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teaching algebra revised

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  • 1. Carlo Magno, PhD De La Salle University, Manila
  • 2. Teaching Principles of School Mathematics
  • 3. Teaching Principles of School Mathematics
  • 4. Teaching Principles of School Mathematics
  • 5. Learning Principles in Mathematics
  • 6. Learning Principles in Mathematics
  • 7. 3 part structure in algebra  Representing the elements in an algebraic form  Transforming the symbolic expressions in some ways  Interpreting the new forms that has been produced
  • 8. Perspectives  Algebra is useful to those who work in the fields like science, engineering, computing, and teaching mathematics.  Algebra is interesting because it affords admirable example of ingenuity.
  • 9. Principles and Standards for school mathematics  Algebra should enable students to:  Understand patterns, relations, and functions  Represent and analyze mathematical situations and structures using algebraic symbols  Use mathematical models to represent and understand quantitative relationships  Analyze change in various contexts
  • 10. Learning algebra  Demonstrating worked examples practice exercises (working for fluency)  Demonstration and practice application  Problem link to topics (working for meaning)  Construct and reflect on the meanings for expressions and equations (working for meaning)
  • 11. Models of teaching  Concept attainment model  Inductive thinking model  Advance organizer  Inquiry training model
  • 12. Concept Attainment Model  Provide students with examples, some that represent the concept and some that do not.  Urge students to hypothesize about the attributes of the concepts and to record reasons speculations. The teacher my ask additional questions to help focus student thinking and to get them to compare attributes of the examples of nonexamples.
  • 13. Concept Attainment Model  When students appear to know the concept, they name (label) the concept and describe the process they used for identifying it. Student may guess the concept early in the lesson, but the teacher needs to continue to present examples and non examples until the students attain the critical attributes of the concept as well as the name of the concept.  The teacher checks to see if the students have attained the concept by having then identify additional examples, say yes or no, tell why or why not their examples, and generate examples and nonexamples of their own.
  • 14. Concept Attainment Model Phase 1 Presentation of data and identification of concept Teacher presents labelled examples Students compare attributes in positive and negative examples Students generate and test hypotheses Students state a definition according to the essential attributes
  • 15. GCF Grouping  5x2 - 25  16x + 8  27y2 - 9  3ax + 6ay + 4x + 4y  2ac + 4ax – 5c + 10x 1. Characterize each set. What do you observe about factoring the equation in the first set? How about in the second set? (characterizing examples) 2. What is the difference between the two sets when they are factored? (comparing attributes) 3. Given the characteristics mentioned, what is factoring using GCF? What is factoring using grouping? (defining)
  • 16. Concept Attainment Model Phase 2 Testing attainment of the concept Students identify unlabeled examples as yes or no Teacher confirms hypothesis, names concept, and restate definitions according to essential attributes Students generate examples
  • 17. Concept Attainment Model  Which of the following requires factoring by GCF? Grouping? (confirming hypothesis)  36x2 – 54  4ax + 16ay + 4x + 4y  20ac + 14ax – 5c + 20x  81 + 18y2  Factor the terms. (confirming the hypothesis)  Give your own examples of terms that requires factoring by GCF? Factoring by grouping? (generating own examples)
  • 18. Concept Attainment Model Phase 3 Analysis of thinking strategies Students describe thoughts Students discuss role of hypotheses and attributes Students discuss type and number of hypotheses
  • 19. Concept Attainment Model  Why will grouping not work for: (describe thoughts)  36x2 – 54  81 + 18y2  Why will GCF not work for: (describe thoughts)  4ax + 16ay + 4x + 4y  20ac + 14ax – 5c + 20x
  • 20. Inductive Thinking Model  Three postulates:  Thinking can be taught  Thinking is an active transaction between the individual and the data  Processes of thought evolve by a sequence that is lawful
  • 21. Inductive Thinking Model  Strategy 1: Concept formation  Enumeration and listing  Grouping  Labelling and categorizing
  • 22. Set A Set B  y=4x + 3  C=10x + 5  D=3x + 2  D = 3x2 + 4x + 5  A = 6c2 + 10c + 3  Y = 3x2 + 4x +6 1.Look at the two sets of data, why are they separated? (listing) 2.What is the difference between them? (categorizing) 3.What do you call the equation in set A? How about for set B? (labeling)
  • 23. Inductive Thinking Model  Strategy 2: Interpretation of data  identifying critical relationships  exploring relationships  making inferences
  • 24. Set A Set B  y=4x + 3  C=10x + 5  D=3x + 2  D = 3x2 + 4x + 5  A = 6c2 + 10c + 3  Y = 3x2 + 4x +6 1.Can the equations be plotted? (critical relationship) 2.What can be produced for each set of data? (exploring relationships) 3.Will there be difference in the graphs for seta A equation and set B equation? (inferences)
  • 25. Inductive Thinking Model  Strategy 3: Application of principles  Predicting consequences, explaining unfamiliar phenomenon, hypothesizing  explaining and/or supporting the predictions and hypothesis  Identifying the prediction
  • 26. Set A Set B  y=4x + 3  C=10x + 5  D=3x + 2  D = 3x2 + 4x + 5  A = 6c2 + 10c + 3  Y = 3x2 + 4x +6 1. If we give a value for x and c, what will the graph look like? (predicting consequences) 2. Students will plot in a coordinate plane equation for set A and equation for set B. (supporting predictions) 3. Did the slopes turn out the way you predicted? Why or why not? (identifying the prediction)
  • 27. Advance Organizer  Phase 1: Presentation of advance organizer  Clarify aims of the lesson  Present organizer  Identify defining attributes  Give examples  Provide context  Repeat  Prompt awareness of learner’s relevant knowledge and experience
  • 28. Advance Organizer  Phase 2: Presentation of learning task and materials  Present material  Make logical order of learning material explicit link material to organizer
  • 29. Advance Organizer  Phase 3: Strengthening cognitive organization  Use principles of integrative reconciliation  Promote active reception learning  Elicit critical approach to subject matter  Clarify
  • 30. Inquiry Training Model  Suchmans’s theory  Students inquire naturally when they are puzzled  They can become conscious of and learn to analyze their thinking strategies  New strategies can be taught directly and added to the students existing ones.  Cooperative inquiry enriches thinking and helps students to learn about the tentative, emergent nature of knowledge and to appreciate alternative explanations.
  • 31. Inquiry Training Model  Phase 1: Confrontation with the problem  Explain inquiry procedures  Present discrepant event  The period T (time in seconds for one complete cycle) of a simple pendulum is related to the length L (in feet) of the pendulum by the formulas 8T2= 2L. If one child is on a swing with a 10 – foot chain, then how long does it take to compete one cycle of the swing?  What do you need to do with the pendulum to make it swing faster?
  • 32. Inquiry Training Model  Phase 2: Data gathering, verification  Verify the nature of objects and conditions  Verify the occurrence of the problem situation  Go to the playground and try elongating the swing. Take the time in seconds. Record it.  Try shortening the swing. Take the time in seconds. Record it.
  • 33. Inquiry Training Model  Phase 3: Data gathering, experimentation  Isolate relevant variables  Hypothesize causal relationship  Set up your own simulated swing getting pieces of string and a yoyo.  Record the time of swing for each length: 15 in, 12 in, 10 in, 8 in, 6 in, 4 in  What do you think is the relationship between time and length?
  • 34. Inquiry Training Model  Phase 4: Organizing, formulating an explanation  Formulate rules or explanations  If time and length are related, what explains this?  Phase 5: Analysis of the inquiry process  Analyze inquiry strategy and develop more effective ones  What other situations can the relationship between time and length be applied?
  • 35. Watch video
  • 36.  Role playing  Concept attainment model  Inductive thinking model  Advance organizer  Inquiry training model
  • 37. Insights

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