The American High School Mathematics Examination:
A 50 year Retrospective
By 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 critical 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 'guessingnegative'. 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 1960’s counting
problems began to appear. In the early 1970’s trigonometry and geometric probability problems
were introduced. In the 80’s 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 lead 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
All problems
Geometry
Logarithms
Logic
Combinatorics
Probability
Statistics
Trigonometry
1950-59
500 (100%)
203 (40.6%)
24 (4.8%)
7 (1.4%)
7 (1.4%)
0 (0%)
0 (0%)
0 (0%)
Number Theory 14 (2.8%)
Absolute Value, 4 (0.8%)
floor, ceiling
Function notation
Function composition
6 (1.2%)
4 (0.8%)
1960-69
390 (100%)
215 (55.1%)
18 (4.6%)
8 (2.1%)
4 (1.0%)
0 (0%)
0 (0%)
0 (0%)
20 (5.1%)
11 (2.8%)
1970-79
320 (100%)
178 (55.6%)
8 (2.5%)
4 (1.3%)
7 (2.2%)
10 (3.1%)
0 (0%)
11 (3.5%)
41 (12.8%)
24 (6.2%)
1980-89
300 (100%)
168 (56%)
10 (3.3%)
3 (1.0%)
20 (6.7%)
20 (6.7%)
14 (4.7%)
17 (5.6%)
25 (8.3%)
14 (4.7%)
1990-99
300 (100%)
100 (33.3%)
8 (2.7%)
6 (2.0%)
32 (10.7%)
10 (3.3%)
7 (2.3%)
8 (2.7%)
21 (7.0%)
5 (1.7%)
4 (1.0%)
2 (0.5%)
10 (3.1%)
4 (1.3%)
12 (4.0%)
3 (1.0%)
13 (4.3%)
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 1950’s 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 AMC10 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 AMC8 (formerly the AJHSME) will continue to serve students in grades 8 and
below, the new AMC10 will serve students in grades nine and ten as well as middle school
students who score well on the AMC8. The AMC12, formerly known as the AHSME, will
continue to be the flagship contest for US high school students. The new exam AMC10 will be
a 25-question, multiple choice contest, with 1 hour and 15 minutes allowed. The AMC12 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 AMC12.
Additionally, the top 1% of AMC10 participants will qualify for the AIME.