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Excerpts from an article from "Mathematics Competitions", Vol 14, No 2, 2001

THE AMERICAN HIGH SCHOOL MATHEMATICS EXAMINATION: A 50 YEAR RETROSPECTIVE

Stephen B. Maurer, Harold B. Reiter, and Leo J. Schneider

On February 9, 1999 students across America participated in the American High School Math Exam. The first such exam was given in 1950. Thus, the 1999 version is the 50th. Perhaps this is a good time to look at the history of the exam, its sponsorship, and its evolution--and important changes to begin in the year 2000. We conclude this article with a Special Fiftieth Anniversary AHSME, which includes one question from each of the first 50 editions of the AHSME. The AHSME is constructed and administered by the American Mathematics Competitions (AMC) whose purpose is to increase interest in mathematics and to develop problem solving ability through a series of friendly mathematics competitions for junior (grades 8 and below) and senior high school students (grades 9 through 12). As you read below how the AMC exams have evolved, you will see that they have moved towards greater participation at many grade levels, much less emphasis on speed and intricate calculation, and greater emphasis on crtical thinking and the interrelations between different parts of mathematics.

HISTORY

Name and sponsors

The exam began in 1950 as the Annual High School Contest under the sponsorship of the Metropolitan (New York) Section of the Mathematical Association of America (MAA). It was offered only in New York state until 1952 when it became national under the sponsorship of the MAA and the Society of Actuaries. By 1982 sponsorship had grown to include Mu Alpha Theta, NCTM, and the Casualty Actuary Society. Today there are twelve sponsoring organizations, which, beside the above, include the American Statistical Association, the American Mathematical Association of Two-Year Colleges, the American Mathematical Society, the American Society of Pension Actuaries, the Consortium for Mathematics and its Applications, Pi Mu Epsilon, and the National Association of Mathematicians. During the years 1973-1982 it was called the Annual High School Mathematics Examination. The name American High School Mathematics Examination the better known acronym AHSME, were introduced in 1983. At this time, the organizational unit became the American Mathematics Competitions. Also in 1983 a new exam, the American Invitational Math Exam (AIME) was introduced. Two years later, the AMC introduced the American Junior High School Mathematics Examination (AJHSME).

Scoring

The scoring system has changed over the history of the exam. In the first years of the AHSME, there were 50 questions with point values of 1, 2 and 3. In 1960 the number of questions was reduced from 50 to 40 and in 1967 was again reduced from 40 to 35. It was finally reduced to the current 30 questions in 1974. In 1978 the scoring system was changed to the formula 30+4R-W, where R is the number of correct answers and W is the number of wrong answers. Ever since 1986 the formula has been 5R+2B where B is the number of questions left unanswered. There has been a distinction between wrong answers and blanks since the beginning, first with a penalty for wrong answers, and later with a bonus for blanks. For several reasons, in 1986 the award for blanks was made large enough to make the exam `guessing-negative'. In other words, random guessing will in general lower a participant's score. Previously, the exam had been `guessing-neutral.'

Participation

Previous to 1992, the scoring of the exam was done locally, in some states by the teacher-managers themselves and in other states by the volunteer state director. Beginning in 1992, all the scoring was done at the AMC office in Lincoln Nebraska. Beginning in 1994, each student was asked to indicate their sex on the answer form. The following table shows the degree of participation and average score among females versus that for males.

Year Females Average Males Average Unspecified Average
1994 104,471 68.8 120,058 76.0 6,530 70.6
1995 115,567 72.3 133,523 78.5 6,877 73.7
1996 124,491 65.8 142,750 71.2 6,659 67.8
1997 120,649 63.8 140,359 69.8 7,944 65.5
1998 108,386 66.5 128,172 71.9 7,438 67.8

Related Exams

Until the introduction of the AIME in 1983, the AHSME was used for several purposes. First, it was supposed to promote interest in problem solving and mathematics among high school students. But it was also used to select participants in the United States of America Mathematical Olympiad (USAMO), the 6 question, 6 hour exam given each May to honor and reward the top high school problem solvers in America and to pick the six-student United States Mathematical Olympiad team for the International Mathematical Olympiad competition held each July and. The introduction of the AIME, to which the primary role of selecting USAMO participants was passed, enabled the AHSME question writing committee to focus on the primary objective: providing students with an enjoyable problem-solving adventure. The test became accessible to a much larger body of students. Even some 7th and 8th graders, encouraged by their successes on the AJHSME, were participating.

THE PROBLEMS

How has the AHSME changed over the years?

In the early years, there were some computational problems. See the 1950 problem on the Special Fiftieth Anniversary AHSME for a rationalizing the numerator problem. Note that each problem is numbered by year together with its position on the test in its year of appearance. For example, the problem above is listed as [1950-10], which means that it was problem number 10 on the 1950 exam. Many early problems involved the simplification of complex fractions, or difficult factoring. In the 1960s counting problems began to appear. In the early 1970s trigonometry and geometric probability problems were introduced. In the 80s problems involving statistical ideas began to appear: averages, modes, range, and best fit. Problems involving several areas of mathematics are much more common now, especially problems which shed light on the rich interplay between algebra and geometry, between algebra and number theory, and between geometry and combinatorics.

Calculators

In 1994 calculators were allowed for the first time. The AMC established the rule that every problem had to have a solution without a calculator that was no harder than a calculator solution. In 1996 this rule was modified to read `every problem can be solved without the aid of a calculator'. Of course the availability of the graphing calculator, and now calculators with computer algebra systems (CAS) capabilities has changed the types of questions that can be asked. The allowance of the calculator has had the effect of limiting the use of certain computational types of problems. Referring to the Special Fiftieth Anniversary AHSME, problems [1954-38], [1961-5], [1969-29], [1974-20], [1976-30], [1980-18], [1981-24], and [1992-14] would all have to be eliminated for this year's contest, either because of the graphing calculator's solve and graphing capabilities or because of the symbolic algebra capabilities of some recent calculators. But the AMC has felt, just as NCTM feels, that student must learn when not to use the calculator as well. Thus questions which become more difficult when the calculator is used indiscriminately are becoming increasingly popular with the committee. For example, consider [1999-21] below: how many solutions does cos(log x)=0 have on the interval (0,1)? Students whose first inclination is to construct the graph of the function will be led to the answer 2 since in each viewing window, the function appears to have just two intercepts.

CHANGES

It is interesting to see the how the test has changed over the years. Has there been greater or less emphasis on geometry, on logarithms, on trigonometry? Have arithmetic problems become less popular? How about counting problems, geometric probability? The table below shows how many problems of each of ten types appeared in each of the five decades of the exam and the percent of the problems during that decade which are classified of that type.

Classification of Problems by Decade

Classification
1950-59
1960-69
1970-79
1980-89
1990-99
All problems
500 (100%)
390 (100%)
320 (100%)
300 (100%)
300 (100%)
Geometry
203 (40.6%)
215 (55.1%)
178 (55.6%)
168 (56%)
100 (33.3%)
Logarithms
24 (4.8%)
18 (4.6%)
8 (2.5%)
10 (3.3%)
8 (2.7%)
Logic
7 (1.4%)
8 (2.1%)
4 (1.3%)
3 (1.0%)
6 (2.0%)
Combinatorics
7 (1.4%)
4 (1.0%)
7 (2.2%)
20 (6.7%)
32 (10.7%)
Probability
0 (0%)
0 (0%)
10 (3.1%)
20 (6.7%)
10 (3.3%)
Statistics
0 (0%)
0 (0%)
0 (0%)
14 (4.7%)
7 (2.3%)
Trigonometry
0 (0%)
0 (0%)
11 (3.5%)
17 (5.6%)
8 (2.7%)
Number Theory
14 (2.8%)
20 (5.1%)
41 (12.8%)
25 (8.3%)
21 (7.0%)
Absolute Value,
floor, ceiling
4 (0.8%)
11 (2.8%)
24 (6.2%)
14 (4.7%)
5 (1.7%)
Function
notation
6 (1.2%)
4 (1.0%)
10 (3.1%)
12 (4.0%)
13 (4.3%)
Function
composition
4 (0.8%)
2 (0.5%)
4 (1.3%)
3 (1.0%)
5 (1.7%)

Some of the entries above need some elaboration. For example, a problem was considered a trigonometry problem if a trigonometric function is used in the statement of the problem. Many of the geometry problems have solutions, in some cases alternative solutions, which use trigonometric functions or identities, like the Law of Sines or the Law of Cosines. These problems are not counted as trig problems. A very small number of problems are counted twice in the table. Many problems overlap two or more areas. For example, a problem might ask how many of certain geometric configurations are there in the plane. The configurations might be most easily defined using absolute value, or floor, or ceiling notation (greatest and least integer functions). Such a problem could be counted in any of the three categories geometry, combinatorics, or absolute value, floor and ceiling. In cases like this, we looked closely at the solution to see if it was predominantly of one of the competing types. This situation often arises in the case of number theory-combinatorics problems because many of these types of objects that we want to count are defined by divisibility or digital properties encountered in number theory, but often invoke binomial coefficients to count. A few problems of this type are double counted. Many of the early problems are what we might call exercises. That is, they are problems whose solutions require only the skills we teach in the classroom and essentially no ingenuity. With the advent of the calculator in 1994, the trend from exercises (among the first ten) to easy but non-routine problems has become more pronounced. Note that even the hardest problems in the early years often required only algebraic and geometric skills. Many of the recent harder problems in contrast require some special insight. Compare, for example [1951-48], one of the three hardest that year with number [1996-27]. The former requires a few applications of the Pythagorean Theorem, whereas the latter requires not only Pythagorean arithmetic, but spatial visualization and manipulation of inequalities as well.

THE FUTURE

The new exam! In the 1950s AHSME was intended for our best and brightest high school students. With the increasing need to enable all students to learn as much mathematics as they are able, the AMC has moved away from encouraging only the most able students to participate. Especially in the past six years, the problems committee has attempted to make the first ten problems accessible even to middle school students. But the test continues to use problems involving topics most students encounter only after grade 10, topics such as trigonometry and logarithms. With this in mind, the American Mathematics Competitions will introduce in February 2000 the AMC10 aimed at students in grades 10 and below. In fact, the American Mathematics Competitions will offer a complete set of contests for middle and high school students. The AMC8 (formerly the AJHSME) will continue to serve students in grades 8 and below, the new AMC10 will serve students in grades nine and ten as well as middle school students who score well on the AMC8. The AMC12, formerly known as the AHSME, will continue to be the flagship contest for US high school students. The new exam AMC10 will be a 25-question, multiple choice contest, with 1 hour and 15 minutes allowed. The AMC12 will also be a 25-question, 75 minute exam. Correct answers will be worth 6 points and blanks will be worth 2 points, so the top possible score is still 150. Students will qualify for the American Invitational Math Exam in the usual way, that is, by scoring at least 100 on the AMC12. Additionally, the top 1% of AMC10 participants will qualify for the AIME.

 

 
 
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