Institutes"><U><B>Institutes</B></U></A></div> <div> <A HREF="lesson_plans_prepared_by_esmis_participants_.htm" TARGET="_top" TITLE="Lesson Plans"><U><B>Lesson Plans</B></U></A></div> <div> <A HREF="id53.htm" TARGET="_top" TITLE="Links"><U><B>Links</B></U></A></div> <div> <A HREF="id138.htm" TARGET="_top" TITLE="Photo Albums"><U><B>Photo Albums</B></U></A></div> <bR> <div> <A HREF="http://www.ed.mtu.edu/search.htm" TARGET="_top" TITLE="http://www.ed.mtu.edu/search.htm"><U><B>Search</B></U></A><A HREF="http://www.ed.mtu.edu/search.htm" TARGET="_top" TITLE="http://www.ed.mtu.edu/search.htm"><U><B> ESMIS</B></U></A></div> <bR> </td> <td valign="top" height=424 BGCOLOR="#EEEEEE" TEXT="#080000"> <div> Writing in the Mathematics Classroom</div> <bR> <div> Maureen Schick</div> <div> ED 529, Ecology of Isle Royale</div> <div> Instructor:  Mary Hindelang</div> <div> August 11, 2000</div> <bR> <div> <I><B>Introduction</B></I></div> <div>      Literacy and writing are important skills that are crucial to any students success. It is no longer acceptable to sign your name with an “X.”  In Michigan, students are confronted with a writing-skills portion on the Michigan Educational Assessment Program (MEAP) test.  Yet last year, I discovered that many of the supposedly “good” writing students in our district could not write an acceptable paper for my geometry class.  After reading through the papers, I realized that students are accustomed to a much different style of writing.  Nearly 95% of their writing is narratives or opinion papers, not scientific reports.  They have never had to synthesize data, calculations, procedures, and results into one paper and consequently were turning in reports that I found unacceptable.</div> <div> One way of introducing writing into mathematics is to tell students to pick a mathematician or scientist and write a biography of them.  While this is not a bad idea, I think it is often superfluous: one that may have great use in a social studies or history class, but one that is not really related to all the mathematics and research that students are doing.  I have noticed that when the “Writing Across the Curriculum” committee in our district passes out writing ideas, mathematics seems to get short shrift with few unimaginative writing ideas.  Mathematics education has become so disconnected from science and applications that many people don't know how to develop ideas aside from the basic “pick a topic/person and report on them.”</div> <div>      The problem I find with most mathematics writing assignments is that they are disconnected from the “complete picture.”  One typical writing assignment is…”explain (concept/idea) in your own words.”  Students may discuss how they figured out the slope of an object but do not think about the uses of slope and what further applications of slope there may be. This is an excellent way to evaluate what the student knows about the particular topic, but I think we shortchange our students if we do not go beyond that question on a more regular basis.  It does not encompass the whole idea of what mathematicians or scientists write about.  When most mathematicians or scientists report on a project or study, they must include the process used, the results obtained and then a summary of their findings.</div> <div> In our district, students must eventually write scientific papers, but usually it is in either their chemistry or physics class.  This means that less than 10% of our population is getting the kind of rigorous writing that is necessary to produce good scientific/mathematical writers.  </div> <div> Unsure about what to do, I thought about scrapping the writing assignment.  While reading the papers, I was having trouble deciding how to grade them so I set out to develop a rubric.  I returned the papers along with the rubric to the students and had them make revisions based on that.  The final papers were not really what I wanted, but at least I saw improvement and felt the students had more of a feeling for the scientific writing process.  </div> <bR> <div> <I><B>Lesson </B></I></div> <div> It is important that instead of teaching mathematics as a separate subject, we incorporate it more with the sciences/writing and show students how to produce reports that coherently and completely describe and explain what they have been learning.  My lesson involves both our geometry and biology class and the use of our 400-acre school forest.  The forest contains five tree plots, established by the Federal Forestry Service, which must be inventoried every five years with the results reported to the Forestry Service.  This year, three of the plots are scheduled for evaluation. </div> <div> Students will be organized in teams with each team responsible for a different part of the study.  Some of the teams will be directly involved in the tree inventory.  They will measure and identify each tree, then calculate the growth and note any special changes in the last five years.  A second group will be using our global positioning satellite (GPS) receiver to identify the locations of specific trees, boundaries and other landmarks.  This information will be passed along to a third group which will be using our geographic information software (GIS) to map all the resources identified.  Depending on the size of the classes, and because it has not been catalogued before, we may also set up groups that will gather baseline data for plants, birds, and wildlife.</div> <div> The students involved in the surveys will be doing practical mathematics; counting, measuring, mapping, and collecting data.  They will also be using their knowledge of biology and science by identifying and classifying objects.  The information gathered may be useful, but it will only be useful if students can properly report the results so others can read and understand the results.  Students will be required to report the results of their portion of the project both orally and in writing.  It is here that many students (and this teacher) will have more problems.</div> <div> As we begin the study and students become aware that written reports will be required, it is important to start discussing the differences and similarities between scientific reports and other types of writing.  From there, each different part of the report will be introduced and described so students get a better understanding about what each section entails.  Then, during following days, students will be allowed some class time to work on their report sections, get teacher and classmate feedback, and finally put together their final report.</div> <div> After reviewing scientific/technical reports, I have come up with a list of sections that would be pertinent to high school student reports.  The following sections are:  title page, abstract or hypothesis, procedure, calculations or important computations, results, conclusions, and bibliography or resources.  Depending on the content, some sections may not be necessary.  Essentially, this covers the “who, what, when, where, why and how” questions that most students are taught when they begin writing; it is just in a different format than most students are used to seeing.</div> <div> If these are the main sections or topics that I am requiring, then the report should be graded according to them.  My goal is to develop a rubric that I can use to grade the reports both effectively and efficiently. While math teachers can and do use rubrics to grade, based on personal observation and discussion with teachers, they are not used as extensively as in some of the other disciplines or subjects.  I also think it is important to share the rubric with the students when the report is assigned so that they will know exactly what is expected of them.  They can also use the rubric in peer review of other reports.</div> <div> Examples of two possible rubrics are listed below:</div> <bR> <bR> <div> <I><B>Rubric 1 </B></I></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="707" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 21 WIDTH="206"><div> <I><B>Title Page</B></I></div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="21" ALIGN="bottom" WIDTH="193" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="21" ALIGN="bottom" WIDTH="274" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> 1</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> 2</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 57 WIDTH="206"><div> no page</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> information missing or not done according to format specified</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> complete information in format requested</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 ><NOBR><div> <I><B>Abstract or Hypothesis </B></I></div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="193" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="274" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> 1-2</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> 3-4</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 38 WIDTH="206"><div> no abstract</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> incomplete description or improper use of terms</div> </tD> <tD VALIGN=TOP COLSPAN=4><NOBR><div> complete description using proper terms</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 19 WIDTH="150"><div> <I><B>Procedure</B></I></div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="178"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="178" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="190"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="190" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="152"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="152" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 19 WIDTH="150"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="178"><div> 1-3</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="190"><div> 4-6</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="152"><div> 7-9</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 76 WIDTH="150"><div> no procedure</div> </tD> <tD VALIGN=TOP COLSPAN=3><NOBR><div> poor description, unclear</div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="190"><div> better description, but does not describe events clearly</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="152"><div> good description, uses correct terms, method easily understood</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 ><NOBR><div> <I><B>Calculations/Computations</B></I></div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="193" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="274" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> 1-3</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> 4-6</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> no calculations</div> </tD> <tD VALIGN=TOP COLSPAN=4><NOBR><div> incorrect calculations</div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> correct calculations</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 19 WIDTH="150"><div> <I><B>Results</B></I></div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="178"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="178" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="190"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="190" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="152"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="152" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 19 WIDTH="150"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="178"><div> 1-3</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="190"><div> 4-6</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="152"><div> 7-9</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 38 WIDTH="150"><div> no results</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="178"><div> results do not agree with expected results</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="190"><div> weak description of results, need more information</div> </tD> <tD VALIGN=TOP COLSPAN=3 WIDTH="152"><div> excellent description, complete, thorough</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> <I><B>Conclusion</B></I></div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="193" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="274" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> 1-3</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> 4-6</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 38 WIDTH="206"><div> no conclusion</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> conclusion not backed up by results/information</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> conclusion backed up results and information</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 ><NOBR><div> <I><B>Bibliography/Resources</B></I></div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="193" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="274" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 19 WIDTH="206"><div> 0</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> 1</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> 2</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=4 HEIGHT= 38 WIDTH="206"><div> none provided</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="193"><div> incomplete or incorrect format</div> </tD> <tD VALIGN=TOP COLSPAN=4 WIDTH="274"><div> correct and thorough</div> </tD> </tR> </TABLE></div> <bR> <div>      With this rubric, grading is done on a section by section basis.  A teacher who does not require a particular section can easily remove that section from the grading scheme.  A teacher may also decide that instead of assigning a range of points as are listed for some of the sections, one discrete value can be used instead.  Points can be changed for those sections that a teacher feels more important than others: for example, the title page is worth two points maximum while the procedure section is worth up to nine points, illustrating the value the teacher places on the procedure section.</div> <bR> <div> <I><B>Rubric 2 </B></I></div> <div> <TABLE BORDER="2" CELLPADDING="1" CELLSPACING="1" WIDTH="485" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><div> Format</div> </tD> <tD VALIGN=TOP><NOBR><div> Procedure</div> </NOBR></tD> <tD VALIGN=TOP WIDTH="91"><div> Results</div> </tD> <tD VALIGN=TOP><NOBR><div> Term Usage</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="91"><div> Strength</div> </tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 WIDTH="91"><div> OK</div> </tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 19 ><NOBR><div> Needs Work</div> </NOBR></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="91"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="19" ALIGN="bottom" WIDTH="91" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <bR> <div>      This rubric uses a general scale for all the sections.  A teacher will list those sections that they think are most important for the report.  The general scale uses the category Needs Work, which means the students “missed the boat” with that particular section.  There is an OK category, which means the students showed most things but there were some errors.  Finally there is a Strength category, which means it was near perfect; it was one of the students’ “strengths” in the paper.  </div> <div> If I were to develop a rubric based on the project I want my students to do, I would probably have a format section under which layout, spelling, etc., would go.  Since I think the Procedure and Results sections are important, I would include separate sections for them.  I also think it is important to have a section labeled Term Usage since it is important for students to use proper terminology throughout.  Space would be left so that a grader could write comments about the sections if necessary.  A teacher could decide that if all four categories were a strength, then the students would receive an A, three strengths and one OK would mean an A-, and so on.</div> <div> Conclusion </div> <div> As a teacher who is trying to incorporate more comprehensive writing into my class, this is only a start.  By providing an account of what a scientific or technical report should be, I cannot only help my students, but also myself in teaching/learning the writing process.  By knowing what I expect in the beginning and providing the students with those expectations, I should be able to get reports that show what the students know and can do, not reports that leaves me disappointed or frustrated.  </div> <div> While my focus is on writing, many other standards for science, mathematics, and technology are covered.  Mathematics content strands included: Patterns, Relationships, and Functions - Standards 1, 2; Geometry and Measurement -Standards 1, 2, 3; and Data Analysis and Statistics- Standards 1, 2, 3 (Michigan Mathematics Curriculum Framework, June 1998).  Constructing New Scientific Knowledge – Standard 1, Reflecting on Scientific Knowledge – Standard 1, and Using Scientific Knowledge in Life Science – Standards 2, 5, are the content strands covered in Science (Michigan Science Curriculum Framework, June 1998).  The Technology standard, Using Information Technologies, is also covered (Michigan Technology Curriculum Framework, June 1998).</div> <div> The Language Arts standards addressed are Meaning and Communication: Reading, Meaning and Communication: Writing, Skills and Processes, Voice, Inquiry and Research, and Critical Standards (Michigan Language Arts Curriculum Framework, June 1998).</div> <div> While mathematics is taught as a separate subject in education, most mathematics done is not highly theoretical work, but it is done within the context of other fields and professions.  People use mathematics to describe, explain, and predict events.  My goal is to eventually move to a classroom in which not only are students writing to learn about mathematics, but also students learning to write about mathematics.</div> <bR> <div> <I><B>Bibliography</B></I></div> <bR> <div> Michigan Department of Education.  1998.  <I>Michigan Curriculum Framework:  Interactive Links to Content Strands, Standards, and Benchmarks</I>.  State of Michigan, Department of Education, Lansing, MI.</div> <bR> </td> </tr></table></body>