Bruce & Levin
National Council of Teachers of Mathematics (NCTM) has described five standards for mathematical processes through which students should acquire and use their mathematical knowledge (October 1998). These standards are:
Problem Solving
Reasoning and Proof
Communication
Connections
Representation
These five standards are similar to Bruce &Levin Taxonomy uses in education that I have decided to apply in mathematics classroom activities.
During the last decade of twenty century, economical agents, industries, most of social services have employed new technology in their daily affairs. They are working with media by sophisticated communication tools. Teachers and educationalists are looking at this newborn (world, wide, web) very carefully. Using computers in teaching and learning is a big issue today, educationalist and teachers have stated many approaches, during last decade, and many procedures and strategies have invented, in order to use computers in learning.
Bruce & Levin professors of University of Illinois have introduced a new approach to teachers, which is based on John Dewey’s ideas, one of the famous U S educationalist. Our peer students during last years have found this taxonomy as a tool, that all educational activities fit on it. In other hand National Council of Teachers of Mathematics has accepted most part of the taxonomy as a mathematical processes through which students should acquire and use their mathematical knowledge. By regarding to classroom activities as an important issue I am trying to apply Bruce&Levin's taxonomy in the classroom activities, and open a new path in teachers training area. The project consists nine parts.
5-Constuction of knowledge and discipline
6-Creating new cognition and expressing
8- Summary
The Main Goals of the Project
Taxonomy Education uses of Technology in mathematics teaching can provide a best learning condition in the classroom. New technology especially using web site in teaching mathematics encourage students to inquire, communicate and construct thoughts and cognition to improve their ability to express knowledge. This study is going to apply Whole procedure in classroom activities.
Planning to teach consists three major parts each part, also have different sections, that, teachers are planning for every session.two major resources for teachers are: 1-Ask Eric (http://ericir.syr.edu/)and 2- California State University North ridge web sites and resources for teachers (http://www.csun.edu/~vceed009/math.html). Both sources have different lesson plans and teachers can use any conventional plan. After many years of experiences the following lesson plan is recommended to teachers and teachers candidates.
Systematic Instructional Planning
1- pre instructional activities (Lesson Plan)
11 The Subject matter
12 The Main Goal
13 Instructional objectives in detail
1-4Behavioral objectives
1-5Determining Prerequisites
1-6Writing Diagnostic Test
1-7Predicting Diagnostic Test Results
1-8Instructional Steps and classification of rules, concepts and so forth
1-9General lesson Framework (overview)
1-10Foregoing Motivational Activities
1-11Summary
1-12Homework
1-13Final Evaluation
1-14Remedial instruction
2-Classroom Activities
2-1 Preliminary Activities
2-2 Lesson Presentation
2-3 Extra activities
2-4Final Activities
3- modifying the lesson plan
In the first part of the activity any teacher is going to make a plan to teach. The plan consists all activities that teacher and students intended to do in the classroom. Most important part of the plan is purposes of instruction that should have been written in detail. We have different approaches to apply in the classroom. It depends on content of subject matter, student’s needs and many other educational factors. In order to have a successful teaching foregoing instructional technology tools is necessary also decision of teacher to create interaction between students and teacher students would be sufficient and benefitful action in any lesson plan arrangement. There are a variety of lesson plan formats that teachers can use in her or his Professional life. Most of teachers are able to write down very useful lesson plan, that is more flexible, appropriate and conventional to apply in the classroom activities.
A plan should give opportunity to teacher to create a learning environment in the classroom conventional plan gives time to students interact, participate and cooperate in all school programs. The inquiry, communication, construction and expressing should embedded in instructional planning.
NCTM has introduced the technology as a Principle for school mathematics.This principle consists: "Mathematics instructional programs should use technology to help all students understand mathematics and should prepare them to use mathematics in an increasingly technological world".
The principle focuses on electronic technologies, including calculators,computers and especially web browsers, because these technologies fundamentally affect education today. second grade children can figure out many mathematical facts by calculator,for example by pressing five times on +2 and repeatedly pressing the (equals) can get product of multiplication. Students can easily practice and learn facts by calculators.
Second and third graders can search the worldwide Web for games easily today.most of kindergartners are playing game on computer. Computer is one of the important tool in family life as car.Schools must provide access to electronic technology for all students and teachers should have adequate technological and mathematical knowledge to provide appropriate support for their students.
There is increasingly demands in the society for technological skill people in the mathematics.If schools do not have a level of technology equivalent to the level found outside schools,if they do not prepare students appropriately with this electronic movement then they are placing their students at a serious disadvantage.
Teachers, who are going to apply inquiry method in the classroom activities, would create some problems in term of his or her lesson. Inquiry learning classrooms give much more opportunities to children to think and find out appropriate strategies to solve problems. The processes of problem solving involve students with logical reasoning and constructing cognitive knowledge.
. Teachers have access to a variety of sources in the World Wide Web. They can choose their materials from different teachers who are publishing and working into Internet. Internet is not only a local library, it is also a device that teachers can benefit and use the references. The Maryland Collaborative for teacher preparation sites (http://www.inform.umd.edu:8080/UMS+State/UMDProjects/MCTP/www/MCTPHonePage.html)This includes information on using technology. In any domain of the science teachers gradually making plans and with comments from other teachers the plan is modifying and completing. Today each individual teacher can publish in the web site and also is able to complete the other teachers work. Some other Internet sources for teachers are: Teachers Edition Online (http://www.teachnet.com/) and Teachers network (http://www.teachernet.org) offers a data base 500 classroom tested projects and a "Let’s talk"bulletin board where teachers can exchange ideas.
Teacher’s cooperation is going to enhance the standards of teaching tools as lesson plan. In any classroom in across the world. It is not only a teacher who is teaching in one classroom, actually they are thousands of teachers in any classrooms.
In inquiry learning students must to think and react in order to respond to questions. Interaction between students is necessary condition in classroom activities. Teachers have different approach in term of student’s interaction. Normally students are working in small groups, and in each group the whole responsibilities have divided between students. Electronic E-mail without time limitation gives opportunity to students to interact at any time they need. Sometimes students have on-line discussion or networking. It means that each group of students without attending in the classroom can get together and discuss. Internet technologies have eliminated the time and place limitation in student’s interaction. Educatinist and some experts were worried about communication between students by using technology in classroom activities. Now Internet communications have solved this problem. In other word students can have classroom session at any time and any places that they want.
New technology helps teachers to open channels of communication and students can cooperate with each other and different mathematical sources in learning groups.World Wide Web is a important source of the mathematical tasks that students can communicate with web publishers. For example (http://www.mathpro.com/math/mathCenter.html) include Mathematical reasoning and problems that students can communicate with this Web publisher.
5-Construction of knowledge and Discipline
The outputs of the classroom activities in cognitive domain are scientific knowledge and disciplines. Productive teaching run on the students’ abilities to perform thoughts, conceptualize and make rules. These are internal operations in learning conditions. New technology as animation, computer simulation, web publishing and networking are a variety of tools which students can get benefit in the way of constructing thoughts and ideas.
Making connections between mathematical concepts and procedures is facilitated by new technology particularly through World Wide Web. Students make connection between formal mathematical knowledge and their existing knowledge in the Web sites. Many Web sites have published tutorial and encourage students to work in mathematical sites without getting tired or hating of mathematics and having mathematics anxiety.
Students make connections among various representations models and also different mathematical topics and with its applications to other content area such as practical science, social science and everyday life.
Constructing meaningful mathematical knowledge is a gradual process,it is important beside of lesson plan ,teachers should have a long-term teaching plan for mathematics instruction. A patient teacher will help children construct a more solid understanding of mathematics.
Lesson plan should include a variety of Web resources in term of mathematical tasks to introduce to the students. Making connection among knowledge and producing new concepts and procedures would be discovery learning in mathematics.
6-Creating New Cognition and expressing
World Wide Web is not only a hypertext tool for obtaining information from the Internet, it is also a tool for publishing, creating new knowledge and understandings. Web documents can combine texts, graphics, sounds and video. Use of the web can promote learning and flexibility.students are interested to have own publishes in web sites. Publishing in the web site motivates students to promote their knowledge and help them to introduce their findings to others.
Expressing,organizing and recording data helps students make sense of complex problems and enables them to find patterns.For example if students meet a problem like the following:
Students printing association got two different bids on the cost of printing books. First one charges $200 to set up the presses plus $4 for each book printed. the second one $150 to set up the presses plus $7 for each printed book. Who has the best prices? First step is organizing data, analyzing and recording them. Some students make a table:
# of books 0 1 2 3 4
FIRST $200 $204 $208 $212 $216
SECOND $150 $157 $164 $171 $178
Some students organize data as: First bid=$200+M $4
Some of them make a graph or represent with symbols.
Encouraging students to talk with one another about the
graphs, pictures and symbols they have used (communication). This kind
of activities might begin with whole class demonstration using a computer
or calculator. Representation of data and mathematical thoughts have a
variety of approach and students can create own approach in any kind of
expressing mathematical concepts. Traditional teaching is introducing (y=ax+b)
as a formula and most of students do not know what is the main goal of
solving this kind of the problem. Computer and calculator help students
by using graphs and predicting a new graph in any step of problem. Starting
from y=a*4 and then changing
the (a) respectively until whole formula.
7-Research
report
Abstract:
In a pilot project as a case study, I have evaluated
the hypothesis in the middle school mathematics classes. The task was an
algebra equation (a+b)2= a2+b2+2ab. The
teacher has been planned the lesson plan as predicted in this project.
She taught by using new technology tools. After testing student’s achievements
the result was extremely excellent comparing with control group which had
traditional instruction. Only technology using class spends a lot of times
in the learning processes.
Method
Subjects:
There were 37 students including 14 boys. They were studying
two different mathematics classes. One of the classes had traditional teaching
as a control group and the next one was experimental group. Access to the
computers was identical for both groups. The experimental group only has
got instruction by new technology tools.
Materials:
1-computers
2-calculators
Procedure
The algebra problem (3+5)2=? Were introduced
to all class members in the experimental group and were asked three questions:
1-How many ways we have to solve this problem?
2-How we can generalize these kinds of problems as (a+b)2=?
3-How we can apply this formula in solving similar problems.
Students were allowed to use calculator and computer.
Students hint to go some addresses in web sites and get examples also problem
solutions similar to problems, which they are trying to solve.
Students have been recommended to have time account when
they are using new instructional technology tools. Method of learning and
reasoning were inductive. Students could get conclusion and make rule then
applied this rule to solve mathematics problem.
Results
After evaluating experimental and control groups by teacher
made tests the results were different. Experimental group could apply the
rule in problem solving processes. Control group has learned the task in
one hour by traditional method and had difficulty in term of problem solving
related to task. Learning time for experimental group was in average 7
hours; instead control group learned only in one hour. actually control
group memorized the formula and they could not solve related problems as
experimental group. It means experimental group learned very deeply and
applicable.
Discussion
New instructional technologies help students to learn
mathematics as an applied knowledge in every day life. Students could use
reasoning in problem solving processes inductively and also deductively.
Students learned to create new rules and new models from mathematical knowledge.
They learned to apply the rule in problem solving. Students understood
new technology helped them to solve mathematical problem communicate with
others and express the knowledge with different ways. Problem solving by
new technology is providing opportunity to communicate with others across
the web sites and Internet sources. The products of this communication
is connecting to each other gradually and make strong body of mathematical
knowledge.
New technology especially worldwide web shows a variety
of new approaches to express mathematical knowledge, and to be a publisher
or author in ground of mathematics.
Evaluation
This case study and its result did not give me much more satisfaction
because of small sample and limited population. I talked to some mathematics
teachers and I explained my project in detail. They had some comments that
I am trying to summarize the most important ones.
1-Teachers are not familiar to use technology especially web sites
and they need in service training.
2-All students do not have opportunity to access to internet.
3-world .Wid.Web have many interesting sites especially for young students
than mathematical web sites so students who have access to web sites they
are browsing in different sites not mathematics pages.
I talked about my project to one of the University of Illinois mathematics
Professor. She encouraged me to make research in a broad field. I hope
to research again in future times.
Textbooks are not only source of the schooling knowledge today schools are not only place to learn and time of learning is not only school time. New technology has changed the old fashion school system. The educational principals are still existing. Bruce & Levin Taxonomy education uses of technology based on John Dewey’s Theory in the field of education with employing new technology is still working. Four important stages of learning development (Inquiry-communication-construction and expressing) are adopted by new technology. Students without any time and place limitation have access to variety of sources on time.
1-The educator’s guide to the Internet resources and classroom activities (developed by Virginia Space Grant Consortium and © 1997 by Addison Wesley)
2-Internet Adventure: A Guide to educational Resources on the Internet (published by Allyn &Bacon).
3-Ask Eric (http://ericir.syr.edu/)
4- Bookmarks online (http://www.connect-time.com)
5- California State University-Northridge Web Sites and resources for teachers (http://www.csun.edu/~vceed009/math.html)
6-Cyber School Magazine (www.cyberschoolmag.com).
7-Guided Educational Tours (http://www.newnorth.net/get).
8- Index of wwwServers on mathematics (http://www.math.psu.medu/Math lists/Contents.html).
9-Teacher’s Edition Online ((http://www.teachnet.com/)
10-Teachers network (www.teachernet.org)
11-Wed Ed K-12 curriculum Links (http://badger.state.wi.us/agencies/dpi/www/webed.html).