Assessing Elementary Algebra with STACK
Christopher J Sangwin
School of Mathematics, University of Birmingham, B15 2TT
+44 121 414 6197 (phone) +44 121 414 3389 (fax)
April 6, 2006
Abstract
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students’ responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught
in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/,
which uses the CAS Maxima, is taken as a case study with which to test the implementation of the ideas developed in this paper. The general characteristics needed from
a CAS for the application of computer aided assessment, or computer based learning is
discussed. In particular the need for consistently implemented noun forms of elementary
arithmetic operations, together with their traditional verb forms is identified.
Keywords: Computer algebra, computer aided assessment, elementary algebra.
1
Introduction
This paper concerns computer aided assessment (CAA) in which a computer algebra system
(CAS) is used to help assess students’ responses, i.e. answers, to elementary algebra questions.
As a simple illustration, imagine a student interacting with a CAA system and in some way
entering a mathematical expression as an answer to a question. The CAA system uses a CAS
to algebraically subtract the student’s response from that of the teacher and to simplify the
resulting expression. If the result of this algebraic simplification is zero, then the system has
established an algebraic equivalence between the student’s answer and the teacher’s answer.
The key point is that the student’s answer contains mathematical content.
General purpose computer aided assessment systems predominantly use multiple choice
questions, or modifications of these. Such modifications provide a significant range of variations, and include multiple response, and matching questions. What all these have in common
is that they are provided response question (PRQ) types. That is to say, questions in which a
student is provided with a list of potential answers and asked to make a selection, match-up,
rearrange or perform various other kinds interactions. Contemporary CAA systems usually
include numerical entry although these have only very limited use in mathematics. Free text
entry is a feature of many systems although tools to assess these are primitive.
The use of a PRQ is almost always a constraint dictated by the software, and not the
preferred choice of the user. Indeed there are well-aired fundamental problems with provided
response questions which are (i) question distortion, (ii) assessment only of lower order skills,
and (iii) strategic learning. While there is evidence these can be ameliorated to some extent
by carefully designing questions, the problems associated with question distortion are severe
1
in mathematics. This is particularly the case with questions involving an essentially reversible
process, where one direction is significantly more difficult. For example, factor vs. expand,
integrate vs. differentiate and so on. Questions which ask students to solve a problem are also
difficult to set as a PRQ since it is too easy for a strategic student to interpret the question as
asking ‘which of these are solutions?’. Since a student may not actually perform the process
asked but do something else, the question may well not test what the teacher intends. In
fact, we might be quite disappointed if students did not take the mathematically sensible, if
strategic, approach. This would at least demonstrate an understanding of the reversibility of
the process, and their relative difficulties. Many of these problems are fundamental to the
PRQ type and are difficult, if not impossible, to remove. What is pedagogically more honest
is to ask for, and assess, a mathematical response from the student.
1.1
Background to CAS-enabled CAA
Since 2000, a number of specialist computer aided assessment systems have used an existing
CAS to provide tools to assess mathematical expressions as answers. Perhaps the first system
to make a mainstream CAS a central feature was the AiM System, first described by [8], with
subsequent technical developments described in [16]. This system operates using Maple, as
does the Wallis system of [9] and the proprietary system MapleTA. Other systems have access
to a different CAS, such as CABLE [10] which uses AXIOM and the STACK system which
uses the CAS Maxima, see [14]. From private correspondence, the author is also aware of
systems which use Derive and Mathematica in a similar way.
In these systems the CAS supports various ancillary functions, such as the display of
mathematical expressions (via perhaps LATEX) and the input of expressions using the typed
linear syntax supplied by the CAS. Indeed, in all the CAA systems mentioned above the
student uses the syntax of the CAS itself to express their response to a question. The syntax
issue is particularly important for elementary algebra assessment. If a learner is only just
meeting concepts such as gathering like terms, or multiplication of brackets then they cannot
be encumbered by the need to learn syntax. As a result, STACK also provides the liberalised
mathematical syntax described in [13].
However, in all these systems the primary purpose of the CAS is to provide the teacher
with a ready made library of useful mathematical functions. This allows the generation
and manipulation of expressions to create structured random questions. These functions
also allow a student’s answer to be manipulated and tested objectively against a variety
of mathematical criteria beyond the prototype of establishing algebraic equivalence with a
teacher’s expression. It is the ability to make fine grained distinctions in elementary algebra
which is the topic of this paper. Distinguishing between different possible responses is central
to the assessment process, whether in CAA or by more traditional means. Only once this has
been done can various outcomes be assigned, which might include a mark (or score), feedback
and so on. While humans are able to extract information from the question context to guide
the assessment of students’ responses, in CAA explicit instructions are needed. Furthermore,
these decisions need to be made in advance of the question being posed to the student. The
functions are also invaluable for generating feedback based on mathematical properties of the
student’s answer and also for constructing worked solutions.
While a CAS provides access to a large and well supported library of mathematical functions, the mainstream CAS cited above have been designed for the research mathematician.
When attention is paid by the designers of CAS to students, as it often is, the envisaged use
2
is almost always as a sophisticated calculator, see for example [6]. The experiences of the
author in using a variety of the CAA systems mentioned above while teaching undergraduate
students since 2000, suggests that CAA as an application is quite different.
In this paper we address the following questions.
1. How do we recognize and describe the characteristics of a ‘correct answer’ to an elementary mathematics question?
2. To what extent can the CAS Maxima establish these characteristics?
Being able to answer the second research question presupposes an understanding of the distinctions a teacher might make when assessing students’ answers in the traditional way. Hence,
we will begin by looking in some detail at the academic subject to which CAA is being applied.
There are significant differences between various CAS, and a comparison from the point
of view of a research mathematician are given in [17] together with other general comments.
Maxima has been chosen for this paper since it underpins the STACK CAA system currently
being developed by the author. We shall comment on some of the differences between Maxima
and other CAS in the discussion.
1.2
Caveats
One philosophical objection to this research is that mathematical work is not about obtaining
the ‘correct answer’. Indeed, the author does not believe that obtaining an isolated expression
is sufficient. In learning and teaching the method used forms essential evidence for a student’s
understanding of the processes involved. Nevertheless, being able to automatically assess a
logically connected sequence of mathematical statements is predicated upon the ability to
examine the properties of a single expression. Nevertheless, establishing the mathematical
properties of a single expression is a significant advance from the use of PRQ question types
currently used by contemporary CAA. Furthermore, practice is a legitimate and necessary
activity and CAA systems such as STACK have been used most often for this purpose.
In this paper we assume that only the student’s final answer is available to the CAA
system. We have no working, logical connections or method comments from which to draw
inferences. Feedback would be richer is this information were available, and in practice
students do complain at the lack of partial credit. Again, this assumption is in line with
contemporary CAA. In this paper we consider only an assessment function, rather than online
intelligent tutoring. In practice the immediacy of feedback, together with the opportunity
for the student to make repeated attempts, can be used to encourage the student to check
their own incorrect working carefully to identify any errors for themselves. Partial credit and
‘follow through’ marking are contrived devices to recognize and reward students’ achievement
in the context of asynchronous marking of traditional paper based work. With CAA the
importance of accuracy in achieving a correct final answer could be reinforced in a way which
does not unfairly penalize the inevitable and minor mistakes which virtually everyone makes
from time to time.
In many cases a general purpose CAS is not required to manipulate the response of a
student to establish its mathematical properties. For example, algebraic equivalence between
two expressions can be automatically established by choosing a dozen or so incommensurate
random values, substituting these into the respective expressions, evaluating numerically and
comparing the results. While this method does not guarantee equivalence, it does have a
3
firm basis in numerical analysis and has been found to be sufficiently reliable for the kinds
of expressions which occur in learning and teaching. Other tests, whether an expression is
in factored or expanded form for example, would require bespoke libraries of mathematical
functions. Systems which take this approach to CAA are the CALM system of [1], the Metric
system of [12] and the Aplusix system of [11], to cite but three examples from an extensive
field. We consider such libraries of functions to be ‘computer algebra’, even if they are not
traditional general purpose systems or explicitly conceptualized as such by their authors. Our
discussion of which mathematical distinctions a teacher might wish to make is just as relevant
to these systems.
In this paper we do not comment on how a student actually interacts with a CAA system
to express themselves mathematically. We assume that through a process of interactions the
student has communicated a syntactically valid expression, and that the CAS has interpreted
this in the way the student intended. This is not trivial. For example, in many CAS the string
4^1/2 is interpreted as two, not because the CAS takes the positive square root, but because
41
1
2 = 2. Since we tell our students that ‘a half’, 2 , as a rational number is a single entity, they
might expect the division order of precedence in this case to exceed that of exponentiation
so that the input string is interpreted as 4^(1/2) rather than (4^1)/2. One alternative to a
linear syntax is to use an ‘equation editor”, and various handwriting recognition systems are
being developed to facilitate interactions with CAS which could also be used in CAA, such
as that of [4]. The option used by the majority of existing CAA systems is for the student
to adopt a typed linear syntax usually identical or very close to that of the underlying CAS.
The issue of entry of mathematical expressions is an important but logically separate topic
from that we consider here, and is addressed in [13].
Simply being able to make a mathematical distinction does not mean that feedback must,
or should be given. We do not comment on what the teacher should do, rather we only
indicate the distinctions which could be, or are likely to need to be, made. For example,
while it may be perfectly acceptable for an engineering student to write the roots of x2 − x − 1
as x = 1.618 and −0.618, we would naturally expect an exact answer from a mathematics
undergraduate. Exactly what are considered appropriate outcomes are context dependent,
and hence will vary. However, being able to make such distinctions is a necessary precursor
to being able to provide educationally useful feedback.
2
Methodology
We begin with an examination and description of what is taught and assessed in elementary
algebra. There is certainly not an unique way of teaching elementary algebra, however there
is surprising uniformity in the approaches taken by contemporary text books and curricula.
This uniformity is reflected in both the ordering of material, and the activities (i.e. questions
or problems) required of the students. We do not wish to dwell on larger scale curricula design
issues, since our research pertains to automatic assessment of individual questions. Hence,
we have taken [2] as a summary of what might be viewed as the mainstream approach. This
is supplemented by questions from the algebra textbook [3]. While these two works are
broadly in agreement differences do occur since [3] is designed as a revision aid. As a specific
example, expansion of brackets and factorization are taken together which is not usual in a
first encounter with elementary algebra.
The methodology adopted is to work through these two sources systematically, matching
4
questions in [3] with topics and concepts in [2]. For each grouping of elementary algebra
tasks, exemplar questions have been selected from [3] and the following considered.
• What is being asked by the question?
• What mathematical processes does the student employ?
• What criteria are being used to mark an answer? i.e. what are legitimate answers and
how do we recognize them?
• What distinctions are possible for answers which are partially correct?
Following from this theoretical analysis, examples of each of the questions from [3] were
automated using the STACK CAA system. Looking at these questions and implementing
marking schemes with CAS code highlights the issues and focuses the discussion. While the
focus is thus naturally on the CAS Maxima, in Section 4 we shall draw on examples from
Maple and other CAS which have been employed for assessment by the author. While Maxima
underlies STACK the results are presented in such a way as to highlight the essential issues
which are likely to be encountered in mathematical CAA, regardless of which implementation
platform is adopted.
3
Results
In this section we give examples of questions from elementary algebra which illustrate the
issues involved when using CAS to assess answers to them in the CAA context outlined above.
In each case the question numbers refer to [3], where for example 3.2a would be Section 3,
question 2a. Algebra is an extension of arithmetic and so it is natural to begin elementary
algebra with arithmetic questions.
Question 1.1. Express each of the following as a fraction in its simplest form.
a)
20
45
b)
16
36
c) −
42
21
d) · · ·
The mathematical task is to remove the highest common factor from each integer in the
fraction. The student is expected to factor the numerator and denominator, cancel common
factors and if necessary multiply the remaining factors together to fully simplify the fraction.
Essentially this is an algorithmic process. To recognize the answer the teacher is looking for
either a single fraction with particular numbers in the numerator and denominator, or the
special case where the denominator is 1, which should not be written as a fraction. In the case
of a rational number greater than one, it may not be clear to the student if ‘simplest’ means
a single fraction or a mixed fraction. Indeed, [2] separates the cases of Reduction of fractions
to lowest terms from Change of improper fractions to mixed fractions and vice-versa.
3
Question 1.3a. Calculate 45 × 16
.
The mathematical task here can be thought of in two stages. The first is to perform a
multiplication calculation, the second is to simplify the result of this calculation, essentially
replicating the task of Question 1.1. Alternatively the student might consider the question
as presenting a partial factoring of the result and work from there, canceling first before
performing a multiplication.
5
These two questions typify many in which the teacher expects the student to (i) perform
an algorithmic numeric calculation and then (ii) express the result in a particular simplified
exact form.
There are other purely numerical tasks. When asking for a prime factorization of an
integer, the expected form of the answer is a product of powers preferably in ascending
order. When finding the highest common factor or least common multiple of two integers,
the expected answer is an integer.
Algebra is signified with the introduction of indeterminate quantities, e.g. x, into expressions. The task of substituting numbers for letters appears algebraic, but from the point of
view of automatic assessment this immediately reduces to numerical work. The beginning of
algebra proper occurs with the manipulation of algebraic expressions. While [2] begins with
collecting like terms as an explicit task, [3] omits exercises on this topic and moves straight
to manipulation of indices. For example.
Question 2.1. Simplify the following algebraic expressions.
a) x3 × x4
b) y 2 × y 3 × y 5
c) · · ·
The object of the question is to write an algebraic expression in a particular syntactic form.
The task to be performed involves applying the rules of indices, which include an arithmetical
calculation similar to that above. Negative or fractional powers can add complexity, without
fundamentally changing the question from the point of view of automatic assessment.
Surds, and their interaction with arithmetic operations including powers,
√ √ appear
√ at this
stage. First is the simplification of products and quotients of surds (eg 2 3 = 6). With
indeterminant quantities the teacher needs to make some choices about what simplifications
√
are permissible. In the early stages √
of elementary algebra it is usual to assume x2 = x. For
√
example, [2] gives the general rule n am = ( n a)m , and [3] confirms this by giving 5y as the
answer to the following question.
p
Question 2.13e. Simplify: 25y 2 .
Similarly, when examining an answer a decision needs to be taken about whether fractional
indices or surd notation is acceptable or perhaps required, depending on the purpose and
phrasing of the question itself.
The next significant topic is that of expanding algebraic expressions. For example
Question 3.1. Write the following expressions without using brackets:
· · · f ) (am)n
g) (a + m)n h) · · ·
We shall include within the discussion quadratic and more complex manipulations such
as the following.
Question 3.1. Write the following expressions without using brackets and simplify where
possible:
a) (2 + a)(3 + b) b) (x + 1)(x + 2) c) · · ·
Again, the question is asking the student to manipulate an expression and to write it
in a particular syntactic form. In this case as a sum of terms, rather than as a product,
which is to say expanded. Fundamentally the student has to distribute multiplication over
addition. Doing this may entail sub-processes of multiplication of numbers, gathering like
terms (i.e. counting or addition), re-ordering and using rules of indices. Indeed all of the
algebraic processes a student has learned up to this topic may be subsumed, in the sense of
[7], into this task.
6
The primary criteria are that the student’s expression is algebraically equivalent to that
in the question and that it has been written as a sum, where each summand is fully simplified,
perhaps a product of numbers and indeterminant quantities. Numerical calculations should
be performed, fractions rendered to their lowest terms and any rules of indices applied. In
addition, traditionally terms are ‘gathered’ and ‘sorted’. For a single variable polynomial
powers usually ordered with the highest first. Hence, recognizing a fully correct answer
requires the application of a number of quite separate criteria.
The reverse process mathematically to expanding out expressions is that of factoring.
Rather then consider this immediately, the topic of algebraic fractions occurs next. The
numerous cases of operations distinguished by [2] can be summarized by analogy to those
with numerical fractions. For example, canceling terms, adding, multiplication and division
and so on. In all these cases the answer is (i) a single fraction algebraically equivalent to that
in the question, and (ii) fully simplified. From the point of view of computer aided assessment
there is little technical variation. Again, a number of separate criteria are applied.
1
2
Question 4.7a. Simplify x+2
+ x+3
.
So far we have dealt only with the manipulations of expressions. Either we have performed
some calculation or manipulated an algebraic expression to render it into a particular syntactic
form. Next we shall deal with mathematical equations. Both [2] and [3] distinguish two types:
solving linear equations and the transposition of formulae.
Question 6.15. Solve the equation 2(y + 1) = −8.
nE
.
Question 7.4. Make n the subject of the formula J = nL+m
In the former the answer is a number, whereas in the latter we would expect an algebraic
expression. The student could state the answer either on its own, or in the form of an equation,
such as y = −5. We have already commented on how a teacher recognizes these forms, and
so there is little new from the point of view of computer aided assessment.
Another important topic is solving simple simultaneous equations. Since we would expect
a list of equations, e.g. y = 1, x = 4, recognizing answers to such questions is a small
incremental step from linear equations.
The next topic of importance is factoring algebraic expressions. Essentially this involves
changing a representation where the first operation is addition, to one in which the first
operation is multiplication and work naturally starts with simple linear factors. For example
Question 3.5a. Factorise 5x + 15y.
Again, such questions ask the student to re-write an algebraic expression in a particular
syntactic form. Much like when expanding out, we need to confirm both algebraic equivalence with the expression given in the question and the form of the answer. Here we make
comparisons in the context of commutative addition and multiplication operations, although
the student is expected to perform any calculations.
Work on factoring linear expressions moves onto factoring quadratic expressions. The
processes used by the student may not be more sophisticated that ‘spotting’ the correct
answer, or even a ‘guess and check’ method. Traditional algebra books contain large numbers
of practice questions, subdivided into a variety of cases. From the point of view of automatic
marking there is little difference between these.
Question 3.7b. Factorise x2 + 6x − 7.
Once factoring of quadratics has been taught, algebra opens up in a number of possible
directions. Hereafter there is little if any uniformity between textbook authors in the order of
topics. The most important topic is solving quadratic equations. Of course, solving quadratics
7
can be, and often is, taught by three methods: (i) factoring; (ii) completing the square; or
(iii) using the formula.
Question 8.1. Solve the following equation by factorisation x2 − 3x + 2 = 0.
Here the answer consists of one or two solutions, either integers or real numbers involving
quadratic surds. Hence, recognizing solutions relies on the ability to recognize numbers.
Solving higher order polynomial equations naturally follows, although from the point of view
of recognizing answers using computer aided assessment there is almost no difference from
the quadratic case.
Question 10.4. Solve the equation x4 − 2x2 + 1 = 0.
Question 10.5. Factorise x4 − 7x3 + 3x2 + 21x + 20 given that (x + 2)2 is a factor.
At this stage the ability to factor quadratics allows one to revisit algebraic fractions of
more complex types, such as the following.
x+2
Question 4.17. Simplify x2 +3x+2
.
Surds may also be revisited with a view to rationalization, to remove surd terms from the
denominators of fractions. Again, a particular syntactic form is required.
Partial fractions may perhaps also be introduced around this time.
x−3
Question 11.4.a. Express the following in partial fractions x2 −2x+1
.
It is the final form of the answer which is important here. The extent to which the
numerator of the fraction needs to be ‘simplified’ to satisfy the teacher, is of course, a matter
of taste.
3.1
A question taxonomy: syntax; calculations; properties
The above analysis of elementary algebra questions has highlighted three broad categories of
questions.
The first category are questions which ask the student to manipulate the written syntax
of a given algebraic expression. Essentially, the mathematical object remains algebraically
equivalent to that in the question and the student is expected to manipulate this. Forms of
numbers include prime-factored, rational numbers can be in single fraction or mixed fraction,
with all rational parts in lowest terms. Fractional indices can be converted to surds, or vice
versa. A teacher needs to take a view on whether decimals are acceptable when they are
exact (eg 0.5 for 12 ) or approximations (eg 0.333 for 13 ). The accuracy of any acceptable
approximations also needs to be considered. Any particular requirements for the form of
numbers applies to those occurring in algebraic expressions. Important algebraic forms are
gathered (including rules of indices), sorted, expanded, factored over the integers, reals (and for
completeness the complex numbers). Forms for algebraic fractions mirror those for rational
numbers: lowest terms, single fraction or mixed fraction. In addition we have partial fraction
form and rationalization of surds. A special form unique to quadratics is the completed
square. Rearranging an equation may be included within the category of syntax questions.
For example, solving a linear equation is essentially a syntax issue since the process involves
manipulating the equation by performing identical operations to both sides of the equal sign.
Here the student may be expected to give the answer in a form such as x =?.
The second category are questions which asked the student to perform a calculation. Here
a functional characteristic of the elementary arithmetic operations are used by the students.
For example, adding two rational numbers is a calculation. However, reducing the result of
this calculation to its lowest terms is itself a syntax question applied to the intermediate step
in this calculation. Calculation can and does occur when manipulating algebraic expressions
8
to render them into a particular syntactic form. However, the primary or explicit purpose
of the question is not to assess the ability to calculate. This illustrates the particular serial
nature of learning mathematics, where earlier topics become prerequisites for later ones.
The third category asks the student to establish some property of a number or algebraic
expression. Finding the highest common factor of a pair of integers, or the set of prime factors
is included here. Also, questions which ask the student to identify part of an expression. For
example, selecting the denominator from a fraction or selecting the coefficient from the highest
order term in a polynomial. More typically we might require the student to find the roots of
an equation. In such a question the form of the answer may have no obvious relationship to
that of the question. In the case of solving a quadratic, for example, the question contains
an equation, but the answer is a list or set of solutions which may be numbers or equations
of the form x = 1.
This taxonomy has been found to be useful when implementing questions in the STACK
computer aided assessment system, particularly when deciding on feedback and issues of
partial credit. For example, in syntax questions it may be appropriate to condone calculations
which have not been fully performed: gathering like terms in a complex expansion for example.
Similarly there is a loose hierarchy of simplifications and, as we shall see in the next section,
in a CAS all of these are performed in a single step. When providing feedback it is useful to
be able to decompress these processes somewhat. In a calculation, one might condone a lack
of full simplification. When asking for properties it may be permissable to condone answers
which are not fully simplified, or in which all calculations are not fully performed. Hence,
the author has found these distinctions to be quite useful when deciding which automatic
simplification regimes to apply to the answer of a particular question when implementing
them in CAA.
While we are discussing taxonomies for describing algebraic tasks, it is worth noting that
in reading through textbooks as part of this study, the word ‘simplify’ is often synonymous
with ‘do what I’ve just shown you in the previous worked example’. It is even possible to
find occurrences where the instruction to perform both directions of a reversible process is
‘simplify’ in different sections of a particular book. These are rare in elementary algebra, but
are more common in trigonometry or work with logarithms, which are not included here. By
adopting more explicit instructions the students can begin to build a vocabulary for describing
mathematical objects, and the syntactic forms they take.
4
Discussion
In the previous section we have mapped out the considerations which need to be made when
assessing answers to elementary algebra questions. In this section we comment in detail on
the ability of the CAS Maxima to implement these, and briefly compare Maxima to other
mainstream general purpose CAS.
Fundamentally the task in computer aided assessment is to establish the properties of
an expression provided by the student. The prototype property is when this expression
is the same as an algebraic expression supplied by the teacher. Either we compare the
student’s expression with the ‘correct’ answer, or establish that it is the same as something
not quite correct in order to be able to provide formative feedback. From the strict computer
science point of view, ‘same’ can have only one meaning: two objects have identical internal
representations in the data structure of the CAS. As a result of this requirement all CAS
9
perform simplifications, which is the process by which a representation is transformed into
a canonical form to enable such comparisons to take place. However as we have already
established, from a pedagogic standpoint there are multiple senses of ‘same’. For example,
there is algebraic equivalence and that two expressions have the same syntactic forms. So, for
example, we might wish to say that a student’s expression is correctly in completed square
form, the numbers occurring in the expression are correct but are themselves in the wrong
form, e.g. decimal approximations rather than fully simplified rational form.
The default behaviour of Maxima, like all general purpose CAS, is to automatically perform simplification and render expressions into a canonical form. This includes performing
all numerical calculations and simplifying all rational numbers to their lowest terms whenever
they appear in expressions. The evenness of cosine is used to simplify cos(−x) to cos(x). For
this application such behaviour is quite inappropriate, since we need to deal with what the
student actually gave, rather than the canonical form of this expression.
In Maxima, simplification can be switched off using the command simp:false, where
in Maxima’s language the colon is used to signify the assignment of a value to a variable.
Maxima is unusual in its ability to suppress automatic simplification completely, and few
other CAS have this ability, making is difficult or impossible when using them to distinguish
between potential responses to elementary algebra questions.
We begin the specific and detailed discussion with Maxima’s representation of numbers.
The following remarks might appear to be trivial wrinkles, and indeed they do not bother
the intelligent user employing CAS as a sophisticated calculator. However, they need to be
resolved for the application of CAA to avoid incorrect marking of student’s work.
Firstly, in Maxima, the unary minus is a prefix operation. Hence, a sum such as 3 − 4 is
represented internally1 as
((MPLUS) 3 ((MMINUS) 4))
literally as 3 + (−4). Similarly, a division such as ab is represented internally essentially as
a × b−1 . Both design decisions result in fewer operations, and remove potential duplication
of operations, significantly reducing the technical complexity of the internal simplifier.
These design decisions have implications when trying to recognize numbers in the absence
4
of simplification. For example, −4
3 and − 3 have quite different internal representations.
4
Hence, without any simplification −4
3 and − 3 are not equal, where here we take equality
to be that of the internal data structure representing the mathematical objects. Complex
numbers are similarly not single entities a + ib, but rather an expression involving addition
and multiplication. Floating point numbers are considered to be a different data type.
Maxima is not a strongly typed language which has distinct advantages over say Maple or
Axiom for CAA. For this application a CAS should tolerate mixed data types within a single
algebraic expression. For example, it is common for students to write expressions containing
both floating point and rational coefficients, such as x3 + 0.5, even if discouraged to do so.
Many CAS have a data structure which only allows a single type of coefficient, and coerces
all to a single type. The example above would be coerced to 0.33333x + 0.5 in Maple, Axiom
and some other systems. Maxima is unusual in allowing mixed types in expressions and when
a coercion is unavoidable the type of the result can be controlled. The default is to coerce to
float, although floats can be coerced to rational numbers using the command ratsimp().
We assume from this point onwards that tests can be applied that flag up the following
issues when they occur. Outcomes can then be assigned by the teacher as appropriate.
1
The command ?print(ex) can be used to view Maxima’s internal representation of the expression ex.
10
1. Whether arithmetic calculations with numbers have been performed by the student.
Hence, no arithmetic ‘simplification’ is required by the CAS.
2. Whether rational numbers are expressed in the lowest terms.
3. Whether floating point representations are used and whether these are exact.
With surd/radical
terms design decisions need to be made. Firstly, automatic simpli√
fication of x2√can result in three different results. If we assume the
√ variable is real and
2
non-negative, x = x, and if √
we assume only that x is real then x2 = |x|. When we
make no assumptions about x, x2 should√remain un-simplified, as it does in Maple. The
default behaviour of Maxima is to simplify
x2 = |x| which can be overridden by a command
√
2
such as assume(x>0), in which case x is simplified to x. For elementry algebra the second
option is more useful, although both are necessary for learning and teaching. Within Maxima
even finer degrees of control over the simplifier can be achieved via the variable radexpand.
The second design decision is simply a matter of how to display expressions involving radical
√
signs. Whether a traditional
symbol is used or fractional powers given instead can be
controlled with the Maxima variable sqrtdispflag. Flexibility such as this is invaluable for
this application.
It is questions which ask the student to expand an expression which present the first
significant challenge. For example, we would like to turn off any automatic simplification so
that a distinction between
x2 + 3x + 2,
and x2 + 2x + x + 2
can be drawn. In this latter expression terms have not been gathered, which automatic
simplification by the CAS would perform. However switching off all simplification does not
suffice. By default in Maxima variables in an algebraic expression are reverse lexicographically
ordered by the simplifier so that a+b would be simplified to b + a. Comparing a + b with b + a,
literally
if a+b=b+a then true else false;
results in true. However, if simplification is switched off this comparison results in false.
As a result, if we assume that we have simp:false, then an automatic comparison procedure
will reject perfectly good answers, such as 1+ x which when treated as a polynomial is usually
represented as x + 1.
To automatically recognize answers we would like some of the algebraic properties, but
not the functional properties of the arithmetic operations to be active. That is to say, we
would like to assume that addition and multiplication are commutative and associative and
perform our comparison in that context. However we do not want to actually perform any
additions or multiplications.
Currently all mainstream CAS treat basic arithmetic operations as ‘verb’ forms, rather
than ‘noun’ forms. For example, 1 + 2 is a calculation to do rather than representing an
addition. The work of [5] develops this distinction via the notion of a procept to capture the
dual process/concept nature of basic arithmetic operations. They comment on the ambiguities
in using the same symbol, e.g. +, for both as follows.
By using the notation ambiguously to represent either process or product, whichever
is convenient at the time, the mathematician manages to encompass both — neatly
side-stepping a possible object/process dichotomy. [5]
11
The supremacy of the verb over noun forms is quite natural when one remembers the origins
of CAS as a sophisticated calculator. However this does not accurately reflect mathematical
usage and furthermore, for our application, we need a much finer degree of control than is
currently available.
The approach taken in STACK is to define parallel noun forms of addition and multiplication operators, and define these as commutative and associative operators. These are
indistinguishable from the usual forms to the student, but their internal behavior is quite
different. The provision of such operations gives the teacher a much needed and finer degree
of control.
Interestingly, many computer algebra systems acknowledge the need for noun forms of
calculus operations. For example, in Maple a capital initial letter is used to denote an inert
differentiation or integration operation. e.g. Diff(y,x). Such a mechanism is necessary
when entering a differential equation for example. After all, here we certainly do not want to
perform the differentiation with respect to x, say, on a symbol y. Unfortunately, the provision
of matching noun and verb forms does not appear to the author to have been implemented
with a consistency which would allow them to be extended to the elementary arithmetic
operations in a natural way. Indeed, their presence seems to be the result of evolution to
allow, for example, differential equations to be expressed, rather than conscious planning
from the proceptual point of view.
Practical experience also suggests that selective simplification is invaluable. One particular
example is identity simplification in which x + 0 is simplified to x and 1 × x to x. Similarly
at this elementary level 0 × x should simplify to 0, and we assume that x1 = x and x0 = 1.
In elementary algebra there are a large number of forms of algebraic expressions. As part
of the CAA system we need functions which will establish that a particular expression is of
this form. Ideally, where an expression is not of a correct form some feedback should be
generated for the student. These functions are not available in general purpose CAS, and as
an example of those developed for the STACK system we consider in detail how to establish
that an algebraic expression is factored over the integers.
Recognizing that an expression is factored is quite different from using a CAS to factor an
expression over, say, the integers. Hence, it is not sufficient to compare a student’s answer with
the expression obtained from the factor function provided by a CAS, even if this comparison
is made in the context of commutative multiplication and addition. To illustrate this in more
detail consider the following factored forms of x2 − 4x + 4, to which a CAA system would
need to respond.
(x − 2)2 ,
(2 − x)2 ,
(x − 2)(x − 2),
(2 − x)(2 − x),
x 2
and 4 1 −
.
2
(1)
We assume here that simplification is off, so that repeated terms in a product are not automatically converted by the CAS to powers. There are additional complexities with even
powers which do occur even when asking students to factor a quadratic expression, as can be
seen in the example above.
The most effective approach is an elementary one: an expression is factored if it is the
product of terms, or powers of terms, each of which are irreducible over a particular field.
Hence it is necessary to pick apart the structure of the expression to establish this. We might
also establish algebraic equivalence of the student’s and teacher’s expressions, however that
is logically quite separate from deciding that an expression is factored.
12
Figure 1: Feedback from the STACK CAA system
Such a function might also return feedback, perhaps highlighting which terms in a product
could be further factored as required. For example, a student may well partially factor an
expression, specifically they may not pull out an integer factor as in
(3x − 6)(x + 5)
vs
3(x − 2)(x + 5).
(2)
One could well argue that both of these forms are legitimate factored quadratics, where the
focus is on the quadratic nature, not the linear integer component. If the teacher wished to
automatically provide tailored feedback to the first expression, CAS tools need to be adapted
for this purpose. This has been done in STACK, and is illustrated in Figure 1. While it is
not claimed that this feedback is optimal, it illustrates what is possible automatically.
The problems of recognizing when an expression is in either partial fraction or rationalized
surd form are very similar to that of recognizing a factored expression. Here again we need
functions which pick apart an expression. Similar issues occur with even powers, and with
pulling out integer factors. The unary minus also causes significant problems, where we might
have expressions such as
1
−1
1
,
, or
.
−
1−x
1−x
x−1
5
Other mathematical topics and associate CAS issues
This paper has concentrated exclusively on algebraic considerations. However trigonometry is
another important topic encountered in elementary mathematics. All mainstream computer
algebra systems assume radian angular measure. This design decision has implications for input, processing and display of mathematical expressions involving elementary trigonometrical
functions. In terms of automatic simplifications, most mainstream CAS understand that sine
is an odd function, and cosine is an even function. Hence, they simplify expressions such as
cos(−x) to cos(x). For CAA the ability to control such simplifications would be advantageous.
A similar issue is related to the design decisions for logarithmic and exponential functions.
While issues relating to input syntax have been dealt with in [15], briefly, all mainstream
computer algebra systems assume log(x) refers to the natural logarithm, and no system by
default accepts the input string e^x as the exponential function. Maxima and Maple both
use exp(x).
The topic of sets is often included at an elementary level. Since duplicate elements are
removed from a set automatically by the computer algebra system there is an implicit notion
of when two elements are considered to be the same. As we have seen this is not quite so
13
simple as it first appears. The ability to impose a various notions of sameness when removing
duplicate elements from a set would be advantageous for this application.
6
Conclusions
This paper seeks to explain the difficulties of using a general purpose computer algebra system
to assess elementary algebra questions. These issues are mostly associated with very simple
mathematics, and in particular when making very fine grained distinctions in elementary
algebra.
If designers of CAA choose not to adapt an existing CAS, but rather write a bespoke
library of mathematical functions for this application then all the issues raised in this paper
will still need to be resolved. Certainly for more advanced work CAS has proved to be
powerful, reliable and useful. Hence, the benefit gained from using an existing CAS far
outweighs the difficulties described in this paper. Maxima’s weak concept of data type; the
ability to suppress all simplification; and the ease with which new operations can be included
make it particularly suited for this application.
The STACK CAA system has been built upon the basis of the principles described in this
paper and a number of tests and functions developed to implement the kinds of questions
detailed in Section 3. Given the complexities of the subject, it is probably the case that
writing reliable questions is a job for an expert. Certainly with the inherent complexities any
question author needs to understand what choices are available to them, and what decisions
have already been made. STACK has been designed very much with the teacher in mind, and
details of the question authoring process are given in [14] where colleagues are the ‘neglected
learners’ referred to in the title.
This system has been used successfully with groups of approximately two hundred undergraduate mathematics students with weekly assignments since September 2005. Ongoing
work will ensure the full implementation noun forms of arithmetic operations in Maxima,
and mechanisms to control simplification to facilitate assessment of all areas of elementary
algebra.
References
[1] H. Ashton, C. E. Beevers, A. A. Koraninski, and M. A. Youngson. Incorporating partial
credit in computer aided assessment of mathematics in secondary education. British
Journal of Educational Technology, 2005.
[2] T. Barnard. A Pocket Map of Algebraic Manipulation. The Mathematical Association,
1999.
[3] A. Croft. An Algebra Refresher. LTSN Maths, Stats & OR Network, The University of
Birmingham, Birmingham, B15 2TT, 2002.
[4] M. Fujimoto and M. Suzuki. Infty Editor — A Mathematics Typesetting Tool with
a Handwriting Interface and a Graphical Front-end to OpenXM Servers, volume 1335
of Computer Algebra – Algorithms, Implementations and Applications, pages 217–226.
RIMS Kokyuroku, 2003.
14
[5] E. M. Gray and D. O. Tall. Duality, ambiguity and flexibility: A proceptual view of
simple arithmetic. Journal for Research in Mathematics Education, 26(2):115–141, 1994.
[6] D. Guin, K. Ruthven, and L. Trouche. The Didactical Chellenge of Symbolic Calculators: Turning a Computational Device into a Mathematical Instrument, volume 36 of
Mathematics Education Library. Springer, 2005.
[7] D. Hewitt. Mathematical fluency: the nature of practice and the role of subordination.
For the learning of mathematics, 16(2):28–35, 1996.
[8] S. Klai, T. Kolokolnikov, and N. Van den Bergh. Using Maple and the web to grade
mathematics tests. In Proceedings of the International Workshop on Advanced Learning
Technologies, Palmerston North, New Zealand, 4–6 December, 2000.
[9] M. Mavrikis and A. Maciocia. Wallis: a web-based ILE for science and engineering
students studying mathematics. In Workshop of Advanced Technology for Mathematics
Education in the 11th International Conference on Artificial Intelligence in Education,
pages 505–512, Sydney, Australia, 2003.
[10] L. Naismith and C. J. Sangwin. Computer algebra based assessment of mathematics
online. In Proceedings of the 8th CAA Conference 2004, 6th and 7th July, The University
of Loughborough, UK, 2004.
[11] J. F. Nicaud, D. Bouhineau, and H. Chaachoua. Mixing microworlds and CAS features
in building computer systems that help students learn algebra. International Journal of
Computers for Mathematical Learning, 9(2):169–211, 2004.
[12] P Ramsden.
Fresh Questions, Free Expressions:
METRICs Web-based
Self-test Exercises.
Maths Stats and OR Network online CAA series
http://ltsn.mathstore.ac.uk/articles/maths-caa-series/, June 2004.
[13] P. Ramsden and C. J. Sangwin. A liberalised mathematical syntax for computer-aided assessment. In Proceedings of the International Mathematica Symposium, Perth, Australia,
2005.
[14] C. J. Sangwin and M. J. Grove. STACK: addressing the needs of the “neglected learners”.
In Proceedings of the WebAlt Conference, Eindhoven, 2006.
[15] C. J. Sangwin and P. Ramsden. Linear syntax for communicating elementary mathematics. (Submitted), 2006.
[16] N. Strickland. Alice interactive mathematics. MSOR Connections, 2(1):27–30, 2002.
http://ltsn.mathstore.ac.uk/newsletter/feb2002/pdf/aim.pdf (viewed December 2002).
[17] M. Wester. Computer Algebra Systems: a Practical Guide. Wiley, 1999.
15