Investigating the Affordances and Constraints of SimReal for

Mathematical Learning: A Case Study in Teacher Education

Said Hadjerrouit

Department of Mathematical Sciences, University of Agder, Kristiansand, Norway

Keywords: Affordances, Mathematical Learning, SimReal, Teacher Education, Visualization.

Abstract: Visualizations tools are one of the most innovative technologies that emerged the last few years in

educational settings. They provide new potentialities for mathematical learning by means of dynamic

animations and representations, interactive simulations, and live streaming of lessons. Moreover,

visualization tools have the potential to foster a visual, dynamic, distributed, and embodied mathematics

rather than individual achievements and static representations. This paper uses the visualization tool

SimReal in teacher education to explore the affordances of the tool for learning mathematics. It proposes a

framework that captures the affordances of SimReal at a technological, pedagogical, and socio-cultural

level. The aim of the article is to investigate the extent to which SimReal afford students’ mathematical

learning in teacher education. Based on the results, recommendations for future work are proposed.

1 INTRODUCTION

SimReal is a new visualization tool that is used to

teach a wide range of mathematical topics both at

the university and school level. SimReal uses a

graphic calculator, video lessons, video live

streaming, video simulations, and interactive

simulations to teach mathematics (SimReal, 2018).

In contrast to other digital tools such as GeoGebra,

SimReal has more than 5000 applications and tasks

in various areas of mathematics (Brekke and

Hogstad, 2010). The tool can be divided in small

subsets, while keeping the same structure and basic

user interface. A subset of SimReal called Sim2Bil

provides 4 windows for visualizations: simulation,

graph, formula, and menu window (Hogstad et al.,

2016),

There is a huge interest in visualization tools, but

there are few research studies that address learning

issues in authentic educational settings (Presmeg,

2014). Some research studies on SimReal focus on

teaching mathematics at the undergraduate

mathematical level (Brekke and Hogstad, 2010;

Gautestad, 2015; Hogstad, 2012). The aim of these

studies is to report on students’ attitudes using

SimReal as a supplement tool to ordinary teaching,

and its usefulness in difficult and abstract

mathematical areas. Hogstad et al. (2016) studied a

subset of SimReal called Sim2bil that aims at

exploring how engineering students use

visualizations in their mathematical communication.

Furthermore, Hadjerrouit and Gautestad (2018) used

the theory of instrumental orchestration to analyze

teachers’ use of SimReal in an engineering class.

Other studies were carried out in teacher education.

Firstly, Hadjerrouit (2015) evaluated the suitability

of the tool in teacher education using usability

criteria. Secondly, Hadjerrouit (2017) addressed the

affordances of SimReal and students’ perceptions of

the tool in teacher education. The present study is a

continuation of these two studies. Based on the

results achieved so far, the purpose of this work is to

explore the affordances of SimReal and their impact

on students’ mathematical learning in teacher

education.

The article is structured as follows. First, the

theoretical framework is described, followed by the

methodology. Then, the results are presented.

Finally, a summary of the results, future work and

recommendations conclude the article.

2 THEORETICAL FRAMEWORK

2.1 The Concept of Affordances

Among a wide range of theoretical approaches that

can be applied to explore the impact of digital tools

Hadjerrouit, S.

Investigating the Affordances and Constraints of SimReal for Mathematical Learning: A Case Study in Teacher Education.

DOI: 10.5220/0007588100270037

In Proceedings of the 11th International Conference on Computer Supported Education (CSEDU 2019), pages 27-37

ISBN: 978-989-758-367-4

on mathematics learning (Geiger et al., 2012), the

theory of affordances provides the most appropriate

framework to address the impact of SimReal on

learning mathematics in teacher education.

The term “affordance”, originally proposed by

the perceptual psychologist James J. Gibson in his

book “The Ecological Approach to Visual

Perception” (Gibson, 1977), refers to the

relationship between an object’s physical properties

and the characteristics of a user that enables

particular interactions between user and object.

More specifically, Gibson used the term

“affordance” to describe the action possibilities

offered to an animal by the environment with

reference to the animal’s action capabilities

(Osiurak, et al., 2017)

The concept of affordances was introduced to the

Human-Computer-Interaction community by Donald

Norman in his book “The Psychology of Everyday

Things” (Norman, 1988). Accordingly, the term

“affordance” refers to the perceived and actual

properties of the thing, primarily those fundamental

properties that determine just how the thing could

possibly be used.

A number of research studies used Norman’s

ideas to implement the concept of affordances in

various educational settings. For example, Turner

and Turner (2002) specified a three-layer

articulation of affordances: Perceived affordances,

ergonomic affordances, and cultural affordances.

Likewise, Kirchner et al. (2004) described a three-

layer definition of affordance: Technological

affordances that cover usability issues, educational

affordances to facilitate teaching and learning, and,

social affordances to foster social interactions. In

mathematics education, Chiappini (2012) applied the

notions of perceived, ergonomic, and cultural

affordances to Alnuset, a digital tool for high school

algebra.

De Landa (2002) emphasized that affordances

are not intrinsic properties of the object. Rather

affordances become actualized in specific context,

e.g. the socio-cultural context of the classroom. In

other words, affordances emerge from the

relationship between the object and the particular

environment with which it is interacting. From this

perspective, the specific context of the mathematics

classroom may include several artifacts or tools that

interact with the user. Accordingly, the artifacts

being used in a mathematics classroom have

affordances and constraints. These may include

paper-pencil techniques, the blackboard, Interactive

White Board (IWB), Power Point slides, and diverse

digital tools, such as Smart phones, IPad, GeoGebra,

SimReal, and mathematics itself by means of

symbols, notations, representations, etc. Artifacts

with their affordances and constraints interact with

the user.

2.2 SimReal Affordances

Based on the research literature described above and

the specificities of mathematics education, this study

proposes three categories of affordances and

constraints at six different levels (Figure 1):

a) Technological affordances that describe the

functionalities of the tool

b) Pedagogical affordances at four levels:

 Pedagogical affordances at the student level or

mathematical task level

 Pedagogical affordances at the classroom level

or student-teacher interaction level

 Pedagogical affordances at the subject level,

that is the area of mathematics being taught

 Pedagogical affordances at the assessment

level

c) Socio-cultural affordances that cover

curricular, cultural, and ethical issues

Figure 1: Three categories of SimReal Affordances at six

different levels.

CSEDU 2019 - 11th International Conference on Computer Supported Education

There are two types of technological affordances:

Ergonomic and functional affordances. From the

ergonomic point of view, these are ease-of-use, ease-

of-navigation, accessibility at any time and place,

accuracy and quick completion of mathematical

activities. From the functional point of view,

SimReal helps to perform calculations, draw graphs

and functions, solve equations, construct diagrams,

and measure figures and shapes. Technological

affordances are a pre-requisite for any digital tool

and provide support for pedagogical affordances.

There are several pedagogical affordances that

can be provided at the student level, e.g., using the

tool to freely build and transform mathematical

expressions that support conceptual understanding of

mathematics, such as collecting real data and create

a mathematical model, using a slider to vary a

parameter or drag a corner of a triangle in geometry

software, moving between symbolic, numerical, and

graphical representations, simulating mathematical

concepts, or exploring regularity and change (Pierce

and Stacey, 2010). At this level, the motivational

factor is important, especially when using

visualizations to engage students in mathematical

problem solving. Furthermore, feedback in various

forms to students’ actions may foster mathematical

thinking. Programming mathematical tasks may also

be a way of using SimReal for learning and

understanding.

Likewise, several pedagogical affordances can be

provided at the classroom level (Pierce and Stacey,

2010). Firstly, affordances that result in changes of

interpersonal dimensions, such as change of

teachers’ and students’ role, less teacher-directed

and more student-oriented instruction. Secondly,

affordances that create more learner autonomy,

resulting in students taking greater control over their

own learning, and using SimReal as a “new”

authority in assessing learning. Other affordances at

this level are change of social dynamics and more

focus on collaborative learning and group work, as

well as change of the didactical contract (Brousseau,

1997). Variation in teaching and differentiation are

other affordances offered by digital tools at this level

(Hadjerrouit and Bronner, 2014). This may result in

flipping the classroom, which is another way of

using SimReal at this level.

Furthermore, three types of pedagogical

affordances can be provided at the mathematical

subject level (Pierce and Stacey, 2010). The first one

is fostering mathematical fidelity, looking at

congruence between machine mathematics and ideal

or paper-pencil mathematics, and promoting

faithfulness of machine mathematics (Zbiek et al.

2007). The second affordance is amplifying and

reorganizing the mathematical subject. The former is

accepting the goals to achieve those goals better.

Reorganizing the mathematical subject means

changing the goals by replacing some things, adding

and reordering others. For example, in calculus there

might be less focus on skills and more on

mathematical concepts (Pierce and Stacey, 2010). In

geometry, there might be emphasis on more abstract

geometry, and away from facts, more argumentation

and conjecturing (Pierce and Stacey, 2010).

Likewise, it may be useful to support tasks that

encourage metacognition, e.g., starting with real-

world applications, and using SimReal to generate

results.

Affordances at the assessment level consist of

summative and formative assessment. Summative

assessment is important for testing, scoring and

grading, and it can be provided in form of statistics

that the tool generates. Formative assessment is

equally important for the learning process. Feedback

is an essential condition for formative assessment. It

can take many forms, e.g., immediate feedback to

students’ actions, a combination of conceptual,

procedural, and corrective information to the

students, or asking question types, etc.

Finally, several socio-cultural affordances can

emerge at this level. Firstly, an important affordance

is that SimReal should provide opportunities to

concretize the mathematics subject curriculum in

teacher education. Secondly, SimReal should be tied

to teaching mathematics in schools, and support the

learning of mathematics at the primary, secondary,

and upper secondary level. In other words, SimReal

should take the requirement of adapted education

into account. Finally, other socio-cultural

affordances can also emerge at this level, in

particular those related to ethical, gender, and multi-

cultural issues.

3 THE STUDY

3.1 Participants

Fifteen teacher students (N=15) from a technology

and mathematics-based course in teacher education

participated in this work. The students were

categorized on the basis of their knowledge level in

mathematics associated with their study

programmes: Primary teacher education level 1-7,

primary teacher education level 5-10, advanced

teacher education level 8-13, and mathematics

master's programme.

Investigating the Affordances and Constraints of SimReal for Mathematical Learning: A Case Study in Teacher Education

The recommended pre-requisites were basic

knowledge of ICT (information and communications

technology) and experience with standard digital

tools like text processing, spreadsheets, calculators

and Internet. No prior experience with SimReal was

required.

3.2 Activities

A digital learning environment centered around

SimReal was created over two weeks, starting from

25 August to 8 September 2016. An example of

SimReal utilization is given in figure 2.

Figure 2: Example of SimReal utilization in mathematics

education.

The environment included video lectures,

visualizations, and simulations of basic, elementary,

and advanced mathematics, and diverse online

teaching material. Basic mathematics focused on

games, dices, tower of Hanoi, and prison.

Elementary mathematics consisted of multiplication,

algebra, Pythagoras and Square theorems, and

reflection. The topics of advanced mathematics were

measurement, trigonometry, conic section,

parameter, differentiation, and Fourier.

To assess experiences on specific mathematical

topics that are of considerable interest for students,

two specific mathematical tasks were chosen. The

first one was Pythagoras theorem (Pythagoras

theorem, 2018). There are many ways of

representing Pythagoras. The theorem has also been

given numerous proofs. These are very diverse,

including both geometric and algebraic proofs, e.g.,

proofs by dissection and rearrangement, Euclid's

proof, and algebraic proofs. Thus, Pythagoras is

more than just a way of calculating the lengths of a

triangle. An example of representing the theorem is

given in the following figure (Figure 3).

Figure 3: An example of representation of Pythagoras

theorem.

The second task was the Square theorem (Square

theorem, 2018). Like Pythagoras, there are many

ways of using and representing the theorem (Figure

4).

Figure 4: An example of representation of the Square

theorem.

3.3 Methods

This work is a single case study in teacher

education. It aims at exploring the affordances of

SimReal for mathematical learning in teacher

education. The study is exploratory in nature. Both

quantitative and qualitative methods were used to

collect and analyze students’ experiences with

SimReal. The following methods were used:

a) A survey questionnaire with a five-point

Likert scale from 1 to 5, and quantitative

analysis of the results

CSEDU 2019 - 11th International Conference on Computer Supported Education

b) Students’ comments in their own words to

each of the statements of the survey

questionnaire

c) Students’ written answers to open-ended

questions

d) Qualitative analysis of students’ comments on

point b and answers to open-ended questions

to point c

e) Task-based questions on Pythagoras and

Square theorem, and programming issues

The design of the survey was guided by the

theoretical framework and the research goal. To

measure the students’ perceptions of SimReal, a

survey questionnaire with a five-point Likert scale

from 1 to 5 was used, where 1 was coded as the

highest and 5 as the lowest (1 = “Strongly Agree”; 2

= “Agree”; 3 = “Neither Agree nor Disagree”; 4 =

“Disagree”; 5= “Strongly Disagree”). The average

score (MEAN) was calculated, and the responses to

open-ended questions were analyzed qualitatively.

The survey included 72 statements that were

distributed as follows: Technological affordances

(12), pedagogical affordances at the student level

(11), classroom level (19), mathematical subject

level (9), assessment level (10), and finally socio-

cultural level (11). The students were asked to

respond to the survey using the five-point Likert

scale and to comment each of the statements in their

own words. In addition, the students were asked to

provide written answers in their own words to open-

ended questions. The responses to students’

comments to the survey and open-ended questions

were analyzed qualitatively.

Of particular importance are task-based

questions on Pythagoras and Square theorems to

collect data on affordances when students engage

with these mathematical tasks. An additional

question on programming issues was given to the

students to assess the affordances of programming

languages for the learning of mathematics. Asking

task-based questions provides supplementary

information on the affordances of SimReal. This

method also provides more nuanced information

about the students’ experiences with SimReal. The

analysis of the data was guided by the specified

affordances of the theoretical framework, and open-

coding to bring to the fore information that was not

covered by the theoretical framework.

4 RESULTS

4.1 SimReal Affordances

The results achieved by means of the survey

questionnaires and open-ended questions show that

the affordances of SimReal emerge at different

educational levels. The affordances are less evident

for students from the study programme 1-7. They

become more visible at the middle level associated

with the study programme 5-10. Affordances come

to the fore at the advanced level, and for

mathematics education students.

Globally, the vast majority of the students

pointed out that SimReal still lacks an easy-friendly

interface and that it is not easy to use, to start and to

exit. For many students, the tool was accessible

anywhere and anyplace, but the navigation through

the tool is still not straightforward. On the positive

side, SimReal has a ready-made mathematical

content, and that the video lessons, simulations,

animations, and live streaming are of good quality.

This is reflected in many students’ responses.

In terms of pedagogical affordances at the

student level, many students think that SimReal

provide real-world tasks, which engage them in

mathematical problem solving, particularly when

using visualizations to simulate mathematical

concepts. Most students think that visualizations are

useful to gain mathematical knowledge that is

otherwise difficult to acquire, and they liked very

much the combination of live streaming of lessons,

video lectures, simulations, and animations. Most of

the advanced mathematical exercises (trigonometry,

differentiation, and conic section) were not difficult

for them to understand. Likewise, SimReal provided

affordances to explore variation and regularities in

the way mathematics is taught, e.g., vary a

parameter to see the effect of a graph. The students

also think that SimReal is congruent with paper and

pencil techniques. On the negative side, most

students think that SimReal is not helpful to refresh

students’ mathematical knowledge.

In terms of pedagogical affordances at the

classroom level, the majority agreed that they can

use SimReal on their own, and that the use of the

tool is not completely controlled by the teacher, and,

as a result, they do not need much help from the

teacher or textbooks to solve exercises. Likewise,

most students think that the tool can be used as an

alternative or supplement to textbooks and lectures.

The tool also facilitates various activities (problem

solving, video lectures, live streaming), and several

ways of representing mathematical knowledge

Investigating the Affordances and Constraints of SimReal for Mathematical Learning: A Case Study in Teacher Education

(texts, graphs, symbols, animations, visualizations).

In terms of differentiation and individualization,

many students believe that the level of difficulty of

the mathematical tasks is acceptable, but it is

relatively difficult to adjust the tool to the students’

knowledge level. Even though the degree of

autonomy is not very high, it is sufficient to allow

students work at their own pace. On the negative

side, the vast majority of the students think that

SimReal does not support much cooperation or

group work, and it does not have collaborative tools

integrated into it.

In terms of pedagogical affordances at the

mathematical subject level, most students agreed

that SimReal provides a high quality of

mathematical content. Many students also think that

SimReal provides real-world applications and tasks

that foster reflection, metacognition, and high-level

thinking. Likewise, the vast majority found that

SimReal is mathematically sound, and that the tool

can display correctly mathematical formulas,

functions, graphs, numbers, and geometrical figures.

On the negative side, the overwhelming majority

think that the software tool GeoGegra has a better

interface, and it is better to express mathematical

concepts. Finally, the combination of mathematics

and practical applications in physics is evaluated as

useful to gain mathematical understanding.

In terms of affordances at the assessment level,

most students think that SimReal provides several

assessment modes and give directly feedback in

form of dynamic animations. This is a clear

improvement compared to previous experiments.

Likewise, SimReal provides satisfying solutions

step-by-step, but not for all tasks. Still, SimReal

does not provide several types of feedback,

differentiated knowledge on student profiles, several

question types, and statistics. Finally, the degree of

interaction is evaluated as satisfying.

In terms of affordances at the socio-cultural

level, most students think that SimReal is an

appropriate tool to use in teacher education, but it

does not take sufficiently into account the

requirement for adapted education. Furthermore,

most students believed that SimReal is appropriate

to use in secondary schools, and in a lesser degree in

middle and primary schools. On the negative side,

the vast majority will not continue using video

lessons and live streaming to learn mathematics, but

some will still be using video simulations in the

future. Nevertheless, the vast majority of the

students think that the tool enables the teacher to

concretize the mathematical subject curriculum.

Summarizing, it worth noting that affordances do

not emerge in the same degree for all students.

Rather they become actualized in relationship to the

participants’ knowledge level from the 4 categories

of study programmes: Primary teacher education

level 1-7 and level 5-10, master programme, and

advanced teacher education level 8-13.

4.2 Affordances of Pythagoras

Theorem

Students were engaged in 16 different approaches to

exploring Pythagoras theorem. These were divided

in paper-based (1-9) and SimReal-based approaches

(10-16). The students were asked to report on

SimReal affordances and critically reflect on their

impact on learning Pythagoras by responding to 4

specific questions.

a) If you should choose only one of the 16 different

approaches of explaining Pythagoras, which of

them would you prefer?

The students provided a variety of solutions in order

of priority according to the perceived affordances of

the approaches. One suggestion was 2/5/7/12/15,

and 16. Some students chose a sequence of

approaches such as 2/3/5, or 2/3, or 7/14. One

student provided another set of preferences that are

worthwhile to study in details. Firstly, the student

decided to use approach 15 as a brief introduction,

and then Pythagoras 1, both as a simple

presentation of the equation of the theorem and as a

first visual proof of the theorem. Then, he suggested

to use approach 2 as a general formula and 3 as a

more specific or realistic one. The student also

suggested a combination of approach 10 (paper-

based) and 14 (SimReal-based), but without the

written explanation or mathematical formula.

Instead, one can start with a given problem such as

“find the area of the pool or the area of the baseball

field”. After having discussed suggestions to

approaching the solution, the student can then check

the explanation provided by SimReal in terms of

written text or mathematical formula. Finally, the

student would demonstrate approach 4 using a

rigorous proof through the usage of algebraic and

geometrical properties. Summarizing, several

affordances emerged in this situation: realistic task,

pen-paper formulas, SimReal visualizations with

written explanations of the theorem, rigorous

mathematical proofs of the theorem, and a

combination of paper-based and SimReal

visualizations.

CSEDU 2019 - 11th International Conference on Computer Supported Education

b) If you should combine one of the pen/paper

approaches and one of the digital simulations

which of them would you prefer?

The students provided a variety of approaches such

as 3/12 and 9/12. As described above, one student

combined approach 10 (pen/paper-based) with

approach 14 (SimReal-based). After the use of the

pen-and-paper solution and an attempt to calculate

the blue area of figure 14, the students could try to

calculate several areas of the figure. Example 10 and

14 have different approaches and content, but they

complement each other in terms of their affordances.

c) If you should choose the combination of the two

approaches 9 and 12, how would you in detail

explain Pythagoras?

A variety of explanations were provided to explain

Pythagoras using various elements such as the layout

and colors of the figure, the dynamic of the

simulation, and mathematical explanations. A good

combination of 9 and 12 is as follows. The student

starts with 9 (angle A=90°), and notes that the pink

and the blue area of the bottom square is equal to the

other two corresponding square areas (blue and

pink). Furthermore, the sum of the blue and pink

square areas is equal to the bottom square area,

consisting of these two rectangles. Moving on to 12,

it is worth mentioning that their area remains

unchanged. Before using the SimReal-based

simulation, some figures of rectangles on the

blackboard would be useful, revising or presenting

the formula for the area of the parallelogram. The

figures should be of various parallelograms,

including interaction with the student by considering

different bases each time and different heights. The

student concluded that a combination of moving the

scroll bar and considering cases on the blackboard

depending on the position of the parallelogram

and its height. This could be a good reasoning step

to explain why the area remains the same. The

digital simulation could be used to clarify the

question.

d) Do you think that teaching different ways of

Pythagoras by combining pen/paper and

SimReal-based simulations would help in the

understanding of this topic, or do you think it

would be confusing for the students?

The students think that the combination of pen/paper

and SimReal simulations is helpful to understand

Pythagoras depending on time and pedagogical

constraints, and students’ knowledge level as well.

They think that it would be positive to use several

approaches to teach Pythagoras considering various

students’ knowledge levels and learning styles when

solving problems. It is therefore important to present

mathematical tasks using different ways. By

showing a figure describing Pythagoras theorem, the

teacher has a good opportunity to explain the

mathematical formula in his/her own words, before

showing a SimReal simulation of the theorem. This

may motivate the students, and stimulate their

curiosity. Approach 9 or 3 combined with simulation

12 would give a good effect. In most cases, a good

combination of pen/paper and SimReal simulations

is preferable, but there may be some confusing cases

that make the understanding of the topic more

difficult. In those cases, the task should be either

pen/paper solution or SimReal simulation, but not

both approaches, even though the teaching may be

less efficient. As a result, a good way of teaching

Pythagoras is a combination of SimReal affordances

with paper-pencil solutions.

4.3 Affordances of Square Theorem

Students were engaged in 6 different approaches to

exploring the Square theorem. These are divided in

paper-based (1-3) and SimReal-based approaches (4-

6). The students were asked to study them and report

on their affordances by responding to 4 specific

questions.

a) Pen/paper proofs (1, 2, 3) versus SimReal-based

proofs (4,5,6) of the Square theorem

Most students preferred a combination of pen/paper

with SimReal proofs, but those participating in the

task should not just passively read the proofs. They

should rather take advantage of the dynamic

visualizations provided by SimReal. Regarding the

Square theorem, the pen-paper approaches 1-2-3 do

not necessarily promote students' understanding,

because these are based on a more mechanical

calculation method. SimReal simulation 4 is good

approach for visualizing the theorem. However, the

second and third approaches are somewhat tricky to

understand geometrically, but still better than just

formulas. Therefore, approaches 4-6 should be used

to create dynamic images of the Square theorem.

Another student preferred pen/paper proofs (1-3)

and think that these methods are mostly used to

describe algebraic operations and expressions.

However, these approaches are important only if the

teacher takes a more practical approach to the

theorem, and the geometrical SimReal-based

approaches could be used to enhance the

understanding of the theorem.

Investigating the Affordances and Constraints of SimReal for Mathematical Learning: A Case Study in Teacher Education

b) In what way do you think the use of SimReal can

provide a better understanding of the Square

theorem?

The students think that SimReal provides

affordances to improve the understanding of the

Square theorem by visualizing mathematical

concepts. More specifically, one student suggested a

quiz, and another a "fill the blanks" exercise, where

a student could get a direct result or feedback if the

answer is correct or not. Globally, the students think

that the digital visualizations are beneficial for

visually strong students, considering the fact that

upper secondary mathematics becomes more

theoretical the higher up the grade, and, as a result,

there is less focus on conceptual understanding, and

why and how to carry out calculations. Digital

simulations can have therefore a positive effect on

student learning and help them to see how

mathematical formulas work.

c) Give some comments about how you could think

to improve either by pen/paper or SimReal the

understanding of the Square theorem

Students provided many ways of improving the

understanding of the Square theorem. One solution

is giving exercises both with symbols and numbers,

but also allowing the use of expansion like (a + b) ^2

= (a + b) (a + b), until the student becomes familiar

with the theorem. The paper-and-pen exercises 1-2-3

show specific procedures on how a student can

change and calculate the Square theorem tasks, but

the procedures would have been clearer if there was

a headline for each example to show how the

theorem works. SimReal solutions 4-5-6 have digital

simulations with explanations, color coding, and

reference to formulas. These cover the Square

theorem quite well, and there is no need for

improvement. Likewise, SimReal simulations can

make it easier for students to see how the formulas

work, and this is especially true for the 1st and 2nd

approach to the Square theorem.

d) Do you prefer learning the Square theorem in

one way or do you feel a better understanding

learning it in different ways?

As already stated above, most students think that a

combination of different approaches is the most

appropriate way to provide a better understanding of

mathematics, while also being careful not to use

several approaches at the same time as this might be

counterproductive. They also argued that it is

important to see mathematics from different angles.

Using new methods to explain the solution to a

single problem will give new perspectives about the

problem and the corresponding solution, and how

these are interrelated. A good example is the figure-

based and algebraic proofs of the Square theorem.

Showing different point of views of the theorem

(like a geometrical one) and applications of the

theorem could indeed be very efficient.

Summarizing, a comparison of the results in

terms of affordances achieved by means of task-

based questions reveal that these are globally in line

those achieved by the survey questionnaire and

open-ended questions in terms of pedagogical

affordances at the student level. The issues that

correspond very well are the usefulness of

visualizations for understanding the Square and

Pythagoras theorem, the congruence of SimReal-

based visualizations with paper-pencil techniques,

and a combination of different representations and

approaches to the theorems.

4.4 Programming Affordances

Programming has rapidly grown as an innovative

approach to learning mathematics at different levels.

The topic will become compulsory in schools from

the study year 2020. As a result, it is expected to

improve SimReal by including programming tasks

using Python and other programming languages.

Given this consideration, it was worthwhile to ask

the students about the affordances of programming.

a) Would it be of interest for you to program your

own simulations in teaching mathematics?

The study reveals that SimReal can provide more

affordances in terms of programming mathematical

concepts. Basically, most students think that

teachers with experience in programming

mathematical simulations and visualizations will

open a new way of teaching mathematics. For

example, a teacher could focus on subjects and tasks

that are difficult for the students to comprehend.

Another possibility is to program tasks that are not

already covered by SimReal, but that are already

available online. Most importantly for teachers is the

use of different methods to promote understanding

and making new connections. Hence, it may be

worthwhile to take advantage of simulations and

explanations combined with some programming

examples so that the knowledge to be learned is

presented with various methods.

b) Do you think it would be of interest and help

that students program their own simulations?

The participants think that students would be

interested in programming visualizations if they

CSEDU 2019 - 11th International Conference on Computer Supported Education

have acquired sufficient skills in this matter. This

would contribute to enhanced motivation and

increased understanding of mathematics, because the

students will be forced to fully comprehend

mathematics before they could program

visualizations. Likewise, it could be of help for the

students if they could program their simulations by

themselves. However, it is crucial that they focus on

the mathematical part of the task rather than

programming issues alone. Programming their own

simulations could be motivating for those students

who are both interested and knowledgeable in

programming. This presupposes, however, that the

students have understood the mathematics before

getting started with programming. Students having

difficulties in mathematics should rather spend their

time on it. Hence, programming would be helpful if

it contributes to the learning of mathematics.

Likewise, advanced mathematics requires a higher

level of programming knowledge, and it may

therefore be necessary to evaluate whether students

have sufficient understanding of mathematics to be

able to program themselves. Finally, only one

student pointed out that he would not spend time on

programming, even though he sees an advantage in

it. Summarizing, programming mathematical tasks

can contribute to the understanding of mathematics,

but it is demanding in terms of efforts and time for

novice students.

5 DISCUSSION

The purpose of this work is to assess the impact of

SimReal affordances on students’ mathematical

learning in teacher education. The study provided an

important amount of empirical data on what students

perceived as affordances of SimReal and their

impact on learning mathematics at different levels.

Although this study does not aim to capture all

potential affordances, it is possible to make

reasonable interpretations of the results and draw

some recommendations for using SimReal in teacher

education.

In terms of technological affordances, there is a

need for a better and intuitive user interface and

navigation for different types of users. From a

pedagogical affordance point of view, SimReal

affords many students to do mathematics both at the

student and classroom level. It provides variation in

teaching mathematics, and visualizations are

considered as useful to gain mathematical

knowledge that is otherwise difficult to acquire. The

combination of live streaming of lessons, video

lectures, simulations, and animations is highly

valued. Students can work at their own pace, without

much interference from the teacher. SimReal also

facilitates various activities and several ways of

representing knowledge. In terms of affordances,

SimReal needs to be better adjusted to the student

knowledge levels, and it should provide a better

support for group work. Furthermore, there is a need

for feedback and review modes, more differentiation

and individualization, including the possibility of

programming their own videos and visualizations.

At the mathematics subject level, the tool has a high

quality of mathematical content. Moreover, the

mathematical notations are correct and sound.

The study shows that the affordances of SimReal

make mathematics easier to understand, because

these provide a concrete way of making

mathematical concepts more dynamic. In addition,

SimReal provides a huge variation of visualization

examples for the teacher to use in classroom, e.g.,

SimReal can support the understanding of Square

and Pythagoras theorems by visualizing the dynamic

behavior of the theorems. Nevertheless, a

combination of pen and paper, digital visualizations,

and chalk-blackboard could be more efficient to

teach mathematics than just SimReal alone.

Moreover, the students think that videos can speed

up the interest and motivation for doing

mathematics. Videos could be used as a supplement

to mathematics on the blackboard and paper-pencil,

and as an alternative way of sharing knowledge and

explaining mathematics. Videos are especially

important because these are one of the main sources

of information for young students. Most students

also think that programming mathematical tasks can

provide more affordances for the learning of

mathematics.

Moreover, many students think that the tool is

appropriate to use in teacher education and upper

secondary school level, and it enables to concretize

the curriculum. At the assessment level, works need

to be done to improve the feedback function.

Summarizing, the theoretical framework has

proven to be useful to address the affordances of

SimReal and their impact on mathematical learning

in teacher education. Nevertheless, the research

literature reveals that the concept of affordances can

be reconceptualised and extended by considering

ontological issues (Burlamaqui and Dong, 2015). As

already stated, affordances are not properties that

exist objectively. Rather affordances emerge in the

socio-cultural context of the classroom, where a

number of other artifacts and their affordances

interact with SimReal, e.g., paper-pencil, black

Investigating the Affordances and Constraints of SimReal for Mathematical Learning: A Case Study in Teacher Education

board, textbooks, Smart Phones, Power Point slides,

mathematical tasks and their representations, etc. A

reconceptualization of the concept of affordances

needs to take in consideration new and more

powerful theories such as Actor-Network Theory

(ANT), which does not consider technology simply

as a tool, but rather as an actor with agency that

serves to reorganize human thinking (Latour, 2005).

In this regard, Wright and Parchoma (2011)

criticized the value of affordances, and proposed

Actor-Network-Theory as an alternative framework

that may contribute to greater critical consideration

of the use of the concept “affordances”. The theory

of assemblage may also contribute to the

understanding of affordances and its relationship to

mathematical learning, which is understood as “an

indeterminate act of assembling various kinds of

agencies rather than a trajectory that ends in the

acquiring of fixed objects of knowledge” (De Freitas

and Sinclair, 2014, p. 52). Moreover, Withagen, et

al. (2017) argued that affordances are not mere

possibilities for action, but can also have the

potential to solicit actions. Hence, the concept of

agency can contribute to a better understanding of

affordances.

6 CONCLUSIONS

The purpose of this article is to assess the

affordances of SimReal for mathematical learning in

teacher education by asking students to respond to a

survey questionnaire and open-ended questions. In

addition, the students had the opportunity to

comment the items of the survey in their own words.

Task-based questions were also used to provide

more nuanced information about the students’

engagement with the Pythagoras and Square

theorem, and their views on programming

affordances as well. The data collected by means of

these methods provided an important amount of

information that gave a better sense and

interpretation of the results achieved in this study.

Even though, the results are promising, it is still

difficult to generalize the findings because of the

small sample size (N=15). In fact, amongst this

sample size there is already variance with regard to

the different primary teacher education levels.

However, it would have better for the research study

to have less variance with such a small sample size

and ensure that one or two of those groups have a

larger representation.

In future studies, students’ recommendations will

be considered to improve the teaching of

mathematics with SimReal. In terms of

technological affordances, there is a need for a better

and intuitive user interface and navigation for

different type of users. In terms of pedagogical

affordances, there is a need for better feedback and

review modes, more differentiation and

individualization, and the possibility of

programming their own videos and visualizations.

The concept of affordances will be refined by

considering other theories, such as Action-Network

Theory, agency, and assemblage theory. It is also

planned to look at students’ learning styles, for

example between visual and verbal students. Finally,

the data collection and analysis methods will be

improved to ensure more validity and reliability.

AKNOWLEGMENTS

This research was supported by MatRIC – The

Centre for Research, Innovation and Coordination

of Mathematics Teaching, project number 150401.

I would like to express my special appreciation and

thanks to Per Henrik Hogstad, and his involvement

and great support in the design and teaching of

mathematics using SimReal.

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