Joseph G. Rosenstein is Director of the New Jersey Mathematics Coalition and a Professor of Mathematics
at Rutgers University.
"How do you multiply 255 by 28?", my third-grade daughter Neshama asked. I naturally assumed
that this was one of her homework problems, but soon discovered that was not the case. Although she
knew how to multiply 55 by 8, she had not yet learned how to multiply numbers with more digits.
On the cover of her dictionary it said "250,000 words and definitions" and Neshama was checking
that out. She saw that there were 255 pages, counted 28 words that were defined in a column, and
understood multiplication well enough to realize that the number of words in the dictionary should be 255
times 28. But she hadn't yet learned how to compute the product.
Kids are curious about numbers and how they work; they ask interesting mathematical questions --
ones which they often cannot answer with the tools they have mastered. But the study of mathematics has
always been sequential, based on mastery of previous topics, and so discussion of interesting questions has
often been delayed until all the tools have been learned, by which time the interesting questions may have
Suppose that there was a way for Neshama to figure out easily the answer to 255 times 28, and
other such questions. We could imagine that after discussing why we would multiply numbers and after
doing simple examples like 4 x 5 , her teacher might ask each student to bring in a real-world question
which would be answered by multiplying. One student might suggest using multiplication to count the
number of apartments in her building, another the number of words in a book, or the cost of a whole box
of candy bars, or the number of tiles on the bathroom floor, or the number of minutes in a day, or ... .
If they were able to answer such questions easily, the children could compare the numbers of apartments
in different buildings, or the costs of different boxes of candy, long before they mastered the computational
skills of multiplication. And it is not difficult to imagine that such problems might provide children with
motivation to learn the intricacies of multiplication.
Of course, it is now possible to do such problems, and many other types of problems, very easily --
by using a calculator. The calculator has made it possible for us to teach mathematics in a very different
way, one where we can discuss questions involving various skills long before those skills are mastered.
Put differently, the calculator allows children to acquire mathematical skills by talking mathematics first
and later filling in the details of computation, much like they acquire language skills, by speaking first and
later filling in the details of grammar.
A parenthetical remark. Readers who carried out the multiplication undoubtedly noticed that 255
times 28 is far short of the 250,000 words claimed by the publisher, even taking into account that each
page had two columns; perhaps the publisher counted all the words occurring in the definitions, but clearly
gave Neshama (and me) the impression that 250,000 words were being defined. When we discovered this,
Neshama and I had a discussion of the importance of thinking critically about what one reads.
The example given above involves arithmetic at the elementary school level, but there are similar
examples at all levels. In the tenth and eleventh grades, one of the major themes is mastering the
intricacies of various kinds of functions. The traditional approach focuses on learning the rules involving
these functions. Many students take naturally to precalculus topics, but many other students are turned off;
they are unable to master these manipulations, and give up on mathematics. The "graphing calculator",
whose miniature video screen creates the impression of a hand-held computer, provides a new approach
since it provides a vivid picture of the graph of any function in a fraction of a second -- a picture the
student and the teacher might not be able to create easily. How do you solve an equation? With a graphing
calculator, you can start by drawing a picture. Many college students fail calculus because they can't
connect the algebra they have learned with geometry; they can't connect the symbols with the graphs.
Graphing calculators can help students make that connection.
The use of calculators (and other technology) in the schools has been endorsed by all of the major
professional groups of mathematics educators and by all of the recent national reports, including the
Curriculum and Evaluation Standards for School Mathematics of the National Council of Teachers of
Mathematics. Everybody Counts, a report prepared by the National Research Council of the National
Academy of Sciences, notes that:
"Calculators enable students to explore a wider variety of examples; to witness the dynamic nature
of mathematical processes; to engage realistic applications using typical, not oversimplified, data;
and to focus on important concepts rather than routine calculation."
The New Jersey Mathematics Coalition, an umbrella organization involving the corporate, education,
public policy, and public sectors of the New Jersey community, has recently adopted the following position
"The New Jersey Mathematics Coalition strongly endorses the use of calculators in mathematics
instruction and assessment at all levels of schooling. Further, it encourages the Department of
Education, and local school districts, to adopt policies and procedures which will result in optimal
use of calculators in both instruction and assessment, and to do so in a matter that assures equity
and access for all groups of students."
The concern that is most often expressed -- by educators and parents -- is that the use of calculators
will somehow diminish the child's mathematical skills. Over the past twenty years a number of studies
have been done about this question, and the results are quite uniform. Students who use calculators
develop computational skills just as well as students who do not use calculators. But they are also better
at solving problems and have a more positive attitude toward mathematics.
Does that mean that students don't need to be able to do arithmetic on their own? Of course not.
They will often need to work without a calculator, and they can solve simple problems much faster without
a calculator. They also must be able to check the accuracy of the results they get from the calculator, since
we all push the wrong buttons sometime; for this they need to be able to estimate the result of a calculation
and compare it to the "answer" on the calculator. In the example above, multiplying 255 by 28, we expect
a student at the 5th grade level to say that her answer should be about 250 times 30, which is 7500, and
do that in her head.
Mathematics is more than the pencil-and-paper calculations which are often the focus of textbooks
and instruction. In an increasingly technological environment, students and graduates will need to analyze
a problem, determine what mathematical models and techniques are appropriate, carry out the necessary
calculations efficiently and effectively, use estimation to determine whether their answers make sense, use
reasoning to determine whether the answer is acceptable, and, if not, modify the problem and start all over
again. They will need to apply these skills if they're building a house or buying one, if they're creating
a product or selling it. The calculator enables instruction to focus on these skills, rather than dwell
exclusively on the computational aspect of mathematics.
As with instruction, so with assessment. If we want to test students' knowledge of mathematical
facts and ability to do simple problems, then they should not be using calculators. But if we want to test
students' problem solving ability, using real world problems, then they should be using calculators. They
will be using calculators outside of the classroom -- at home and at work -- and it doesn't make sense to
keep them from using them in school. On the job, they will be evaluated on the answers they get, not on
how they get them; no one today, outside of school, is ever asked to multiply two three-digit numbers
without a calculator.
In our educational system, like it or not, the curriculum is often determined by the tests students
take. As a result, in order to emphasize that students should do more complex real-world problems, and
should use a calculator when appropriate, we need tests which include those kinds of problems. We need
to test the skills that we value. All of the national reports emphasize that the simple "shopkeeper"
mathematics that was adequate 50 years ago is no longer enough. What is valued in today's technological
society is the ability to solve a variety of problems using a variety of techniques and tools. And that is what
the assessments should reflect.
The College Board has announced that students applying for college will be allowed to use
calculators on the SAT (Scholastic Aptitude Test) in 1994. The SAT-II (Achievement Test) permitted the
use of calculators last year and will require their use in 1994; the test is expected to be modified so that
for a number of problems the use of calculators will actually be necessary.
On the state level, the tests now in place do not permit the use of calculators. The New Jersey
Mathematics Coalition, in urging the Department of Education to change this practice, noted that "a
requirement that calculators be used on the new High School Proficiency Test (HSPT) would send the
message clearly throughout the state that calculators belong in all mathematics instructional activities."
This recommendation applies to both the High School Proficiency Test (HSPT) given in the 11th grade and
the Early Warning Test (EWT) given in the 8th grade.
The proper use of calculators on these tests will lead to improvement in mathematics curricula
throughout the state and should be encouraged wholeheartedly. "Proper use of calculators" includes
ensuring that teachers are trained in the instructional use of calculators, that students have and use
calculators extensively before taking the EWT and HSPT, that all students use calculators which have
substantially the same abilities, and that problems on the test are creatively designed to take full advantage
of the technology.
In 1986, the National Council of Teachers of Mathematics issued a position statement which notes
"Although extensively used in society, calculators are used far less in schools, where they could
free large amounts of the time students currently spend practicing computation. The time gained
should be spent helping students to understand mathematics, to develop reasoning and problem
solving strategies, and in general, to use and apply mathematics."
The calculator is at present an underutilized instructional tool; with adequate support, more teachers will
find better ways of using it to enhance children's understanding of mathematics. The New Jersey
Mathematics Coalition supports and encourages this trend.
The New Jersey Mathematics Coalition is an organization dedicated to the improvement of
mathematics education in the state of New Jersey. Through the active participation of educators, industry,
government, and the public, it seeks to convey the message that mathematics is accessible to all. It informs
the community about the recommendations in the recent national reports and encourages and facilitates
their implementation, building on the many activities and programs taking place throughout the state. For
further information about the New Jersey Mathematics Coalition, the reports mentioned in this article, or
about its position on calculators, your are invited to write to the Coalition at P.O.Box 10867, New
May 18, 1992