"It is the consensus of virtually all the men an women who
have been working on curriculum projects that making material interesting
is in no way incompatible with presenting it soundly; indeed, a correct
general explanation is often the most interesting of all."
(Jerome Bruner, The Process of Education, 1960, p 23.)
In this paper we offer those teachers who want to enhance their teaching of mathematics with science one class that uses a PSL in the exploration of the relationship between distance and time in two particular situations, in an environment that supports cooperative learning.
The paper is organized in three main sections. First section presents the relevance of this project inside our discipline. The second section presents the class activity that we have designed; this section is divided in four subsections: Objectives, Methodology, Mathematical Content and Assessment; the last section presents our conclusions, and suggestions for further improvement of the class. We include five appendix with some detailed information about PSL, two worksheets of PSL activities, an evaluation check list, and pair test.
We think that the use of technology will make a significant contribution to teaching of mathematics. We encourage teachers to get a chance to see that the new generation of teaching of mathematics with technology is more than a dream.
Why this topic?
According to Kieran (1992, pp. 408) the topic of Functions has been presented to the students from a structural perspective ­p;using the set theory definition­p; and using principally the symbolic representation and promoting translations from the symbolic to the graphical representation. As a consequence:
· The constant function, the piece-wise function and the function represented as a discrete set of points are the ones that causes more difficulties among students.
· Students neglect domain and range.
· Students do not understand completely the concept and representation of images and pre-images, both in symbolic and in graphical representations.
· The variety of examples students have is limited; with a high preference of linearity.
· Students find easier to translate from symbolic to graphic representation, than from graphic to symbolic representation.
Nevertheless, the development of the hand held technology and of microcomputers' software, has changed the perspectives for teaching functions in the school. According to the standards, today is a requirement that students:
· understand the concepts of variable, expression and equation;
· represent situations and number patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of these representations;
· represent situations that involve variable quantities with expressions, equations, inequalities, and matrices;
· use tables and graphs as tools to interpret expressions, equations, and inequalities (National Council of Teachers of Mathematics, 1989, p. 102, p. 150).
So the concept of function seen as an integrator in the curriculum can be taught in a meaningful way.
In the society of twentieth century, people needs cooperative skills more to be successful in their jobs and in their daily lives. The significance of cooperative learning lies in the fact that this method provides a social environment where people freely can exchange ideas, ask questions, help one another to understand the ideas in a meaningful way, and to express what they feel and think. On the other hand, research on cooperative learning indicates that in order to become successful and confident problem solvers and in order to work on mathematics projects, students need to work cooperatively (Johnson and Johnson, 1990).
In this activity, our objective using cooperative learning technique is to help students gain these skills, and to help them learn how to use these skills. They will learn and practice basic characteristics of cooperative learning when they are working as a team to attain their common goal.There are two explicit reasons for using this methodology in our class.
Learning by different ways.
People learn by talking, listening explaining, and thinking with others, as well as by tehmselves. In addition to that a test of understanding is often the ability to communicate to others (that is also what we have done in oral type or essay type of examinations assessing the students knowledge) and this communication act is itself often the final and most crucial step in the learning process.
Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) addresses the same issue:
Small group work, large-group discussions, and presentation of individual and group reports -both written and oral- provide an environment in which students can practice and refine their growing ability to communicate mathematical thought processes and strategies.
Small groups provide a forum for asking questions, discussing ideas, making mistakes, learning to listen to other's ideas, offering constructive criticism, and summarizing discoveries in writing. (p.79)
Handling richer and more challenging situations.
In mathematics, there can be created many opportunities for creative thinking, for exploring open-ended situations, for making conjectures and testing them with data, for posing intriguing problems, for solving non-routine problems.
Students in groups can often handle challenging situations that are well beyond the capabilities of the individuals at the developmental stage. As an individual attempting to explore those same situations, students in general often make little progress and experience unnecessary frustration (Davidson,1980).
We are going to present our class design talking about its four main components: Objectives, methodology, content, and assessment (Rico, 1991).
Students will appreciate the value of mathematical representation by deriving and understanding corresponding mathematical representation for a physical event.
Students will be able to predict and verify the corresponding physical structure for a given mathematical representation.
Students will be able to explain the relationship between motion and mathematical representation.
Students will be able to use a graph for describing a motion and communicate their mathematical thinking.
Students will be able to mathematize a given physical event and give reasoning to their representation.
Students will use the PSL for understanding physical motion. We assume that students are familiarized with coorperative learning environment since the activities will be performed groups of three or four members.
The followings are the descriptions of the two tasks proposed. The idea is that more than one group work on each task, so at the end, the class will be able to compare different approaches and results. Activity 1 comes from the PSL Teacher's Edtion Book (IBM, 1990, p.1-18). The goal is to find both, the physical explanation and the symbolic representation from a given graph that represents a motion. The goal of Activity 2 is to produce the graph that corresponds to a description of a motion and to resepresent symbolically the graph.
Activity 1: A Challenge for You Cats.
The educational environment is much like that of CBL (Calculator Based Laboratory). A cat-like graph is given to students (IBM, 1990, p. 1-18). Observing the graph is the initial and fundamental thinking process by which information is acquired.
The aim of the activity is to find and verify mathematical representation when the physical representation is given. Students are expected to improve their intuition about the phenomena being studied and mathematical knowledge being involved. This activity has a potential since many different concepts of functions studied in mathematics class should be used in doing PSL activity. See Appendix 2 for the detailed text of the Activity 1.
After studying the graph, students open a copy of the graph on PSL's screen. When they describe the movement that would produce the graph each student can compare and relate the similarities and differences of his or her thought with the other students. This experience will provide the students with the opportunity for active participation. They will talk about the speed , time, and direction of the movement one another within a group. The application of physical science to mathematics is truly realized since the importance of the quantitative nature of science is stressed out in the exploration in many ways. Students need to be very sensitive to the speed, time, and movement that they do within the experiment. Changing the range of values on the axes is also understood conceptually when they want to have a better fitted graph because students are taking care of real data from their activity.
Activity 2: Trace the Target.
The second activity is reversed form of the first activity. We ask students to graph a given physical movement description.
The aim of the activity is to find the mathematical representation of a described motion. Students will be able to understand and analyze the corresponding movements required for a shown graph. Students will be able to mathematize the movement and construct the corresponding piece-wise functions. See Appendix 2 for the detailed text of the activity.
After studying the description, students work on the movement that will correspond to the given graph. Then student should create the corresponding graph on the computer's screen. After analysing the graph that would produce the movement described, each student can compare and relate the similarities and differences of his or her thought with the other students. Such an experience will foster active participation and mathematical communication for each student. They will talk about the speed, time, and direction of the movement one another within a group. The application of physical science to mathematics is truly realized since the importance of the quantitative nature of science is stressed out in the exploration in many ways. Students need to be very sensitive to the speed, time, and movement that they do within the experiment. Approximation and graph fitting will also be understood conceptually, because during experiment they will be trying to fit to their predicted equation and they will also be modifying their graph if necessary.
Every student will have a PSL program built in an IBM computer, PSL equipment, a worksheet, and a check-list for cooperative work asessment. The use of graphing calculators are allowed when the students feel the need of them to find a graph. Time for the group work is 30 minutes. After the group investigation students should present to the other groups their result. A teacher will act as a facilitator of students' active exploration. The teacher have to explain to the students the main goal of the activities. He or she will monitor the students' activity and make a note for further feedback to the students.
We have two categories for the content of the PSL activities, the mathematical and the science content.
The mathematical content covered by these two activities includes: Linear, quadratic, and constant functions; Cartesian graphs, meaning of x and f(x); piece-wise functions; domain and range of a function; slope of a line; maximum of a parabola and linear equations. Further, this mathematical content will include system of linear equations, symbolic forms of linear, quadratic, and constant functions, graphs of linear, quadratic, and constant, translation from symbolic to graphical representation and vice versa, mathematical models for verbal situations, and description of mathematical situations.
The science content includes movement, speed and time, measurements, and their mutual relationships.
We have designed five different assessment tools for group and individual achievement, attitude, and performance. We want to emphasize that these tools are designed to provide the teacher information on those aspects, but that the final grade should be the same for every student in the group, since this is one of the main characteristics of the cooperative learning strategy. This implies that the teacher has to establish a percentage of each part, the individual component and the group component, in such a way that that final grade assigned to the group reflects not only the group achievement but each student's achievement as well.
We have designed five tools for assessing these components, Table 1 shows the name of each tool with a brief description. Table 2 shows how the tools can be used for assessing the components.
Table 1: Description of the assessment tools.
Name of the tool Description
Answer Sheet (AS) This sheet has the answers of the group to the worksheet, and a group reflection about the activity. (See Appendix 2)
Oral Presentation (OP) This corresponds to the presentation that one person of the group, selected by the group, presents the results of the exploration. Each group has 5 minutes and one transparency. They have to decide what to present. In a further discussion, teacher may ask students the rationale for choosing the topics of the presentation.
Cooperative Work Checklist (CL) This sheet contains the opinion of each student about the group and individual performance as a group in the particular task. Teachers can fill their own checklist for each group, according to their perception of the groups' work. (See Appendix 3)
Reflection (RF) This corresponds to the writing that each person in the group produces after the activity. In it students will explain how the activity helped them to gain a better understanding of mathematics and science concepts as well as comments for improving the activity and the group work.
Pair Test (PT) This corresponds to the answer to the open question designed for assessing the objectives of the class. Students will work in pairs during 50 minutes producing a two page paper that shows how they arrived to the solution and what are the connections to real life. They have to show a deep understanding of the concepts and a good communication strategy. (See Appendix 4)
Table 2: Use of the tools for assessing each component.
Components Individual Group
Achievement RF, PT AS, OP
Performance CL AS, OP, CL
Attitude RF CL
The purpose of this last table is to show teachers how to use each of the tools for obtaining information about the different aspects covered in the task. Observe that self-assessment is considered when students answer the checklist. We agree with the Assessment Standards For School Mathematics (NCTM, 1995, p. 2) in the importance of having more than one source of information for assessing our students. Hence the five tools proposed.
Conclusions and Suggestions
One thing which is really great for students is that mistakes are allowed. Students don't have be discouraged by mistakes. If their movement doesn't fit to the graph the PSL equipment and PSL software make it easy for them to repeat an experiment. Students can try again and again until the result is satisfactory.
Communication skills are very important when they express their ideas and insights to write the movement that they need to produce the graph. Writing skill is also required when they describe the movements for the graph. The application of the concept or information gained in the exploration should be combined with their mathematical knowledge of functions when they try to find a piece-wise function which can explain the graph in symbolic mathematical representation.
A graphic calculator would be allowed in finding the functions. Students will be able to use the coordinate system if they want to find an exact form of piece-wise functions. Students are expected to have the higher-level of ability in applying and analyzing the graphs. Students will be able to appreciate the effectiveness of a scientific activity in learning of mathematical concepts.
There are several ways to implement cooperative learning method in classroom. Among the possible cooperative activities that can be used in mathematics classroom, the following list presents those characteristics that define a good cooperative task:
one that will suit all ability levels,
one that needs many contributions,
one that involves manipulatives,
one that involves challenging activities,
one that involves pencil and paper activities,
one that involves games,
one that includes experiments.
These type of cooperative tasks help the process of cooperative learning and thereby, provide a perfect atmosphere and a social context to improve mathematical communication and to study mathematics.
Doing this project we have discovered that Bruner was true when he said that
" making material interesting is in no way incompatible with presenting it soundly; indeed, a correct general explanation is often the most interesting of all."
The access to different technologies in our classrooms together with a positive attitude towards change will make possible a new generation of mathematics teaching. We are in a privileged position as practitioners and as researchers, since we can see the two sides of the same coin: we can make interesting material, but not only can present it soundly, but in a way that is, we hope, really profitable to our students. We hope that this contribution will give many teachers the possibility of taking the risk of doing new things with a high probability of success: teachers and students not only will have fun but will begin to find new ways to go across the school curriculum in an easier way. We encourage the teachers that have read this contribution to give us a chance to see that this dream, the new generation of teaching of mathematics with technology, is more than a dream.
Bruner, J. S. (1960). The process of education. New York: Random House.
Davidson, N. (1990). Cooperative Learning In Mathematics. New
York: Addison Wesley.
Johnson, D.W. and. Johnson, R. T. (1990). Using Cooperative Learning in Math. In: N. Davison,
(Ed.). Cooperative Learning In Mathematics. New York: Addison
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grows, (Ed). Handbook of
Research on Mathematics Teaching and Learning.(p.p. 390-419) New York:
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1995). Assessment Standards for School
Mathematics. Reston, VA: Author.
Rico, L. (1991). Los tetraedros del curriculo. Diseño, desarrollo y evaluacion curricular. [The
tetrahedra for the curriculum. Curriculum design, development, and evaluation]. Unpublished
document. Granada: Universidad de Granada.
APPENDIX 1: The PSL
The Personal Science Laboratory, PSL is a microcomputer-based laboratory (MBL). This means that it allows students to collect data in situ, to manipulate variables, to analyze data that can be presented in both graphical and tabular modes, and to see immediate results from one experimental procedure. It was released by the IBM in 1990.
The PSL operate on any IBM PS/2 with monochrome or color display and graphics capability or on any IBM compatible, ­p;286 or higher­p; with a 3.5" or 5.25" diskette drive, a graphics display and an asynchronous adapter card with a 25-pin connector (a serial port). It supports printer and fixed disk drive. The PSL requires DOS 3.0 or above.
Besides the publications that support the operation of the PSL ­p;the PSL Hardware User's Guide and the PSL Technical Reference­p;, there are three publications that are of pedagogical interest:
· PSL Student Version, contains instructions for running experiments and raise questions that provide learning opportunities to the students.
· PSL Annotated Teacher Version contains the answers to the questions posed to the students and the result of the completed experiments.
· PSL Student Versions Blackline Master Edition, offers worksheets that can be freely photocopied and distributed in the school that uses the PSL.
Components of the PSL
Roughly speaking a PSL is composed of one base unit to which four modules can be connected. There are two types of modules the Motion and Mechanics and the TLp (Temperature, Light, and pH) modules. The first is used to connect the Motion and Distance probe, and the second to connect the Temperature, Light, and pH probes (one jack for each). As an interesting feature Temperature and Light probes can be used in any of the jacks of the TLp module; this gives the possibility of collecting data from 12 different sources, for example from 12 different students, with only one base unit.
The PSL is controlled by a very powerful software, the 'PSL Explorer'. The software offers the following functions:
· to create, run, and retrieve experiments
· to analyze data of one experiment
· to export collected data to a spreadsheet
· to import data from a spreadsheet
An experiment consists of a set of parameters and the data collected during a defined period of time. The probes are used for sending the computer the value of the measured variable (distance measured in meters, temperature measured in Celsius, light intensity measured both as absolute or corrected in a way similar to the human eye and pH in pH units) at specific intervals of time, or distance, or volume. After running the experiment the software presents a graph of the two variables used in a Cartesian plane. A table of the data collected is produced too. The analysis option allows the manipulation of both the graph and the table; it offers functions for altering the graph ­p;zoom, regression line drawing, change of coordinates­p; and for altering the data, ­p;add, subtract, multiply or divide by a constant, perform log and antilog operations, compute power operations (by rational numbers); take reciprocals, calculate sines and cosines and differentiate and integrate the y-axis with respect to the x-axis.
The PSL is an interactive tool that provides teachers a large amount of possibilities for exploring scientific ideas in relationship with mathematics. Contrary to what can be expected setting up the hardware and the software is an easy task. The package offers a variety of pre-designed experiments for using each of the probes. There are some concerns about its cost: It may be unaffordable for small schools.
APPENDIX 2: Activity 1: A Challenge for You Cats
The aim of the activity is to find and verify mathematical representation when the physical representation is given.
Students are expected to improve their intuition about the phenomena being studied and mathematical knowledge being involved.
This activity has a potential since many different concepts of functions studied in mathematics class should be used in doing PSL activity.
a. Study the following graph, then use the Disk option to load the target file TAE1C11.PSL and display a copy of the graph on PSL/s screen.
b. Write a description of the movement that would produce the graph.
c. Use PSL to measure the movement. Sketch PSL's graph with individual data.
d. Write a description of the graph. i.e. find a piece-wise function for the graph.
e. Discuss the difference with your group members if your result does not match the target graph.
APPENDIX 3: Activity 2: Trace the Target
The aim of the activity is to find the mathematical representation of a described motion.
Students will be able to understand and analyze the corresponding movements required for a shown graph.
Students will be able to mathematize the movement and construct the corresponding piece-wise functions.
a. Read the following description of a movement:
Description of the Movement:
I am a turtle who stands 1.5 meters from the probe for 5 seconds. Then I begin to walk with a decreasing speed for 5 seconds and reach three meters zone form the probe. When I get there, at the tenth second, I jump 40 cm backwards and then I jump 30 cm forwards. From the 11th seconds to 15th I stay there, and then I jump backwards 30 cm and forwards 40 cm. Without waiting, at the 16th second, I began to walk towards probe with an increasing speed and I suddenly stop when I reach 1.5 meters zone at the 21st second. In the rest time, I stay there.
b. Describe the graph
c. Think about the mathematical representation of the given graph. Discuss the curves that will corresponds to the movement described above. Discuss the piece-wise functions that corresponds to the described movement.
d. Corresponding piece-wise function (only for time interval 0-13 seconds.
e. Plot the corresponding graph.
f. Use PSL to perform the described movement and obtain the graph of your movement.
g. Compare your graph you have found in PSL and the described graph below. Discuss.
APPENDIX 4: Evaluation Check List
The following items are designed to assess individual and group cooperative work. Each member of the group should complete this assessment sheet by the end of the activity and should submit it to the teacher. Give your response to each item and mark the appropriate box. In the response scale, 6 denotes excellent (Exc) and 0 denotes Not Applicable (NA).
Response Characteristics of My Cooperative Work
NA 1 2 3 4 5 6 (Exc)
1. I give help to the others ( ) ( ) ( ) ( ) ( ) ( ) ( )
2. I get help from the others ( ) ( ) ( ) ( ) ( ) ( ) ( )
3. I work with the others, not alone ( ) ( ) ( ) ( ) ( ) ( ) ( )
4. I share/compare answers ( ) ( ) ( ) ( ) ( ) ( ) ( )
5. I share/compare approaches ( ) ( ) ( ) ( ) ( ) ( ) ( )
6. I split up the work ( ) ( ) ( ) ( ) ( ) ( ) ( )
7. I give/get opinions and ideas ( ) ( ) ( ) ( ) ( ) ( ) ( )
8. I talk with others about the task(s) ( ) ( ) ( ) ( ) ( ) ( ) ( )
9. I get along with another ( ) ( ) ( ) ( ) ( ) ( ) ( )
Characteristics of Our Group
I think that...
NA 1 2 3 4 5 6 (Exc)
1. Group members like each other ( ) ( ) ( ) ( ) ( ) ( ) ( )
2. Everybody does equal work ( ) ( ) ( ) ( ) ( ) ( ) ( )
3. Group is well organized ( ) ( ) ( ) ( ) ( ) ( ) ( )
4. People work well together ( ) ( ) ( ) ( ) ( ) ( ) ( )
5. Group remains on task ( ) ( ) ( ) ( ) ( ) ( ) ( )
6. there is no fooling about/fighting ( ) ( ) ( ) ( ) ( ) ( ) ( )
7. Group completes the work ( ) ( ) ( ) ( ) ( ) ( ) ( )
8. People help each other ( ) ( ) ( ) ( ) ( ) ( ) ( )
9. People share/listen ( ) ( ) ( ) ( ) ( ) ( ) ( )
10. There is a good atmosphere/ fun ( ) ( ) ( ) ( ) ( ) ( ) ( )
11. People talk about mathematics ( ) ( ) ( ) ( ) ( ) ( ) ( )
Our Benefits from Cooperative Learning
Each member of the group can...
NA 1 2 3 4 5 6 (Exc)
1. Ask others if he/she doesn't know ( ) ( ) ( ) ( ) ( ) ( ) ( )
2. Learn to work well with others ( ) ( ) ( ) ( ) ( ) ( ) ( )
3. Get different ideas about the work ( ) ( ) ( ) ( ) ( ) ( ) ( )
4. Help each other ( ) ( ) ( ) ( ) ( ) ( ) ( )
5. Get to understand different people ( ) ( ) ( ) ( ) ( ) ( ) ( )
6. Work with friends ( ) ( ) ( ) ( ) ( ) ( ) ( )
7. Check his/her answers ( ) ( ) ( ) ( ) ( ) ( ) ( )
APPENDIX 5: Pair Test
Think about how to produce one experiment that involve periodic functions. You have to find different situations that produce periodic functions and different explanations for graphs of periodic functions.
Produce a two paper report at the end of the assessment time. In writing this paper assume that the reader is a lay parson, that is interested in learning about the relationship between the movement of an object and the graph that may mathematize the movement.
Mathematical content, creativity, organization and depth of the analysis will be used for grading.