1. Algebraic and Symbolic Reasoning

2. Today’s Agenda 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.
You should expand a bit on the agenda shown here so participants know what to expect. After a review and wrap up of the PLC activities since your last meeting, today’s session will focus on Algebraic and Symbolic Reasoning.

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.
Depending on the size of your group, it will likely be best to split them into small groups for this discussion. After some time has passed you can have a large group discussion to compare experiences and wrap up the material on mathematical reasoning. 3

4. Symbolic Reasoning

5. Arithmetic 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.
This section of the presentation is based on (Matz, 1982), which analyzes common high school algebra errors. Interestingly, the article did not appear in a book about mathematics instruction but rather a collection called “Intelligent Tutoring Systems” in which authors considered how computers might use artificial intelligence to act as interactive and adaptive tutors for students. Identifying common student errors and why they arise is an important component of designing any such system.
One conclusion: artificial intelligence and computing power would have to increase dramatically before this dream becomes reality.
Even in 2009, nearly 30 years later, we are not there. Online homework and quiz systems like WeBWorK and WebAssign provide some automated interaction for students, but could hardly be considered the equivalent of a human tutor.

6. Algebraic 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:
Matz points out that most existing taxonomies of high school algebra errors start by labeling the errors as arithmetic or algebraic, and then subdivide each category further: cancellation errors, factoring errors, collecting like term errors, sign errors, etc. The implication is that algebraic is simply a list of rules to be memorized, and a student who repeatedly makes a mistake simply hasn’t learned the appropriate rule yet. Sometimes we are tempted to write off such students as being incapable of algebraic thinking.
With certain students that might be true, but Matz suggests that many of these students are actually making very reasonable errors as they try to generalize arithmetic.

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:
For example, Matz suggests that this common error can arise as follows:
If a and b were numbers, the student knows he can use arithmetic to add them together and then find their square root. Because he does not know what a and b are, however, he tries to think of a rule from arithmetic that might apply in this situation. Having distributed one operation (multiplication) across another operator (addition) a thousand times, he tries to apply the same “pattern rule” in this new situation.
Matz’s theory can also explain why even good students who know this is incorrect will occasionally make the same error: when they work fast, they might subconsciously slip back into the ingrained pattern from arithmetic.
An aside: some teachers might feel students are less likely to make this error if the square root symbol is draw all the way across (a+b). That is probably true, and meshes with Matz’s ideas: using that notation, it looks less like the distributive law and students would be less likely to (incorrectly) use the distributive “pattern rule.”
The notation here, which used to be more common, was chosen to emphasize the similarity to the distributive law.

8. Generalized Distribution
[For technical reasons, this table is reproduced from (Schoenfeld, 1985), but Schoenfeld reproduced Matz’s Table 1 verbatim.]
This table contains many more examples of ways that students might naively attempt to generalize the distributive property, but correctly and incorrectly. Point out that, when teaching students to avoid the kinds of errors on the bottom, it is more subtle than saying, “Certain operators or functions distribute, and others don’t.” In particular, the square root and exponentiation both appear in the top and the bottom portion of the table.
In short, it depends on both the outside and the inside operation.

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.
Examples are given for each type of error in the following slides.

10. Matz’s Classification of High School Algebra Errors
Errors generated by an incorrect choice of an extrapolation technique
Example:
We have already discussed this type of error. Students who are presented with a new situation might compare it to known problems and apply their solution techniques to the new problem, even if it does not apply.

11. Matz’s Classification of High School Algebra Errors
Errors reflecting an impoverished (but correct) base knowledge
Examples:
Both of the errors shown here result from the change in notation as a student transitions from arithmetic to algebra. In arithmetic, concatenation of two digits results in a larger number whose place values are given by the original digits. Similarly, the concatenation of a number and a fraction implies addition – commonly called the mixed form of a fraction.
In the two examples on this slide, a student has used correct knowledge about concatenation in arithmetic and attempted to apply it in algebra, where the notation means something different.

12. Matz’s Classification of High School Algebra Errors
Errors in execution of a procedure.
Example:
These types of errors might occur in the planning of the procedure: an inadequate plan, for example, might result in a page full of algebraic steps, all of which are correct, but do not get the student any closer to the solution. Alternatively, as shown here, the student might have a correct plan but make a mistake in the actual execution. The student correctly knew to clear the denominators, but simply forgot to multiple the 4 on the right hand side by the required expression.
(This is known as the “Lost Common Denominator” error.)

13. Algebra Errors – Your Turn
Look at the common algebra errors on your handout. With your group members, classify them according to Matz’s framework.
Some of the errors are in a gray area and could rightly be classified as different types of errors in Matz’s scheme. That’s ok, but be sure to discuss as a class what the issues are and why different teachers view the errors differently. For your reference, Matz describes the errors as follows:
is a blind application of the cancellation procedure. Students might visually separate the fraction in two: on the left they can cancel an X, and on the right they can cancel a Y. One might argue that this is an execution error, but Matz uses it as a basic example of extrapolating a technique to a new situation where it does not apply.
is an execution error, similar to the Lost Common Denominator error.
is a bad extrapolation of a known – and correct – procedure for the case when the constant on the right is 0.
is another extrapolation error; it was shown before as an incorrect generalization of distributivity.
is an error based on correct but insufficient knowledge. In arithemtic = always indicates equality or a tautology. In algebra might mean those things – consider [4(x+3)=4x+12] – or it might represent a constraint -- 4(x+3)=20 is true only when x=2. In this error the student uses the arithmetical view of = to conclude that x+1 must be 5 and x+4 must be 6.

14. Symbolic Reasoning Find the error in the argument below.
How would you describe this type of error?
Having analyzed one- or two-line algebraic errors from Matz’s article, tell your participants it’s time to apply that framework to longer problems. These are similar to problems from the book “1=0” by Dale Seymour Publications (1998) and would be worth analyzing even without Matz’s framework; students often ask teachers why a certain proof that 1=0 or 1=-1 is invalid.

15. Symbolic Reasoning Why does the last statement contradict the first?
What is the error?
How would you describe this type of error?
Another invalid proof similar to the problems in “1=0.”

16. Reasoning with Magic (Squares)
Up until now, you have worked almost entirely with symbolic manipulations and reasoning. Algebraic reasoning uses more than symbols, however. The following activity, adapted from (Schoenfeld, 1991), blends numeric and symbolic reasoning with exploration and deductive reasoning.

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.
Give teachers 20 minutes to work alone.
Many will try multiple combinations before noticing that the center square plays an important role.
A big leading question if they struggle at the beginning: What can go in the middle square?
One solution is the following.
8 1 6
3 5 7
4 9 2

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?
You can do or not do as time and participants’ fortitude permits.
Answers:
1. No. For example, multiplying every term by 5 gives a new sequence which works also. Or adding 4 to every term works also.
Many others.
No. The mean of the numbers must be the center square and the mean is not a value in the set.
One third of the magic sum.
The general form is (if they struggle, try having them notice the symmetry along the diagonals first in the magic squares they have already made)
a+x a-x-y a+y
a-x+y a a+x-y
a-y a+x+y a-x

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?
Alan Schoenfeld used magic squares in his college classroom and was able to get students so motivated about them that they were making conjectures and proving/disproving them by the end of the day. Some of the students asked him if their results were publishable! The point is that if students are using reasoning well then it should feel like original mathematics to them – because it is.

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.
If you are interested and wish to have extra material prepared, there are many extensions to magic squares mentioned in (Schoenfeld, 1991) and the NCTM’s 2008 Navigations book on Reasoning and Proof in grades 9-12.

21. Reasoning with Magic Squares
What sort of mathematics did you use?
What types of mathematical reasoning did you use? (Think about last session)
Small group and/or large group sharing on these. Refer them to the types of mathematical reasoning used in Day 2.

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
This part of the day helps teachers to see that there is algebraic reasoning at play prior to students even seeing any symbols. In creating representations of a problem situation, as opposed to being given an equation or symbols at the outset, students are making decisions informed by their algebraic understanding.
Rule of comes from the Harvard Calculus Textbook series which was a popular reform textbook. The “3” are table, graph, equation and the “1” is verbal or real-world. Another example of multiple representations is the Lesh Translation model from Lesh & Doerr (2003). Beyond Constructivism.

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.
From NCTM’s The Open-Ended Approach (1997) which is an English translation of a 1977 Japanese math ed publication. The book advocates open-ended activities to promote the reasoning and generalizing abilities of students, K-12. 24

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”.
Some background:
The “student-professor problem” was first reported by Kaput and Clement (1979), who found that many students displayed particular types of errors when constructing equations to describe a linear relationship between two variables. While translating between words and symbols in “student-professor-type” situations are themselves interesting mathematical tasks, Kaput (e.g. Kaput and Sims-Knight, 1983) has suggested that these errors can be most useful as a lens to investigate the underlying mental processes that students use while solving this problem, because students these same processes while working on other math problems. The goal of this study is to investigate the student-professor problem from several new perspectives and, in doing so, describe ways we might help students work more successfully with related algebraic ideas. Nearly thirty years ago, Kaput and Clement (1979) (and later Clement, Lochhead and Monk (1981)) noted that students had difficulty with the following problem: Write an equation using the variables S and P to represent the following statement: ‘There are six times as many students as professors at this university.’ Use S for the number of students and P for the number of professors. (p. 288) In multiple surveys of college students, roughly 40-60% solved the problem incorrectly.

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?
The most common error made by students was the reversal error: 6S=P. This reversal error was even more common when both variables in the linear relationship have coefficients other than 1 (e.g. when the corresponding equation is 4C=5S). This reversal error arises not only when students are constructing an equation based on words, but also when they attempt to construct an equation based on a table of values or a diagram. MacGregor and Stacey (1993) found that similar reversal errors occurred as well in problems with an additive structure and they concluded that students “were not matching the symbols with the words in an item but were expressing features of some underlying cognitive model of a mathematical relationship.” (p. 228)

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
The rough visualization on the bottom of this slide is correct, but because there are six students and one professor it is all to easy to translate the diagram into the equation 6S=P.

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?

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.
Kaput, Sims-Knight and Clement (1985) suggested that there are two components necessary for success with “student-professor-type” problems: “understanding variables (and the underlying notion of mutual variation of two quantities) and understanding the syntactical features of the algebraic representation of variables.” (p. 58) While Rosnick and Clement (1980) noted that a misconception of the equals sign could be associated with incorrect responses, few other researchers have focused on equality misconceptions as a potential source of error. Not only does the reversal error appear in many situations, but it has also proven difficult to remediate.
The background notes for the last few slides is directly cited from an article called NEW PERSPECTIVES ON THE STUDENT-PROFESSOR PROBLEM by Aaron Weinberg, Ithaca College. It appears in
Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29
The annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Stateline (Lake Tahoe), NV: University of Nevada, Reno.

30. Recall Day 1 Proofs Let’s prove the following:
… + (2n-1) = n2
Proof that proves (only):
Use Mathematical Induction
Next: Proof that explains (and proves)…
Present on overhead or board, one step at a time.
Use Mathematical Induction.
Step 1: Show that the statement is true for n = 1.
It is, since 1 = 12
Step 2: Assume the statement is true for n = k.
So, assume … (2k-1) = k2
Step 3: Using the assumption for n = k, show that the statement is true for n = k + 1.
If n = k + 1, then starting with the left side,
… + (2(k+1)-1) = … + (2k + 2 – 1)
= … + (2k + 1)
= … + (2k - 1) + (2k+1) by writing out the last two terms
= k2 + (2k+1) by the induction assumption
= k2 + 2k+1
= (k + 1)2
So … + (2(k+1)-1) = (k + 1)2 . That is, we have shown that the statement is true for the n = k + 1 case. Then, by the principle of mathematical induction, the statement must be true for all positive integer values of n. QED.

31. Proof That Explains (and proves):
From Proofs Without Words by Roger B. Nelsen MAA publication.
This book has many other similar proofs which are great fun to decipher. Some are quite deep, mathematically.
… + (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?
Another from Proofs Without Words, Volume 1, by Roger Nelsen.
Identify Algebra tile users so they don’t spoil it for others.
You might give the diagram, without any symbols or labels to half the group, and give them the equation and ask them to relate the two together. Then the other half of the group could do the reverse (try to determine the equation from the diagram and symbols) as seen on the slide above.

34. Baseline Assessment
Before finishing the day, go over the baseline assessment with participants. Attach a copy of it to their handout and give everybody minutes (or as needed) to complete it, before discussing it problem by problem. Information about expected student responses is contained in the Microsoft Word document with the assessment. 34

35. Baseline Assessment
Explain why the following argument results in a false statement. Be sure to identify which step(s) are untrue. 
They should work through this again, and look at the rubric to see how to score student responses.

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.
They have already worked this exact problem, so just go over the rubric.