Algebraic and Symbolic

Reasoning

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

Reasoning ReviewReasoning ReviewPass 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.1. Which problems did your students perform well on?

2. Which did they struggle with?

3. What classroom activities did you use to foster reasoning?

4. Describe one situation where you saw students exhibit growth in their reasoning abilities.

Symbolic Reasoning

Arithmetic with SymbolsArithmetic with Symbols

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.

Algebraic ErrorsAlgebraic Errors

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:

€

a+ b( ) = a + b

Algebraic Errors… are Algebraic Errors… are Reasonable?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:

€

a+ b( ) = a + b

€

3(a+ b) = 3a+ 3b

−(a+ b) = −a−b

Generalized DistributionGeneralized Distribution

Matz’s Classification of Matz’s Classification of High School Algebra ErrorsHigh School Algebra Errors

1. Errors generated by an incorrect choice of an extrapolation technique

2. Errors reflecting an impoverished (but correct) base knowledge

3. Errors in execution of a procedure.

Matz’s Classification of Matz’s Classification of High School Algebra ErrorsHigh School Algebra Errors

1. Errors generated by an incorrect choice of an extrapolation technique

Example:

€

a+ b( ) = a + b

Matz’s Classification of Matz’s Classification of High School Algebra ErrorsHigh School Algebra Errors

2. Errors reflecting an impoverished (but correct) base knowledge

Examples:

€

If 4X = 46, then X = 6

If X4

3=

5

3, then X =

1

3

Matz’s Classification of Matz’s Classification of High School Algebra ErrorsHigh School Algebra Errors

3. Errors in execution of a procedure.

Example:

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Solve for x :

5

2 − x+

5

2 + x= 4

5(2 + x) + 5(2 − x) = 4

Algebra Errors – Your TurnAlgebra Errors – Your Turn

Look at the common algebra errors on your handout. With your group members, classify them according to Matz’s framework.

Symbolic Reasoning

x xx 1 1x 0

(1 x) 1(x 1)1

(1 x) 1 (x 1)1

1 x 1 x 11

0 4

• Find the error in the argument below.

• How would you describe this type of error?

Symbolic Reasoning

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

x x

(x)2 (x)2

(x 5)2 52 (x)2

x 2 10x 25 25 x 2

x 2 10x x 2

10x 0

x 0

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.

 

Reasoning with Magic Squares

More Questions on Magic Squares

1. Do the entries in a magic square have to be 1, 2, …, n2 ?

2. What other sequences could you use?3. 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}?

4. In a 3X3 square, the center square is always ___________ of the magic sum.

5. What is the general form of a 3X3 magic square if a is the value of the center?

Reasoning strategies (Schoenfeld, 1991):A. Establish subgoals

• What can you say about the sum of any diagonal, row or column?

• What is the important square to find first?B. Working backwards

• Assume the sum of any column is S. • Can you find the value of S?

C. Exploit extreme cases• Try the large and small values around the

perimeter – what works?

Reasoning with Magic Squares

Reasoning strategies (Schoenfeld, 1991):D. Exploit symmetry

• Balance large values with small ones on either side of the center.

E. Work forward to try solutions.

F. Be systematic about trying cases.

 

Reasoning with Magic Squares

Reasoning with Magic Squares

What sort of mathematics did you use?

What types of mathematical reasoning did you use? (Think about last session)

Algebraic Reasoning

Algebraic ReasoningPrior 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

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.

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”.

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?

Student Professor ProblemUsing 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:

1. A direct translation of the words into symbols. (Six times students is professors)

2. Students may visualize the following “classroom”: d

S S S e ProfessorS S S s k

Student Professor ProblemIf 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.

Student Professor Problem Clement (1982) suggested that

the two essential competencies for solving the problem are:1. Recognizing that the letters

represented quantities (notion of variable)

2. Creating a “hypothetical operation” to make the two quantities (such as the number of students and professors) equal.

Recall Day 1 ProofsLet’s prove the following:

1 + 3 + 5 + … + (2n-1) = n2

Proof that proves (only):◦Use Mathematical Induction

Next: Proof that explains (and proves)…

Proof That Explains (and proves):

1 + 3 + 5 + … + (2n -1) = n2

The Geometry of AlgebraGeometric representations of

algebraic processes can be helpful

For example, interpreting “n2” as a square of dimensions, n x n

Let’s try another…

Completing the SquareCompleting the Square

What equation is represented by the shapes below?

BaselineBaselineAssessmentAssessment

Baseline Assessment

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

 

xxx 2x 2

4(x 2)4(x 2)

4(x 2)4(x 2 3) 3

4(x 2)4(x 1) 3

4x 84x 4 3

8 1

90

Baseline Assessment

2. 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.