Algebraic and Symbolic Reasoning

1. Algebraic and Symbolic Reasoning

2. Mathematical Reasoning: Review & Sharing Analyze student errors in symbolic reasoning. Look at reasoning with Magic Squares. Investigate algebraic reasoning: Among different representations By using geometry Discuss baseline assessment. Today’s Agenda

3. Reasoning Review • Pass around and look at samples of student work from the Mathematical Reasoning baseline assessments and student interviews. • Based on this work, share your condensed summaries of your students’ reasoning abilities. • Which problems did your students perform well on? • Which did they struggle with? • What classroom activities did you use to foster reasoning? • Describe one situation where you saw students exhibit growth in their reasoning abilities.

4. Symbolic Reasoning

5. Algebra can be described as a generalization of arithmetic so that equations can be solved once for all numbers. It allows us to make general statements about a process without being tied to one specific example. The use of a symbol for a changeable value is a hard conceptual leap for many students. Arithmetic with Symbols

6. Many, if not most, errors in algebra involve incorrect manipulation of symbols. When we see a student make these errors, we often think, “This student has no number sense,” or “This student can’t handle the abstract thinking required for algebra.” Example: Algebraic Errors

7. Algebraic Errors… are Reasonable? This error results from the assumption that the square root function is linear. That indicates a basic misunderstanding of the square root funtion… …but is understandable if you think about the hundreds, or even thousands, of times students have used the distributive property:

8. Generalized Distribution

9. Matz’s Classification of High School Algebra Errors Errors generated by an incorrect choice of an extrapolation technique Errors reflecting an impoverished (but correct) base knowledge Errors in execution of a procedure.

10. Matz’s Classification of High School Algebra Errors Errors generated by an incorrect choice of an extrapolation technique Example:

11. Matz’s Classification of High School Algebra Errors Errors reflecting an impoverished (but correct) base knowledge Examples:

12. Matz’s Classification of High School Algebra Errors Errors in execution of a procedure. Example:

13. Look at the common algebra errors on your handout. With your group members, classify them according to Matz’s framework. Algebra Errors – Your Turn

14. Symbolic Reasoning • Find the error in the argument below. • How would you describe this type of error?

15. Symbolic Reasoning • Why does the last statement contradict the first? • What is the error? • How would you describe this type of error?

16. Reasoning with Magic (Squares)

17. Reasoning with Magic Squares Place the digits 1-9 in the box so that the sum along any row, column, or diagonal is the same.

18. More Questions on Magic Squares Do the entries in a magic square have to be 1, 2, …, n2 ? What other sequences could you use? Can the set of numbers {1, 3, 5, 7, 8, 10, 12, 14, 16} be used to form a 3 X 3 magic square? Or {0, 2, 3, 4, 5, 7, 8, 9, 11}? In a 3X3 square, the center square is always ___________ of the magic sum. What is the general form of a 3X3 magic square if a is the value of the center?

19. Reasoning with Magic Squares • Reasoning strategies (Schoenfeld, 1991): • Establish subgoals • What can you say about the sum of any diagonal, row or column? • What is the important square to find first? • Working backwards • Assume the sum of any column is S. • Can you find the value of S? • Exploit extreme cases • Try the large and small values around the perimeter – what works?

20. Reasoning with Magic Squares • Reasoning strategies (Schoenfeld, 1991): • Exploit symmetry • Balance large values with small ones on either side of the center. • Work forward to try solutions. • Be systematic about trying cases.

21. Reasoning with Magic Squares What sort of mathematics did you use? What types of mathematical reasoning did you use? (Think about last session)

22. Algebraic Reasoning

23. Algebraic Reasoning Prior to manipulating symbols, students must use reasoning in determining appropriate representations to set up, manipulate, and solve problems Many ideas in algebra can be expressed with multiple representations i.e. Rule of 3 + 1 Reasoning is often used in shifting among representations in algebra

24. Reasoning about Different Algebraic Representations The graph and table below show how values of two functions change. Choose functions from below that have a property in common with one or both of the functions above. Explain your point of view. Find as many viewpoints as you can.

25. Student Professor Problem Take a minute and answer the following question: Using the letter S to represent the number of students (at a university) and the letter P to represent the number of professors, write an equation that summarizes the following sentence “There are six times as many students as professors at this university”.

26. Student Professor Problem • Using the letter S to represent the number of students (at a university) and the letter P to represent the number of professors, write an equation that summarizes the following sentence “There are six times as many students as professors at this university”. • How many wrote 6S = P? • Why is this incorrect? Discuss at your tables. • Why might students answer this way? What reasoning are they using?

27. Student Professor Problem • Using the letter S to represent the number of students (at a university) and the letter P to represent the number of professors, write an equation that summarizes the following sentence “There are six times as many students as professors at this university”. Two possible explanations for incorrect answer of 6S = P (Schoenfeld, 1985: • A direct translation of the words into symbols. (Six times students is professors) • Students may visualize the following “classroom”: d S S S e Professor S S S s k

28. Student Professor Problem • If a student answered 6S = P, how might you go about helping this student? • Discuss at your tables. • What areas do you think the student needs help with? • Discuss at your tables.

29. Student Professor Problem • Clement (1982) suggested that the two essential competencies for solving the problem are: • Recognizing that the letters represented quantities (notion of variable) • Creating a “hypothetical operation” to make the two quantities (such as the number of students and professors) equal.

30. Recall Day 1 Proofs • Let’s prove the following: 1 + 3 + 5 + … + (2n-1) = n2 • Proof that proves (only): • Use Mathematical Induction • Next: Proof that explains (and proves)…

31. Proof That Explains (and proves): 1 + 3 + 5 + … + (2n -1) = n2

32. The Geometry of Algebra • Geometric representations of algebraic processes can be helpful • For example, interpreting “n2” as a square of dimensions, n x n • Let’s try another…

33. Completing the Square What equation is represented by the shapes below?

34. BaselineAssessment

35. Baseline Assessment • Explain why the following argument results in a false statement. Be sure to identify which step(s) are untrue. 

36. Baseline Assessment The graph and table below show how values of two functions change. Choose functions from below that have a property in common with one or both of the functions above. Explain your point of view. Find as many viewpoints as you can.