Educ. Humanismo, Vol. 17 - No. 29 - pp. 286-307 - Julio-Diciembre, 2015 - Universidad Simón Bolívar - Barranquilla, Colombia - ISSN: 0124-2121

http://publicaciones.unisimonbolivar.edu.co/rdigital/ojs/index.php/educacion

Open access web technology for mathematics

learning in higher education*

Mari Carmen González-Videgaray

Rubén Romero-Ruiz

María del Rosario Hernández-Coló

Universidad Nacional Autónoma de México

DOI: http://dx.doi.org/10.17081/eduhum.17.29.1258

Recibido: 28 de enero de 2015 Aceptado: 21 de abril de 2015

Tecnologías web de acceso abierto para el

aprendizaje de matemáticas en la educación superior

Abstract

Problems with mathematics learning, “math anxiety” or “statistics anxiety” among

university students can be avoided by using teaching strategies and technological

tools. Besides personal suffering, low achievement in mathematics reduces terminal

efciency and decreases enrollment in careers related to science, technology and

mathematics. This paper has two main goals: 1) to offer an organized inventory of

open access web resources for math learning in higher education, and 2) to explore

to what extent these resources are currently known and used by students and teach-

ers. The rst goal was accomplished by running a search in Google and then classi-

fying resources. For the second, we conducted a survey among a sample of students

(n=487) and teachers (n=60) from mathematics and engineering within the largest

public university in Mexico. We categorized 15 high-quality web resources. Most of

them are interactive simulations and computer algebra systems.

Resumen

Los problemas en el aprendizaje de las matemáticas, como “ansiedad matemática”

y “ansiedad estadística” pueden evitarse si se usan estrategias de enseñanza y he-

rramientas tecnológicas. Además de un sufrimiento personal, el bajo rendimiento

en matemáticas reduce la eciencia terminal y decrementa la matrícula en carreras

relacionadas con ciencia, tecnología y matemáticas. Este artículo tiene dos obje-

tivos: 1) ofrecer un inventario organizado de recursos web de acceso abierto para

aprender matemáticas en la universidad, y 2) explorar en qué medida estos recursos

se usan actualmente entre alumnos y profesores. El primer objetivo se logró con un

perl de búsqueda en Google y una clasicación. Para el segundo, se condujo una

encuesta en una muestra de estudiantes (n=487) y maestros (n=60) de matemáticas

e ingeniería de la universidad más grande de México. Categorizamos 15 recursos

web de alta calidad. La mayoría son simulaciones interactivas y sistemas de álgebra

computacional.

Key words:

Math anxiety, Web resource,

Free software, Video, Statistics.

Palabras clave:

Ansiedad matemática,

Recurso web, Software libre,

Video, Estadística.

Referencia de este artículo (APA): González-Videgaray, M. C., Romero-Ruiz, R. & Hernández-Coló, M. R. (2015). Open access web

technology for mathematics learning in higher education. En Revista Educación y Humanismo, 17(29), 286-307. http://dx.doi.

org/10.17081/eduhum.17.29.1258

* Artículo vinculado al proyecto “Vientos de cambio: Estrategias y buenas prácticas en el uso de ambientes virtuales para la educación

superior” que hace parte del Programa de Apoyo a Proyectos para la Innovación y el Mejoramiento de la Enseñanza (PAPIME) de la

Dirección de Asuntos del Personal Académico de la Universidad Nacional Autónoma de México.

1. Actuaria por la UNAM, Maestra en Educación y Doctora en Ingeniería Industrial por la Universidad Anáhuac. Realizando postdocto-

rado en Independencia Intelectual en la Universidad Anáhuac. mcgv@unam.mx

2. Ingeniero Eléctrico y Electrónico e Ingeniero en Computación por la UNAM, especialista en Gestión Informática y Maestro en Tecno-

logías de Información por la Universidad Anáhuac.

3. Licenciada en Enseñanza de Inglés por la UNAM, Maestra en Educación por la UNID, Doctorante en Investigación y Docencia por el

CEPES.

287

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1. Introduction

Problems with mathematics learning have

negative consequences among university stu-

dents. These problems tend to undermine per-

ception of achievement, lower self-esteem, di-

minish welfare within school, and affect career

selection. Worse than that, there are even clini-

cal states identied as “math anxiety” (Ashcraft,

2002; Geist, 2010; Meece, Wigeld and Eccles,

1990) and “statistics anxiety” (Chew and Dillon,

2014; Onwuegbuzie and Wilson, 2003; Zeidner,

1991), with their own form of diagnoses, etiol-

ogy and treatment. These conditions have been

reported in Mexico (Garcia-Santillan, Moreno-

Garcia, Carlos-Castro, Zamudio-Abdala and

Garduño-Trejo, 2012; Tyteca, Parrilla and Mar-

tinez, 2013), and extend to undergraduates in

mathematics and related areas (Cardoso Espi-

nosa, 2012; Escalera-Chavez, Garcia-Santillan

and Venegas-Martinez, 2013; Espinosa, Lopez

and Mercado, 2012), where also students who

have chosen this area of knowledge struggle

with learning.

Personal suffering is not the only conse-

quence of math learning problems. Besides all

the above, scientic and technological knowl-

edge construction depends greatly on math

thinking (Andrade-Arechiga, Lopez and Pulido,

2013). Most scientic research has its basis on

statistical theory. Rigorous mathematical rea-

soning is fundamental to propose new ideas in

several scientic elds, and even qualitative re-

search makes use of mathematical notions. Prob-

lems in math learning hinder student selection of

science and technological careers, keeping them

away from scientic research. So, the fact of

being left behind in math learning has negative

consequences for any country’s development.

Mathematics learning is a global problem

and Mexico is not the exception. At the end of

2013, PISA 2012 results were published, show-

ing data that generates concern to our country.

On one hand, Mexico decreased from 419 points

in math in 2009, to 413 points in 2012. On the

other hand, it was found that “55 % of Mexican

students did not get the basic competence level

in mathematics” (OCDE, 2013). These are clear

indicators that math learning is still an important

challenge. Within international statistics, Mexi-

co is the last ranked country in the Organization

for Economic Co-operation and Development

(OECD), as shown in Figure 1.

Figure 1. Math average in PISA 2012, by country

Source: (OECD, 2013)

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As a consequence of the above, only 0.4598 %

of the undergraduate students in 2011-2012 were

matriculated in mathematics and statistics (AN-

UIES, 2013) in Mexico. Students commonly

avoid this eld of study because they anticipate

strong difculties with it. Henceforth, there are

few professionals in this area and many univer-

sity teachers belong to other related disciplines,

but mathematics. This also has consequences in

the teaching-learning process.

As this is a worldwide problem with different

magnitudes and perspectives, there are multiple

initiatives directed to attend this issue. Some of

them involve the use of technology and, above

all, new pedagogical ideas such as “active learn-

ing” (Freeman, et al., 2014), “ipping the class-

room” (Alvarez, 2012; Rath & Chung, 2013;

Tucker, 2012) and “minimally invasive educa-

tion” (Mitra & Rana, 2001) can be found. The

general approach of these views consists of hav-

ing teachers’ guidance and fostering students’

practice in class, while leaving lectures and oral

explanations to technological tools, such as vid-

eos or readings.

These pedagogical innovations require a sub-

stantial change of the teacher’s role. She or he

must be a kind of coach, capable of designing

challenging and meaningful activities. A teach-

er as well must help with errors, guide through

misconceptions, motivate, praise success, create

learning environments where students’ partici-

pation and discussions take place comfortably,

and develop a pleasant learning experience for

everyone.

As it was mentioned before, lectures, oral or

written explanations, and repetitive exercises

can be left to technology. In fact, some tools that

might help do this work are already in the web.

All the boring, stressing and intimidating aspects

of learning can be lessen if students can get into

them whenever and wherever they decide to

do so. Learners can listen to clear explanations

and good examples in comfortable and private

spaces. Videos and animations can be stopped,

rewound or fast-forwarded as needed. It is also

possible to practice with a wide variety of exer-

cises and try them out as many times as each stu-

dent wishes or needs. As they are working alone,

they will not feel ridiculous or silly in front of

their teacher and classmates. They will not re-

ceive offensive commentaries or suffer from bul-

lying.

For instance, Lloyd and Robertson (2012)

have shown that screencasts or videos on how to

use statistics software, are effective in teaching

statistics to psychology students. Web tutorials,

examples, lectures (Kay & Kletskin, 2012) and

animations (Taylor, Pountney, & Baskett, 2008)

tend to be perceived as attractive by learners.

Garcia et al. (2014) propose the incorporation of

algebra computer systems or CAS, not only to

the learning processes, but to assessment as well.

They suggest that this incorporation has a posi-

tive effect on developing valuable competences

among students. Other authors (Shaffer, 2006)

advise the use of immersive games or digital en-

vironments to teach math. These novelties can

engage students and make learning more attrac-

tive.

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In fact, Garcia et al. (2014) point out that

there has been a qualitative leap in math teach-

ing in the last thirty years, within the Euro-

pean Community, thanks to new technologies,

of course, but also to appropriate legislative

frames. This is not Mexico’s situation because

most teachers continue using old teaching prac-

tices, even though the existence of an interesting

and diverse offer of web resources designed to

support math learning.

High cost is one of the barriers for the use

of technology in class. However, due to recent

worldwide initiatives as the Budapest Open Ac-

cess Initiative (Open Society Institute, 2002),

and the Free Software Foundation (Europe,

2014), among some others, there is a trend to

offer high-quality open access web resources.

They are usually free and their developers either

receive grants or charge users for additional ser-

vices such as consulting, customizing or expand-

ing benets.

In the case of mathematics learning, there are

several open access web resources aimed to sup-

port learning. These resources are free, accessi-

ble via internet and easy to use, but their quality

has to be assessed in order to use them appro-

priately. Once we have a categorized inventory

of this kind of technology, it will be possible to

study the extent they are currently being used.

Thus, we propose two main hypotheses. The rst

one is that there is an interesting set of open ac-

cess web resources with the potential to promote

math learning. The second one is that university

teachers and students are not yet conscious that

the use of these tools can change math teaching

and foster math learning.

So, this paper has two main goals. 1) To build

an inventory of open access web resources that

can help undergraduate mathematics learning

in different ways. These resources are catego-

rized by their kind of learning support, topics

covered and language (English and/or Spanish).

2) To conduct a survey of the use of these re-

sources among students and teachers from the

most important public university of Mexico,

Universidad Nacional Autonoma de Mexico or

UNAM. We only selected students and teach-

ers of mathematics and engineering, because we

suppose that they should be best informed about

these resources than people from other areas,

and could, eventually, become intensive users of

them. In other words, our results could probably

be an over estimation of the characteristics of the

population of all undergraduates.

Thus, our aim is to produce a synthetic –al-

though probably not exhaustive– view of open

access web resources intended to foster math

learning. Even though these tools change and

get better every day, the inventory could be used

as a starting point to assess the possible benets

of this technology. Also we offer a panorama of

how these resources are used in one of the larg-

est and most important public universities in

Mexico.

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2. Material and Methods

2.1. Open Access Web Resources Review

In order to locate open access web resources

aimed to promote math learning, we ran the fol-

lowing search in Google: (online OR web OR

digital OR internet OR computer) AND (learn-

ing OR teaching OR instruction OR knowledge)

AND (math OR algebra OR calculus OR “dif-

ferential equations” OR statistics OR geometry).

The search was run in Spanish and in some vari-

ants of both languages (English and Spanish) as

well.

To be selected in our inventory, web resourc-

es had to:

• Have an educational purpose.

• Offer undergraduate mathematical high-qual-

ity content.

• Be free and open access.

• Be functional within the web browser.

• Contain or show few and far between com-

mercial ads.

• Show image and text balanced content.

• Be displayed in English and/or Spanish.

Once collected, web sites were categorized by:

• The kind of learning support: a) computer al-

gebra system or CAS; b) interactive simula-

tions; c) graphic visualizations; d) videos.

The topic covered: a) Algebra; b) Linear

algebra; c) Numeric analysis; d) Calculus; e)

Differential equations; f) Statistics; g) Analytic

geometry; h) Applied mathematics; i) Discrete

mathematics; j) Mathematics for computer sci-

ence; k) Modeling and simulation; l) Combina-

torial optimization; m) Probability.

• Their language: a) English; b) Spanish; c)

Both English and Spanish.

2.2. Survey Sample Frame

Survey sample sizes (Table 1) were deter-

mined based on population sizes during the

autumn-winter cycle 2013-II at Facultad de Es-

tudios Superiores Acatlán (Acatlan Faculty of

Higher Studies), the largest of UNAM’s multi-

disciplinary campi, located in the north zone of

Mexico City. Students and teachers who were at

the Faculty at that time and who were willing to

respond answered the questionnaires. So, sam-

ples were not completely randomized.

We sampled the three careers that belong to

the area of Mathematics and Engineering: Actu-

ary, Civil Engineering and Applied Mathematics

and Computing (MAC). Sample sizes were re-

lated to our practical possibilities. They provide

a margin of about 6 % of error and a condence

level of 90 %, which are acceptable for explor-

atory studies such as this.

Table 1. Study sample frame

Major

Student

population

size

Student

sample

size

Teacher

population

size

Teacher

sample

size

Actuary 1061 160 82 20

Civil

Engineering

1084 160 104 20

MAC 1386 167 99 20

Total 3531 487 285 60

2.3. Questionnaires

Two questionnaires were designed, one for

students and one for teachers. The following

data were collected:

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• Students: Career; semester; shift (morning or

afternoon); sex; level of knowledge and use

of web resources within a Likert scale with

four options: 0 = I do not know it, 1 = I know

it but I have never used it, 2 = I have used

it from time to time, 3 = I use it frequently;

other web resources that you want to add.

• Teachers: Career studied by teacher;

subject(s) taught; level of knowledge and use

of web resources within a Likert scale with

four options: 0 = I do not know it, 1 = I know

it but I have never used it, 2 = I have used

it from time to time, 3 = I use it frequently;

other web resources that you want to add.

2.4. Survey Administration

To administer the survey to students, we

showed up in the classrooms and asked each

teacher for permission. All of the teachers

granted permission. Students attending class an-

swered the printed version of the questionnaire.

As for teachers, they were asked to willingly

respond the questionnaire. Even though partici-

pants remained anonymous, it is possible that

teachers did not feel comfortable answering the

questionnaire because the administrator could

easily identify them.

2.5. Survey Descriptive Statistics

A database was built in Excel with the ques-

tionnaires’ answers. Likert scale average was

calculated for each web resource, within each

sample group. These averages were presented in

bar charts. A list of other suggested web resourc-

es was added to the end of the results.

3. Results

3.1. Open Access Web Resources Review

Fifteen open access math web resources with

the characteristics explained in Section 2.1 t the

search prole. They are shown in Table 2 in al-

phabetical order, with their correspondent URL.

Table 2. Open access math web resources in alphabetical order

Web Resource URL

1. Descartes http://recursostic.educacion.es/descartes/web/

2. GapMinder http://www.gapminder.org

3. GeoGebra http://www.geogebra.org/cms/

4. Khan Academy http://www.khanacademy.org/

5. Math 2 Me http://math2me.com/es/

6. Maxima http://maxima.sourceforge.net/

7. MIT Mathlets http://math.mit.edu/mathlets/

8. PhET Interactive Simulations http://phet.colorado.edu/

9. Rice Virtual Lab in Statistics http://onlinestatbook.com/rvls.html

10. The Number Empire http://www.numberempire.com/

11. OCW Videos http://ocw.mit.edu/index.htm

12. Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/

13. Web Interface for Statistics Education http://wise.cgu.edu/

14. WIRIS http://www.wiris.com/

15. Wolfram Alpha http://www.wolframalpha.com/

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Table 3. Web resources by their kind of learning support

Web Resource CAS Interactive Simulation Graphic Visualization Video

1. Descartes X

2. GapMinder X

3. GeoGebra X

4. Khan Academy X

5. Math 2 me X

6. Maxima X

7. MIT Mathlets X

8. PhET Interactive Simulations X

9. Rice Virtual Lab in Statistics X

10. The Number Empire X

11. OCW Videos X

12. Virtual Laboratories in Probability and Statistics X

13. Web Interface for Statistics Education X

14. WIRIS X

15. Wolfram Alpha X

Total 5 6 1 3

Table 4. Web resources by the topic(s) they cover

a) Algebra; b) Linear algebra; c) Numeric analysis; d) Calculus; e) Differential equations; f) Statistics; g) Analytic geometry; h) Applied

mathematics; i) Discrete mathematics; j) Mathematics for computer science; k) Modeling and Simulation; l) Combinatorial optimization;

m) Probability

Web Resource a b c d e f g h i j k l m Total

1. Descartes X X X X X 5

2. GapMinder X 1

3. GeoGebra X X 2

4. Khan Academy X X X X X X X 7

5. Math 2 me X X X X X 5

6. Maxima X X X X X X X X 8

7. MIT Mathlets X 1

8. PhET Interactive Simulations X X 2

9. Rice Virtual Lab in Statistics X 1

10. The Number Empire X X X X 4

11. OCW Videos X X X X X X X X X X X X 12

12. Virtual Laboratories in Probability and Statistics X X 2

13. Web Interface for Statistics Education X 1

14. WIRIS X X X X X X X X 8

15. Wolfram Alpha X X X X X X X X 8

Total 9 6 4 8 6 12 8 2 1 1 1 1 8

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Web resources were classied by their kind of

learning support and organized in Table 3. Most

of them are interactive simulations and CAS.

Table 4 shows web resources organized by

the topics they cover. OCW Videos cover more

topics than other resources, followed by three

CAS: Maxima, WIRIS and Wolfram Alpha. In

turn, Statistics is the most covered topic, fol-

lowed by Algebra.

Finally, we present the categorization by lan-

guage, where we can see that there are two web

resources in Spanish, ve in English and Span-

ish, and eight in English.

Table 5. Web resources by language

Web

Resource

Spanish English &

Spanish

English

1. Descartes X

2. GapMinder X

3. GeoGebra X

4. Khan Academy X

5. Math 2 me X

6. Maxima X

7. MIT Mathlets X

8. PhET Interactive

Simulations

9. Rice Virtual Lab in

Statistics

10. The Number Empire X

11. OCW Videos X

12. Virtual Laboratories in

Probability and Statistics

13. Web Interface for

Statistics Education

14. WIRIS X

15. Wolfram Alpha X

Total 2 5 8

3.2. Student’s Survey

As Figure 2 shows, Actuary students are

relatively familiarized with Wolfram Alpha and

somewhat less with GeoGebra, Maxima and

Math 2 me. The other web resources are practi-

cally unknown to them; they have indexes near

zero.

Regarding Civil Engineering students, they

practically do not recognize nor use any of the

web resources listed (Figure 3).

Within MAC career students’ results, Wol-

fram Alpha and GeoGebra stand out (Figure 4).

The rest of the resources present low values.

3.3. Teacher’s Survey

According to Figure 5, Actuary teachers are

somewhat familiarized with GeoGebra and Wol-

fram Alpha. The third more known value corre-

sponds to Descartes.

Civil Engineering teachers report, like the

students’ case, that they are unaware of almost

all math web resources (Figure 6).

Finally, MAC teachers know about Wolfram

Alpha and GeoGebra (Figure 7). They practi-

cally do not know or use other resources.

3.4. Other Resources Suggested by Respon-

dents

With respect to the open question where they

could add some other web resources which can

contribute to mathematics learning, participants

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Figure 2. Average knowledge of math web resources of Actuary students

Figure 3. Average knowledge of math web resources of Civil Engineering students

Figure 4. Average knowledge of math web resources of MAC students

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Figure 5. Average knowledge of math web resources of Actuary teachers

Figure 7. Average knowledge of math web resources of MAC teachers

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Figure 6. Average knowledge of math web resources of Civil Engineering teachers

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mentioned: Octave, a high-level interpreted lan-

guage, primarily intended for numerical compu-

tations; Cinderella, an interactive geometry soft-

ware; self-teaching tools developed by UNAM;

Scilab, an open source software for numerical

computation; Winplot, a general-purpose plot-

ting utility, which can draw (and animate) curves

and surfaces presented in a variety of formats;

Mathcad, proprietary software for engineering

calculations; Scientic Workplace, a commer-

cial mathematical word processor; Statgraphics

Online, a commercial statistical package that

runs within a web browser; EViews, solutions

for forecasting, econometrics and simulation;

GrafEq, a program for producing graphs of im-

plicit equations and inequalities; NumPy, a pack-

age for scientic computing with Python; SciPy,

a Python-based ecosystem of open-source soft-

ware for mathematics, science, and engineering;

SymPy, a Python library for symbolic mathemat-

ics; Scholastic, with solutions for prime math-

ematics; and some other educational software

products as Prezi or Hot Potatoes.

4. Discussion

4.1. Open Access Web Resources Inventory

In order to accomplish one of the objectives

of this paper, and with the help of our search

strategy, we built an open access math web re-

sources inventory. It includes fteen high-qual-

ity items, useful for learning mathematics in

higher education, Most of them are interactive

simulations and CAS, and the rest are videos and

graphic visualizations. This is really interesting

because all of them can support the idea of active

math learning as stated in Freeman, et al. (2014).

Development of HTML5, Java applets and new

programming languages has contributed to the

creation of this kind of resources.

There is an almost natural relationship be-

tween interactive simulations and mathematics.

Simulations are based on mathematical probabi-

listic processes. Maybe this is due to the fact that

most open access web math resources are inter-

active simulations. Besides, math learning im-

plies comprehension and practice, and both can

be promoted by engaging simulations, where the

student can experiment, devise causal relation-

ships and even play with mathematical artifacts.

Some studies (Ray, 2013; Siddique, Ling, Rob-

erson, Xu & Geng, 2013) report that interactive

simulations can be effective in promoting math

learning and high order thinking, but more work

is needed in this area.

The use of CAS in math teaching is an im-

portant –and somewhat polemic– issue. As it can

be seen, ve open access CAS, which are very

useful and well designed, were found. Recently,

Marshall et al. (2012, p.423) stated that “emerg-

ing technologies such as computer algebra sys-

tems (CAS) brought with them the potential to

shape new directions in the teaching and learn-

ing of mathematics”. Part of our study is directed

precisely to see if these new teaching directions

are being considered in UNAM FES Acatlan.

In particular, Wolfram Alpha is, as indicated

in its web site, a very powerful “computational

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knowledge engine”. It can solve analytic equa-

tions, not only numerical operations, with a

very easy and intuitive notation, from arithme-

tic operations and basic algebra, to differential

equations, everything within a web browser. It

displays solutions, as well as the required pro-

cesses to get to them. It creates nice and useful

graphics. It even works in an iPhone, paying a

small fee for the service. All these advantages,

make us reect on our actual teaching and learn-

ing processes. As a result, we need to give a seri-

ous thought to the idea of including Wolfram Al-

pha in our daily practice. Even though this might

sound appropriate or even needed, it is an idea

that requires further discussion.

Regarding other resources, it is remarkable

that OCW and Khan Academy provide videos

of almost every math undergraduate topic. In

the rst case, we can nd complete lectures, one

or two hours long, with speakers of the highest

quality at the Massachusetts Institute of Technol-

ogy. It is a really nice opportunity to take some

lessons directly from the very authors of classic

textbooks, such as Linear Algebra with Professor

Gilbert Strang. In the second case, Salman Khan

has developed a novel teaching approach, with

short and clear videos of each particular topic.

This kind of presentation is consistent with the

idea of “learning objects” and their desirable

properties (Gonzalez-Videgaray, Hernandez-

Zamora & Del-Rio-Martinez, 2009), since they

are brief and self-contained. This allows students

to select the exact video they need to fulll a cer-

tain goal, and to watch it many times. The oppor-

tunity to watch it as much as needed, provides

the students with extra practice, and in some

cases, this might help avoid exasperating the

teacher or suffering bullying from other class-

mates. Besides, learners can watch them when-

ever and wherever they want (Parslow, 2012), in

a private and comfortable environment, from a

computer or a mobile device. Another advantage

of this kind of videos is that they can reach a

large number of students (Heckel, Bach, Richert

& Jeschke, 2012), which is generally the case at

Mexican public universities.

Maybe a special instance within web resourc-

es is GapMinder, an initiative of Hans Rosling,

a Swedish professor of global health. He con-

structed a site that combines statistics with dy-

namic animation, beauty and social awareness.

He provides visualizations of important country

indicators, making easy to identify causal rela-

tionships (Lodha, Gunawardane, Middleton &

Crow, 2009) and their behavior through time.

This site presents real and signicant data, which

can contribute to engage students in statistics

learning. Rosling’s software, called Trendalyzer,

is now used by the most important statistics or-

ganism in Mexico to display demographic data:

the National Institute of Statistics and Geogra-

phy (INEGI, 2014). Even though trend animation

seems to be exciting and enjoyable for students,

it is possible that sometimes they lead to inter-

pretation errors (Robertson, Fernandez, Fisher,

Lee & Stasko, 2008). These ideas deserve to be

further researched.

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We found that there are web resources for

the most common topics of undergraduate math-

ematics programs. So, we denitively believe

that it is possible to combine them with other

teaching strategies. In particular, OCW Videos

cover almost all the topics that are usually found

in a mathematics program. This is understand-

able because they intend “to publish all of our

course materials online and make them widely

available to everyone”, as it is written in their

web site (MIT, 2014). This is splendid, but as

the videos are actual lectures, anyone could ar-

gue that they are not technically nor pedagogi-

cally designed to be watched via internet with

a learning objective in mind. In spite of that,

these resources have beneted learners, includ-

ing “informal learners” and “independent learn-

ers” (Carson, Kanchanaraksa, Gooding, Mulder

& Schuwer, 2012).

The results showed that the topic with more

available web resources is statistics. Our ndings

are consistent with the meta-analysis of Sosa et

al. (2011) and with Symanzik & Vukasinovic

(2003), who consider that the high number of

web resources about statistics is due to the large

offering of data, statistics calculators, software

and other internet tools. This, in turn, can be ex-

plained because this topic is included in most

majors and careers. One reason of this inclusion

is that this subject is required to learn about re-

search techniques and methods, especially quan-

titative research. Another reason is, perhaps, that

statistical thinking is an important skill essen-

tial to be a more prepared and complete citizen

nowadays. As Australians say in their Statisti-

cal Declaration (Australian Bureau of Statistics,

2008): “The ability to understand and evaluate

statistics that permeate our daily lives is an es-

sential building block of basic numeracy skills.

Improving statistical skills should be part of the

curriculum in all levels of education, as well as

in training programs in government, business

and the wider community”. It is remarkable that

in the entire world a big effort has been done

to improve statistical literacy (Schield, 2004).

In Latin America and Spain there is important

research on this issue, conducted by Batanero

(2004) and some others.

Statistics is followed by algebra, which is not

surprising because it is also part of many curri-

cula in higher education. Since 2003, an effective

performance in algebra through a combination

of self-regulated learning and the use of CAS has

been reported (Kramarski & Hirsch, 2003) and,

of course, CAS were designed to precisely make

improvements on this topic. Today they even

have a greater potentiality. Algebra is considered

a “foreign language” to some students. They

struggle with mysterious characters that repre-

sent numbers, but simultaneously, can be oper-

ated as numbers considering certain rules that

are usually memorized without comprehending

them. Certain students write numbers instead of

letters, in order to get a better understanding, so

it is now challenging to have an “algebraic cal-

culator” for a change.

In respect of the languages in which the web

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resources are available, it was a pleasant surprise

to nd that there are two in Spanish and ve in

English and Spanish. The rest are in English and

that is in many cases an obstacle for some poten-

tial users in Latin America. The Mexican govern-

ment has made some efforts to include English

as a mandatory subject in public schools, but it

will be taught in all public schools until 2018 (La

Redaccion, 2011). However, within our study, it

is important to point out that all mathematics and

engineering university students must display an

English prociency diploma as a requirement to

obtain their B.A. degrees; unfortunately, many

undergraduate students do not get their language

diplomas on time. By the time these students n-

ish school and incorporate to real life jobs, they

cannot make any improvements in their profes-

sional practice because they have not fullled

this requirement.

Several OCW Videos have Spanish subtitles.

Statistical machine translation continues its

development (Koehn, 2010) and it surely will

help with this matter soon. Besides, Carlos Slim

–Mexican entrepreneur and one of the wealthiest

men in the world– is now supporting the Spanish

translation of Salman Khan videos (Fundacion

Carlos Slim, 2013). Considering this, it is pos-

sible now to really ip our classroom’s dynamic,

by leaving some of the traditional blackboard

explanations to these videos. Obviously, this can

only be possible if there is access to computers

and internet. About 38.4 % of the population in

Mexico (Grupo del Banco Mundial, 2014) has

this opportunity, and almost all of undergraduate

students are considered in this percentage.

In addition to our inventory, the survey re-

spondents suggested some interesting resources.

We are conscious that our search prole took

only a still photograph and captured just a few

items of a richer world of possibilities. More and

better math web resources are being developed

constantly and some of them are open access.

Maybe it is important to be aware of this and

stay permanently updated. This represents addi-

tional work for both teachers and students and

implies to develop a lifelong learning capacity,

which seems to be a hallmark of the 21

century.

We can conclude by mentioning that the in-

ventory of math web resources with open access

is rich, diverse, useful and interesting. We must

not forget that there are a lot of commercial or

proprietary resources that can also help promote

mathematics learning, but the existence of free

and accessible tools, available through inter-

net, represents good news to our country and to

Spanish-speaking countries in general.

4.2. Use of Web Resources

Once we have reviewed the inventory of

valuable web resources to foster math learning,

it is advisable to analyze how they are used in

our universities, if so. Survey results suggest

that, despite the variety of tools such as CAS,

video lectures, short video explanations, interac-

tions, visualization and animations, all of them

free and easily available, there is little knowl-

edge about them amongst teachers and students.

Therefore, their use is very limited, even in

mathematics and engineering careers, where one

could expect to be more acquainted with them.

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It seems that students and teachers from Ac-

tuary and Mathematic careers are somewhat

more familiarized with these resources than

those from Civil Engineering, possibly because

the rst ones make a more intensive use of com-

puters in their scholar work. Data does not pro-

vide enough evidence of a signicant difference

between teachers’ and students’ knowledge, as

all items got low values.

Wolfram Alpha and GeoGebra stand slightly

over other resources. As we noted before, Wol-

fram Alpha is a powerful knowledge engine,

capable of doing symbolic operations with sim-

ple or partial information. As for the second,

its acquaintance is possible due to the fact that

our Faculty hosts the GeoGebra Institute of the

UNAM (GeoGebra, 2014), but we then would

have expected much more. This resource is an

interactive free CAS, with thousands of free

resources and learning materials, from basic to

higher education, and there are reports about its

suitability to support learning in combination

with interactive white blackboards (Lavicza &

Papp-Varga, 2010) Cambridge, England [Papp-

Varga, Z] Eotvos Lorand Univ, Dept Comp Sci,

Budapest, Hungary Lavicza, Zsolt (re-

print author. Students and teachers at our Faculty

are aware of its existence but do not use it.

It is clear that open access math web re-

sources are poorly known and used within our

Faculty. There can be several reasons that might

explain these ndings. One possibility is that

teachers are using, instead, commercial tools

designed to promote math learning. However,

as far as we could establish by asking the cor-

respondent department, there are no software li-

censes or agreements of this kind in the Faculty.

Of course, it is possible –not at all desirable– that

some teachers and students are using illegal cop-

ies, but this practice would be eliminated with

the use of open access web resources.

Marshall et al. (2012) have stated that com-

mercial cost and difcult syntax are important

obstacles for the use of CAS. It is interesting to

see that both problems are practically inexistent

in Wolfram Alpha, which is free and allows a

very intuitive way of writing formulas. For in-

stance, it is possible to type “derive x2+4x+4”

to instantly get the result shown in gure 8; or

“plot normal” to watch the screen displayed in

Figure 9. This can explain why this web resource

is the most used by students. If one wants to see

step-by-step solutions, a Pro version is offered

with a cost of $4.99 USD per month or $45 USD

per year.

Figure 8. Derivative produced with Wolfram

Alpha

Figure 9. Normal distribution variants elaborated

with Wolfram Alpha

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Our results could also be linked to a general

lack of information literacy (American Library

Association, 1989) at the university. De Arenas

et al. (2004, p.459) have reported deciencies in

this kind of literacy in Mexico and Spain, that

they attribute “to educational bureaucracy, to

anachronistic ways of teaching that validate re-

liance on note-taking by students, on textbook-

based learning and on absorbing outdated infor-

mation”. Maybe teachers and learners are still

relying on traditional practices and have not

been interested in looking for new alternatives,

or maybe they have not developed good search

strategies that could help them nd these specif-

ic resources. As Rowlands et al. (2008) say, the

idea that digital natives and Google generation

are expert searchers is a “dangerous myth” and

there is “increasing evidence that students do not

use technology effectively when they conduct

research or communicate” (Katz, 2013, p.4).

Possibly, even when they are conscious of their

problems with mathematics, they do not con-

sider it a global problem and neither that there

are interesting learning alternatives within their

reach on the web.

Besides, our ndings can also be related to

deciencies of digital literacy as established by

Ng (2012): “Unless taught explicitly to use other

(educational) technologies, it is unlikely that

digital natives would think about educational

technologies”. Ng points out that students can

easily manage technologies if they are aware of

them and want to include them in their learning

process, but they do not look intentionally for

them. So, if teachers do not know about these re-

sources, it is almost impossible that students do.

Another possibility is that mathematics

teachers are slowly getting aware of the poten-

tial advantages of these resources. Maybe there

is some resistance to change and certain fear

of using these tools. We heard recently a math

teacher stating the importance of knowing how

to solve a square root manually. He said that if

a student were in a desert island, how would she

or he solve a square root without a computer?

The question we make is, would a student fac-

ing such circumstance be interested in solving

square roots? Maybe the central discussion point

is related to the concern expressed by Reed, Dri-

jvers & Kirschner (2010, p.1): “students may

learn only to manipulate the tools rather than

master the underlying concepts”. Of course,

the main goal of math learning is that the stu-

dent masters concepts, develops skills and feels

comfortable with mathematical thinking. Prob-

ably deeper work is required to show how web

tools can effectively foster math learning before

teachers actually adopt them. These studies will

let them realize the potential benets that em-

ploying mathematical computer tools in a class,

have.

Within the course of this research, we had the

opportunity to see students’ and teachers’ reac-

tions to the survey. Some students took photo-

graphs of the questionnaire, in order to store the

resources inventory for subsequent use. Some

wondered about so many sites they did not know.

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Some others were worried because they did not

want their teachers to be conscious of this kind

of tools: students currently use them –particular-

ly Wolfram Alpha– to solve homework and they

want the teacher to think that they are still work-

ing with pencil and paper. In fact, they seemed a

bit upset with the research and the researchers.

Besides, several of them said they have passed

exams thanks to Khan Academy, and not thanks

to the teachers’ classes.

Teachers, in turn, took a defensive position.

They thought it was a kind of test or an assess-

ment on their actualization. Some of them were

curious, but some others were worried for not

knowing the web resources listed in the ques-

tionnaire. It will be interesting to apply the sur-

vey again soon and see if there are any changes.

Also, more research is needed on the teacher’s

perspective concerning these web resources

uses.

4.3. Limitations of our study

Our study was circumscribed to one campus

of the most important public universities in Mex-

ico; it would be valuable to replicate it in other

campi and considering other similar careers. The

questionnaires could be enriched with some of

the resources mentioned by the respondents and

with a more exhaustive search. Our research

was mainly quantitative and did not consider the

teachers’ and students’ positions on this subject.

4.4. Conclusions

Our study conrms the assertion of Marshall

et al. (2012, p.423): “In spite of the fact that a

growing number of international studies have

demonstrated that CAS-based instruction has

the potential to positively inuence the teaching

and learning of mathematics at various levels of

the education system, CAS has not been widely

implemented in schools and higher education in-

stitutions”. This seems to be true not only about

CAS, but other videos, interactions, animations

and simulations for undergraduate math learning

in the web, at the UNAM and probably at some

other universities in our country.

We have seen that there is a large amount of

varied, attractive and useful open access web

resources that can promote undergraduate math

learning, especially in English but also in Span-

ish. In order to exploit this opportunity, there are

some pre-requisites. We must promote informa-

tion and digital literacies, so every teacher and

student can nd effective tools to support math

learning. This is very important because there

are always novelties, so one must research for

them every now and then, to stay updated in this

subject. Besides, it is indispensable to be capable

of assessing and selecting the best tools for each

requirement.

It is also essential, for our teachers and stu-

dents, to achieve prociency in English as a

second language. This will open up the range of

possibilities and allow a more active participa-

tion in learning. Of course, it is also desirable

that efforts should continue to translate or sub-

title existing resources, besides producing them

in Spanish.

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In any case, it is imperative to subject to aca-

demic discussion the use of these web resources

within math undergraduate education processes.

With more than twenty years of experience, Gar-

cia et al. (2014) propose a model for the incorpo-

ration of CAS to the math learning process. This

model asks for three requirements: a) integrate

CAS to all learning and assessment activities,

b) motivate students to create tools and prob-

lem solutions of their own with the CAS, and c)

promote the use of CAS out of the specic math

subject, for instance, to solve chemistry or phys-

ics problems. Probably, the use of CAS in assess-

ments and tests continues to be the most polemic

issue (Marshall, et al., 2012) among teachers.

More discussion and research is needed about it.

The kind of dynamic visualizations offered

by these resources can beyond doubt help to

improve the experience of math learning, be-

cause they contrive to a better comprehension,

remembrance, inspiration and motivation in stu-

dents (Burewicz & Miranowicz, 2002). Besides,

it seems that there is no way back in the use of

these tools, which are each day more attractive,

efcient, powerful and easy to use.

We plan to continue studying why these web

resources are not properly disseminated, how

they are currently used, and what should be done

to promote their intelligent and critical use. We

need to reect on the implications of this use not

only for teaching strategies, but for the selec-

tion of contents that should be taught, as well.

In other words, these web resources compel us

to evaluate how we should teach, but also what

we should teach in mathematics. Maybe we must

reduce procedure and mechanical aspects in or-

der to enhance reection, approach and problem

solving, as well as real signicant applications

of math skills.

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