Computer Modeling and Programming in Algebra

Arnulfo Perez

, Kathy Malone

, Siva Meenakshi Renganathan

and Kimberly Groshong

Department of Teaching and Learning, The Ohio State University, 1945 N. High, Columbus, U.S.A.

Department of Computer Science and Engineering, The Ohio State University, Columbus, U.S.A.

Keywords: Mathematical Modeling, Computer Programming, Python, Project-based Learning, Computational

Thinking.

Abstract: This paper introduces a novel approach to providing high school students with access to computer science

experiences as part of an Algebra unit on linear functions. The approach is being developed and tested as

part of a funded National Science Foundation study. The unit piloted in the study integrates computational

thinking and computer modeling into a project-based Algebra unit on linear functions. Literature on

computational thinking, access to computer science in secondary settings, modeling approaches, project-

based learning, and design-based research is described to provide a rationale for the study design. The

ultimate goal of the study is to develop a paradigm for integrating computer science experiences into

algebra as a way to increase engagement in STEM and computing among students from all backgrounds.

1 INTRODUCTION

Whereas algebra has long been discussed as a

“gatekeeper” to college-preparatory mathematics

tracks (Spielhagen, 2006), we argue for

repositioning algebra as a gateway both to college

and to STEM and computing careers. This paper

describes a novel approach for incorporating

computer modeling and programming into a project-

based exploration of algebra using engineering

applications. The approach is currently being

evaluated through a two-year National Science

Foundation funded pilot study that proposes to

increase student understanding of functions and

integrate 21

-century skills into classroom

experiences through the strategic infusion of

computational thinking (CT).

Although computational thinking has been

defined in various ways (Grover and Pea, 2013), in

this pilot study, teachers and students develop an

understanding of computational thinking as a way of

creatively approaching tasks using fundamental

concepts from computer science (Barr et al., 2011).

This study leverages the power of computational

thinking for 21

-century learning by piloting a

manageable yet compelling integration of modeling

and computer programming into a project-based

exploration of linear functions using engineering

applications.

2 CONCEPTUAL FRAMEWORK

The approach taken in the study’s unit reflects the

benefits of context-based experiences with

mathematics (Bickmore-Brand, 1993) and follows a

Guided Inquiry and Modeling Instructional

Framework (EIMA) with the following progression:

Engage, Investigate, Model, and Apply (Schwarz

and Gwekwerere, 2006). Although based on earlier

research on learning progressions and teaching

cycles (Bybee, 1997), EIMA goes beyond discovery

to position the creation, revision, and application of

models as the focus of inquiry. Further, EIMA was

explicitly developed to pave the way for student

engagement with “computer models and simulations

that are central in modern science and engineering”

(Schwarz and Gwekwerere, 2006: 160).

EIMA effectively sets the stage for students’

encounters with computer science. The project is

developing an Interactive Computer Modeling

(ICM) platform based on SAGE (Software for

Algebra and Geometry Experimentation) and the

Python computer language. SAGE, an open-source

program by Stein (2008), can be used to generate

models using data gathered from their observations.

SAGE is a powerful tool for approaching

mathematical tasks from a computational

perspective. Its web-based platform (Notebook)

allows users to enter equations and data using a

Perez, A., Malone, K., Renganathan, S. and Groshong, K.

Computer Modeling and Programming in Algebra.

In Proceedings of the 8th International Conference on Computer Supported Education (CSEDU 2016) - Volume 2, pages 281-286

ISBN: 978-989-758-179-3

281

command-line interface, and a graphical interface

allows users to visualize and interact with data

(Gray, 2008). The project incorporates into SAGE a

set of user-friendly interfaces that will allow

students to easily manipulate variable amounts.

Consequently, when making predictions, they can

quickly observe the link between algebraic and the

graphical representation of functions. A robust body

of research suggests that creating and manipulating

dynamic models may enhance both student

understanding of mathematical concepts (in this

case, linear functions) and their ability to use

modeling strategically in a mathematics context

(Borba and Villareal, 2006; Zbiek and Conner,

2006).

Historically, efforts to improve mathematical

problem solving have been limited by an

overemphasis on heuristic strategies at the expense

of the metacognitive skills that are needed to

manage the application of these strategies (Lester,

1983; Schoenfeld, 1983). By contrast, EIMA

provides an ideal context for engaging

computational practices (such as effective

abstraction and iterative approaches to problem-

solving) as well as computational perspectives that

encompass the attitudes and dispositions of

programmers, including confidence and persistence

in the face of complex problems, tolerance for

ambiguity, resourcefulness in the face of open-ended

problems, and a capacity for cooperation with others

in the pursuit of a common goal (Barr et al., 2011).

The need for growth in this area is demonstrated by

U.S. students’ relatively weak performance on

international assessments such as PISA where they

are asked to model real-world situations in multi-

step problems (Organization for Economic Co-

operation and Development, 2012).

3 PROJECT GOAL

The research project seeks to construct a learning

environment that effectively integrates computer

modeling and programming into a project-based

algebra unit on linear functions. To accomplish this

goal, we are developing a project-based algebra unit

that uses computer modeling and programming to

explore engineering applications involving linear

functions. Next, we are designing a 10-day summer

STEM+C Institute to support the unit’s

implementation by math educators from secondary

schools. Researchers and graduate research

assistants will document teachers’ engagement in the

summer institutes, gather data on the implementation

of the unit, and assess teacher and student outcomes

using pre- and post-tests, interviews, and other data

sources. Following the pilot implementation of the

unit, participating teachers will engage in a 5-day

STEM+C Institute II to explore data from the study

and examine student work. Researchers will assess

the effect of the modeling and programming unit on

teachers’ and students’ understanding of functions,

problem-solving practices, persistence, and

computational thinking.

4 THE UNIT DESIGN

The proposed unit opens with an engagement

activity that allows the students to discover the

needed components of a circuit by attempting to use

a battery and wires to light a bulb. This pre-activity

focuses the students on the observation of a

phenomenon in the world before they consider a

mathematical representation (Sullivan, 1997). Once

students determine how a circuit must be physically

connected, they can be introduced to electrical

meters that will allow them to generate a table of

findings focusing on voltage and amperage. The

students will be prompted to enter their data into the

project’s newly ICM platform (see figure 1).

Figure 1: Data Entry Table.

The platform will produce a graph of the situation

based on the data collected (see figure 2). The

students will be asked to use these findings to model

their observations by developing a function that will

allow them to be predictive of what is happening in

the circuit. When prompted to model their

observations by developing linear equations (Ohm’s

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law), some students will produce equations that keep

voltage constant while varying 1/resistance (slope).

Others will produce equations where resistance is

the constant and voltage varies (slope). Students

then enter the equations into ICM platform to

generate interactive graphical representations that

allow them to modify slope or resistance using

sliders to further develop their understanding of the

relationship between observed phenomena, algebraic

and graphical representations of these phenomena

(see Figure 2).

Figure 2: Graph representing the data.

The final stage of the opening experience is for

students to discuss and make observations about how

the ICM platform produces these graphs by

examining the “backside” code in Python that makes

them possible. The unit will scaffold the students in

production of coding sequences starting with

iterative loops that will allow the students to solve

real-world engineering application problems that

would be tedious to solve by hand. This will give the

students the opportunity to experience coding in an

unintimidating environment using a platform that is

used for science and engineering applications.

The next phase of the unit involves students in

further explorations using the ICM platform and

scaffolded programming in Python using

application-oriented exploratory STEM exercises

and tasks modified from Python programming: an

introduction to computer science (Zelle, 2010).

These tasks are framed within the context of

engineering applications further exposing students to

STEM careers though engineering based scenarios

that drive the students towards solving the problem

based scenario using mathematical modeling and

computer programming. Materials for these

exercises build context for exploration of

programming environments by highlighting actual

uses for Python in the real world. As the students

explore the engineering application activities, they

will need to make predictions based upon their

mathematical models thus showcasing and

developing elements of computational thinking.

Ultimately, the unit task shows how decision making

in math can be used in real world applications to

make educated decisions.

5 METHODS

The project employs design-based research methods

that deliberately intertwine the design of innovative

learning environments (in this case, the

programming-infused algebra classroom) and the

development of a theory of learning to generate

relevant implications for practitioners and other

research designers (DBRC, 2003). In this

exploratory study, we follow the approach of

progressive refinement (sometimes called iterative

design) to revise both the learning environment and

the theory of learning through cycles of design,

implementation, analysis, and revision (Cobb, 2001).

5.1 Instruments

As shown in Table 1 and Table 2, we will collect

data from multiple sources, including videos of

teachers during STEM+C Institutes, classroom

videos, semi-structured interviews, pre- and post-

tests with open-ended questions designed to provide

insights into learners’ thinking, and Likert-scale

surveys to assess perceptions of programming and

STEM fields (Bannan, 2007). Triangulating findings

among various data sources and conducting

preliminary analysis after each implementation cycle

will enhance the reliability and validity of the

study’s findings (Cobb and Gravemeijer, 2008).

When possible, we are using already constructed and

validated instruments. However, in several

instances the research team is constructing and

validating instruments for specific purposes. In the

case of the unit for students, the team will be

designing minimally worked problems to use during

semi-structured interviews. The minimally worked

Computer Modeling and Programming in Algebra

283

problems are incomplete representations of real

world engineering problems that also have

incomplete Python programming representations.

The students will talk aloud as they work though

problems, allowing researchers to observe their

computational thinking.

Table 1: Data Sources by Student Participants.

Data Sources Focus of coding/analysis

Pre/Post content tests Understanding of function

and programming – yet to

be developed

CT STEM Attitudinal

Survey (

Weintrop et al.,

2014)

5-pont Likert scale survey

focused on attitudes

towards CT and STEM;

confidence in these

subjects; and interest in

fields related to

computation

Semi-Structured

interviews

Focus on perceptions of

math and computer

programming and

knowledge of linear

functions, CT and

modeling through the use

of Minimally Worked

Problems (MWPs)

Videos of student work

groups

Discourse around

functions, CT, and

modeling

Teachers will take a series of assessments that

not only focus on their understanding of the content

being covered but also their self-efficacy towards

mathematical modeling, computer programing and

functions. There are few instruments that are already

validated that suit the needs of the study. The

project team is currently working on validating a 46-

question Likert scale survey focusing on teacher

understanding of what constitutes a mathematical

modeling task and on teacher perceptions of

obstacles and supports that either discourage or

encourage teachers’ use of mathematical modeling

tasks within the classroom. The survey was based on

work done by Schmidt (2011). The survey includes

organizational, student-related, and teacher-related

obstacles, which influence teachers’ decision-

making about incorporating mathematical modeling

tasks in their lessons (Blum, 1996). The validation

study of this instrument should be complete by early

April 2016.

6 ANALYSIS

Preliminary analysis of these data sources will occur

at each stage in the design process followed by a

final retrospective analysis after all phases of the

project are completed (Molina et al., 2007).

Triangulating findings among various data sources

and conducting preliminary analysis after each

implementation cycle will enhance the reliability

and validity of the study’s findings (Cobb and

Gravemeijer, 2008).

Table 2: Data Sources by Teacher Participants.

Data Sources Focus of coding/analysis

Pre/Post content tests Understanding of function

and programming – yet to

be developed

Semi-Structured

interviews

Focus on pedagogical

content knowledge in

algebra, ideas about the

nature of mathematics and

computer science, beliefs

regarding problem-solving

and real world applications

as part of their curriculum

Algebra Teachers’ Self-

Efficacy Instrument

(ATSE) (Gupta et al.,

2015)

Likert survey that focuses

on PCK, modeling and

functions

Modeling Survey – in the

process of being

validated.

Likert survey that focuses

on teachers’ motivation to

use mathematical

modeling tasks

7 CONCLUSIONS

This work-in-progress report frames how students

explore computational thinking as a way of

creatively approaching mathematics using

fundamental concepts from computer science. The

presentation will evaluate concrete strategies for

incorporating computer modeling and programming

into algebra and examine real-world applications

that can be used when exploring linear functions

with learners.

The potential value of computational thinking

and computer modeling for learning in secondary

mathematics will be explored. The draft lessons

explored during the presentation will provide an up-

close look at an approach that has the potential to

increase equity in education and broaden access to

STEM careers.

The validation study of the Modeling Survey will

be explored as well as its potential as a tool for

studying teacher beliefs.

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ACKNOWLEDGEMENTS

We thank the reviewers of the draft of this document

for their helpful feedback. This material is based in

part upon work supported by the National Science

Foundation under Grant Numbers 1543139. Any

opinions, findings, and conclusions or

recommendations expressed in this material are

those of the authors and do not necessarily reflect

the views of the National Science Foundation.

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