IMPROVING MATHEMATICAL KNOWLEDGE THROUGH MODELING IN ELEMENTARY SCHOOLS
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IMPROVING MATHEMATICAL KNOWLEDGE THROUGH
MODELING IN ELEMENTARY SCHOOLS
Nicholas Mousoulides, Marios Pittalis & Constantinos Christou
Department of Education, University of Cyprus
The study presents the results of a 6th grade class (11 year olds) working on a
modeling activity. Traditional mathematics textbooks mostly provide single and
straightforward solution problems at which students only apply a formula to reach a
solution. On the contrary, students’ work on modeling activities focus on analysing a
problematic situation, setting and testing conjectures and model construction. In
modeling activities students work in small groups and they are actively engaged in
fruitful discussions with their peers and teacher. The results of the study showed that:
(a) students with no prior experience in modeling activities applied effectively their
informal mathematical knowledge to solve an authentic problem, and (b) social
interactions in groups enhanced the discovery of mathematical knowledge.
INTRODUCTION
The economy and work force demand for school graduates to be able to work
collaboratively in demanding projects, to effectively use new technological tools and
to possess more flexible, creative, and future-oriented mathematical skills.
Professional organizations (AAAS, 1998; NCTM, 2000) address the need for a change
in the school mathematics and propose reforms in mathematics education. Most of
these reforms emphasize a critical need for students to study mathematics in real world
contexts and to construct models in exploring and understanding problem situations
(Greer, 1997). In mathematical modeling students develop important mathematical
processes, such as describing, explaining, predicting, representing, and organising
data (NCTM, 2000). Mathematical modeling that explores interesting and non trivial
situations for students, can become an effective medium for students to be actively
engaged in acquiring mathematical knowledge (Blum & Niss, 1991) in experientially
real contexts (Gravemeijer, Cobb, Bowers, & Whitenack, 2000).
THEORETICAL FRAMEWORK
In the present study, a model is defined as a construct consisting of elements,
relations and operations that can be used to describe, explain or predict the behaviour
of some other familiar systems (Doerr & English, 2003). Models focus on the
structural characteristics of the systems that they are referring to, and are expressed
using a variety of representational media, including written symbols, diagrams or
graphs (Lesh, & Lehrer, 2003; Lesh & Doerr, 2003). Models constructed by students
for a problematic situation may also inform teachers and researchers about students’
2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the
International Group for the Psychology of Mathematics Education, Vol. 4, pp. 201-208. Prague: PME. 4 - 201Mousoulides, Pittalis & Christou mental models and conceptions about mathematical constructs (Greer, 1997; Lesh & Doerr, 2003). Modeling activities move beyond traditional problem solving experiences, by addressing adequately the knowledge, the processes, and the social developments that students require in dealing with increasingly sophisticated systems (Lesh, & Lehrer, 2003; English & Watters, 2005). Modeling activities for young learners, designed for group work, are inherently social experiences (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003) and provide the basis of effective communication and teamwork. Students need to effectively communicate with their peers to develop and refine models that can be applied in a range of contexts (English, 2003). Modeling activities involve mathematizing – by quantifying, dimensioning, coordinating, categorizing, algebraizing, and systematizing relevant objects, relationships, actions, patterns, and regularities (Lesh et al., 2003). Modeling activities aim to help students externalize their understanding of situations by developing models which can serve in conceptualizing mathematical ideas and processes (Lesh et al., 2003). These models focus on significant mathematical structures, patterns, and regularities and the development of such products requires multiple cycles of interpretations, descriptions, conjectures, explanations and justifications (Schorr & Amit, 2005; Lesh et al., 2003). Current research in mathematics education is demonstrating that young learners can be benefited from working with authentic modeling problems (English & Watters, 2005). In particular, it has been argued that modeling activities can help students to build on their existing understandings, to engage in thought-provoking, multifaceted problems within authentic contexts that allow for multiple interpretations and different approaches (Schorr & Amit, 2005; Doerr & English, 2003). Specifically, in a project with modeling activities, 10 year old students were able to work successfully with mathematical problems when presented as meaningful, real-world case studies. Students were also able to discover relationships and patterns in data and applied their learning in working with similar problems (English, 2003). Finally, Doerr and English (2003) showed that students working with modeling activities developed their abilities in planning and revising their models by challenging one another’s assumptions and claims, and asking for clarification and justification for problem solutions. THE PRESENT STUDY The Purpose of the Study The aim of the present study is to explore the ways in which students work in modeling activities to develop the concept of average. To this end, it is expected from students to work with authentic mathematical problems, using their prior mathematical knowledge to investigate, make sense and understand specific problems which lead to a conceptual understanding of average. 4 - 202 PME30 — 2006
Mousoulides, Pittalis & Christou Participants and Modeling Activities Twenty students (12 females and eight males) from an intact 6th grade class in an urban school in Cyprus participated in two modeling activities. None of these students had prior experience in solving problems in a mathematical modeling context. In this study two modeling activities were presented to students, namely the “Drug Industries Golden Award” and the “Summer Camp Job”. Both activities derived from a list of problems found in Lesh and Doerr (2003). The purpose of the first activity was to provide opportunities for students to organize and explore data, to use statistical reasoning and to develop appropriate models for solving the problem. The “Summer Camp Job” activity provided opportunities for students to apply the models and new learnings they had developed in the content of the “Drug Industries Golden Award” activity. The second activity also provided a setting for students to focus and work with the notions of ranking, selecting, aggregating ranked quantities and weighting ranks. The application of the “Drug Industries Golden Award” activity (see Figure 1) followed three stages: (a) the warm-up stage in which students read an article about Ian Fleming with the purpose to familiarize themselves with the context of the modeling activity, (b) the readiness stage which involved the discussion of the article, and (c) the modeling stage in which students were engaged in constructing a model to answer the basic questions of the problem. Procedure Students spent two 40 minutes sessions in completing each of the two modeling activities. Each activity started with a whole class discussion on the warm-up task and readiness questions. Then students worked in groups of three or four to provide solutions for the activity. After completing their work, each group presented its models to the rest of the class for questioning, comparing with others’ models and constructive feedback. Students again worked back in their groups to revise and refine their models. Finally, a whole class discussion focused on the key mathematical ideas and processes that were developed during the modeling activity. Data Sources and Analysis The data for this study were collected through (a) videotapes of students’ responses during whole class discussions, (b) audiotapes of students’ work in their groups, (c) students’ worksheets and final reports detailing the processes used in developing models, and (d) researchers’ field notes. Videotapes and audiotapes were analyzed using interpretive techniques (Miles & Huberman, 1994), for evidence of students’ mathematical developments towards the statistical concepts appeared in the modeling activity. Due to space limitations, we mainly present the results of one group of students, working on the “Drug Industries Golden Award” activity. PME30 — 2006 4 - 203
Mousoulides, Pittalis & Christou
Use the reaction times in the table below to rank the four drugs by their effectiveness. Write
a letter, explaining and documenting your results, to the chairman of the Drug Industries
Association.
Kanatol Saracetamol Ralpol Kefapol
20 10 12 10
18 19 14 12
19 13 15 17
22 11 15 17
15 11 7 17
14 12 9 19
23 10 9 22
12 9 8 22
11 8 8 21
10 8 15 10
7 14 19 7
9 13 10 7
10 12 10 7
17 17 23 19
13 11 24 18
12 11 23 14
14 13 10 12
14 20 8 10
8 25 17 10
9 13 19 10
Figure 1: The Drug Industries Golden Award Activity
RESULTS
Students’ purpose in the modeling activity was to provide a reasonable answer of
how to select the best drug. The results are presented on the basis of the modeling
cycles developed in students’ work after the analysis of the transcripts.
Cycle 1: Focusing on Subsets of Information
The first purpose of students’ work was to rank the four pain-relief drugs, according
to their effectiveness. In this first attempt, students’ efforts focused on subsets of
information, as they only concentrated on the smallest reaction time for each drug.
The transcript below shows how students perceived the solution of the problem.
These initial solutions created the need for students to search for more justifiable
solutions.
Helen: I believe that Saracetamol is the most effective since it has the least
reaction time. Other drugs need more time.
Alex: Yes, but what about Kefapol? In three cases it needs the least time to act.
4 - 204 PME30 — 2006Mousoulides, Pittalis & Christou
Alice: Check over here! Kefapol’s times are 17, 17 and 17 while Saracetamol’s
are 11, 11 and 12 respectively.
Alex: You are right, but Saracetamol has also 20 and 25 minutes reaction time.
The students engaged in debates over how to generate a comprehensive model which
could handle both small and big reaction times. This first discussion led them to
approach the problem in a more systematic way. Students used their informal
knowledge to make a number of conjectures and justify their claims:
Helen: Let’s circle in each line the drug with the smallest reaction time.
Researcher: How will you rank the drugs?
Alice: We will count the number of circles for each drug.
The above process did not lead to an appropriate answer to the problem; however,
this approach forced students to argue about its usefulness since there were drugs
with the same reaction time. Alex suggested circling both drugs. Based on this idea,
students ranked the drugs in the following order: Saracetamol, Ralpol, Kefapol and
Kanatol. A similar but more “refined” approach was used by a second group of
students. Their solution was to circle both the smallest and biggest reaction times and
then subtracting the two numbers. That group ended with a different drug ranking:
Saracetamol, Kefapol, Ralpol and Kanatol.
Since none of the groups used a systematic approach to tackle the problem, there
were long debates during students’ presentations. Different approaches and
contradicting results led students to face the need to mathematize their procedures.
Thus, students began to use two main mathematical operations to handle the data for
each drug, namely, (a) totalling the amounts of reaction time for each drug, and (b)
finding the average for each drug and comparing the averages.
Cycle 2: Using Mathematical Operations and Processes
The core characteristic of the solutions appeared in the second cycle was the adoption
of more sophisticated mathematical processes. Alex’s group next approach was based
on the assumption that any new development should consider all reaction times and
not only the best or/and worst reaction times.
Alex: We should add all reaction times for each drug.
Alice: Why should we do that?
Helen: Alex is right. By adding all numbers ... find the drug with the least sum.
This one will be the most effective pain relief drug.
This new mathematical approach, based on finding the sum of reaction times for each
drug changed the ranking of the drugs to: Saracetamol, Ralpol, Kanatol and Kefapol.
Students were really surprised since the last ranking was quite different from
previous ones.
The big numbers that students encountered while working with “sums-model” started
a new round of discussion. Helen suggested that they could divide the sums by the
number of the cases to find the average, because “it’s difficult to work out the sum of
PME30 — 2006 4 - 205Mousoulides, Pittalis & Christou
reaction times, especially if we have more cases”. Alex realized that Helen’s model
would produce the same ranking, since “we divide the sums by the same number, so
nothing will change in the ranking”. Alice was a little bit confused and remained
unconvinced; however, she asked for more clarifications as shown in the transcript
below:
Helen: First we add all times and divide by ... (she was interrupted by Alice)
Alice: Four. We have four drugs.
Alex: No, this is not correct. We do not find the average like this. We need to
divide by twenty, the number of cases.
Alice: You mean that we add all reaction times and divide by 20?
Alex: No, there is no reason to add all reaction times. We only add the reaction
times for each drug because we need to calculate the average for each
drug.
Helen: We do not always divide by 20 but with the number of cases. We need
one average for each drug to find the differences between the four drugs.
We could also calculate one average for all drugs, but only to compare
these four drugs with other ones.
Most groups used a systematic approach to find a solution to the problem, either by
finding the sums of reaction times or by working with the average. It should be noted
that not all students approached the problem using a mathematical approach. For
example, James, even after the discussion and the groups’ presentations of
appropriate solutions, believed that the “best drug” was the one having the smallest
reaction time in one case.
A representative and interesting snapshot of students’ work appeared in the letter they
sent to the Chairman of the Drug Association. They made evident that they spent a lot
of time searching for the best solution, and they were confident that their final
solution was correct. Quite impressive was a group’s comment on the transferability
of their model:
“Comparing the averages is also appropriate for similar competitions you will have in
the future. Our solution can be used to find the most effective drug, even in cases with
more than four drugs. You also can use average to compare other products, like day
skin creams. Be careful though, since in other cases, you might need to find the highest
and not the lowest average”.
The final presentation of students’ suggested models and solutions resulted in one
more round of arguments and a discussion on the meaning of average. One student
pointed that a drug’s average is the time needed for the drug to react in most of the
cases. Alex disagreed with her, mentioning that: “the average actually shows the
reaction time of a drug if that time is the same for all cases”.
DISCUSSION
An important conclusion of the present study is that the participating students were
able to work successfully with mathematical modeling problems when presented as
meaningful, real-world case studies. The framework, within the problem was
4 - 206 PME30 — 2006Mousoulides, Pittalis & Christou presented, helped students to realize and to get familiar with the problematic situation and thus enhanced their statistical understandings (English, 2003). At the same time, the activity did not narrow students’ freedom and autonomy to approach and analyze the problem taking into account their prior and informal knowledge. On the contrary, students’ work was impressive; they analyzed the problem using different viewing angles, set and test hypotheses, evaluate, modify and refine their models and solutions, just like professional mathematicians! On the problem presented here, the students progressed from focusing on subsets of information which resulted in not suitable models to applying the appropriate mathematical concepts and processes that helped them finding an effective mathematical model. This new model was reusable, shareable and could serve to construct more sophisticated models for solving even more demanding problems (Doerr & English, 2003). It was also clear that many students identified the structural elements of the problem in developing their final model, in such a way that they could easily transfer and modify their models in the second activity. At the same time it is impressive that students’ models took place in the absence of any formal instruction, and involved the children in describing, analyzing, explaining, justifying, checking, and communicating their ideas with peers and teachers (Lesh et al., 2003). Quite important were also students’ efforts in documenting their solutions in a letter to the Chairman of the Drug Association. Few problems in traditional textbooks generate learning of this nature and quality (English, 2003) and students appeared to be successful and productive. An important aspect of students’ work is the communication and social interaction that took place naturally within the groups. These interactions engaged students in analyzing, planning and revising courses of action, challenging one another’s assumptions and claims, and ensuring the group worked as a team. Given the importance of communication and sharing of ideas in mathematics education (NCTM, 2000), there is evidence that modeling activities can successfully serve in this direction. Finally, a possible direction for future research in the area could be the investigation of students’ ability to transfer effectively their constructed models in solving similar structured problems and to modify ready made models (by peers) in solving problems. Quite interesting would also be the examination of the role of new technological tools in solving non routine authentic problems like the one presented here and to investigate whether the use of technological tools can provide alternative approaches and developed models. References American Association for the Advancement of Science (1998). Blueprints for reform: Science, mathematics, and technology education. New York: Oxford. Blum W., & Niss M. (1991). Applied mathematical problem solving, modeling, applications, and links to other subjects – State, trends and issues, Educational Studies in Mathematics, 22, 37-68. PME30 — 2006 4 - 207
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