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The Effect of Math Modeling on Student’s Emerging Understanding

Andrzej Sokolowski

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Abstract
This study investigated the effects of applying mathematical modeling on revising students’preconception of the process of optimizing area enclosed by a string of a fixed length. A group of
28 high school pre-calculus students were immersed in modeling activity that included direct
measurements, data collecting, and formulating algebraic representation for the data. The lab
conduct was enriched by scientific inquiry elements such as hypothesis stating and its verification.
While 86% of the students (N=24) falsely hypothesized that the rectangular areas enclosed by astring of a fixed length will remain constant before engaging in the lab, the subsequent tasks ofthe modeling activity prompted the students to correct their ways of thinking. The study showedthat the modeling processes provide ample means of revising students’ perception to establish
firm conceptual background for inducing a more rigorous algebraic approach to solving problems
in math classes. Suggestions for further studies follow.
Keywords: Mathematical modeling; optimization; problem solving; high school.

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Introduction
Misconceptions are defined as strongly held, stable cognitive structures that must be overcome oreliminated for students to achieve expert understanding (Hammer, 1996). Many researchers
object the term misconception because from the student’s viewpoint, the ideas are logical andinstead the terms preconception, naïve theories, and alternative framework have been proposed(Sneider & Ohadi, 1998). Since the emphasis of this study is to help students uncover, bythemselves, the mathematical underpinning of the optimization process and its correctinterpretation, the author prefers to use the term preconception whose correctness will be
attempted during the process of modeling.
Mathematical modeling (MM) has been described in many ways; for instance, Lesh and Harel(2003) defined MM as finding quantifiable patterns of a phenomenon and its generalization,
while Blum, Galbraith, Henn, and Niss (2007) defined mathematical modeling as a process of“learning mathematics so as to develop competency in applying mathematics and building
mathematical models for areas and purposes that are basically extra-mathematical” (p. 5).
Although initiated several decades ago, MM recently gained substantial popularity in mathresearch and education; in fact, in the Common Core Curriculum, MM is recognized as one of theeight fundamental standards (Porter, McMaken, Hwang, & Yang, 2011).
MM activities can be organized in many ways: by asking open-ended questions, bymathematizing given situations, by using simulations as a means of supporting context, or bysolving word problems (Haines & Crouch, 2007). Thus MM serves as an embodiment of problem
solving, and the learning methods are interwoven. A problem in mathematics is defined as asituation carrying open questions (Blum & Niss, 1991). As such, it usually follows four steps:
understanding the problem, devising a plan, carrying out the plan, and looking back (Polya, 1957).
MM offers a different perspective; it focuses learners on the process of transferring given
information presented in a real-world situation by data gathering and model formulating and thenrequires the learner to use the model to develop new knowledge or solve context-related realworld problems (Crouch & Haines, 2004). By converting problems into the process of taking
quantifiable data and modeling formulation, MM activities can “meet the individual abilities of(many) more students” than traditional teaching methods (Kaiser, 2007, p. 104). MM can be
guided by different types of inquiries, ranging from deductively situated, authentic problemmodeling activities (English & Sriraman, 2010) to inductively organized inquiries that havestudents make plausible arguments that take into account the context (Lesh & Zawojewski, 2007).
Prior Research Findings
Optimization is a field of applied mathematics whose principles and methods are used to solve
quantitative problems in mathematics and other disciplines including biology, engineering,
physics, and economics (Rardin, 1997). Optimizing is a process of making the best possible
choice from a set of candidate choices (Boyd & Vandenberghe, 2009). It consists of maximizingor minimizing a real function by systematically choosing input values from within an allowed setof computed values of the function. Optimization problems are very common in high school andundergraduate mathematics curricula, and they are an essential component of problem solving in
calculus courses (National Council of Teachers of Mathematics [NCTM], 2000). Despite thewide applicability range, the process of optimization is not often investigated in mathematics
education; in fact, a search for prior research findings using university search engines such as
ProQuest Educational Journals, Science Direct, and Google Scholar returned only several suchstudies, whose summaries follow.

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Troxell (2002) pointed out certain pitfalls in using technology while teaching optimization andsuggested relying more on analysis and validation. Poon and Wong (2011) proposed an adoptionof Polya’s (1957) problem-solving model to investigate and solve optimization problems ingeometry. They highlighted benefits of inter-disciplinary learning and investigation of multiple
solutions while working on such problems. Schuster (2004) discussed the role of combinatorial
optimization activities to solve optimization problems in high school mathematics and concluded
that such activities provide opportunities not only for mathematical modeling of real world
problems, but also for discovering, constructing, and investigating algorithms. The process of
optimizing areas enclosed by a string of a fixed length was investigated by Brijlall and Ndlovu(2013), who found out that when taught by traditional methods, students used “isolated facts andprocedures” (p. 16) while solving these problems, which showed their lack of understanding ofunderlying principles. They further suggested that “teachers needed to be aware of learners
learning conflicts so as to reinforce the new concepts they encounter” (p. 17). Ledesma (2011)
investigated how to identify a solid with a maximum volume built by cutting squares of various
side lengths from a cardboard and folding the shape. She concluded that calculus students faceddifficulties understanding the optimization process, which she considered a main obstacle insolving these problems. As a means of improving, she suggested strengthening the visualizing
process by using simulations. A similar optimization problem was investigated by Lowther(1999). Although she intended the students to find the maximum value algebraically, the students
chose to determine the answer using a method of trial and error. Lowther noted, “I was sure thatsomeone would use an algebraic approach, but no one did” (p. 764). By not formulating algebraic
function, this investigation did not help the students solve text problems, as was initially intended.
Although all of the above scholars provided valuable suggestions to improve understanding of the
optimization processes, the idea of verifying students’ preconceptions on these processes andapplying MM to provide learners with tangible experiences was not proposed within this domain.
While optimization problems can be classified as typical math problems, the process oftransferring one dimensional geometrical object (length) into two dimensional geometrical object(area) is not that apparent. This study proposes an activity where this process is made moretransparent to students.
Research Questions
While typical problems of optimization focus students’ attention on finding unique solutions due
to formulated algebraic representation that maximizes the scenario (e.g., see Demana, Waits,
Foley & Kennedy, 2008; Stewart, 2007; Sullivan & Sullivan III, 2009), an MM activity seeks todiversify the process by having students investigate possible outputs and identify a pattern before
formulating an algebraic representation. The purpose of the study reported herein was todetermine if immersing students in an MM activity would help them with understanding thealgebraic underpinning of optimization. In this line, the following research questions were
formulated:What are the students’ preconceptions on the output of area enclosed by a string of a
fixed length?

Will modeling environment help learners understand the process of optimization?

Which of the MM stages—data collecting, table generating, graph plotting, or model
formulating—have the most significant effect on understanding optimization?

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Theoretical Framework of the Treatment Design
The theoretical framework of the study was supported by constructivist learning theory thatsuggests learner’s construction of their own knowledge based on received impulses (vonGlasersfeld, 1995). This theory has been selected because constructivism is suited for activities
that focus on meaning – making, concept construction, or elucidation of alternative concepts(Fergusen, 2007) which encompass the process of finding an optimum of a quantity due to given
constrains. Minstrell and Smith (1983) have recommended several teaching methods and learning
settings for eliminating student’s misconceptions.
Experiment or experience that will allow students testing their thinking is one of learning setting.
While typical problems on optimization focus students’ attention on finding unique solutions due
to formulated algebraic representation that maximizes the scenario (e.g., see Demana, Waits,
Foley & Kennedy, 2008; Stewart, 2007; Sullivan & Sullivan III, 2009), an MM activity seeks todiversify the process; have students investigate possible outputs, identify a pattern before
formulating an algebraic representation and finding its optimum value. Considering research
recommendations and MM learning outputs, modeling was selected as framework for the activity
design.
Development of the Modeling Process
MM usually follows a cycle (e.g., see Blum & Leiss, 2007; Geiger, 2011; Lesh & Lehrer, 2003;
Yoon, Dreyfus, & Thomas, 2010). Although the MM activity in this study intended to follow ageneral modeling process in which a real-world problem was abstracted, mathematized, solved,
and evaluated, special attention was given to having students hypothesize the problem’s output
before attempting to solve it and then revise or confirm their hypotheses after eliciting andvalidating the mathematical model. It was anticipated that inserting this element of scientificinquiry (Hestens, 2013) would not only help assess the research questions but also allow thestudents to realize the importance of mathematical processes in modifying or confirming theirprior thinking. Since the activity involved quantification, the hypothesis had a twofold nature:qualitative and quantitative.
The quantitative part, called a prediction (see Figure 1), involved having students predict a
numerical output of their investigations, and the qualitative part involved having themhypothesize a general property or output of their investigations. While a typical modeling activity
is initiated by asking learners to write a problem statement (Lesh & Zawojewski, 2007), in thisstudy’s activity, the problem was formulated and the students’ initial task was to hypothesize
possible outputs. Another element of the activity design was to decide about the type of inquiry.
There were two main options to consider, deductive and inductive, which are the main reasoning
inquiries used in science, mathematics, and engineering (Prince & Felder, 2006). As deductive
inquiry characterizes the process of reasoning from a set of general premises to reach a logicallyvalid conclusion, inductive inquiry is a process of reasoning from specific observations to reach ageneral conclusion (Christou & Papageorgiou, 2007). In this activity, students were given a stringof a fixed length and asked to formulate various rectangles and search for patterns; thus, an
inductive inquiry was adopted. All of the design constraints were guided through the scheme
depicted in Figure 1.

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Real Context

Qualitative

Problem Statement

Quantitative

Hypothesis/Prediction
Data Gathering
Analysis
Model Formulation

Qualitative

Model Verification

Quantitative

Model Validation

Hypothesis Verification

Figure 1. Adopted modeling process based on Sokolowski (2015)
The real context, in this activity, will be provided by material and described scenario that will befollowed by a problem statement. Note that the multiple arrows indicate stages when students candecide to modify or revise concluded patter on algebraic model. Such established theoreticalframework satisfied also the six principles for problem-solving activities developed by Lesh and
Kelly (2000).
Context Development
The following problem was used as a contextual basis for converting into a modeling activity:“Among all the rectangles, whose perimeters are 100 ft, find the dimensions of the one with amaximum area” (Demana, Waits Foley, & Kennedy, 2007, p. 184). Converting the problem into amodeling activity required inducing all of the elements of the modeling cycle described in Figure
1 as well as considering research recommendations concerning inducing metacognitive changes
in students’ minds. For instance, Bonotto (2007) suggested that well designed MM activity (a)emphasizes the strong interactivity of the instructional techniques, and (b) eliminates ready-madesolution processes. Bearing the adopted modeling process and the recommendations, an outline of
instructional support was developed.
Methods
This study can be classified as an empirical one-group quasi-experimental (Shadish, Cook, &Campbell, 2002). Randomization of participants was not possible due to a low school population
where the study was conducted. Quasi –experimental study shares though many similarities withexperimental design.

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Participants
A group of 28 students (12 females, 16 males, Mage = 16.5 years, age range: 16-17 years) enrolled
in a pre-calculus course in a suburban high school was assigned as a treatment group. The groupof students was selected due to availability. 25% of the group constituted minorities.
Evaluation Instruments
Analysis of students’ hypotheses, assigned prior the lab conduct, along with verification of the
hypotheses, reflections, and survey were used to evaluate learners’ prior thinking and the effectsof the treatment. A content analysis (Hsieh & Shannon, 2005) that characterized a systematicclassification process of coding and identifying themes and patterns was used to evaluate students’
free responses.
Treatment, Setup, Materials, and Procedure
A physics lab was the designated site for the activity because the students could be seated ingroups, which allowed for collaboration and discussions. There were seven groups of four
students. The treatment lasted for one class period (55 minutes). Each group was given a 74 cm
long string, a Styrofoam board of 40 cm by 60 cm, four pins, a metric ruler, and a set of French
curves to sketch a smooth curve for generated data. Each student was given instructional support
in the form of a lab outline. The instructor initiated the activity by explaining that the students
would investigate the area enclosed by a string of a fixed length and pointing out the importanceof reading the problem statement and hypothesize the answer. The students were asked to state
their hypothesis individually, without discussing the context with group partners. As suggested byprevious research (Lesh & Lehrer, 2003), the instructor then took on the role of a guider,providing suggestions when questions arose rather than offering direct solutions. Although thestudents took data in groups, each student was responsible for completing the lab analysis
individually. Students then began collecting data by formulating various rectangles due toprearranged lengths (see Figure 2).

Figure 2. The process of data taking

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In order to systematize the process, the participants recorded their data in a table (see Figure 3):

Figure 3. Generated table of values
The lengths of the bases of the rectangles were prearranged. The students measured resulting
heights and computed area for each variation. The analysis of the data was initiated by generating
the area of the rectangular polygons versus height graph (see Figure 4).

Figure 4. Area versus height graphNote, in the Figure 4, the students called the height width which did not change the datainterpretation. Having the points plotted and the graph sketched, the students identified its general
algebraic structure. There pre-calculus students did not have difficulties with identifying
underpinning function as quadratic. The next stage referred to formulating the function equation
for the graph and discussing the graph properties such as the extreme value, its coordinates andinterpretation (see Figure 5).

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Figure 5. The process of formulating the area function
This student identified the height of 17 cm for which the area was a maximum and equal to 344
cm2. She further used this coordinate to compute the leading coefficient, a, of the parabola.
Once the model was developed, it underwent a twofold validation process—firstly by using theelicited function to further identify and confirm the maximum area, and secondly by usinggraphing technology to generate the graph and calculate the maximum value. After the validation
process was finalized, the students were asked to reflect on their hypotheses. Lab reflections andsurvey completions concluded the activity.
Results and Discussion
The analysis of students’ hypotheses revealed that the majority (N=24, 86%) claimed that themagnitudes of the areas enclosed by a string of a fixed perimeter would remain unchanged.
Following are examples of students’ hypotheses:•

The area will not change because the total length of the string is the same.

•

The area will not change because it is enclosed by a length that cannot change.

•

The area will not change because the length of the string as a whole never changes.

Representative responses of students who claimed that the area would change were as follows:•

The area will change because if you have a perimeter of 30 cm, then, for example, 5
cm x 10 cm = 50 cm2, 8 cm x 7 cm = 56 cm2, etc.

•

The area will change because it is dependent on its dimensions, which will change.

After the lab was completed and the model confirmed, the above-mentioned majority group of
students agreed that the area enclosed was changing and that it reached a maximum value when
the formulated polygon was a square of about 17 cm by 17 cm, and the remaining students (N=4,14%) confirmed their hypotheses.
Students were asked to answer the following reflection question after the experiment: “Describe
your learning experience while working on the lab. What did you learn?” In their responses, 54%of the students (N=15) referred to their falsely stated hypotheses and commented on revision ofthis preconception that was prompted by the lab outcomes. Specific verbal responses wereclustered and quantified into three categories, as illustrated in Table 1. Sample student responses
are provided following the table.

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Table 1. Summary of Students’ Verbal Reflections
Category

Frequency of Response /Percent

Commented on revising their hypothesis

15 (54%)

Commented on properties of heights and formulation of the maximum area

11 (39%)Made general comments

2 (7%)The majority of the students reflected on their incorrect hypotheses, indicating that the lab helped
them discover the correct process. Responses included the following:•

The lab taught me that even though something is of a fixed length, it can be made into
different shapes that will result in different areas.

•

On the lab my original hypothesis was wrong so I learned that even with the same sizeof the string there are certain shapes and dimensions that will have a larger area.

•

While working on the lab, I learned that a fixed string can have various areas
depending on its length and width.

•

I learned that even though the perimeter between two rectangles is the same, the areas
do not have to be the same. They can be different.

•

I proved my hypothesis about the string correct. I learned that the area of the stringchanges when the base length is changed.

•

I learned that if you have a fixed perimeter, then it does not mean you have a fixedarea. I never truly thought about it.

The majority of the students pointed out the interface of transitioning from one-dimensional
geometrical object (length of the string) to a two-dimensional geometrical object (area). They
noted that the product of the lengths can take different values despite a constant perimeter.
Eleven of the students (39%) described the behavior of the elicited function, focusing on the
maximum area and on the properties of the dimensions that produced the maximum area.
Responses included:•

We learned that changing bases and heights changes the area but in a pattern creatinga parabola. The area increase reaches the max and then the area begins to decrease.

•

I learned that a perfect square will have the greatest amount of area. I also learned
how functions apply to the real world.

•

It was fun and interesting working with classmates. I learned that to get the maximum
area, the length and width of the rectangle should be as close in measurement as
possible.

•

I learned that the area increases as the width and height become more similar.

•

I learned that as the width got bigger, the area would increase then reach its peak, thenslowly decrease, making a parabola.

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General comments describing the learning experience were provided by two students (7%), as
follows:•

I learned how to apply the equations in real-world situations. This helped meunderstand some of the purposes for these equations and rules.

Since mathematical modeling is organized by sequentially organized phases, the question ofinterest was what phase of the MM process had the most significant impact on changing students’
perception on the optimization process. The following multiple choice survey questions sought toprovide insight on this inquiry: “What phase of the lab convinced you the most that the areas
enclosed by a fixed length perimeter were not constant? Circle the phase that applies to you:sketching the area vs. length graph, measuring the lengths of rectangles and calculating the areas,or finding the function equation and its maximum value and generating a table of values.” Theorder of the written phases did not follow the actual order of the activity during the lab in order toreduce the chance of bias. The question was addressed to all students, even to those whose
hypotheses were correct. Their responses are summarized in Table 2.
Table 2. Summary of Modeling Phases That Affected Students’ Change of Thinking
Phase Description

Frequency of Response/Percent

Sketching area vs. length graph

3 (11%)

Measuring the lengths of the rectangles and calculating the areas

8 (28%)

Finding the function equations and the maximum area

2 (7%)

Generating a table of values

15 (54%)

The table shows that most of the students concluded the pattern and revised how the area changed
during systematizing their data in a table of values, followed by the phase of sketching area
versus length graph. It is to note, that all of the processes are part of scientific modeling (Schwarz& White, 2005). This finding further supports the process of blending scientific modeling withmathematical modeling advocated by Sokolowski (2015).
Conclusions
The results support the research hypothesis that immersing students in MM activities can help
revise their preconceptions and help them understand the underlying principle of mathematics.
The study also revealed that the tangible experiences of taking measurements and generating atable of values were the phases that altered the students’ thinking the most. Thus, the systematicway of gathering data and subsequent analysis led the students to discover an embedded principleof optimization. One can conclude that inducing elements of measurement that are typical of thescientific inquiry process (Hestenes, 2013) benefits learners in mathematics classes as well. Thisfinding further supports integrating scientific modeling with mathematical modeling in one
coherent structure advocated by Sokolowski (2015).
This study also supports the conclusion reached by other scholars (e.g., see Brijlall & Ndlovu,
2013; Ledesma, 2011) that the source of difficulties with optimization problems might notnecessarily be rooted in the mathematization of the processes but in the difficulty ofunderstanding the underlying mathematical principle. The ultimate extension of this study would
be to investigate whether the lab experience helped students with formulating the solution
processes to solve textbook problems on optimization. More specifically, another study might

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explore how to use reached conclusions to induce more rigorous mathematical apparatus to solve
typical textbook problems on optimization, or might try to determine if removing the students’
misconception is sufficient to have them succeed with problem solving. It seems that both of thevenues are worthy undertaking.

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References
Blum, W., Galbraith, P. L., Henn, H. W., & Niss, M. (2007). Modeling and applications inmathematics education: The 14th ICMI Study. New York, NY: Springer.
Blum, W., & Leiss, D. (2007). How do students and teachers deal with modeling problems. In C.
Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling (pp. 222-231).
Chichester, England: Horwood.
Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications,and links to other subjects—State, trends and issues in mathematics instruction. Educational
studies in mathematics, 22(1), 37-68.
Bonotto, C. (2007). How to replace word problems with activities of realistic mathematicalmodelling. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modeling(pp. 185-192). Chichester, England: Horwood.
Boyd, S., & Vandenberghe, L. (2009). Convex optimization. New York, NY: Cambridge
University Press.
Brijlall, D., & Ndlovu, Z. (2013). High school learners’ mental construction during solving
optimisation problems in calculus: A South African case study. South African Journal of
Education, 33(2), 1-18.
Christou, C., & Papageorgiou, E. (2007). A framework of mathematics inductive reasoning.
Learning and Instruction, 17(1), 55-66.
Crouch, R., & Haines*, C. (2004). Mathematical modelling: Transitions between the real world
and the mathematical model. International Journal of Mathematical Education in Scienceand Technology, 35(2), 197-206.
Demana, F., Waits, B. K., Foley, G., & Kennedy, D. (2008). Precalculus: Graphical, Numerical,
Algebraic. New York, NY: Addison Wesley.
English, L., & Sriraman, B. (2010). Problem solving for the 21st century. In B. Sriraman & L.
English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 263-290).
Berlin, Germany: Springer Science.
Ferguson, R. L. (2007). Constructivism and social constructivism. Theoretical frameworks forresearch in chemistry/science education (pp. 28-49). Upper Saddle River, NJ: Pearson
Education, Inc.
Geiger, V. (2011). Factors affecting teachers’ adoption of innovative practices with technology
and mathematical modeling. In G. Kaiser, W. Blum, R. Borromeo, & F. G. Stillman (Eds.),
Trends in teaching and learning of mathematical modeling (pp. 305-314). New York, NY:
Springer.
Haines, C., & Crouch, R. (2007). Mathematical modelling and applications: Ability andcompetence frameworks. In G. Kaiser, W. Blum, R. Borromeo, & F. G. Stillman (Eds.),
Trends in teaching and learning of mathematical modeling (pp. 305-314). New York, NY:
Springer.

154

The IAFOR Journal of Education

Volume III - Issue II - Summer 2015

Hammer, D. (1996). Misconceptions or P-Primes: How may alternative perspectives of cognitive
structure influence instructional perceptions and intentions? The Journal of The Learning
Science, 5, 97-127.
Hestenes, D. (2013). Remodeling science education. European Journal of Science andMathematics Education, 1(1), 13-22.
Hsieh, H. F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis.
Qualitative Health Research, 15(9), 1277-1288.
Ledesma, E. F. R. (2011). Representation registers in the solution of calculus problems. Creative
Education, 2(3), 270.
Lesh, R., & Harel, G. (2003). Problem solving, modeling, and local conceptual development:
Revealing activities for students and teachers. Mathematical Thinking and Learning: An
International Journal, 5(2/3), 157-190.
Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students
and teachers. Mathematical Thinking and Learning, 5(2-3), 109-129.
Lesh, R., & Kelly, E. (2000). Multi-tiered teaching experiments. In E. Kelly & R. Lesh (Eds.),
Handbook of research design in mathematics and science education (pp. 591-646). Mahwah,
NJ: Lawrence Erlbaum.
Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling. Second Handbook of
Research on Mathematics Teaching and Learning, 2, 763-804.
Lowther, M. (1999). Optimization: A project in mathematics and communication. Mathematics
Teacher, 9(92), 764-767.
Kaiser, G., & Maaß, K. (2007). Modelling in lower secondary mathematics classroom—problems
and opportunities. In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematicalmodeling (pp. 99-108). Chichester, England: Horwood.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards forschool mathematics. Reston, VA: Author.
Polya, G. (1957). How to solve it. Princeton, NJ: Lawrence Erlbaum.
Poon, K. K., & Wong, H. C. (2011). Problem solving through an optimization problem ingeometry. Teaching Mathematics and Its Applications, 30, 53-61.
Porter, A., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core Standards: The new US
intended curriculum. Educational Researcher, 40(3), 103-116.
Prince, M. J., & Felder, R. M. (2006). Inductive teaching and learning methods: Definitions,comparisons, and research bases. Journal of Engineering Education, 95(2), 123-138.
Rardin, R. L. (1997). Optimization in operations research. Upper Saddle River, NJ: Prentice Hall.
Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Experimental and quasi-experimental
designs for generalized causal inference. Belmont, CA: Wadsworth Cengage Learning.

155

The IAFOR Journal of Education

Volume III - Issue II - Summer 2015

Schuster, A. (2004). About traveling salesmen and telephone networks—Combinatorial
optimization problems at high school. ZDM, 36(2), 77-81.
Schwarz, C. V., & White, B. Y. (2005). Metamodeling knowledge: Developing students'
understanding of scientific modeling. Cognition and instruction, 23(2), 165-205.
Sneider, C. I., & Ohadi, M. M. (1998). Unraveling students' misconceptions about the earth's
shape and gravity. Science Education, 82(2), 265-284.
Sokolowski, A. (2015). Mathematical Modelling - Synthesis of Research. Saarbrücken, Germany:
Scholar’s Press.
Stewart, J. (2007). Single variable calculus: Concepts and contexts. Belmont, CA: Brooks/Cole.
Sullivan, M., & Sullivan III, M. (2009). College algebra enhanced with graphing utilities (5thed.). Upper Saddle River, NJ: Pearson Prentice Hall.
Troxell, D. S. (2002). Optimization software pitfalls: Raising awareness in the classroom.
INFORMS Transactions on Education, 2(2), 40-46.
Yoon, C., Dreyfus, T., & Thomas, M. O. (2010). How high is the tramping track? Mathematisingand applying in a calculus model-eliciting activity. Mathematics Education Research
Journal, 22(2), 141-157.
Von Glasersfeld, E. (1995). A constructivist approach to teaching. In L. P. Steffe & J. Gale (Eds.),
Constructivism in Education (3–15). Hillsdale, NJ: Erlbaum.

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