Teaching Mathematics for Understanding

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Teaching Mathematics for Understanding

• Barb Whyte
• Nick Hable
• Marlis Downs

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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.

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The Fundamental Theorem of Math Education? It
Cant Be Remembered
But Perhaps if it Was Understood
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Why is understanding needed?

• WKCE-CRT (other tests)
• Recall from year to year
• Apply to

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What does it mean to understand?

• Fractions
• Congruence of Figures
• Density

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Framework for Procedural Understanding

• 1) What is the goal of the procedure?
• 2) What sort of answer should I expect?
• 3) How do I carry out the procedure?
• 4) What other procedures could I use?
• 5) Why does the procedure work?
• 6) How can I verify my answer?
• 7) When is this the best procedure to use?
• 8) 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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Skill Vs. Understandingof Adding Fractions

• Skill
• Find a common denominator, add numerators, and
  keep the same denominator.

• Understand
• Is my answer reasonable? (Self-check)
• Why can a fraction be changed?
• How and how else?

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Quadratic Equations

• Name the methods you can use to solve a quadratic
  equation.
• Square Root Method
• Factoring
• Quadratic Equation
• Completing the Square

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Quadratic Equations

• What about the equation tells me which one I
  should use?
• Do all of them work all of the time?
• Is one better than another?

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Skill Vs. Understanding

• What would an understanding lesson look like?
• Order of Operations and Exponents

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My Experience

• Nick Hable
• Mauston High School
• Algebra (A/B)
• Trigonometry
• Calculus

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Goals

• Help students learn better
• Raise Standards
• Apply to one lesson a unit

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Pitfalls

• Time
• Students
• Is it helping?

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Positives

• Less re-teaching
• Deeper knowledge base for future applications
• Connects math together

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Reflections

• I know this will help
• Changing norms
• Sprinkle in

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My Experience

• Barb Whyte
• Mauston High School
• Algebra I, A/B
• Geometry
• Trigonometry

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Goals

• Help students learn better
• Increase MAP scores
• Sprinkle in.

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Pitfalls

• Time
• Making it a habitverbal Vs. written
• Switching classes

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Examples

• What error did the student make?
• Solve it correctly.

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Examples

• Anthony has saved 40. He wants to buy a CD
  player for 129 in about 4 months. To find out
  how much he should save each week, he wrote
• 40 16x 129

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Examples

• 40 16x 129
• What does the 40 stand for?
• The x?
• The 129?
• The 16?

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Examples

• Hexagonal cells are common among cell building
  insects (e.g. honeybees) and there are good
  reasons for using this shape. Round, octagonal
  or pentagonal cell arrangements leave empty
  spaces between cells. Why do you think it is
  important not to have empts spaces between cells?

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Positives

• Students think more.
• Getting inside a students head.
• I did it different (but right)
• Students improved their writing.
• Teacher Collaboration.

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Reflections

• Sprinkling in works, need to continue.
• Expand on lessons done this year.
• Notes to myself on what worked and what didnt
  (and why).

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My Experience

• Marlis Downs
• Pittsville High School
• Physical Science
• Physics

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Goals

• Manipulate equations in science.
• Students would have better understanding of
  applications of math.

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Pitfalls

• Time
• Keeping the routine

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Positives

• Student Skills Levels Visible
• Great Review Technique
• (Science to Math Relationships)
• Incorporated math in to science seamlessly.

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Reflections

• Framework questions can be applied to science.
• Natural questioning fit to the framework.
• Successful equation usage.
• Freshman struggled, Upperclassmen achieved.

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Wrap Up

• Mission Accomplished?

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Acknowledgements

• Dr. Jon Hasenbeck
• Dr. Jennifer Kosiak
• Classmates
• Rebecca, Doug, Stephanie, Jamie, Shane, Chris

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Questions