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Assessment criteria and tools to evaluate students' performance in the course (E2)
Models of an "A" and a "B" paper for the mid-term exam

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EXAMPLE OF "A" PAPER

Midterm paper -- Lampert's article -- Exploring exponentiation

Over the course of the semester, I have gained many new perspectives of what makes for effective teaching. I have decided that re­examining a classroom story from early on in the course in light of my more recent discoveries of the inquiry approach and the NCTM recommendations would be an informative pursuit. In particular, I selected the instructional episode Magdalene Lampert uses to illustrate her arguments in the essay "When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching."

The lesson described involved 18 fifth grade students in learning basic properties of exponentiation. Lampert speaks of having two simultaneous agendas. The first was in straightforward gaining of mathematical skills, especially understanding the effects of adding exponents. Further, she hoped for students to acquire knowledge about mathematics itself, to practice skills and dispositions important for engaging in an intelligent discourse within the subject. Thus, as she neatly describes it, she hoped that students would "learn not only that they could divide or multiply by subtracting or adding exponents and how to use the technology of exponents, but also that the warrant for doing so comes from mathematical argument and not from a teacher or a book."

These goals do strike me as worthwhile. I think largely of the NCTM curriculum standards and the emphasis on math as communication and as reasoning. Lampert clearly wants her students to come at solutions via reasoning, both deductive and inductive, by posing arguments rather than merely absorbing methods and facts. They will be asked to communicate amongst themselves and clarify their own thinking while reflecting carefully on others' ideas. Also, we can expect favoring of problem solving activities instead of routine application of formulas from the teacher or text. A potential weakness may lie in the lack of concern for establishing connections among various aspects of math and with real life applications. There is no indication that the children were led to understand or even question why learning exponentiation will be of value to them later and outside of class.

Lampert's intentions also coincide fairly well with those of an inquiry approach. Students are to be actively involved in constructing a view of math. The role of the teacher lies in posing questions and guiding discussion without standing as a final authority on what is officially correct. Yet, again, the exploration isn't necessarily geared to responding to student interests for motivating appreciation of exponents.

I would like to examine the specifics of the lesson more closely with relation to the NCTM professional teaching standards. It seems logical to me to focus on each of those six standards in turn and draw on specific examples from the vignette as appropriate. Before proceeding, I offer a brief summary of the major events. First, students were allowed to create, using calculators, tables of squares from 12 to 102 and uncover patterns within them. They were then presented with the questions "What is the last digit in: 54? 64? 79?" and allowed to debate them independently before class discussion. Language conflicts were resolved, then exponentiation with base 5 was generalized and compared with base 6. 74 quickly agreed upon, debate on the fifth power of 7 opened up various differing views and led towards the familiar law of multiplying by adding exponents.

First, by NCTM advising a teacher needs to pose worthwhile mathematical tasks. These should draw on significant math concepts as well as student backgrounds and learning styles. The patterns in last digits of squares of numbers is something I was never fully aware of, but I believe it a powerful task for making exponents more accessible. Indeed, the students were considerably engaged throughout the lesson, and 12 distinct voices can be counted within the discussion. Lampert was wise to start with the simpler examples of powers of 5 and 6 and allow students to work alone to start. She recognizes from previous experience the likelihood of confusing exponentiation with multiplying by the power; consequently, she checked in with certain students to ensure they were clear enough to participate constructively. The question about powers of 7 was most effective of all, opening up conflict between various viewpoints and stimulating progressive communication. I wonder primarily about the students' prior mathematics experience, for I doubt the activity itself was sufficient to provoke as much enthusiasm as was displayed. I'd be inclined to suggest further connections to prove usefulness of the topic.

Lampert does assume a teaching role that effectively promotes discourse. Overall, she is careful to listen to all students, acknowledge contributions, and seek justifications. Early on, she decides to step in declaring a "language problem" upon perceiving the ambiguity of Alianna's articulation to "square 5 two times." Consistently, whenever contradictory answers are given, all are written down to invite further consideration, notably when 1, 9, and 7 are offered for the last digit of the fifth power of 7. One striking decision by Lampert was to recognize Sam's confidence in the correct answer, then quickly turn to Arthur for clarification of his incorrect answer rather than letting Sam "give it away." The message is that everyone's thoughts should be valued. She also asks Sarah why she thought 9 and gives a clearer account of Arthur's viewpoint. She jumped in a bit hastily to point out that 7 3 ends in 3 when the pattern 1,7,9, . . . was assumed, although this was based on a previous declaration by Gar. Her final injections point an arrow closer to the adding exponents law, guiding students toward their own belief in that official result.

It is also becomes clear that the students have been well directed towards their appropriate roles in discourse, and Lampert comments upon these herself. Arthur's concession, "I want to revise my thinking," is a readily apparent example to that effect. Not what I normally expect to hear from a student, the statement acknowledges previous incorrectness without reason for doubt in oneself. Sam's discounting of Martha's suggestion for 8 as last digit of 7~ is not derisive and is backed up with facts: "it's odd number times odd number and that's always an odd number." Car also offers his decision that it can't be 9 as a proof and doesn't direct it towards anyone else as showing him or her wrong. Debates are handled with reference to mathematical evidence instead of deference to authority.

The use of tools to enhance discourse, the NCTM's fourth standard, is a bit thin in this particular selection. Most noticeable is the use of calculators to simplify multiplication, a fully logical approach for the complexity of exponents. In her discussion before detailing the lesson, Lampert mentions some concrete examples of comparisons of distances to illustrate usefulness of having exponents, and we can hope she used such models at other points in the unit. The table to elucidate possible patterns in the squares is a valuable tool, but there were no occasions to use graphs or other pictures, metaphors and stories, different symbols and terms, or writing down of thoughts. Admittedly, the concept here doesn't naturally invite much hands­on activity; still, creativity has definite merits for making the whole learning experience memorable.

While we are missing many of the earlier details of how classroom dynamics were established, we can give a rather positive assessment of the learning environment that determines this exponential lesson. The students are obviously accustomed to both independent work and collaborative discussion. As illustrated above, they feel competent enough to offer comprehensive mathematical arguments for their beliefs and revise their thinking without embarrassment. This we can assume resulted largely from Lampert's consistent use of tactics previously praised. Everyone's thoughts are valued upright and equally laid out for the scrutiny of all. Sam's correct assertion that a power of 5 must end in 5 was not simply acknowledged as right; the whole class was invited to consider its validity. Students had to grapple with themselves over conflicting answers for the 7 question, all ideas left on level ground until convincingly disproved or justified. Communication flows so as to propel the whole group forward.

The NCTM also recommends an ongoing analysis of teaching and learning while it takes place. Doubtless, Lampert engaged in such examination throughout as her official role leans toward researching rather than pure teaching. Some of her observations led to adjustments within this lesson, such as the attention in checking student's individual progress to those who had confused exponentiation and multiplication previously. It appears also that her experience with Sam's quicker competence led her away from directly asking him to explain his answers, thus to allow other students opportunity to express themselves. Lampert's analysis has led her to isolate five major potential obstacles to true mathematics learning for all in a classroom: relying on a teacher or other authority as source of correctness, using given rules as sufficient justification without true argument, silence due to self­doubt, discounting others' ideas by force or political power, and stubbornness and face­saving behaviors. She thus advocates changes in teaching to create a more useful view of what it is to know and do math.

The teaching session I have elaborated upon did proceed in many ways as an inquiry. Student took responsibility for problem solving and critiquing each other while the teacher acted as a manager and clarifier. At one point, a group posed a problem of their own to investigate, namely to show that positive powers of 5 all end in not only 5 but 25. Students were further able to observe that while powers of 6 all end in 6, they don't all end in 36, and the convenient property of powers of 5 and 6 did not extend generally to other numbers. Asking why may lead to some interesting discoveries; it is a shame more time wasn't given to such problems. The lesson falls short of truly being an inquiry on terms of relevance to student interest ­ it lacks in providing worthwhile connections with everyday life and among mathematical disciplines as well as in allowing students options on what questions to spend more time considering within the class.

Lampert's lesson can be considered more effective than more traditional approaches on multiple levels. The students in it built up towards a law that would have normally been just given to them. Thus, they have more reason to value and remember it, having appreciation of how mathematical reasoning may have established it. The experience for them was another element in an ongoing focus on cooperative discourse. Within it, they learned further to respect each others' ideas and to support their own ideas with reasoning. Math was discovered not as unwieldy system of rules, but as a dynamic and intriguing development of human logic and debate.

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Instructor's evaluation for the "A" paper

Good identification of the instructional goals of the classroom episode at the beginning. Although your description of the episode is very brief, your whole paper demonstrates your thorough understanding of this instructional event.

Excellent -- especially in your analysis with respect to the NCTM Professional Standards. All your points are well supported with illustrations from the episode.

Your analysis of the episode with respect to the NCTM Professional Standards was very thourough and excellent! In constrast, you did not develop a similar analysis with respect to the "content goals" of the episode -- although at the beginning you make some good references to the NCTM Curriculum Standards as you discuss the significance of Lampert's "process goals."

I agree with your conclusion that Lampert's teaching is consistent with the principles of an inquiry approach -- I wish, however, that you had better substantiated this conclusion by examining systematically each of the features of inquiry we discussed in class, just as you did with the NCTM Professional Standards. I thought you made a very good point about the importance for students to "value" what they are learning; this does not need to coincide, however, with recognizing practical applications for the mathematical topics studies: it is not clear from your paper where you stand about this point.

Your paper was clearly written, understandable and to the point: good job!

General comments:

Excellent paper. Your analysis was very thorough and demonstrated a very good understanding of the pedagogical principles and instructional approaches examined in the course so far.

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EXAMPLE OF "B" PAPER

Midterm paper -- Area story -- Episode 1 ("Rocket activity")

Being most familiar with the area topic after the many class discussions, I decided to concentrate on the beginning of that classroom story. Since the first three days of that unit was also used as an example on the assignment sheet, I figured that would be a reasonable chunk of material to work with.

The main goals of the lesson are:

  1. Encourage logical thinking skills on the part of of the students. The idea here is to have them form their own tactics for problem solving rather than being given meaningless numerical problems to compute.
  2. Have the students realize that ``area'' is something of value to them. The problem they were trying to solve is one that involves real objects and therefore has meaning beyond the textbook. This is not something explicitly stated, but clearly important.
  3. Have the students realize that precise answers are often unnecessary, and in some cases can't be found. In this problem, they were trying to decide how much wood and paint is needed to construct this rocket. Some of the properties of this are that measurements are imprecise, having extra paint isn't a bad thing, and we can't tell exactly how much paint is needed anyway.
  4. Distinguish area from perimeter. This was an area of confusion for some students. ``Size'' is used to discuss a variety of characteristics of an object. Using a student as a measuring device shows one example where only one dimension (height) is used. Other dimensions exist, and an internal measurement (area) can be found based on all of the external dimensions. The various ``sizes'' need to be made clear for many tasks involving the real world.
  5. Talk about the unit size and relate that to the ultimate scale of the object to be built. Doubling the length and width changes the area by a factor of four.
  6. Be able to recognize and use various tactics for finding areas in general problems of this nature. This is a useful skill for any problem solving task. Recognizing the problem and components of the solution will often carry you most of the way to a solution.
  7. Understand where the formula for area of a rectangle comes from. Simply knowing that you multiply width by height isn't sufficient. An understanding for why this is the case (we are looking at multiple rows of boxes where each row is the same length) is necessary.

The goals all seem appropriate, both for the "age/experience" of the students, and for the task they are trying to solve.

The activities involved individual thinking in some cases, pairs of students, and eventually the class as a large team. The focus on individual work was only in the beginning to ensure that the task was clear to everyone. The initial task was simply presented to students to individually find the area of the rocket. The purpose was explained (to help the primary students build something for their carnival), providing everyone with a reason for solving the problem. This is the first and most important point in designing an inquiry unit. A problem of interest to the students needs to be presented. Since no other information was provided to start with, they were on their own to use whatever skills they had to find a solution. This opened up the activity to each student's creativity.

While the general approach of inviting each student to solve a problem is good, they could easily become frustrated. The next step was to have students work together in pairs to discuss what they came up with. Eventually the whole class got involved in sharing ideas. A key point in the planning was to have students share what they know with others to receive feedback. Unfortunately in this case it appeared that students did not do a good job of presenting their ideas, and the teacher ended up participating as an interpretter. This could be damaging to the goal of having a student centered learning environment since the teacher's interpretations of student descriptions is likely to contain her own knowledge of the problem. Her behavior in probing students for further information was at least consistent with the inquiry approach. In most cases she did not volunteer information that had not been presented by the students in some form.

An important strategy used here was to not tell the students what the correct answer was until a sufficient class effort was made. Aside from the fact that ``correct'' had not yet been defined, the students needed a reason to stay interested in pursuing a solution. Answering their question about which answer was correct would only end their questioning before the topic had been exhausted. Unfortunately we see students all too often that only want ot find an answer and not understand it. Once an answer is found, the problem is finished, whether there is more to explore or not.

Each day, at the start of class, students are reminded of what they had discussed previously. This review is based on what the students found, not what the teacher intended for them to know at this point. This is important in allowing the students to guide the lessons. This review can then open up further questions that students had from the previous day, or act as a stepping stone to new material or outlooks on the problem.

Clearly, in this class, discussion and student opinion are placed at a very high level of importance. The students came out of this having developed solutions on their own, and therefore probably knowing the material they did much more thoroughly. My interpretation of the reading suggests one shortcoming, and that's that many of the students still didn't seem to grasp the idea that the ``correct'' answer could be determined by group consensus. They all wanted to know what Ms. Anthony's solution was, as if that was sure to be the requireed answer.

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Instructor's evaluation of the "B" paper

Your discussion throughout the paper demonstrates a good understanding of this episode. Your identification of goals in the first page is especially thorough.

You have made several good observations, referring to specific illustrations of the episode to make your points, although your analysis was not very systematic.

Although I can gather from your observations and comments throughout the paper that you are sympathetic with the spirit of the NCTM Standards, you did not make any explicit reference to them in your analysis of this episode.

Same as above!

Your paper is easy to read and overall well written. As indicated in the manuscript, you could have occasionally used more appropriate terminology. A brief description of the episode upfront would also have helped the reader follow your comments better.

General comments:

Your paper suggests an understanding of the pedagogical principles and instructional approaches discussed in the course so far, but your lack of references the readings is problematic. By not examining the episode in light of the NCTM Standards recommendations and the principles of an inquiry approach, you did not fully satisfy what requested of you in this assignment.

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