Teaching Mathematics in Online Courses

An Interactive Feedback and Assessment Tool

Edgar Seemann

Furtwangen University, Villingen-Schwenningen, Germany

Keywords:

Online Courses, Mathematics, Assessment, Grading, Teaching, Tutoring.

Abstract:

Online courses often require students to work self-dependently using books or video material. For abstract

subjects such as mathematics this is particularly challenging. To improve student motivation and learning

results, we propose an interactive feedback and assessment tool tailored to math exercises. Our system is

able to process and analyze mathematical expressions using an underlying computer algebra system. It allows

teachers to create exercises with a much wider range of question types as it is possible with today’s learning

management systems, which are mostly restricted to multiple choice questions. We can provide automatic

individual feedback to students for almost any kind of mathematical exercise. Thus, making it easier for

students to practice and study math in a self-dependent manner.

1 INTRODUCTION

Mathematics is the foundation of every engineering

discipline. That is why teachers want students to

reach a certain level of mathematical understanding.

Meaning, that they should be able to apply learned

techniques to new problems or even real world prob-

lems.

There is a consensus among teachers, that in or-

der to master mathematical problem solving, students

need to practice.They need to practice calculation

methods e.g. how to compute the derivative for a cer-

tain type of function. And they need multiple exam-

ples to internalize the underlying ideas and to be able

to generalize a technique to other tasks or settings.

In on-campus university classes students try to

solve exercise problems and get feedback and assis-

tance from their teachers or teaching assistants. Pro-

viding adequate feedback is one of the most impor-

tant steps in the learning process. Without it, students

often internalize incorrect rules or develop incorrect

notions of mathematical problems and methods.

This feedback is ideally immediate or at least pro-

vided in a timely manner. Students need to be encour-

aged with positive feedback, if they solved a problem

correctly. And they need to be made aware of mis-

takes as soon as possible. Note that this does not nec-

essarily mean that step by step instructions are pro-

vided for each exercise. The teacher should be able to

control the complexity of the exercises and the granu-

larity of feedback to foster deep learning without frus-

trating the students.

For online courses this poses a particular chal-

lenge. On the one hand, communication between the

teacher and students is still and always will be more

difﬁcult than in a face-to-face conversation (even

though virtual classroom software is improving). On

the other hand, the communication is, to greater ex-

tent, asynchronous.

Online courses are often taught through a series of

videos combined with a discussion forum. Feedback

to student questions is therefore typically delayed by

hours or days. Live virtual classroom sessions typi-

cally only take place at the beginning or the end of

larger learning units. Finally, with the advent of Mas-

sive Open Online Courses (MOOC) the teacher to stu-

dent ratio makes individual feedback a challenge.

Without appropriate communication and feedback

many students fail to achieve the required skills. In

fact, the success rate for online courses is in most

cases considerably below success rates of on-campus

courses with a comparable student body. In (Collins,

2013) and (Thrun, 2013) the difference in success rate

is as large 40% for an online course compared to the

respective on-campus class in spring 2013.

In order to improve teaching of mathematical

problem solving skills in online courses, we believe it

is necessary to provide meaningful, individual feed-

back to every student, no matter whether there are 30

or 30000 students in a class. Learning management

415

Seemann E..

Teaching Mathematics in Online Courses - An Interactive Feedback and Assessment Tool.

DOI: 10.5220/0004939204150420

In Proceedings of the 6th International Conference on Computer Supported Education (CSEDU-2014), pages 415-420

ISBN: 978-989-758-020-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

software e.g. (Moodle, 2013; OLAT, 2013) provides

interactive (self-)tests, which allow students to prac-

tice and evaluate their skills. Thus, obtaining feed-

back on their learning progress. Unfortunately, giv-

ing automatic feedback for mathematical exercises is

difﬁcult. Responses may not only involve numeric

values, but expressions may also include functions or

mathematical sets which have many equivalent repre-

sentations (e.g. x(x +1) is equivalent to x

+x). These

mathematic expressions cannot be handled properly

by state-of-the-art learning management systems.

For a meaningful feedback an interactive learning

tool needs to better “understand” these mathemati-

cal expressions. As a ﬁrst step in this direction we

therefore propose an automatic feedback and evalua-

tion system tailored to mathematical exercises. Our

system leverages existing computer algebra software

in order to evaluate student responses and to provide

immediate individual feedback. It allows teachers to

easily create custom content with the granularity of

feedback of their choice.

With the automatic feedback functionality turned

off, the system can also be used for efﬁcient assess-

ment and grading of large numbers of students.

2 ELECTRONIC SUPPORT FOR

MATH TEACHING

In abstract subjects such as mathematics feedback

plays an important role in the learning process. While

self-studying from a book or video is important, stu-

dents require much more support compared to facts-

based subjects (e.g. history).

Traditionally, this support is to a large extent

provided through face-to-face communication with

teachers, teaching assistants or fellow students. When

teaching online, we need to replace this valuable form

of feedback using different means.

2.1 Virtual Classrooms

Virtual classroom software has gained widespread ac-

ceptance in recent years. Universities use commer-

cial services like Adobe Connect or Google Helpouts

for live sessions between students and teachers. Vir-

tual classroom software provides real-time audio and

video communication. With the included whiteboard

and screen sharing capabilities this setup can mimic

the lecture style of large on-campus lectures. That is,

a professor is explaining and students are listening.

http://www.adobe.com/products/adobeconnect.html,

http://helpouts.google.com

Student questions in these virtual classrooms are

often tedious due to organizational issues (who is al-

lowed to talk at which point?) and technical problems

(microphone setup, network delays etc.). More over,

the software or network often only allows for a lim-

ited number of participants. When groups are small,

virtual classroom can, however, be a decent tool to

provide feedback. Particularly, since touchscreens of

today’s notebooks and tablets make it easy to write

mathematical equations and formulas.

2.2 Forums

Forums are another popular tool for online learning.

Communication is asynchronous and thus does not

provide immediate feedback. Students may ask a

question and a member of the teaching staff responds

later on. While in some cases this might result in stu-

dents working harder to solve a problem all by them-

selves, the delayed feedback often also disrupts and

postpones the learning process.

Forums require the teaching staff to spend a large

amount of time formulating written answers to stu-

dent questions, which is tedious and does not scale

well to large amount of students.

2.3 Learning Management and

Tutoring Systems

Existing learning management systems e.g. (Moo-

dle, 2013; OLAT, 2013) and tutoring systems e.g.

(Koedinger and Corbett, 2006; Melis and Siekmann,

2004) offer the functionality to create electronic ex-

ercise sheets. This allows students to practice and to

obtain immediate feedback to their solutions. These

systems may also display context-sensitive informa-

tion, hints and instruction to guide students towards

reasonable next steps.

LMS: Learning Management Systems. Gener-

ally, teachers are required to reformulate mathemat-

ical problems in a way, that is supported by the LMS.

That is, in order to allow automatic feedback, exer-

cises have been in the form of multiple choice or ﬁxed

answer questions.

Since these systems do not understand mathemat-

ical expressions and their equivalent representations

it is impossible to use more complex expressions as a

solution. Exercise questions allowing free-text ﬁelds

are of course possible, but they require manual inter-

vention, which leads to a delayed feedback.

Overall, current LMS limit the selection of possi-

ble exercises considerably, which is why most math

teachers avoid using these systems.

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416

ITS: Intelligent Tutoring Systems. In the context

of this paper, we divide intelligent tutoring systems

into two categories. One category of systems focuses

on building student proﬁles while observing student

performance in online tests and provides hints for

the learner on its personal weaknesses and strengths

(Schiafﬁno et al., 2008; Cheung et al., 2003),. These

systems often build upon simple multiple choice and

ﬁxed answer questions and aggregate their results.

They can tell a student e.g., that she/he has made

many mistakes in the exercises on a certain topic.

However, these systems cannot deal with mathemati-

cal expressions.

The other category focuses on domain speciﬁc

problems. In the area of mathematics, there exist a

number of these tutoring systems (Beal et al., 1998;

Melis et al., 2009; Koedinger and Corbett, 2006). To

our knowledge none of these systems leverage capa-

bilities of symbolic mathematics software to support

students. In fact, most of these systems focus on early

high school level maths.

It is often hard for teachers to develop their own

content for existing intelligent tutor systems. In fact,

very few systems have detailed instructions on how

to do this and in many cases programming skills are

required. The Carnegie Cognitive Authoring Tools

(Aleven et al., 2006) e.g. require authors to know Java

or Adobe Flash. This is certainly one of the reasons,

why the adoption of intelligent tutoring systems has

been rather disappointing.

3 AN INTERACTIVE FEEDBACK

AND ASSESSMENT TOOL

As pointed out in the sections above today’s learn-

ing management systems are not prepared to deal with

mathematical exercises.

The system proposed in this paper tries to combine

interactive exercises with the capabilities of compute

algebra systems. Our goal is to build a practice and

assessment tool which is capable of “understanding”

mathematical expressions and as a result give mean-

ingful feedback and assistance to the students. Our

system allows teachers to use a much wider range of

exercise and question types as it is currently possible.

Contrary to many tutoring systems, we aim for

math problems in general and do not want to restrict

ourselves to a certain subdomain (e.g. dealing with

fractions). That is, whether high school teachers want

to design exercises to train the expansion of simple

mathematical expressions or if university professors

design exercises to solve differential equations, the

capabilities of the underlying mathematical software

should allow to provide meaningful feedback. More

over, We want to make it simple for teachers to de-

velop their own content without programming skills

(see section 3.1).

3.1 Example Exercise

To get a better understanding of how the proposed

system works, we present a short example exercise.

We show how an exercise may be deﬁned by the

teacher and how the system reacts to user input. The

technical details on how an appropriate user feedback

is accomplished will be explained in subsequent sec-

tions.

Questions are displayed in a modern web-based

user interface. Input ﬁelds allow students to enter

their solutions and an appropriate feedback is returned

once response has been submitted. In Figure 1 a stu-

dent has provided an incorrect answer. Obviously

he/she computed the derivative of numerator and de-

nominator independently and the system displays the

feedback directly below the input.

Figure 1: A hint is displayed if the provided solution is in-

correct.

If a correct solution is entered, the input ﬁeld

turns green. Optionally, an exercise may yield points.

Points may be used as a motivation or for grading pur-

poses (see Figure 2).

Figure 2: Correct solutions are marked green.

Finally, since the system recognizes mathematical

equivalences, the user may input a different represen-

tation of the solution. In the above example the rel-

atively complicated formula is, in fact, nothing else

than the constant function 1 for values of x 6= 1 (see

Figure 3).

Note that the system could also display additional

feedback for correct but overly complicated solutions.

TeachingMathematicsinOnlineCourses-AnInteractiveFeedbackandAssessmentTool

417

Figure 3: Equivalent solutions are recognized by the under-

lying symbolic math software.

Thus, hinting the student to a quicker way of solving

the exercise.

Content Creation. Writing custom exercises is in-

spired by the syntax of markdown (Gruber, 2004),

which has become quite popular in recent years. We

use a relatively simple text format, with some con-

trol characters. The text format is later translated to

HTML.

A new exercise is created by starting a line with

three question marks followed by the title. Question

text may contain LaTeX-style formulas (see below).

LaTeX is still the most common format for deﬁning

math exercise sheets and teachers are familiar with it.

Since specifying a solution requires the deﬁnition

of multiple values and properties, we opted for XML-

syntax. A field-tag deﬁnes the expected solution

type. In the example below this is a symbolic math

value, i.e. a function, with the variable x. Embedded

—answer—-tags deﬁne the behaviour depending on

different user inputs. In the case below, the input of

the value 1 results in returning points for the correct

answer. The input 2x results in a customized feedback

for the incorrect answer.

??? Derivatives

a) Compute the derivative of the function

$f(x)=\frac{xˆ2-1}{x-1}$ for $x\neq 1$

<field:symbolic var="x">

<answer="1" points="2">

<answer="2x" feedback="Note that f(x)

is a quotient. Use quotient rule or

transform the function.">

</field>

3.2 Solution Types of Mathematical

Exercises

In order to allow teachers to use a wide range of

different math exercises in an e-learning system, we

need to be able to represent their respective solutions.

In the following, we want to discuss the most common

types of solutions for mathematical exercises. We will

also point out the difﬁculties of representing and han-

dling these solutions in an e-learning system.

Numbers. Numbers are probably the most com-

mon solution type for a math exercise. Even simple

numbers have different ways of representation, which

need special treatment in an e-learning system. The

number 0.75 e.g. may be represented by the fraction

. If we want to allow students to enter either form,

the e-learning system has to be aware of these rep-

resentations. We also have to take care of irrational

numbers like square roots or logarithms (e.g.

√

5).

One could argue that results could be restricted to

rounded decimal numbers. Many teachers, however,

prefer students to be able to solve exercises without

a calculator. Solving a quadratic equation e.g. may

easily lead to an irrational result, thus requiring a cal-

culator to obtain a decimal representation.

Vectors and Matrices. For vector computations,

we need to represent both matrices and vectors. In

fact, since each entry in a vector or a matrix is a num-

ber, we also have to be aware of number representa-

tions as described above.

Intervals. Inequations often lead to number inter-

vals as a solution. The inequation x

−4 < 0 e.g. has

all numbers between −2 and 2 as a solution. Intervals

are essentially a pair of numbers with additional in-

formation of whether the interval is open or closed at

the respective boundary.

Sets. Sets are unordered lists of numbers, which be-

long to a solution. The simplest example is the solu-

tion of a quadratic equation. The equation x

−1 = 0

e.g. has the solutions −1 and 1.

Both sets and intervals can be combined using

union, intersection or complement operators.

Functions. Exercises may also yield functions as a

solution, e.g. when a derivative needs to be computed.

There are many different possibilities to represent the

same function, e.g. x(x + 1) is equivalent to x

+ x.

An e-learning system needs symbolic calculation ca-

pabilities to deal with these.

Geometry and Proofs. Finally, some exercises re-

quire students to sketch a geometric object or graph.

Others require them to proof a mathematic property.

Both of these two types are out of scope of this paper.

4 SYSTEM IMPLEMENTATION

In the following we will discuss how user input may

be veriﬁed leveraging the capabilities of existing math

CSEDU2014-6thInternationalConferenceonComputerSupportedEducation

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software. From the results obtained by this software,

we generate a meaningful feedback for the students.

4.1 Computer Algebra Software

Mathematical software packages have become ex-

tremely capable. Tools like Maple or Matlab (Maple-

soft, 2013; MathWorks, 2013) are widespread in

academia. They are capable of manipulating mathe-

matical expressions (e.g. solving an equation). While

these are also great tools to be used independently of

an e-learning system, they essentially require users to

learn a programming language. In some cases this is

more difﬁcult for students than learning the mathe-

matics itself.

We therefore hide this complexity from the user

and do the manipulation of the user input transpar-

ently in the backend of our system.

4.2 “Understanding” Mathematical

Expressions

Veriﬁcation of user input depends on the solution type

of an exercise. As we have seen in section 3.2 math

exercises may have various different types of answers.

In the following, we will show how to verify answers

for these different types.

Numbers. While numbers have different represen-

tations, we can compare them by their values. In our

implementation this is accomplished by subtracting

the number provided by the student from the actual

solution.

In Matlab this can be expressed with the following

code, which is executed when students submit their

results.

1 i sCor re ct = 0;

2 di ff = abs ( input - s ol ut io n ) ;

3 if ( diff < e ps i l o n ) is Co rr ec t = 1;

Using the absolute value makes sure that we ob-

tain a positive number for the difference. Note that

number representations in a computer are never ex-

act due to the limited number of bits available. Con-

sequently, the difference of ﬂoating point numbers

should never be compared to 0, but rather we need

to make sure, that the difference is smaller than a cer-

tain epsilon. This also accommodates for rounding er-

rors. In our implementation, we use an epsilon value

of 0.001.

Vectors and Matrices. In order to compare matri-

ces, we essentially have to compare the numbers for

the corresponding rows and columns. Fortunately

mathematic software makes this comparison easy. In

fact, when using Matlab we can even avoid imple-

menting nested loops.

1 i sC or re ct = 0;

2 d if fm at ri x = ab s (i n put - s o l u t io n ) ;

3 m ax i mu m = max (m a x ( d i ff ma tr ix )) ;

4 if ( max imum < e ps il on ) i sC or re ct = 1;

Note that we compute the difference matrix and

then ﬁnd the global maximum of rows and columns in

the matrix. If this maximum is smaller then epsilon,

the matrices are considered equal.

Intervals. Intervals are described by their endpoints

and information of whether these are included in the

interval or not. The interval (1, 3] e.g. includes all

numbers between 1 and 3 excluding the number 1,

but including 3.

To compare intervals, we need to parse the ex-

pression. First we look for the type of parentheses

or brackets. Then, we split the left and right and end-

points. Two intervals are considered equivalent, when

their brackets match and when their endpoint numbers

are identical with respect to the criteria described in

paragraph 4.2.

Sets. Sets are unsorted lists of numbers. For veri-

ﬁcation, we sort the numbers to obtained an ordered

list. This ordered list can now be handled in the same

manner as a vector.

An implementation using MATLAB could look

like this:

1 i sC or re ct = 0;

2 d if fv ec to r = ab s ( sort ( inp ut )- so rt ( sol ut io n ) ) ;

3 m ax i mu m = max (m a x ( d i ff ve ct or )) ;

4 if ( max imum < e ps il on ) i sC or re ct = 1;

Functions. Comparing function representations is

the most difﬁcult among the presented solution types.

We need to manipulate the provided expressions to

make them match.

Given a user input of x(x+1) and a solution x

+x,

we need to expand the input expression in order to al-

low a proper comparison. A straight forward expan-

sion is not always sufﬁcient to compare two symbolic

expressions. In many cases, we have to do further

simpliﬁcations, either for the user input and or the so-

lution.

It might e.g. be necessary to simplify logarithms,

exponentials or radicals. In such a manner, we can

simplify an expression like

−1

1+e

to e

. The algorith-

mic implementation details to perform these manip-

ulations are described in (Fateman, 1972). They are

TeachingMathematicsinOnlineCourses-AnInteractiveFeedbackandAssessmentTool

419

readily available in state-of-the-art computer algebra

systems.

In our implementation, we use the math software

Maxima (Schelter, 2013) to do symbolic calculations.

We compare function representations using the fol-

lowing code lines:

1 di ff ( x ) := in put (x ) - so lu ti on (x );

2 s im pl if ie d ( x ) := r ad c an ( diff (x )) ;

3 r e su lt (x ) := exp and ( s i m p l i f ie d ( x )) ;

Finally, we check whether the resulting function

result(x) equals to zero.

5 CONCLUSION

In this paper, we have shown how an interactive feed-

back and assessment system tailored to math exer-

cises can be implemented. The proposed system

provides meaningful immediate feedback to students

while practicing math. Thus, allowing them to bet-

ter learn on their own without requiring human assis-

tance in the form of a teacher. This helps to alleviate

some of the frustrations student encounter when deal-

ing with mathematic problem solving and is particu-

larly important for online courses.

We leverage the capabilities of existing math soft-

ware packages to analyze user input and to decide

which feedback to generate. Thus, making the system

a general math tool, where exercises may range form

simple high school math to university level calculus

and algebra.

For teachers, the system allows the use of a much

wider variety of question types compared to stan-

dard e-learning systems, which are mostly restricted

to simple multiple choice questions. It is easy for

teachers to create their own exercises. Contrary to

most tutoring systems no special programming skills

are required. Exercises may be deﬁned in a simple

text format where formulas may be entered with the

familiar LaTeX syntax.

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