Diverse learners

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PART 1

The term "self fulfilling prophecy" was defined by the American sociologist William Isaac Thomas (1863 - 1947) in the dictum: 'If men define situations as real, they are real in their consequences.' Events tend to turn out as one has hypothesized, not because of some great insight but because one behaves in a manner to achieve this outcome. A large body of evidence exists in various areas of psychology supporting the self-fulfilling prophecy. R. Rosenthal, Experimenter Effects in Behavioral Research (New York, 1976) Research is showing that one of the major influences on student achievement is teacher expectations. . Read and reflect upon the following : INITIAL POST The Pygmalion Effect Definition and Background: What is the Pygmalion Effect? People tend to live up to what's expected of them and they tend to do better when treated as if they are capable of success. These are the lessons of the Pygmalion Effect. Pygmalion first appeared in Greek mythology as a king of Cyprus who carved and then fell in love with a statue of a woman, which Aphrodite brought to life as Galatea. Much later, George Bernard Shaw wrote a play, entitled Pygmalion, about Henry Higgins (the gentleman) and Lisa Doolittle (the cockney flower girl whom Henry bets he can turn into a lady). So the Pygmalion Effect has come to mean "you get what you expect." If you expect disaster, your expectations may well be met in a kind of "self-fulfilling prophecy", yet another catch phrase about the pressure of expectations. Case Study In another classic experiment, Rosenthal and Lenore Jacobson worked with elementary school children from 18 classrooms. They randomly chose 20% of the children from each room and told the teachers they were "intellectual bloomers." I first learned of this story from Zig Ziglar, a renown motivational speaker and positive thinking proponent. He goes into detail about how a set of school teachers were told that their students were geniuses. They've been tested by some new methodology of determining the success of school age children, and THESE kids were the best of the best. In addition, these teachers were told that they were uniquely entrusted with these children's welfare for the coming school year. They explained that these children should show remarkable gains during the year. The children, performed admirably, gaining an average of two IQ points in verbal ability, seven points in reasoning, and four points in overall IQ. At the conclusion of the experiment, the teachers were informed that these students were randomly assigned, much as any others are during any normal school year. And the teachers as well, prior to this year, were nothing special -- they, too, were selected randomly. The participants in his study were not necessarily aware that they were being monitored and that this was an experiment of any kind.

REPLIES: PLEASE REPLY TO A AND B

A. I found the case study really interesting. It's hard to believe that just high expectations for students is all there needs to be for high achievement and students to succeed. It's amazing that the IQ of the students in the study did rise. It seems like high expectations are key, but what does this look like in the classroom? Does it have to do with our behavior, what we assign, or how we teach? After reading the study these questions came to mind. In the past, when struggling students do well on something even if it’s a small thing I have always praised them for their work. These students then often feel proud of them and become motivated to do well but this doesn't always occur. I do agree that the Pygmalion Effect and the "self-fulfilling prophecy" are connected. In the past there have been students who believe they are going to fail no matter what they do so they don't put any effort into their work because they think what the point is. It's important that students believe in themselves and think positively about their abilities. For students who don't its important for teachers to then step in and help them see their potential. In thinking about high expectations, I feel that this means believing that the students are capable of doing well. I find that at my school teachers do a great job of believing in the students and letting them know they are capable of success. However, I think students don't realize that effort also has a great deal to do with success. There is a correlation between asking for help, putting in the effort, and doing well. I believe that along with teacher's high expectations, parents also play an important role in their child's success. I find it frustrating in parent teacher conferences when a parent claims that they were never good in math so that's why their child isn't great either. At home, students should also be treated as if they are capable of success. I do agree that "people tend to live up to what's expected of them." From the study, I saw that the first step towards student success is as a teacher being aware of yourself and how you are perceiving your students.

B. I have come across several tests that prove that teacher expectations affect student performance. However, the research on how teacher interaction depends on their expectations was amazing. When teachers have a higher expectation of their students, they provide more specific feedback, more approval and positive signals. Students sometimes encounter low expectations not only from teachers but also from parents and it is difficult to get the students out of the strong negative self-efficacy.

Teachers can practice some behaviors regularly with every student consciously. They can give specific feedback to student responses with acknowldegement of correct parts in their responses and asking follow-up questions. Another behavior on the teacher's part I think is wait time. Longer wait time not only gives access to all students but also gives the teacher enough time to think about her own reaction and behavior.

Teachers also need to connect with the students and know them better. We mostly see the students who are anxious about math and struggle in class only in the negative light. One of my students who struggled in math had great leadership skills that I had not noticed in class. I once watched him at a basketball game and saw that he was very motivating and brought out the best in the other boys on his team. He was a totally different person - confident, caring with a positive outlook. The other kids respected him a lot. That was a turning point for me. That year I put in effort to get to know my students better, learned about their interests, asked them to show work from their art class or their writings.

We need to be consciously aware of the signals we are sending students through our interaction with them. We have to tell our students that we believe in them through our actions and not just words.

INITIAL POST: REFLECTION/SOLUTION

Students in all 50 states must now participate in the National Assessment of Educational Progress (NAEP). The results of this assessment are published as the Nation's Report Card. The learning standards on which students are assessed are rigorous and the expectation is that all students receive experiences and instruction that align with those standards. Well, how then, can the teacher in an urban or rural district, teaching in a heterogeneous classroom, provide appropriate instruction for each student? What about teachers in village schools wehre there may be multiple age levels in the same classroom? These are the realities that face teachers. Teaching in a diverse environment includes situations that go beyond multi-cultual and multi-lingual. Often learning differences offer multiple challenges to a teacher of mathematics. Think about the following problem given to an algebra class.

Solve the problem yourself. What is the big idea in the problem? Or, what is the purpose of assigning this problem? After solving the problem, think about how you might revise the problem to allow access into the problem by all students without changing the main focus of the problem. Rewrite the problem so that all students have access to the same main concept.

PART 2

Examine the following student responses. Based upon the evidence contained within the student work, what can you say about each student's understanding of the concept and the procedures for solving this problem?

STUDENT A.

STUDENT B

STUDENT C

STUDENT D

REPLIES: PLEASE REPLY TO A AND B

A. PRASAD Solution to problem: x = Zune y = iPhone y + 3x = 1,148.97 → y = 1,148.97 – 3x 2y + 2x = 1,297.98 2(1,148.97 – 3x) + 2x = 1,297.98 2297.94 – 6x + 2x = 1,297.98 2297.94 – 4x = 1,297.98 –4x = –999.96 x = $249.99 y = 1,148.97 – 3x y = 1,148.97 – 3(249.99) y = 1,148.97 – 749.97 y = $399

One Zune costs $249.99 and one iPhone costs $399. Reflection: I found this problem similar to the one last week with the Ipad and Ipad 2. As I was completing the problem, the purpose of assigning this problem seemed be to check students understanding of solving a system of equations. A student could use two different methods to solve this problem, linear combination and substitution. I like that the problem already provides visuals. Without changing the main focus of the problem, I would revise the problem by changing the final costs to simpler numbers without decimals. To revise my problem I worked backwards. I saw that in the original problem the cost of one Zune was 249.99 and one iPhone was 399. I rounded 249.99 to 250 and 399 to 400. Then I found what the total cost would be with the same amount of gadgets. One iPhone and three Zunes would now cost 1,150. Two iPhones and two Zunes would now cost 1,300. This eliminates the decimals in the problem. My revision of the problem would then be: You are given the costs to purchase different amounts of gadgets. One iPhone and three Zunes cost $1,150. Two iPhones and two Zunes cost $1,300. a. What is the price of one Zune? b. What is the price of one iPhone? Be sure to explain your answers. Comparison to the one given: My revision was the same as the one given. I found that the only way to revise the problem without changing the main focus was to adjust the costs to become simpler. To make sure the cost would work I had to figure out what the solutions would be first, which seems similar to what they must have done with the revision shown in the module. Student Responses:

Student A: This student does not demonstrate an understanding of solving a system of equations as he used a guess and check method. The student chose the simpler problem indicating that perhaps the student is not comfortable with decimals. The student does not show an understanding of being able to create a graph based on the equation given. The graphs should be linear graphs and should not represent guesses. However, the student does arrive at the correct solutions using the guess and check method. Student B: Through his calculations, this student demonstrated an understanding of solving a system of equations through the use of the linear combination strategy. The student also chose the original problem with decimals which shows that the student is not afraid to work with decimals. This student obtained the correct solutions and correctly represent the equations on the graph. Overall, this student showed conceptual understanding. Student C: This student does not demonstrate an understanding of solving a system of equations. The student incorrectly used a bar graph to represent the linear relationship represented in the problem. The student, however, made an insightful observation that the change of gadget between the two choices increased the price by $150. Using this information the student did come up with the correct solution. The student 's explanation was clear, easy to understand, and very detailed unlike the explanation from the first two students. The student also used the simpler version of the problem indicating that he might be uncomfortable with using decimals or wanted to take the easier route. Student D: This student demonstrated an understanding of solving a system of equations through algebraic manipulation. The student was able to isolate the variable and then plug it into the second equation. However, the student does not show an understanding of translating equations into a table or a graph and using variables within a table. This student showed evidence of procedural and mechanical understanding. Next Steps: All of the responses differed greatly. Student A needs to practice algebraic manipulations as they still are using a guess and check method. One strategy to use here would be to provide the student with equations where they have to isolate a specific variable. Then construct a table by plugging in values. Both student A and C need to work towards becoming comfortable with using decimals. Student D needs recognize and understand the relationships between tables, graphs, and equations. In 7th grade we don't solve system of equations but work heavily with linear equations. I have used similar templates for recording their work as the one shown in the module except many of them have been used to analyze a single situation. This strategy might be helpful for student D so that they can slowly gain understanding for working flexibly with the multiple representations. Another strategy to use with some of these students is to pair certain students up to discuss their strategy and learn from each other. While student B was able to complete all of the parts for the task they might benefit from being paired with student D so that they could learn another way of manipulating an equation. Student D could learn how to construct a table and graph from Student B. Any type of discourse between students is always beneficial to their learning. After working independently on a task like this, the group discussion and then a whole classroom discussion would be helpful in addressing misconceptions. To see if these strategies have been effective I can give a similar problem to students who struggled with this one and assess them again. For students A and C, I would look for the use of a mathematical strategy in solving the system of equations. For student D, I would look to see if they are able to construct a table and graph either from the data given or through the use of equations.

B. DACEY: This particular problem is looking at a system of 2 equations and essentially trying to determine the cost of both the ipod and zune. This can be done by substitution or elimination, through graphing, and by making a table with guess and check work. For me, this problem is being given to the students not only to test their flexibility and understanding of algebra, but also to see if they could solve to determine the relationship between 2 variables.

Like last week, my revision simply had me choose whole numbers that were easier to work with. My revision was very similar to the alternative listed in the module! I let the ipod equal $100 and the zune to equal $200. Therefore, for the first example:1 ipod + 3 zunes = $700 and in the 2nd example: 2 ipods + 2 zunes = $600. The students had to work to solve for the original price of the ipod and zune.

Student A: Student A seems fairly well versed in “what” to do, but may be lacking some of the understanding behind the work. I think student A’s graph is the most telling piece of his/her work as it clearly shows the intersection of the 2 functions. This student did use guess and check fairly well, and his/her logic in the table made sense. This student’s work seemed fairly procedural. Perhaps, he/she learned how to do these steps with a similar problem.

Student B: Student B seems the most comfortable with this material and the the chart, graph, and equation work were all detailed. This student also knew it was a series of linear equations that he/she was working with. Again, this student demonstrated his/her knowledge most accurately using all 3 strategies. This students has the best conceptual knowledge.

Student C: I don’t understand student C’s work at all. I can see that they seemed to know the cost of each device, and were just doubling it in the chart. I am not sure how he/she got his/her answers. Some of the writing was too small to read.

Student D: Student D definitely had the actual algebraic process down. However, I am not sure if this students knows as much as student B because he/she was only able to demonstrate the work algebraically as a system of 2 equations and not by using a chart or table. This student seems more mechanical.

To address any conceptual problems, I think we need to back up and talk about what we are actually trying to solve and define a relationship in some way. I think we also need to go through what a chart and graph can show, and how we would use these to model the algebra. For me, it does not seem that the biggest issue here was finding the answer, but more about understanding why and how a strategy got us there.