Challenges of Mathematics Teaching Today:
How Can School Leaders Help?

NASSP Bulletin

February 1997
pp. 1-10

Copyright 1997 NASSP

by Edwin M. Dickey

Abstract:

Students in middle and secondary schools require an education in mathematics that goes beyond what was needed by students in the past. This is a significant challenge to today's mathematics teachers. Teachers need the commitment and support of educational leaders as they adopt new curriculum materials and new methods of teaching.

Today's mathematics teachers are experiencing major changes not only in the mathematics content they teach, but also in the way they teach. Nearly all of these teachers came through school when mathematics consisted of a collection of facts and skills to be memorized or mastered by a relatively homogeneous group of students taught using a lecture approach. Now teachers are called on to teach new, more challenging mathematics to a very diverse audience using active learning approaches designed to develop understanding. This is an enormous challenge that coalesced with the publication of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics in 1989 and continues today. To meet this challenge mathematics teachers need the support and encouragement of school leaders at all levels.

Compare where we were to where we are. See if you recognize any of your mathematics teachers in the two classroom vignettes that follow.

In 1978, the National Science Foundation (NSF) commissioned various studies to assess the state of mathematics instruction. One case study provided a snapshot of a mathematics class that was repeated by nearly every observer (Fey, 1979):

"In all math classes I visited, the sequence of activities was the same. First answers were given for the previous day's assignment. The more difficult problems were worked by the teacher or a student at the chalkboard. A brief explanation, sometimes none at all, was given of the new material, and problems were assigned for the next day. The remainder of class was devoted to working on the homework while the teacher moved about the room answering questions. The most noticeable thing about math classes was the repetition of this routine."

In contrast, the NCTM Professional Standards for Teaching Mathematics (1991, pp 47-9) offers this portrait of a high school mathematics class:

"Ms. Chavez has rolled the math department computer into her class for the morning and has connected it to her LCD viewer. Her 28 first-year algebra students, seated at round tables in groups of threes and fours, are working on a warm-up problem. The day before they had had a test on functions. For the warm-up to today's class, Ms. Chavez has asked students to set up a table of values and graph the function y = |x|. ... Ms. Chavez listens to [student] conversations while she takes attendance. After about five minutes, she signals that it is time to begin the whole-group discussion.

"A girl volunteers and carefully draws her graph on a large wipe-off grid board at the front of the room. As she does this, most students are watching closely, glancing down at their own graphs, checking for correspondence. A few students are seen helping others who had difficulties producing the graph. ... Another student suggests that they enter the function into the computer and watch it produce the graph. ... [Ms. Chavez guides the students to discover how the graphs of y=|x| + 1, y=|x| + 3, y=|x| - 2 relate to the original graph y=|x|]

"Ms. Chavez asks the student to write in their journals, focusing on what they think they understand and what they feel unsure about from today's lesson. They lean over their notebooks, writing. A few stare into space before beginning. She gives them about ten minutes before she begins to return the tests. She will read the journals before tomorrow's class.

"At the end of the period, she distributes the homework that she has prepared. The worksheet includes additional practice on the concept of y =|x| c as well as something new, to provoke the next day's discussion: y = |x c|. ... "

It is not clear at this time how many teachers have incorporated into their teaching the vision of the NCTM Standards. A great deal of local, state, and national effort supported by Eisenhower Act and NSF funds has been exerted to familiarize teachers with the Standards and to assist them in developing new teaching strategies and techniques similar to those used by Ms. Chavez in the vignette. Evidence is accumulating that this effort is having some positive effect.

While none of this may directly be attributed to the Standards, statistics from the United States Department of Education (1995) document progress in mathematics achievement. Between 1982 and 1992, the mathematics proficiency scores of 17-year-olds on the National Assessment of Educational Progress (NAEP) increased 9 points. Improvement for 17-year-olds from 1982 to 1992 was roughly equivalent to an additional year or two of learning in high school.

Chart 1: Mathematics Proficiency Scores of Students

(Source: United States Department of Education, Office of Educational Research and Improvement. The Condition of Education. Washington, D. C.: United States Department of Education, 1995.)

Using the mathematics portion of the Scholastic Assessment Test (SAT) test as a measure, even as the number of high school graduates taking the SAT has increased by 8 percent since 1983, the average SAT-Mathematics score has risen from 468 in 1983 to 482 in 1995. Increased participation usually pushes down scores; SAT-Mathematics scores are rising despite increased participation.

Chart 2: SAT-Mathematics Scores

(Source: United States Department of Education, Office of Educational Research and Improvement. The Condition of Education. Washington, D. C.: United States Department of Education, 1995.)

Similar statistics (United States Department of Education, 1995) show that many more high school graduates in 1992 were taking mathematics courses at the level of first-year algebra or higher than were their counterparts in 1982. The bar graph below shows that the percent of high school graduates taking remedial or below-grade-level mathematics has dropped dramatically while the percent taking first-year algebra, geometry, and second-year algebra has risen. This trend to take less remedial work and more courses in the so-called "college-prep" curriculum is found in all ethnic and racial subpopulations as well.

Chart 3: Percent of High School Graduates in Various Mathematics Courses

(Source: United States Department of Education, Office of Educational Research and Improvement. The Condition of Education. Washington, D. C.: United States Department of Education, 1995.)

Despite these positive indicators, parents and educators agree that instruction and student learning in mathematics can and should improve even more. Numerous initiatives and projects are suggesting ideas that are likely to make mathematics teaching even more effective. At the same time, these changes exert even more pressure on teachers. The following list contains some of the challenges facing mathematics teachers today:

  1. higher expectations;
  2. new curriculum materials;
  3. high stakes tests that are not aligned to new curriculum;
  4. integration of technology; and
  5. block scheduling.
Each of these challenges calls for support and encouragement from the educational leaders of these teachers.

Higher Expectations

Following the core curriculum recommendation of the NCTM Standards, many states are expecting all high school graduates to earn credit in first-year algebra. Mathematics educator Cathy Seeley (1993) summarizes this challenge as follows:

"Students who have never experienced success in mathematics will be placed into 'academically challenging' courses ... some students will blossom in the hands of well-trained, caring teachers who teach a different kind of mathematics ... other teachers will continue to teach the kind of algebra they have always taught in ways they have always taught, even though their audience is drastically different ... "

These latter teachers, Seeley points out, will claim that the students could not handle the higher expectations and will issue failing grades confirming their belief that some students cannot deal with some of the mathematics curriculum.

Seeley points to an alternative and equally unfortunate scenario:

"In other districts, compassionate teachers concerned about the potential for students failing .. or teachers pushed by administrators concerned about high failure rates, will see to it that algebra becomes accessible to their students by watering down the content of their course "

Neither scenario will have a positive impact. The key to avoiding the situations described by Seeley is teacher professional development.

How can educational leaders help?

New Curriculum Materials

The current offerings from major textbook publishers are now beginning to address directly the vision put forward in the NCTM Standards. The NSF has also underwritten numerous curriculum development projects that are now reaching the market place. The November 1995 focus issue of the Mathematics Teacher provides a "sampler" of many of these materials.

Video and computer delivered instructional resources are also beginning to show promise. The NSF funded Math in the Middle project is piloting video lessons connecting middle grades mathematics to music, oceanography, design, and other areas. Support materials for the videos are available through the Internet (http://www.scsn.net/serc). Teachers and others are beginning to make learning units available through the World-Wide Web. The Math Forum maintains a collection of units written by teachers (http://forum.swarthmore.edu/web.units.html). Pacific Bell sponsored and maintains a Web-delivered unit entitled "Look Who's Footing the Bill" -- a cooperative learning, interdisciplinary activity about the democratic process, the U.S. budget, and our national debt (http://www.kn.pacbell.com/wired/democracy/debtquest.html).

Teachers need time and guidance to examine these new offerings, consider them carefully in light of the needs of their students, then select wisely. In many cases, it may be best to adopt materials tentatively and make adjustments as teachers gain direct experience while working with students.

With such a growing wealth of materials and new delivery systems, teachers can plan courses by using materials from different sources. While this enhances the students' experience, it makes planning for instruction more time-consuming.

How can educational leaders help?

High Stakes Tests

Imagine your goal as an athlete is to excel in the decathlon. You train in all ten events spending different amounts of time and effort preparing for each based on your individual strengths and weaknesses. Now imagine you arrive at the final meet and learn that the championship will be decided solely on the basis of winning the 100 meter run. You have more to offer but you will be judged on that one event -- perhaps not your strongest. The NCTM Standards have proposed a broad array of topics as indicators of mathematical prowess. Too many of the tests used today to measure success focus on too few areas of mathematics. Many tests fail to assess students' achievements in problem solving, communicating mathematical ideas, connecting mathematics to reality, and reasoning mathematically.

Thomas Romberg and Linda Wilson (1992) analyzed six widely used standardized tests to determine whether the tests reflected recommendations in the NCTM Standards. The authors concluded that the "tests do not adequately cover the range of content described in the grades 5-8 standards ... [and] the tests do not address any of the primary standards that the authors of the Standards document hold are the basis for the authentic learning of mathematics: problem solving, communication, reasoning, and connections." Through assessment we communicate to teachers, to students, and to parents what we value in learning. We are at a time when our curriculum and our instruction speaks in the language of inquiry, constructivism, and active learning while many of our assessment methods listen only to the rapid recall of isolated facts.

Mathematics teachers experience a schizophrenic situation when they attempt to utilize instructional resources that reflect the NCTM Standards and, at the same time, prepare students for the multiple-choice tests that make up many state assessment systems.

There are bright spots emerging: NAEP uses content domains similar to those in the NCTM Standards and includes open-ended, multi-step problems that require written student work. The Advanced Placement (AP) Calculus examinations and the new AP Statistics examination include high-level, open-response questions and actually require the use of a graphing calculator with specific features. Both the SAT and the ACT now allow the use of calculators. The state of Vermont has done pioneering work in the area of state-wide assessment. In 1995 the Vermont mathematics assessment system consisted of a uniform assessment (The New Standards Reference Exam at Grade 10) and a multi-year high school mathematics portfolio (Vermont Institute for Science, Math, and Technology, 1995).

How can educational leaders help?

Integration of Technology

Most mathematics teachers working today learned mathematics without the use of technology. A few were pioneers as computers began entering the classroom teaching programming languages such as BASIC, but in mathematics, as in other subjects, the promise of computer technology has not been fully realized. Unlike other subjects, a particular technology has had and continues to have a profound influence on mathematics instruction.

Hand-held computers with built-in mathematics software (known to most teachers as graphing calculators) are successfully changing how mathematics is taught and learned. According to Currence (1992),

"teachers are using the graphing calculator in visualizing concepts, exploration, experimentation, generalizing, and checking solutions to algebraic problems. In addition, training [has] led to the restructuring of topics taught in mathematics. Many teachers are omitting or giving less attention to certain topics ... also, many teachers of algebra are using real world problems to motivate their students and they are also using the graphing utilities of the calculator to solve equations and inequalities."

Graphing calculator technology has the advantages of low cost and high portability. Teachers can assume that their students will have access to the technology outside of class, even at home. Special computer labs are unnecessary, and with equipment like calculator-based laboratories (CBLs), data gathered in science class can be stored in the calculator for use in the mathematics class.

How can educational leaders help?

Block Scheduling

The increasingly popular move to block schedules offers a particular challenge to mathematics teachers. Classes of students actively discovering concepts while using technology and experimentation (second vignette above) are better suited for longer time periods than more traditional lecture-format classes. Before adopting a change to longer class periods, an assessment of teachers' willingness and ability to lecture less and involve students more is advised. Effective staff development addressing new techniques will likely be necessary and adequate planning time is needed as teachers adopt new methods.

The mathematics curriculum should also be reconsidered when making a dramatic scheduling change. It seems naive to assume that the familiar Algebra I-Geometry-Algebra II sequence of year-long five-day per week 50-minute classes will work as well in a alternating or semester block schedule. Teachers should consider additional courses or topics that address data analysis or discrete mathematics. Integration of these and other topics as well as other subjects, particularly science, should also be considered.

Semester block scheduling offers yet another challenge. Students who take a one semester mathematics class, then pass on math for one semester and a summer are likely to have difficulty when they begin studying mathematics again. It has been conjectured that if the learning is more meaningful, the difficulty will be minimized. There has been no definitive study to document whether the gap in learning is truly a problem.

How can educational leaders help?

Mathematics teachers in the United States are among the most innovative in the world. They reach a broad, diverse audience of students teaching mathematics content that, in many cases, was not part of the curriculum they studied when they were students. They make increasing use of technology and other teaching strategies that they themselves never experienced as students. With the strong support and leadership of school principals and superintendents, mathematics teachers will build on the moderate successes of the past decade and develop our students' potential in mathematics into a wellspring for our nations' future.

References