VIEWS: 4 PAGES: 49 POSTED ON: 8/15/2011
Making Student Thinking Visible: A Close Reading of Online Conversations University of Maryland September 29, 2004 Anita Salem Professor of Mathematics Rockhurst University Online Calculus Conversations: Making Student Thinking Visible A Carnegie Foundation Scholarship of Teaching & Learning Project Anita Salem (Mathematics) Renee Michael (Psychology) Problem Inability of most students to apply methods and concepts used in a practiced problem to a new situation Literature Review Types of Intelligence (Robert Sternberg) Book Smarts (skills, methods, procedures) Street Smarts (using common sense to find new strategies for solving problems) Reasoning Modes (Brown, Collins & Duguid) Students reason with laws acting on symbols resolving well-defined problems producing fixed meaning Practitioners & Just Plain Folks reason with casual stories acting on situations resolving emergent problems producing negotiable meaning Project Goal Increase students’ conceptual understanding of first principles in calculus by creating an activity where students could practice solving problems using a Just Plain Folk approach. Project Description Student-to-Student web-based threaded discussion (participation required & graded) Three questions each focusing on key concepts in Calculus Follow up in-class student-to-student discussion Exam question conceptually related to the key concept but contextually varied from the practice problems. Calculus Conversations Question #1 Was there ever a time since you were born that your weight in pounds was equal to your height in inches ? Provide a mathematical explanation for your answer. The Main Idea Weight (pounds) Height (inches) Age The Conversation Tuesday, 12:24 pm I think we should probably pretend that this takes the form of a graph. Maybe making two lines. One would represent a person’s height and the other line representing a person’s weight. Tuesday, 5:41 pm The graph idea is great. When the two line graphs are shown together in the same graph, the intersections would show the age when the height and the weight are the same. The y-axis would have to have the same calibrations for the height & weight. The x-axis would have the age of the person. Tuesday, 9:05 pm Can you really do this & make it clear to the outside observer what our graph represents or does it have to be done in two graphs and find a common point? Do you need to specify inches & pounds? One graph or two graphs? Weight (pounds) Height (inches) Age Age Tuesday, 9:05 pm We might want to start out with a scatter plot at first for a basis to start with. Label one side weight and the other height. … Then you’d need to take it to the next level and add the age factor. Get rid of the “age factor” ! Tuesday, 9:26 pm Unless people were born in some different way than I am accustomed to, you weigh less than your height at birth. I was 11 inches tall and weighed 6 pounds. This should take care of the question of yes or no. But the answer to where that point takes place, a scatter plot seems to be the best answer. Bingo! We have the first “conceptual” response. However – it is very poorly & incompletely expressed. Tuesday, 9:27 pm I think the idea of having age on the horizontal axis is a great idea. For the vertical axis, I think that maybe it should just be a listing of numbers… you know from 0 to maybe 200. Then just draw two different graphs in two different colors to depict between the two of them. It would be really easy to tell when the two (height and weight) are the same because they will be the same point!!!!!! Tuesday, 11:18 pm I feel that you should make two graphs. One showing the results of the comparison between age and height and the other showing the comparison between age and weight. Then you should lay one graph on top of the other and see if there is a point when the height and weight equal each other. Wednesday, 9:50 am The lines for weight & height aren’t functions since they aren’t continuous. Weight tends to fluctuate in a range of 10 to 15 pounds on most people weekly. This creates quite a bumpy line that cannot be simulated by a function. What about those squiggles? Wednesday, 3:21 pm Just because a line is wavy or bumpy does not mean that the line is not a function. It cannot be a function if you assign two values of height to one age which I agree is hard if you measure age in years. But why not measure age in months? Or days? By zooming in on the graph, and calculating age in days, I believe it is possible to find the point of intersection. Wednesday, 11:02 pm Really, one would not have to draw a graph at all. If one were to make a table with three columns, the same idea could be captured. For instance the first column would be labeled “Time”, the second column “Height” and the third column “Weight” Time should start with birth. Then with every input of time, there should be a height and weight to correspond. Of course, this would take extensive time and a really, really long piece of paper. Let’s dump the graph idea! Age Height Weight Birth 20 inches 7 pounds 9 months 27 inches 18 pounds 48 months 36 inches 35 pounds 60 months 38 inches 40 pounds Thursday, 6:38 pm I believe that yes there will be a time in our lives that we do weigh in pounds as much as we are tall in inches. Now as far as a graph, I think you would have to weigh yourself monthly or bi-monthly until you begin to get very close to the barrier. Then when you got closer you would have to start weighing yourself at least once a day. Also, then maybe you get sick and lose some weight in which you could have crossed back over the barrier. It is possible that you can cross it more than once. Response Continued There could be a legend used for clarification as to whether the increments on the y-axis represent pounds or inches (depending on which line the observer was observing.) We know that babies length in inches will exceed its weight given normal circumstances. Therefore the weight line will start below the height line. But at our current age we know that our weight in pounds exceeds our height in inches. So, at some point in time the weight line grew above the height line, and this is where the two lines crossed. Assessing the Activity Quantitative Analysis Looked for possible relationships between participation in the on-line discussions & performance on the contextually varied exam questions. Qualitative Analysis Examined student approaches to problem solving. Quantitative Analysis Exam Question Average TABLE 2 Exam 1 Question Avg Exam 2 Question Avg Exam 3 Question Avg 64 % 86 % 60 % Scoring the Activity Score = 1 Did Not Participate Score = 2 Contribution Confused the Conversation Score = 3 Contribution Kept the Conversation Level Score = 4 Contribution Moved the Conversation Forward Quantitative Results Exam Question Averages by Activity Scores 100% 90% 80% 70% 60% 1 50% 2 40% 3 30% 4 20% 10% 0% Exam 1 Exam 2 Exam 3 Statistical Correlation A statistically significant relationship exists between student performances on the Calculus Conversation activities and corresponding performances on conceptually related but contextually varied exam questions. Coding the Responses Practical Response (P) Conceptual Response (C) Intercommunication (I) Language Extremes (L) Pose Questions (Q) Qualitative Results 100% 90% 80% 70% 60% P 50% C 40% I 30% L 20% Q 10% 0% Problem Problem Problem 1 2 3 Real Results Activity provided a window into students’ understandings and misunderstandings Observation that students struggle to rise above the details of a problem Implications for Practice Think more carefully about how to get students to use Just Plain Folk problem solving approaches. Look for ways to capitalize on students’ comfort levels with practical matters to help them move into more conceptual practices? Be more aware of the effect of our presence in an activity. Conclusion THIS WAS A CASE OF An attempt to improve students’ conceptual understanding of fundamental ideas in calculus THAT RESULTED IN improved teacher understanding of how students construct their own meaning of fundamental ideas in calculus. Acknowledgements Participating Calculus Instructors John Koelzer Paula Shorter Keith Brandt Julie Prewitt Kramschuster Research Assistance Craig Sasse Tom Jones Professional Changes Better consumer of the Scholarship of Teaching & Learning. Increased respect for how other disciplines ask & answer questions that are of interest to me in becoming a better teacher. Clearly articulate how this work is scholarly in every sense of the word. Student Attitudes 51 % approved 30 % disapproved 13 % no comment Typical Positive Responses Typical Negative Responses Positive Responses to Calculus Conversations Calculus Conversation Questions brought the thinking to a whole new level. It was difficult to understand what people were thinking because it forces us to write about Math. You also have to challenge yourself to get your thoughts across to the rest of the group. It would have been helpful to somehow have learned a way to write about Math and get your point and thoughts across early. It was quite challenging. I found out the importance of weighing in early. When I weighed in early, I got more out of it and was able to take a more active role in the conversation. Though I didn’t always get the answers, I understood the concepts better. The conversations allowed people to think independently, but at the same time work to solve problems in a group. The conversation allowed us to talk out the problem in understandable terms. Negative Responses to Calculus Conversations This helped me some of the time, but most of the time I wished there weren’t so many people doing the same problem. Many times, I would check it the second day and there would be so many students with postings that I was overwhelmed and got it set in my mind that there was no way I could do the problem. I feel the conversation often starts fast and then drags on courtesy of the people at the end who have no idea what they are going to write. They just repeat what other people have already said. This repetition clogs up the discussion which detracts from the usefulness of the activity. I never felt that Calculus Conversations contributed to my understanding of a concept. There have been numerous times that I have totally understood something, then gone to read other peoples’ responses to the topic and gotten so confused that I have no idea what I am supposed to do anymore. Real Results Activity provided a window into students’ understandings and misunderstandings Observation that students struggle to rise above the details of a problem Striking link between learning theories, found in the literature based on studies from K-12 math classes & practice, found in the project Implications for Practice Think more carefully about how to get students to use Just Plain Folk problem solving approaches. Look for ways to capitalize on students’ comfort levels with practical matters to help them move into more conceptual practices? Be more aware of the effect of my presence in an activity. Conclusion THIS WAS A CASE OF An attempt to improve students’ conceptual understanding of fundamental ideas in calculus THAT RESULTED IN improved teacher understanding of what students know and don’t know about those ideas. The Collaboration Collaborator & Colleagues My reflections on the collaboration Reflections of my collaborator By-products of collaboration By-Products of Collaboration Good model for interdisciplinary work “Have always heard about the value of interdisciplinary work but haven’t seen much of it. This experience gave us one concrete example of how it works.” Helped to create a community for the scholarship of teaching “ Provided a strong example of how a teacher thinks about and incorporates scholarship into her courses.” My Reflections on the Collaboration Provided me with a customized roadmap for learning theories and methods of assessment applicable to my project Allowed me to work more comfortably and confidently “out-of-discipline” Sharing responsibility served to keep me on-task and it raised the bar for the project Collaborator Reflections Provided a non-artificial environment for mentorship Allowed me to practice qualitative research skills Learned practical ideas for my own course development Learned about our Calculus courses; deepened my knowledge as a student advisor