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AAA= ------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
Promoting Student <= span style=3D'position:absolute;top:31.25%;left:8.42%;width:83.33%;height:9.25%= '>Understanding of Algebra
Jon Hasenbank =
Jennifer Kosiak= 3;
and An= dy Belter
UW-La Crosse
Wisconsin Mathematics Council
41st=  Annu= al Conference
Green Lake, WI
May 6, 2009
www.uwlax.edu/faculty/hasenbank/talks/wmc2009
3Dteaching
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Overview
•Background= & Teaching for
“Understanding” = ;
•Transformi= ng “Traditional Tasks”
into “Understanding Tasks”
•Sharing Activities and
Assessments
•S= ome Results
–P= re-post test comparisons
–A closer look: state of knowledge
•C= onclusions
3D"math_400"
http://www.uofaweb.= ualberta.ca/compneurolab/images/math_40= 0.jpg
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First, Some Examples
•On your note cards, wr= ite a specific question that involves a math procedure.
–
•Answer your question, = and consider:
<= span style=3D'mso-special-format:bullet;position:absolute;left:-7.34%'>–<= /span>What would an “expert” know and do?
= –<= /span>How does this compare to what your students know and do? =
–<= /span>How can we bridge that gap?
------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
Our Examples
<= span style=3D'font-size:88%'>•Solve: = ;
1.2x - 1 =3D 6
2.|x – 3| < 5
3.x2 + 3x – 4 =3D 0
4.2x + 3y < 6
5.3x + 4y =3D 14, and y =3D -(5/2)x
6.
<= span style=3D'font-size:88%'>•Consider:&= #13;
–What would an “expert” know and do? &#= 13;
–How does this compare to what your students know and do?
–How can we bridge that gap?
------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
Dimensions of Knowledge
•T= ype of Knowledge
–C= oncepts, Procedures, Facts
•O= ther Dimensions:
–S= hallow vs. Deep
–N= ovice vs. Practiced
Shallow vs. Deeper
Novice vs. Practic= ed
Novice
Novice
Practiced
Shallow
Deep
Shallow
Deep
Practiced
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------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0027.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
Another Look…
=
•What questions could we ask to enrich this task to promote deeper knowledge?
–<= /span>Solve: 2x - 1 =3D 6
–<= /span>Solve: |x – 3| < 5
–<= /span>Solve: x2 + 3x – 4 =3D 0
–<= /span>Solve: 2x + 3y < 6
–<= /span>Solve: 3x + 4y =3D 14 and y =3D -(5/2)x
•
------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
7
•
<= ![if !ppt]>
1a) What is the goal of the procedure?
1b) What sort of answer should I expect?
2a) How do I carry out the procedure?
2b) What other procedures could I use?
3)   Why does the procedure work?
4)   How can I ve= rify my answer?
5)   When is this= the “best” procedure to use?
6)   What else ca= n I use this procedure to do?
Procedural Knowledge
3D"Text=
Procedural Knowledge
Framework for Procedural Understanding
(or, Eight Alg= ebraic Habits of Mind)
Solve: 2x - 1 =3D 6
Solve: |x – 3| < 5
Solve: x2 + 3x – 4 =3D 0
Solve: 2x + 3y < 6
Solve: 3x + 4y =3D 14 and y =3D -(5/2)x
Adapted from NCTM’s Navigating through Algebra in Grades 9-12
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------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0008_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhlAAaAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAQAAACM ABgAgQAAAAAAAP///wECAwL/hI+py+0Po3Sh2ouzNbr7D4biSJLCiabqilbsC8fyTNf2ja8BkPf+ DwzadsKi8Yh8EZPMpnPIe0qn0yX1ihW4flYVJrnFBVIXbTg4hlmAXW86XVyL4acwnX3X5W9t8nvv czbkVocEaLb1hXjY1/KnBYno92e35mIHKRfpuFl4JncZWrg4mljaohbFIgpa9+aJGmkqO/s66TYG 90i7mRure0qaqvbpinpXNiqrjGzrKPiIaTYNnCxtK7iqqudl7En3CbzM7Ns53RktTt17HksLrqyz TUjmjh0svqsf357vrOtsHCVL1jTJU6IJVMJKib44bJjhWa6IEOGRm5jNHpQsKRyxWKTRqKNIQwZn hByJcuTJlCw9zmsJ02XMmVl2lLiJM6fOnTwxACgAADs= ------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0034.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
Benefits of Understanding
•If existing knowledge is understood, new = knowledge is easier to learn and understand.   (Hiebert & Carpenter, 1992)
<= span style=3D'font-size:56%;visibility:hidden'>•
•Knowledge that is understood is more robust = and can be used flexibly within and across = different contexts. (Carpenter & Lehrer, 1999; Hiebert & Carpenter, 1992; Van Hiele, 1986)
  •Students should use procedures “flexibly, accurately, efficiently, and appropriately” (National Resear= ch Council, 2001)
<= span style=3D'font-size:56%;visibility:hidden'>•
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Putting “TfU” to the Test=
•When teachers focus on understanding, = what impact does it have on student learning? •Algebra t= ests given to students in 9 participating classrooms and 5 comparison classrooms.
------=_NextPart_01C9D095.49D59B50 Content-Location: file:///C:/F07446F3/2009-05.07-AlgebraicUnderstandingofHSStudents_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" The algebraic understanding of high school students
Timeline of Progress
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