A Deeper Understanding of AlgebraMath

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A Deeper Understanding of Algebra/Math

• Rebecca Chaouki, OMS
• Jon F. Hasenbank, UW-L

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Why Teach for Understanding?
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Without Understanding

• students have trouble applying knowledge,
• over-generalize patterns,
• (ab)2 a2b2
• quickly forget what they learn,
• see math as disconnected and meaningless.
• Evidence?

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Discouraging Data

• Solve for x x2 4x 32

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The Fundamental Theorem of Math Education? It
Cant Be Remembered
But Perhaps if it Was Understood
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With Understanding

• When students learn mathematics with
  understanding, it becomes meaningful for them.
• But what does it mean to understand solving
  linear equations, the slope of a line?

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The nature of knowledge

• Type Concepts vs. Procedures
• or Ideas vs. Skills
• Aptitude Novice vs. Practiced
• or Arduous vs. Automatic
• Depth Shallow vs. Deep
• or Disconnected vs. Connected

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Three Dimensions of Knowledge
Shallow
Deep
Procedure is executed intelligently, understood
Procedure is executed by rote
Procedural
Concepts are well-memorized and well-connected,
understood
Concepts are well-memorized but remain isolated,
disconnected
Conceptual
Procedure is not well-memorized, but is better
connected. Execution is informed, but slow
Procedure is not well-memorized and is isolated
Executed with high cognitive load
Procedural
Concepts are notwell-memorized and are isolated,
disconnected
Concepts are not well-memorized, but connections
are forming
Conceptual
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Aptitude

• Aptitude (Skill, Automation)
• Developed with repetition and practice.
• Aptitude is what we usually try to assess.
• Algebra allows us to think less and less about
  more and more. Bertrand Russell

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Depth

• Depth (Connectedness)
• Develops with reflection and experience.
• Students can see big picture, know how the
  pieces fit together.
• Difficult to assess, but is often deduced from
  aptitude.

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Growth

• Two Dimensions for Growth
• Growth can occur independently along Aptitude or
  Depth dimensions.
• NCTM (PSSM, p. 20)
• The alliance of factual knowledge, procedural
  proficiency, and conceptual understanding makes
  all three components usable in powerful ways.

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Two Dimensions of Growth
Shallow
Deep
Procedure is executed intelligently, understood
Procedure is executed by rote
Procedural
Concepts are well-memorized and well-connected,
understood
Concepts are well-memorized but remain isolated,
disconnected
Conceptual
Procedure is not well-memorized and is isolated
Executed with high cognitive load
Procedure is not well-memorized, but is better
connected. Execution is informed, but slow
Procedural
Concepts are notwell-memorized and are isolated,
disconnected
Concepts are not well-memorized, but connections
are forming
Conceptual
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Understand a Procedure?

• We have defined procedural understanding in terms
  of an 8 question Framework.
• If you understand some procedure, what kinds of
  questions could you answer about it?

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Framework for Procedural Understanding

• 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 verify my answer?
• 5) When is this the best procedure to use?
• 6) What else can I use this procedure to do?
• Adapted from Navigating through Algebra in Grades
  9-12 (Burke, Lott, Erickson, Obert, 2001)

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Timeline

• 2001 Framework printed in Navigating Through
  Algebra.
• 2005 Tested the Framework in college algebra.
• Large gains in understanding.
• No declines in skill.
• 2006 - We asked What about K-12?
• 2007 - We obtained funding to spread the word.

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Experiences with Teaching for Understanding

• Students need to communicate their understanding
  to be valued as something learned

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Skill versus Understanding

• Skill- Find slope
• If they give me two points, I put it in my
  memorized formula with a bunch of xs and ys and
  Ill get slope.

• Understanding- Slope is how one point relates to
  other on a line- The change in the y versus
  change in the x. If I have 2 points, I can
  figure out its slope.

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Students need motivation for understanding

• Grade
• Be useful in their math future
• Be useful in real life

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Long term Benefits

• Retention due to connections made to other math
• Flexibility
• Creation of learning

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What am I seeing in the classroom?

• Interesting conversations/debates about Algebra
• Unexpected connections
• Untrue understandings
• Increased attention to vocabulary due to writing
  requirement

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Are the students benefiting?

• They can explain their understanding with
  appropriate math vocabulary
• They use definitions to create understanding

• They dont like to struggle for understanding
• I need to continue my changing of how Im
  teaching

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Narrowing the Achievement Gap

• Understanding without skills doesnt work very
  well
• Balance is the key!

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How to work in understanding questions
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Thank you!

• For handouts and more info
• http//www.uwlax.edu/ faculty/hasenbank/witq2007/
  wwec
• Your district interested in improving
  understanding?
• Contact Dr. Hasenbank (hasenban.jon_at_uwlax.edu)
• Questions? Comments? (Jokes?)

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Framework for Procedural Understanding

• 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 verify my answer?
• 5) When is this the best procedure to use?
• 6) What else can I use this procedure to do?
• Adapted from Navigating through Algebra in Grades
  9-12 (Burke, Lott, Erickson, Obert, 2001)