International Panel: Bridging Policy and Practice
A Focus on Teacher Preparation

Presentation 8: Pre-service and In-service Education in the United States

Johnny Lott
University of Montana
National Council of Teachers of Mathematics
(42nd President)
Linda Antinone
Paschal High School

view PowerPoint Presentation 8

Pre-service Teacher Education

Johnny Lott

A four-year university degree is universally required for teaching in public primary through secondary schools in the US. Some programs have a four-year degree in mathematics followed by a fifth year where students take education courses. Under this model, mathematics is often in the same program as that of mathematics majors who don't plan to teach. Because of teacher shortages many other ways coming into teaching exist-often with as little as a college degree and two classes in pedagogy

The 1989 and 2000 Standards from NCTM (http://www.nctm.org/standards/) led to changes in state standards, which affected many teacher education programs. Reforms in mathematics at the university level, primarily the calculus reform project (like Harvard Calculus3), led to changes and reforms in some mathematics programs. Secondary and primary school programs have also started changing. Technology has also led to major changes in teacher education programs in some cases. Changes are being made in teacher pre-service and in-service programs throughout the United States to center more on thinking about how students learn and how to shape teaching to enable student learning to take place. A central focus across K-12 mathematics education is to provide equitable opportunities for all students to learn challenging and high quality mathematics.

A pre-service example from the University of Montana

Math Modeling with Technology

This is a third year course that students take after one year of calculus, linear algebra, a statistics course (and possibly introduction to proof). Some students would have also had a computer science course. The details of the math modeling with technology course are as follows:

We want to teach the way we want our students to teach, so we arrange the classroom setting and instruction in specific ways.

Classroom Setting

  • Mathematics Department Computer Lab (16 computers, max of 32 students). Every student has access to a computer, no more than 2 students per computer.
  • A presentation computer with projector

Class Instruction

A mixture of lecture, technology demonstration, group work, group presentation (primary student presentation form), and individual presentation (occasional student presentations).

Types of Technology

  • Graphing Calculators (the department bought them and loans them to students so that they can all have one even if they can't afford to by one. Students were encouraged to purchase their own)
  • Data collection devices with calculators
  • Microsoft® Word (word processing with equation editor)
  • Microsoft® Excel (spreadsheet)
  • Maple™, Mathematica®, or Derive™ (Computer Algebra Systems)
  • Dynamic Geometry Software (Cabri™ and Geometer's Sketchpad®)
  • MATLAB® (Linear algebra software). Most students have used this program in earlier classes, so it is not really a focus in the course
  • Fathom™ (statistics software) program used for demonstration. The program was difficult for us. We are having trouble finding a program that we like.

Types of Assignments

Writing assignments, technology homework, and group projects are the three types of assignments. All assignments must be turned in via the Internet.

Examples of Assignments

Writing Assignment: Rattlesnake Head
The head of a rattlesnake can accelerate 50 m/sec.2 in striking a victim. If a car could do as well, how long would it take for it to reach a speed of 60 mph from rest?
Spreadsheet Assignment: Construction Elevator
An open construction elevator is ascending on the outside of a building at a rate of 8 ft/sec. When the floor of the elevator is 80 ft. above the ground, a ball is dropped from 160 ft. above the floor of the elevator. How long does it take the ball to reach the floor of the elevator?
TI-92™ Assignment: Melting Snowball
A snowball of radius 12 in. is melting so that the radius is changing at the rate of 0.15 in./hr. What is the volume of the snowball when the radius is 12 in.? How long will it take for half of the volume of the snowball to melt away? How long will it take for the entire snowball to disappear?
Maple™/Mathematica® Assignment: Conjecture
Use a spreadsheet to make a conjecture about the following question: For which natural numbers, a, is the expression lim (as x approaches ∞) of e(10-ax) finite? Prove or disprove your conjecture.
Assignments comparing the outputs from different types of technology. (Forces students to examine the limits and advantages of each type of technology and become familiar with the quirks of each).
Use Maple™ /Mathematica®, to graph the following equations. Compare the graphs found with comparable ones found on the TI-92™.
    a) f(x) = 3[x - 2]
    b) f(x) = (x3 - 3)/(x - 1)
(Explain any asymptotic behavior.)
Geometer's Sketchpad® Assignment: Heat Deflection
In the SIMMS module Patty O'Conic, the suggestion is made that a hyperbolic heat reflector in a space heater is a good type of reflector to deflect heat to many parts of a room. If you consider a heat ray to act like a light ray, and a conic section to be locally linear, make a conjecture about how the heat is deflected and prove your answer. This may be done with calculus without all the assumptions listed.

Final Assessment (using open-ended problems)

A good approach to solving open-ended problems involves the use of technology. The following problem has been used with high school students and college students. Many times, the high school students are better at solving it.

Problem

According to a newspaper report, the trees in a certain land area are being cut at a rate of 15% per year. The lumber company claims that it replants 2000 trees every year in this area. Discuss the future tree production of this land area if this plan continues.

An in-service example from the Fort Worth Independent School District Connecting Mathematics and Science Institute

Linda Antinone

Texas requires every student to take one year of algebra, one year of geometry, and one year of either advanced algebra or modeling. When I took algebra in high school, it was mainly for college bound students who were abstract thinkers. Now, many of our students are concrete thinkers. Many of them work from examples to the big picture rather than from the big picture to the specific examples. We are trying to find ways to help students understand symbols and graphs.

We have employed an interdisciplinary approach that features mathematics, physics, and chemistry. The technology we used included Texas Instruments Incorporated Graphing Calculators, a TI Calculator-Based Ranger™ (motion detector), a TI Calculator-Based Laboratory™ System (data collection system with probes), and computers. We used several different instructional strategies during the lessons. For example, we used whole group discussion and data collection, small group data collection and analysis, hands-on activities, week-long investigations, group presentations, and sharing results of group investigations.

One activity we completed was called "Modeling in the Real World." Another was called "Ball Drop." Other activities related to motion, cooling, and sound waves. One of the group projects we did was to determine the accuracy of the common, although counterintuitive, assertion that hot water freezes faster than cold water.

For some of these activities we use a calculator with a motion detector. Based on the motion, it plots a graph that appears on the overhead display. This is important for exploring the meaning of graphs. It provides a real-time example of the relationship between variables (e.g. time and distance) and the features of graphs (axes, slope, maxima, and minima). Another activity involves starting with a graph and then trying to move in a way that would replicate the graph. The cooling activity involves putting a temperature probe in hot liquid and leaving it there as the liquid cools. The calculator graphs the cooling rate. The starting and stopping points on the graph show that there is a relationship between the rate of cooling and the size of the difference between liquid temperature and room temperature. Students are to find a function that describes the relationship. These efforts to get students involved in using and interpreting data helps them make connections between representations and their meanings in the real world, and we hear much less complaining that students are bored.

In the future we expect that changes with respect to the use of technology will happen rapidly. Teacher shortages may cause the biggest changes in teacher preparation requirements. We also predict that in-service will change faster than pre-service because testing is driving practicing teachers to change faster. Tests like the SAT® (a university entrance examination), and some state tests, are allowing-even expecting-calculator use. Since teachers are held accountable for students' performance, they are pushed to learn how to use-and teach with-the technology. Changes to undergraduate mathematics resulting from the Harvard Calculus program have also encouraged teachers to change their attitudes about how using technology in high school will impact college performance.

Promises and Challenges Related to the Approach in the USA

Many seminar participants found the use of technology to be promising, especially to the extent that it facilitates interdisciplinary lessons and the study of commonly neglected topics such as modeling. Some noted the potential of the technology to propel testing towards more meaningful content. Some also mentioned the potential of technology to motivate students to engage with the mathematics and to think about the real-world meaning of the representations and symbols they use. However, some expressed concern that the technology might receive more attention than the mathematics.

Some of the challenges the participants expressed related to the expense associated with the technology, especially in poor communities and highly populated countries. Other concerns related to the difficulty of training enough teachers to teach effectively with calculators and to developing activities that teachers could use to explore important mathematics with students. Some also mentioned the possibility of backlash from parents, resistance from teachers, and discontinuities between students' earlier and later experiences learning mathematics.


3 Harvard Calculus is a colloquial reference to curriculum developed by a consortium based at Harvard University and published in 1994 by John Wiley & Sons, Inc.

4 The SAT was formerly an acronym for Scholastic Aptitude Test, but the test is no longer an aptitude test, and the letters SAT are no longer an considered an acronym. See www.collegeboard.com/newsat/index.html for details about the new format.


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