Running head: JAMES J. KAPUT
James J. Kaput: Mathematician, Researcher, Educator
Mike Mazzarella
George Mason University
JAMES J. KAPUT
James J. Kaput: Mathematician, Researcher, Educator
It is rare to find an individual who has the passion for mathematics, the desire to change
the way it is taught, and the vision to see the future of the subject. One person possessed all of
these qualities while sharing his knowledge as a college professor. James J. Kaput was able to
understand mathematics in a way that was difficult for many others. His contributions to
technology in mathematics, school algebra, and calculus application made him a significant
figure in the field before his passing in 2005.
Personal Life
James J. Kaput grew up and spent his entire life in Massachusetts. English was not his
first language; Kaput’s family spoke Polish and he did not learn English until he began his
schooling (Tall, 2008). Kaput earned his bachelor’s degree in mathematics from Worcester
Polytechnic Institute, and his master’s degree and PhD from Clark University in Worcester, MA.
(Stickgold, 2005; Prial, 2005). His home for most of his life was in North Dartmouth, MA,
which was near the university at which he would spend his entire career. Kaput was a
mathematics professor for 25 years at University of Massachusetts-Dartmouth before his
untimely death in 2005 (Tall, 2008).
Calculus
One of Kaput’s main interests was changing the way that society thinks about calculus.
Kaput essentially defined calculus in two ways. First, “Calculus the Institution” referred to
policy makers and larger use of calculus beyond the classroom. “Calculus the Institution,” or CINST, Kaput argues, is responsible for making calculus material and applications inaccessible for
a large majority of the country (Kaput, 1994). Second, “Calculus the System of Knowledge and
Technique,” or C-KNOWL, refers to the academic material that is taught in high schools and
JAMES J. KAPUT
colleges. The field of C-KNOWL spans from what is included in calculus textbooks to how
many students are taking an AP Calculus course in high school. It is important to note that while
Kaput was certainly interested in improving both aspects, he saw more value in addressing CINST as a whole. He claimed that “it is not enough to toss around names of ideas, procedures,
and relationships that exist in the formal cultural record of mathematical achievement…” (Kaput,
1997). Rather, Kaput wanted to make calculus available and understandable to the 90% of the
population that was not exposed to it. This desire rooted in the C-INST realm drove his work in
the C-KNOWL realm as well.
In an effort to make calculus available to the general public, Kaput launched the SimCalc
program. SimCalc is project that intends to bring the mathematics of change and variation
(MCV) available to students of all ages (Roschelle, Kaput, & Stroup, 2000). One hurdle that
SimCalc attempts to overcome is the way that calculus is generally presented. Since calculus is
typically considered a college-level course, it is thus presented as such. However, Kaput believes
that even students at the elementary level can begin to learn the basic ideas of calculus
(Roschelle & Kaput, 1996). Another important aspect of SimCalc is the emphasis on lived
mathematical experiences, rather than simply manipulation of written symbols. Drawing his
philosophy from that of Dewey, Kaput believes mathematics is more powerful when one can feel
it and learn it kinesthetically. While the SimCalc framework seemed radical in the mathematics
field, Kaput was confident that this project would revolutionize the way that calculus is viewed
and accessed across the world.
Algebra
Kaput’s interest in mathematics was not limited to higher-level material. In addition to
calculus, Kaput was largely focused on the nature of algebra and how it is taught and learned in
JAMES J. KAPUT
the classroom. Based on his research, experience, and observations, Kaput categorized algebraic
thinking into five strands (Kaput, 1998):
1. Algebra as generalizing and formalizing patterns and regularities, in particular,
algebra as generalized arithmetic
2. Algebra as syntactically guided manipulations of symbols
3. Algebra as the study of structure and systems abstracted from computations and
relations
4. Algebra as the study of functions, relations, and joint variations
5. Algebra as modelling
Kaput argued that classroom algebra is ineffective in promoting student algebraic thinking
because only the second strand is emphasized. For example, Kaput stated that many algebra
classrooms are too focused on learning a set of “rules” on how to solve an equation, such as
memorizing the “steps” to solve 2x + 3 = 7 (Kaput, 1998). As a result, many of Kaput’s writing
on algebra called for a reform in the curriculum.
One way that Kaput sought to change the way teachers and students viewed algebra was
the idea of “algebrafying” the curriculum. This idea is demonstrated clearly and effectively in an
example involving a second grade classroom. The problem was as follows:
The second graders at Jefferson School have raised money to visit the Statue of Liberty.
Thirteen friends are planning to go. They are very excited about the trip and worried that
they might forget something! On the night before the trip, they call one another to
double-check what they need to bring. Each friend talks to every other friend once. How
many phone calls are made?
JAMES J. KAPUT
Asked this question alone, students may figure out the answer (12 + 11 + 10 + … + 1) by
individually counting each phone call, but may not understand a pattern or relationship between
the number of people and the number of phone calls. However, Kaput then “algebrafies” the
problem by asking the same task but with 100 people instead of thirteen. Students are then forced
to look for patterns that can be used to solve the problem. Finally, Kaput says that changing the
task once more to n students promotes a way of thinking that students may not be familiar with,
but is important in the grand scheme of algebraic thinking (Soares, Blanton, & Kaput, 2006).
One might argue that this type of algebra problem encourages multiple strands of algebraic
thinking that Kaput (1998) outlines.
James Kaput also found great value in connecting mathematics with other classroom
subjects. In particular, he believed that since science and mathematics are so related, math
teachers should make connections to scientific relevance often. For instance, in the previously
mentioned telephone problem, students could physically act out the “calling” with one another
using paper cups and string. Performing this activity allows teachers to bring in the discussion of
how sound moves from place to place. Additionally, since the aforementioned problem
mentioned a trip to the Statue of Liberty, teachers and students could discuss immigration in the
United States, the significance of the landmark, and how New York City played a role in
immigration (Soares, Blanton, & Kaput, 2006). By doing this, not only do students get exposed
to other subject areas in the math classroom, but they also make realistic and visual connections
between mathematics and other disciplines.
To study how algebraic thinking was realistically being used in the classroom, Blanton
and Kaput (2005) conducted a qualitative study of the effectiveness of a program called
“Generalizing to Extend Arithmetic to Algebraic Reasoning” (GEAAR). The study, which
JAMES J. KAPUT
included observations in a third grade urban classroom, looked at how effectively planned
activities that were intended to promote algebraic thinking. The results were positive in that there
were many more instances of “spontaneous algebraic reasoning” (SAR) than “planned algebraic
reasoning” (PAR). For instance, students took part in a lengthy discussion on the results of
adding two odd, two even, or an odd and even number. Through the SAR event, Blanton and
Kaput (2005) showed that students were able to generalize patterns and relationships among
whole numbers. Furthermore, students were able to learn and understand concepts that were well
beyond the third grade curriculum, such as the representation of an odd number as 2n + 1.
Blanton and Kaput concluded that when regularly incorporated into the classroom, students are
capable of complex algebraic reasoning and critical thinking.
James J. Kaput did, however, see barriers in incorporating meaningful mathematical
problems into classroom. Hancock, Kaput, and Goldsmith (1992) claim that there is a lack of
data modeling in the classroom. Data modeling refers to “using data to solve real problems and
answer authentic questions” (p. 337). This definition also refered to Kaput’s (1998) fifth strand
of algebraic thinking. Specifically, their article mentions computer-based tools as the means for
achieving this goal, which will be discussed more in depth later. They include three themes that
they hope to emphasize in the process: reasoning with the aggregate, the objectification of
knowledge, and the pragmatics of classroom projects. Unfortunately, Kaput and his colleagues
argued that effectively incorporating data modeling into the classroom is a challenge for several
reasons. First, implementation of their vision would take more than one year, particularly with
the logistics of computers in schools. Second, not only would procedures change, but teachers
must make pedagogical changes, many of whom are not willing or able to do so. Nevertheless,
JAMES J. KAPUT
Kaput’s vision for including data modeling in the algebra classroom was a foundation for his
future research on technology in the classroom.
Technology
James Kaput’s research in calculus and algebra led him to promote and develop
technology as a way of learning mathematics. According to Kaput, with the use of technology,
students of all ages have the capacity to use mathematics that is more complex that what is
expected of them. Furthermore, Kaput claimed that computers and other technological devices
allow students to understand the application of mathematics in addition to making complex
connections among different topics (Kaput, 1992). Kaput believed that human cognition has
evolved and developed a type of symbolic processing that was not possible in earlier education.
As a result, he saw technology as a way to give students more mathematical fluency and think
about the subject in a way that was not available before. Kaput envisioned a “virtual culture” that
would use computers to “learn varieties of representational systems, provide opportunities to
create and modify representational forms, develop skill in making and exploring virtual
environments, and emphasize mathematics as a fundamental way of making sense of the
world…” (Shaffer & Kaput, 1999). Thus, Kaput was a huge proponent of computers and
technology, and would spend a large part of his career researching how to best implement certain
programs into various subjects.
As mentioned before, Kaput attempted to use technology as a way to incorporate data
modeling into the classroom. In 1992 Kaput and his colleagues proposed using a software
program called Tabletop to help students investigate, organize, and analyze data. In this program,
students are given data that is randomly scattered across the screen. Students can then click on
the piece of data to learn about it, and then label it accordingly. Additionally, students can create
JAMES J. KAPUT
Venn diagrams, charts, graphs, and tables of the data that shows some type of pattern or trend.
Kaput initially introduced this software to two urban middle school classes. The study consisted
of introducing teachers to the logistics of the software itself, introducing students to the software
and the concept of data modeling, and developing activities the promote organization and
interpretation of data through Tabletop. Through this experiment, Kaput and his colleagues
found that students did not initially find trends in the data, but rather focused on individual cases
of high or low data. Furthermore, more students also used Venn diagrams to organize continuous
data rather than an axes system, even though a set of axes would have been more appropriate to
represent the data (Hancock, Kaput, & Goldsmith, 1992). Overall, though activities could have
been more valuable, the program itself was a positive foundation for promoting data modeling in
the classroom.
Not only did Kaput see the benefit of students using technology, but he also believed that
teachers should be well-versed in even the more simple technological devices. Stylianou, Smith,
and Kaput (2005) conducted a study investigating the use of calculator-based rangers (CBRs) in
pre-service mathematics teachers. CBRs are devices that can measure temperature, motion, or
other numeric values and represent them graphically. Kaput was interested in seeing how preservice teachers would use the data pedagogically; that is, would these teachers use these devices
to effectively teach algebra skills? The findings of this study, according to Kaput and his
colleagues, were very valuable. Not only did pre-service teachers grow as “teachers in
transition,” but they also grew as “learners of mathematics.” For instance, one particular task
asks students to record, graph, and interpret the results of themselves walking along a path. The
students were asked to use data for both position and velocity. As teachers in transition, preservice teachers benefited by “recognizing the value of building on students’ kinesthetic
JAMES J. KAPUT
experience” and “recognizing the need to provide environments that allow for discussion and
communication around the topic of graphs,” among others. As learners of mathematics, preservice teachers benefited by “facing their misconceptions in interpreting graphs ‘iconically’”
and “using graphs as a means for communication,” among others (Stylianou, Smith, & Kaput,
2005). This demonstrates Kaput’s belief that any technology can act as a learning tool for both
teachers and students alike.
One of Kaput’s greatest contributions to the mathematics technology field was his
software program MathWorlds. MathWorlds, first introduced in 1994, was the first software that
was used in conjunction with SimCalc. This program was designed specifically for the
mathematics of motion; such topics include the relationships among position, velocity, and
acceleration (Kaput, 1994). One feature of this software was the ability for students to see an
“actor” (e.g. a clown or duck) move along a plane and a graph simultaneously recording either
the placement, velocity, or acceleration of the “actor” (Roschelle & Kaput, 2000). By doing this,
MathWorlds made a significant breakthrough in allowing students to see MCV develop right
before their eyes.
Using this software is not limited to a calculus classroom. Hegedus and his colleagues
(2015) attempted to implement MathWorlds into an Algebra 2 classroom by replacing the
regular curriculum with a SimCalc curriculum; that is, one that often used technology to generate
scenarios and explain ideas. They found that learning through the software significantly
increased students’ mathematics achievement on an Algebra 2 content assessment. Furthermore,
it was found that SimCalc was also beneficial for low-achieving minority groups, such as
African American and Hispanic students. Finally, the authors found that a non-honors algebra
class that used the SimCalc curriculum significantly outperformed an honors algebra class that
JAMES J. KAPUT
used the regular curriculum (Hegedus et al., 2014; 2015). These studies produced significant
results with SimCalc over twenty years after it was first introduced. Not only does this show how
effective Kaput’s innovative technology programs are, but how long-lasting they are as well. It is
clear that James Kaput’s ideas were revolutionary.
Future and Conclusion
Tragically, James J. Kaput’s life came to an abrupt end in 2005 when he was hit by a
vehicle during a jog in his neighborhood. Kaput was just 63 years old (Stickgold, 2005). Kaput’s
colleagues, however, so valued his work that a special issue in the Educational Studies in
Mathematics journal was published that reflected on his work and envisioned a future in which
Kaput’s ideas were fully implemented. Hoyles and Noss (2008) extensively reviewed Kaput’s
work and found four main themes: attending to representational infrastructures, working for
infrastructural change, outsourcing processing to the computer but attending to the implications,
and exploiting connectivity to encourage sharing and discussion. These themes are not simply a
culmination of what Kaput accomplished in the past, but a path that researchers will follow in the
future.
One example of Kaput’s work being replicated after his passing was by studying the
difference between the quality of mathematics classrooms with and without technology. Hegedus
and Penuel (2008) found that using SimCalc in a high school math class stimulates students in
two main ways. First, mathematical performances are enhanced because the technology
promotes creativity, connectivity, and student interactions. Second, technology promotes
participation within small or large groups, and results can easily be displayed across these
groups. Additionally, as mentioned previously, students have been found to make more complex
mathematical connections in technology-based lessons than in traditional lessons (Hegedus &
JAMES J. KAPUT
Penuel, 2008). Two other examples of following Kaput’s lead in mathematics education are
incorporating science and data modelling. Kaput’s earlier work emphasized the importance of
connecting mathematics curriculum to science material (Soares, Blanton, & Kaput, 2006). He
also saw the utmost importance in introducing authentic mathematical problems to students in
the classroom (Hancock, Kaput, & Goldsmith, 1992). Lesh and his colleagues (2008) contributed
to the special issue dedicated to Kaput by addressing several questions that, according to the
authors, were questions that Kaput himself was wrestling with before he died:
1. What is the nature of typical problem-solving situations where elementary-but-powerful
mathematical constructs and conceptual systems are needed for success in a technologybased age of information?
2. What kind of “mathematical thinking” is emphasized in these situations?
3. What does it mean to “understand” the most important of these ideas and abilities? How
do these competencies develop?
4. What can be done to facilitate development?
5. How can we document and assess the most important achievements that are needed for:
a. informed citizenship, or
b. successful participation in wide ranges of professions that are becoming
increasingly heavy users of mathematics, science, and technology?
In this article, the authors also went on to state that thoroughly addressing these questions will
require new research methods. They mention design studies as a possible way to attempt new
methods and receive constant feedback from students, teachers, and researchers (Lesh,
Middleton, Caylor, & Gupta, 2008). It is clear that Kaput’s own work contributed to significant
progress in mathematics education, but perhaps the more valuable contribution is the vision he
JAMES J. KAPUT
left with those with whom he worked. These colleagues valued his ideas and truly believe that
following through on his passion will lead to even more accomplishments and breakthroughs in
the field.
Despite his life being cut short, James Kaput’s contributions to the field of mathematics
will live well beyond his years. Not only did he change the way that mathematicians view the
field itself, but he changed the way that teachers view how algebra and calculus are taught in the
classroom. James J. Kaput’s legacy will always be one of being a pioneer in technology and
mathematics education.
JAMES J. KAPUT
References
Blanton, M.L. & Kaput, J.J. (2005). Characterizing a classroom practice that promotes algebraic
reasoning. Journal for Research in Mathematics Education, 36(5), 412-446. Retrieved
from http://www.jstor.org/stable/30034944
Hancock, C., Kaput, J.J., & Goldsmith, L.T. (1992). Authentic inquiry with data: Critical barriers
to classroom implementation. Educational Psychologist, 27(3), 337-364.
Hegedus, S.J., Dalton, S., Roschelle, J., Penuel, W., Dickey-Kurdziolek, M., & Tatar, D. (2014).
Investigating why teachers reported continued use and sharing of an educational
innovation after the research has ended. Mathematical Thinking and Learning, 16(4),
312-333. doi: 10.1080/10986065.2014.953017
Hegedus, S.J., Dalton, S., & Tapper, J.R. (2015). The impact of technology-enhanced curriculum
on learning advanced algebra in US high school classrooms. Educational Technology and
Research Development, 63(2), 203-228. doi: 10.1007/s11423-015-9371-z
Hegedus, S.J. & Lesh, R. (2008). Democratizing access to mathematics through technology:
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Comments:
Mike, I really enjoyed your presentation of Kaput's work! He was a prolific researcher and
math educator. I love
the way his research focused on creating more access for students to engage in gateway
courses like algebra
and calculus. I think he was a pretty progressive thinker in developing SIMCALC and
MATHWORLDS and using
simulations and CBRs to make math come alive for students as they engaged in real world
data.
I hope that his work informed your literature review for algebra. I am glad that you found the
way that he broke
down algebra into 5 different constructs very helpful for you as an algebra and secondary
educator.
Excellent job on both the bio paper and presentation. YOu kept us very engaged!