 Automated Checking of Flexible Mathematical Reasoning in the Case of
Systems of (In)Equations and the Absolute Value Operator
Antti Valmari
a
University of Jyv
¨
askyl
¨
a, Faculty of Information Technology, PO Box 35, FI-40014 University of Jyv
¨
askyl
¨
a, Finland
Keywords:
High School Mathematics, Elementary University Mathematics, Automatic Feedback, eLearning Software,
Non-trivial Reasoning Patterns, Higher Order Thinking Skills.
Abstract:
We present an approach and a tool for automatically providing feedback on solutions that involve complicated
reasoning patterns. Currently the tool supports linear systems of equations and inequations that may also
contain the absolute value operator and a restricted form of rational functions. This sufﬁces for designing
problems that are laborious to solve with standard mechanical procedures, but much easier using short-cuts that
students may ﬁnd by creative thinking. Earlier research has found that struggling with important mathematics
promotes conceptual development. Our goal is to encourage students to such struggling. A crucial feature is
to give them great freedom to choose the paths via which they solve problems, and at any time ask the tool
to check the work done so far, no matter what path was chosen. This was implemented by adopting standard
notation from mathematical logic, and developing some new logical notation. The tool has been used in a
course on elementary university-level mathematics. It has worked reliably, but there is not yet any statistics on
the pedagogical merits. The tool is expected to also support quadratic (in)equations in the near future.
1 INTRODUCTION
The book Adding It Up: Helping Children Learn
Mathematics” by the National Research Council char-
acterizes mathematical proﬁciency as consisting of
ﬁve strands: conceptual understanding, procedural
ﬂuency, strategic competence, adaptive reasoning and
productive disposition (Kilpatrick et al., 2001, p. 5).
The extent to which the development of each of these
can be supported by modern computer tools varies.
Automatic answer checkers are widely used for de-
veloping procedural ﬂuency. Conceptual understand-
ing beneﬁts from playing with tools that illustrate and
facilitate experimenting with mathematical concepts.
The present study addresses the use of our com-
puter tool to develop adaptive reasoning, deﬁned
by (Kilpatrick et al., 2001) as capacity for logi-
cal thought, reﬂection, explanation, and justiﬁcation.
(Anthony and Walshaw, 2009) observed that “Ques-
tions that have a variety of solutions or can be solved
in more than one way have the potential to provide
valuable insight into student thinking and reasoning.
We believe that such questions also have the potential
to develop the students’ reasoning skills. To illustrate
a
https://orcid.org/0000-0002-5022-1624
our idea, consider the following problem. It is from
the famous book “How to Solve It” (P
´
olya, 1945).
Solve the following system of equations.
x +7y + 3v + 5u = 16
8x + 4y + 6v + 2u = 16
2x + 6y + 4v + 8u = 16
5x + 3y + 7v + u = 16
(1)
Of course, this system can be solved by following
blindly a mechanical procedure. For instance, one
could eliminate variables one at a time by solving a
variable from one equation and assigning the solution
to the remaining equations. Doing this for x and the
ﬁrst equation produces the following:
x = 16 7y 3v 5u
8(16 7y 3v 5u) + 4y + 6v + 2u = 16
2(16 7y 3v 5u) + 6y + 4v + 8u = 16
5(16 7y 3v 5u) + 3y + 7v + u = 16
Carrying the solution through with this approach
would consist of lots of tedious, error-prone, unmo-
tivating work.
P
´
olya wanted the student not to take that ap-
proach. His book is about developing general mathe-
matical problem solving skills. His intention was that
the student should ﬁnd clever short-cuts via which the
system could be solved with much less work.
324
Valmari, A.
Automated Checking of Flexible Mathematical Reasoning in the Case of Systems of (In)Equations and the Absolute Value Operator.
DOI: 10.5220/0010493103240331
In Proceedings of the 13th International Conference on Computer Supported Education (CSEDU 2021) - Volume 2, pages 324-331
ISBN: 978-989-758-502-9
Copyright
c
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved For instance, one could start by adding the second
original equation to the third, resulting in
10x + 10y + 10v + 10u = 0 (2)
Adding the ﬁrst and last original equation yields
6x + 10y + 10v + 6u = 0 (3)
By subtracting (3) from (2) we get
4x + 4u = 0
from which obviously u = x. Assigning x in the
place of u in (2) or (3) yields 10y + 10v = 0, that is,
v = y. These facilitate immediate simpliﬁcation of
the ﬁrst two original equations to
4x + 4y = 16
6x 2y = 16
Adding them gives 2x + 2y = 0, implying y = x.
This together with 6x 2y = 16 yields 8x = 16.
We conclude x = 2, y = 2, v = 2 and u = 2.
The idea is thus to give students problems that can
be solved with little work if they use some creativity,
but are laborious or impossible to solve by blindly fol-
lowing mechanical procedures. The present paper is
about how to support this idea with automated feed-
back or automated checking of the student’s solution.
Of course, traditional automatic checking of ﬁnal
answers can be employed to check that, for instance,
in the above example, the student ended up with the
correct values x = 2, y = 2, v = 2 and u = 2. How-
ever, as such, it does not sufﬁce. Assume that the stu-
dent made a sign error and, instead of (3), got
6x + 10y + 10v + 6u = 32 (4)
Continuing like above results in x = 8, y = 0, v = 8
and u = 0. Traditional automatic checking can only
tell that it is incorrect and reveal the correct answer.
This does not give the student much indication of
where the error took place. Furthermore, the student
has spent a lot of effort in progressing from (4) to
the incorrect ﬁnal answer. This work is now wasted,
which is frustrating.
We have developed a tool with which the stu-
dent can check each step immediately after writing
it (Kaarakka et al., 2019; Valmari and Rantala, 2019).
If the student asks the tool to check (4), the tool tells
that it is incorrect. So the student becomes immedi-
ately aware of the problem and can focus on ﬁxing
it, instead of ﬁrst doing a lot of futile work in devel-
oping an incorrect ﬁnal answer. We are aware of no
other tool that supports such step-by-step checking of
both arithmetic and logical reasoning.
The students have full control. They choose the
paths via which to solve the problems: which variable
to solve ﬁrst and from which equation; or not to start
like that at all and instead add equations to each other
like was done above; or do something else. They
choose what intermediate steps to write and what to
perform only in their heads. They choose when to ask
the tool to check what they have written.
A student that is ﬂuent with a problem may pro-
ceed in big steps. If the tool accepts the step, great;
and if not, then the student may add sub-steps to make
the tool tell the location of the error more precisely. A
newcomer may proceed in very small steps, pressing
the check button after each step.
The tool is not restricted to systems of linear equa-
tions. Currently it also supports , <, , >, 6=, ab-
solute values, a restricted form of rational functions,
and a range of logical symbols including quantiﬁers.
Univariate quadratic rational inequations with abso-
lute values are being implemented. The tool works by
using advanced algorithms from mathematical logic
to check whether each step written by the student is
logically correct or not. In many cases this means that
the tool need not know in advance anything about the
problem that the student is solving. As a consequence,
a teacher may give students new problems without
feeding in any model solution or similar. This is the
case, for instance, with the problem (1) by P
´
olya.
Alternatively, the teacher may give a mathemati-
cal formulation of a problem to the tool, and present
the problem to the students in natural language. Then
it is a part of the task of the students to translate the
problem from natural language to a mathematical rep-
resentation. The formulation need not be precisely
the same as the teacher’s; it sufﬁces that it is logically
equivalent.
Of course, for this to work, there must be a no-
tation in which the students express their reasonings.
The tool uses symbols from mathematical logic and
a small number of additional symbols. The latter are
needed to express non-trivial patterns of reasoning in
a practical way.
The features described in this paper have been im-
plemented only recently. They have been technically
tested and many of them have been used in a real-
life course. There is, however, not yet any systematic
statistics from the pedagogical point of view.
In the next section we introduce the basic concepts
of our approach, only using a straightforward reason-
ing pattern. Section 3 discusses various kinds of feed-
back that our tool gives. The topic of Section 4 is
how to express complicated claims such as systems
of equations. Reasoning patterns that are not straight-
forward are illustrated in Section 5. The section con-
tains the solution to (1) developed above, in a form
accepted by our tool. Beyond the most elementary
Automated Checking of Flexible Mathematical Reasoning in the Case of Systems of (In)Equations and the Absolute Value Operator
325 |2x-1| = 5
<=> 2x-1 = -5 \/ 2x-1 = 5
<=> x = -2 \/ x = 3
Figure 1: An example input by the student.
Figure 2: The output by the tool for the input in Figure 1.
notions, the notation system for logic has not become
standardized. In Section 6 we compare our notation
to those found in textbooks. Our concluding remarks
are in Section 7.
2 BASIC CONCEPTS
The key technical concepts in this paper are formulas
and reasoning steps. Figure 1 shows a reasoning that
consists of three formulas that have been organized
into two reasoning steps. It has been written in the in-
put language of our tool. Figure 2 shows the response
by our tool to the input in Figure 1. It consists of a
copy of the input translated to standard mathematical
notation, followed by the verdict that the reasoning
was correct.
The ﬁrst formula in Figure 1 (or Figure 2) ex-
presses the equation |2x 1| = 5. We call it the origi-
nal problem, because it is what the student is expected
to solve. The roots of an equation are the values of the
unknown that make the equation hold. For instance,
the roots of |2x 1| = 5 are 2 and 3.
The last formula of a reasoning is called the ﬁnal
answer. In this case it is x = 2 x = 3. It says that
x is 2 or x is 3 ( denotes the logical or). Because it
lists the roots as the possible values of x, it is correct.
The second formula 2x1 = 5 2x1 = 5 is an
intermediate result. A reasoning that solves a difﬁcult
problem may contain many intermediate results.
In general, a formula expresses a claim about the
values of the variables in question. As was illustrated
above, the original problem is represented as a for-
mula, the ﬁnal answer is represented as a formula, and
each intermediate result is represented as a formula.
In this section, we say that a formula is correct if
and only if it is true exactly when the original prob-
lem is true. (This becomes more complicated in Sec-
tion 5.) For instance, every formula in Figure 2 is
true when x is 2 or 3, and everyone is false when x
has any other value, so they are all correct. This no-
tion is different from “tautology” and “valid formula”
in mathematical logic (they mean, roughly speaking,
that the formula is always true). The original problem
is correct by its nature, but intermediate results and
the ﬁnal answer may be incorrect. It is the task of our
tool to check whether they are correct.
A reasoning step consists of two formulas and a
reasoning operator that relates them. Until Section 5,
we only use one reasoning operator , meaning that
the formula on its left hand side is true exactly when
the formula on its right hand side is true. If this holds,
we call the reasoning step correct, and otherwise in-
correct. For instance, if x = 2, then both |2x1| = 5
and x = 2 x = 3 are true; if x = 3, then both
are true; and if x has any other value, then both are
false (and thus not true). Therefore, |2x 1| = 5
x = 2 x = 3 is a correct reasoning step.
A counter-example to an incorrect -step is val-
ues of the unknowns such that one side of the step is
true and the opposite side is not true. Every incorrect
and no correct step has a counter-example. For in-
stance, |2x 1| = 5 x = 3 has the counter-example
x = 2, because it makes |2x 1| = 5 true (indeed,
|2 · (2) 1| = 5) but x = 3 false (because 2 6= 3).
Successive reasoning steps may share a formula.
Figure 1 (and 2) shows two reasoning steps that share
the formula 2x 1 = 5 2x 1 = 5.
If every reasoning step in a reasoning of the form
“original problem formula . . . formula” is
correct (in the sense of correctness of -steps just
introduced), then every formula in the reasoning is
correct (in the sense of correctness of formulas intro-
duced earlier on). This property is known as transi-
tivity.
More reasoning operators will be introduced in
Section 5. Each of them has its own notion of cor-
rectness.
3 FEEDBACK BY THE TOOL
Figure 3 shows an example of the feedback by the
tool to an incorrect input. It says that if a = 3, then
|a 3| = 2a is false (F) but a 3 = 2a a 3 = 2a
is true (T). Indeed, if a = 3, then |a 3| = | 6| = 6
but 2a = 6, so |a 3| = 2a is false. Furthermore,
a 3 = 2a is true, making a 3 = 2a a 3 = 2a
true. That is, the tool gave a counter-example to the
incorrect reasoning step. We see that a 3 = 2a may
hold such that 2a is negative, and then |a 3| = 2a
does not hold because |anything| is non-negative. We
will discuss a correct solution in Section 4.
According to (Hiebert and Grouws, 2007, p. 383,
CSEDU 2021 - 13th International Conference on Computer Supported Education
326 Figure 3: The output by the tool for an incorrect input.
387), struggling with important mathematics is a key
feature that promotes conceptual development. They
emphasize that struggling does not mean “needless
frustration” in front of an impossible task. It means
that “students expend effort to make sense of math-
ematics, to ﬁgure something out that is not imme-
diately apparent. We believe that in many cases,
a counter-example may be a good starting point for
such struggling. For instance, the counter-example
in Figure 3 may lead the student to realize that when
|. . . | is replaced by . . . or (. . .), it may be necessary
to ensure that the substitute is non-negative.
Sometimes every existing counter-example is a
root of the original problem. This can be intentionally
caused by giving F as an intermediate result. Then
the counter-example reveals the correct ﬁnal answer
or a part of it. This is not a big issue, if the inten-
tion of the student is not to cheat but to learn. Fur-
thermore, it is possible to switch off the printing of
counter-examples.
It may also happen that incorrect thinking leads to
a correct result by accident. Consider again |a 3| =
2a. A student may plan to write a3 = 2a a3 =
2a, write a 3 = 2a, and be interrupted by a phone
call. After the call (s)he no longer remembers what
(s)he was doing. There is a 3 = 2a on the screen,
so (s)he may continue by solving it. Altogether (s)he
wrote |a 3| = 2a a 3 = 2a a = 1.
The tool accepts this as correct, because the same
solution could have arisen from correct thinking: An
experienced student sees quickly that the root of a
3 = 2a is a = 3. However, it cannot be a root of
the original equation, because it makes 2a negative.
Therefore, there is no need to write a branch for the
case where a 3 = 2a.
Of course, the tool rejects |a 3| = 2a a 3 =
2a. If the student solves many and diverse enough
exercises, the odds are that most misunderstandings
are caught sooner or later, although not necessarily at
the ﬁrst time when the student makes the error.
Figure 4 shows the reply by the tool when the stu-
dent has not developed the answer to an explicit form.
The formula x = 2 2x = 6 is correct, that is, it is
true precisely when the original formula |2x 1| = 5
is true. However, the root x = 3 is not directly shown
Figure 4: An incompletely solved example.
in the formula, but only indirectly via 2x = 6. Be-
cause the answer is mathematically correct, and the
problem is only that the student has not simpliﬁed it
far enough, the tool shows the remark in magenta in-
stead of red.
In the most typical case, an explicit form is x = c
1
··· x = c
n
, where x is the unknown and the c
i
are constants. Also the opposite direction c
i
= x is
allowed for the reason discussed soon. If the equation
has no roots (e.g., x = x + 1), then the explicit form
is F, and if every real number is a root (e.g., x + 1 =
x + 1), then it is T.
The set of roots may also contain intervals. This
is obvious in the case of inequations (e.g., Figure 5),
but may also occur with equations using the absolute
value operator (e.g., |x| = x x 0). Therefore, also
x c
i
, c
i
< x c
0
i
, etc., are treated as being in the
explicit form. Alternatively, c
i
< x c
0
i
can be written
using the logical and as c
i
< x x c
0
i
.
Figure 5 was obtained with the input
(2x+1)/(5x-2) >= 0 <=> x <= -1/2 \/ x >= 2/5
If the latter >= in the input is replaced by >, the tool
accepts the input as correct.
Figure 5 also illustrates that our tool uses a third
truth value “undeﬁned” (U). This is a long story, but
let us quickly mention some most important aspects.
When x =
2
5
,
2x+1
5x2
is undeﬁned, because the divisor
5x 2 is 0. (In)equalities whose one or both sides is
undeﬁned must not be treated as true, because other-
wise the corresponding value of the unknown would
be a root. Therefore, with only F and T available,
both
2x+1
5x2
0 and
2x+1
5x2
< 0 should be F when x =
2
5
.
On the other hand, we would expect ¬(
2x+1
5x2
0) to
be
2x+1
5x2
< 0, and ¬F is T (¬ is the logical not). So
2x+1
5x2
< 0 should be both F and T when x =
2
5
, which
is impossible.
Figure 5: Intervals of roots, and division by zero.
Automated Checking of Flexible Mathematical Reasoning in the Case of Systems of (In)Equations and the Absolute Value Operator
327 Figure 6: An unnecessarily complicated ﬁnal answer.
This obstacle is removed by treating
x
0
0 neither
as true nor as false, but as undeﬁned. The negation
of undeﬁned is undeﬁned as well. The law ¬(x y)
x < y remains valid even if x or y is undeﬁned.
“Formula1 formula2” still means that formula1 is
true precisely when formula2 is true. This works even
if formula1 yields U and formula2 yields F, because
neither U nor F is T. Because resides not in but
between formulas, it cannot be within the scope of ¬.
Currently the tool makes some attempts to check
that the ﬁnal answer is not more complicated than
necessary. An example of this is shown in Figure 6.
However, the tool could do much more towards this
end than it currently does. This is work in progress.
If the teacher is willing to touch the tool, (s)he can
set an upper bound to the length of the ﬁnal answer.
This has proven a very efﬁcient method of catching
too complicated ﬁnal answers.
4 USE OF LOGICAL AND
The equations in a system of equations are intended
to hold simultaneously. This can be obtained by com-
bining them with the logical and . Similarly, the sub-
formulas that tell the simultaneous values of different
unknowns are combined with . Figure 7 shows an
example of both of these. The ﬁgure also illustrates
that if a system with two unknowns has more than
one root, the alternative roots are separated with .
Let us return to the example in Figure 3. By def-
inition, |x| = x if x is positive or zero, and |x| is x
if x is negative. Therefore, the (or at least a) correct
method to solve |a 3| = 2a is to split it to two cases.
In one case a 3 is positive or zero, and a 3 is used
in the place of |a 3|. It can be written as a 3 0
a 3 = 2a. Solving a 3 = 2a yields a = 3. How-
ever, if a = 3, then a 3 0 does not hold, so 3 is
not a root in the end. In the other case a 3 is nega-
tive, and (a 3) is used in the place of |a 3|. That
is, a 3 < 0 (a3) = 2a. We have (a3) = 2a
a = 1. If a = 1 then a 3 < 0 holds, so 1 indeed is
Figure 7: Two examples of the use of .
Figure 8: A clumsy solution to |a 3| = 2a.
a root. In conclusion, |a 3| = 2a a = 1.
This reasoning can be expressed with , and
, but only in a clumsy fashion. Figure 8 shows it.
On the second, third and fourth line, the two cases
are combined with . The two equations a 3 = 2a
and (a 3) = 2a are solved in lockstep, although
it would be more convenient to solve them one at a
time. The problem is that with no other reasoning
operator available than , each intermediate result
must reﬂect the original problem in full; that is, it is
not possible to say that for a while, we focus on one
aspect of the original problem and ignore the rest. In
the next section, we introduce notation that removes
this restriction and facilitates handy representation of
many reasoning patterns.
Our tool also knows the symbols ¬ (not), (if
. . . then) and (if and only if). For instance, the
tool accepts the reasoning step ¬(x < 0) x 0.
The second line in Figure 8 can also be represented as
(a 3 0 a 3 = 2a) (a 3 < 0 (a 3) =
2a), and some people ﬁnd it more natural.
5 MORE REASONING PATTERNS
In the previous section we only used a straightfor-
ward type of reasoning, where the formulas are sepa-
rated with , and each formula must be true precisely
when the original problem is true. In this section we
discuss more general patterns of reasoning. They are
important for giving the student the freedom to choose
the path via which to solve the problem.
The key concept is logical implication, expressed
with and in reverse direction with . The rigorous
deﬁnition is that “formula1 formula2” is a correct
reasoning step if and only if all those value combina-
tions of the unknowns that make formula1 true, also
make formula2 true. For instance, Figure 9 tells that
|x| + y = 6 3x + 5y = 22 x = 4 y = 2 is incor-
rect because of the counter-example x = 1 y = 5,
but the opposite direction is correct.
There is evidence that the asymmetry of is very
difﬁcult for many students. For instance, the ACM &
IEEE curriculum recommendation speciﬁes the cor-
responding topic with exceptional detail: “Notions
CSEDU 2021 - 13th International Conference on Computer Supported Education
328 Figure 9: An incorrect implication whose converse is cor-
rect.
of implication, equivalence, converse, inverse, con-
trapositive, negation, and contradiction” (ACM and
IEEE., 2013, p. 78). A hopefully helpful way to think
of it is that if and only if “formula1 formula2”,
then formula2 contains the same or strictly less in-
formation about the values of the unknowns than for-
mula1. For instance, 3 < x 5 contains less informa-
tion about the value of x than 2 x 4, because the
latter speciﬁes the value of x more precisely.
Consider again (1), with added to the end of
its ﬁrst three lines. Intuitively, picking two equations
from among the four amounts to throwing some in-
formation away and keeping the rest. Therefore, we
should have “all four in (1) the 2nd and 3rd in (1)”.
The ﬁrst two lines of Figure 10 show that our tool is
of the same opinion. (The ﬁgure also illustrates that
the student may write comments in a solution, and the
tool copies them to the output in brown colour.)
As a matter of fact, “formula1 formula2” ap-
plies to any reasoning that does not use other situ-
ation-speciﬁc information than what is in formula1.
(General laws such as x + 0 = x are not situation-spe-
ciﬁc information.) Adding two equations only uses
information in those equations (in addition to general
laws), so also the third line in Figure 10 is correct.
The fourth line uses the keyword original to
handily refer to the original problem. It derives the
sum of the ﬁrst and fourth original equation without
showing any intermediate results. Because the formu-
las on the third and fourth line follow from the orig-
inal problem, also the result of subtracting one from
the other does. This is shown on the ﬁfth line, fol-
lowed by the solving of u in terms of x.
The solution described in Section 1 could be car-
ried all the way through only using original,
and , but we chose to use also other features, to il-
lustrate them. An arbitrary reasoning (that is within
the capacity of the tool) may be enclosed between
subproof and subend. The ﬁrst such instance in Fig-
ure 10 takes the results from the third and ﬁfth line,
and reasons v = y from them.
The second instance starts by lifting u = x v =
y to the status of a general law that is valid through-
out the instance. This makes it possible to compare
x + 7y + 3v + 5u = 16 8x + 4y + 6v + 2u = 16 to
4x + 4y = 16 6x 2y = 16 with instead of
, although only the former uses u and v. As a con-
sequence, the tool checks fully the correctness of the
elimination of u and v, while would have allowed,
for instance, 4x + 4y = 16 as such (that is, without
6x 2y = 16) on the right hand side.
This is because allows throwing any informa-
tion away. It is used to intentionally throw some infor-
mation away. However, the logic and the tool do not
know which pieces of information were thrown away
intentionally and which accidentally, so must al-
low throwing any information away.
This issue does not make the tool accept incorrect
ﬁnal answers. If the reasoning as a whole fails to es-
tablish the -relation between the original problem
and the ﬁnal answer (for instance, only establishes the
-relation), then the tool says in magenta that “Im-
plication was used without returning to the original”.
However, it can cause the error message occur much
later than where the actual error was, potentially con-
fusing students.
The second subproof in Figure 10 continues in an
already seen fashion to the value of y in terms of x, and
to the numeric value of x. Here “original” refers to
the ﬁrst formula of the subproof after the assumption.
Having derived u = x, v = y, y = x and x = 2,
the ﬁnal answer is easy to write.
We have now presented, in a form that our tool
was able to check, the reasoning in Section 1 that
solved P
´
olya’s problem via short-cuts. The reasoning
could be made shorter still by replacing the subproofs
with original -structures, but we chose to use this
opportunity to illustrate subproofs and assumptions.
Let us discuss some further examples that illus-
trate other aspects of solving equations. Figure 11
shows an example of using assumptions to express the
conditions of the cases of an absolute value operation.
In the ﬁrst branch, thanks to the assumption x 0,
we have |x| = x and thus |x| + y = 6 x + y = 6
x = 6 y. Similarly in the second branch |x| + y = 6
x + y = 6 x = y 6. The advantage of this so-
lution is that it avoids altogether. It thus avoids the
risk of postponed error messages mentioned above.
A solution by cases can be written without sub-
proofs by starting the ﬁrst case with and the re-
maining cases with original . This works because
each case contains at least the same information as
the original problem: all (in)equations in the original
problem apply, and also the (in)equation that speci-
ﬁes the case applies. Analogously to , runs the
risk of accidentally putting too much information to a
Automated Checking of Flexible Mathematical Reasoning in the Case of Systems of (In)Equations and the Absolute Value Operator
329 Figure 10: Solving P
´
olya’s problem (1).
case, causing an error that the tool detects later than
where it actually occurred.
With the next version of the tool,
x
2
9x+18
x
2
+2x15
= 0
can be solved by solving x
2
9x + 18 = 0, and then
rejecting those roots that make x
2
+2x 15 zero. The
roots are 3 and 6, of which 3 must be rejected. The
advantage is that x
2
+2x 15 = 0 need not be solved.
This solution can be presented with or a subproof.
By assuming x
2
+ 2x 15 6= 0 in the latter, can be
avoided. (All this already works on a prototype.)
6 ON ALTERNATIVE NOTATIONS
Although our use of , and agrees with fairly
common informal practice, the reader is warned that
Figure 11: Exploiting assume in a solution by cases.
textbooks on mathematical logic tend to disagree with
both our use and each other. We make a distinction
between that resides in a formula and produces a
truth value; and that resides between two formu-
las and expresses a correct or incorrect reasoning step
(and similarly with and ). Most if not all authors
introduce the former notion. Many denote it with ,
but many others use instead. Most authors avoid
the latter notion altogether, use ad-hoc notation, or
confuse it with the former notion.
For instance, (Hammack, 2018) uses , and =
in the roles of our , and (e.g., p. 51: P Q
= (P Q) ( P Q)). It apparently lacks our .
(Hein, 1995) uses similarly to us, similarly to
our , and lacks our and . (Huth and Ryan,
2004) use similarly to us, do not use , and use `
and a` somewhat similarly to our and .
Our and are relational: P Q R is cor-
rect if and only if P Q is correct and Q R is cor-
rect. This is crucial for handy writing of reasonings,
which in turn is central to our goal of supporting the
development of adaptive reasoning.
Unfortunately, the existence of different conven-
tions can lead to dramatically different meanings
for the very same expression. For instance, (Stan-
ford University., 2021, Section 2.2) uses a right-
associative in the role of our . Under that con-
vention, P Q R means the same as P (Q R),
that is, Q R is evaluated ﬁrst, and its result is then
used as the X in the evaluation of P X. Consider
CSEDU 2021 - 13th International Conference on Computer Supported Education
330 x = 2 x = 1 x = 0. When x 6= 2, it is T because
it is then of the form F something. When x = 2, it
is T because x = 1 x = 0 is then of the form F F
which is T, so the expression as a whole is of the form
T T, and thus T. That is, although in our conven-
tion x = 2 x = 1 x = 0 is just plain wrong, under
the Stanford convention it is T for every value of x!
In our convention, P (Q R) is a syntax error.
The notations original, subproof, subend and
assume are by us, but of course the ideas are next to
trivial. Together with assume, original is essen-
tially the same as Γ |= in hard-core formal logic.
The undeﬁned truth value U is almost never men-
tioned in textbooks. Its behaviour in the case of ¬,
, , and is fairly unproblematic; the tool fol-
lows (Kleene, 1952). Beyond that, in particular re-
garding and , the issue becomes complicated
indeed. A number of different approaches were dis-
cussed by (Schieder and Broy, 1999). The tool uses
our own approach (Valmari and Hella, 2017).
7 CONCLUDING REMARKS
We illustrated how our tool can automatically check
various patterns of reasoning in a restricted domain
of problems, and give feedback that the student can
use as a starting point for improving the solution. Al-
though the domain is restricted, it can meaningfully
accommodate many non-straightforward patterns of
reasoning. The example by (P
´
olya, 1945) is certainly
challenging enough to bring forward the beneﬁts of
creative thinking over blindly following mechanical
procedures. It was successfully dealt with by the tool.
It is worth emphasizing that our tool facilitates
verbal problems that the students have to translate
to (in)equations themselves. The teacher gives the
(in)equation (or anything logically equivalent, such as
the roots) to the tool as the original problem and asks
the tool to keep it hidden from the students.
An earlier feature of our tool has been found to
statistically signiﬁcantly improve performance in an
examination (Kaarakka et al., 2019). The implemen-
tation of the features discussed in this paper started
in spring 2020. Therefore, there is not yet much ped-
agogical experience. In a course, students had little
problems with the tool itself, but had big problems
with mathematics that they were supposed to already
master: absolute values, pairs of equations, and so on.
They also had big problems in modelling verbally ex-
pressed problems mathematically. Their earlier stud-
ies had not developed these skills to the promised
level.
Until now it has not been possible to make stu-
dents solve this kind of problems in great numbers,
because of the lack of teachers to check the answers
and provide feedback. Now we have a tool for this.
REFERENCES
ACM and IEEE. (2013). Joint Task Force on Computing
Curricula, Association for Computing Machinery and
IEEE Computer Society, Computer Science Curric-
ula 2013 : Curriculum Guidelines for Undergraduate
Degree Programs in Computer Science. ACM, New
York, NY, USA.
Anthony, G. and Walshaw, M. (2009). Effective pedagogy
in mathematics. UNESCO/IBE.
Hammack, R. (2018). Book of Proof. Open access.
Hein, J. L. (1995). Discrete Structures, Logic, and Com-
putability. Jones and Bartlett Publishers, Inc., USA.
Hiebert, J. and Grouws, D. A. (2007). The effects of class-
room mathematics teaching on students’ learning. In
Lester, F. K., editor, Second Handbook of Research on
Mathematics Teaching and Learning, pages 371–404.
Information Age Pub.
Huth, M. and Ryan, M. D. (2004). Logic in Computer Sci-
ence - Modelling and Reasoning About Systems (2.
ed.). Cambridge University Press.
Kaarakka, T., Helkala, K., Valmari, A., and Joutsenlahti,
M. (2019). Pedagogical experiments with MathCheck
in university teaching. LUMAT: International Journal
on Math, Science and Technology Education, 7(3):84–
112.
Kilpatrick, J., Swafford, J., and Findell, B., editors (2001).
Adding It Up: Helping Children Learn Mathematics.
The National Academies Press, Washington, DC.
Kleene, S. C. (1952). Introduction to metamathematics. Van
Nostrand, New York.
P
´
olya, G. (1945). How to Solve It. Princeton University
Press.
Schieder, B. and Broy, M. (1999). Adapting calcula-
tional logic to the undeﬁned. The Computer Journal,
42(2):73–81.
Stanford University. (2021). Stanford introduction to logic:
An online course on symbolic logic. http://intrologic.
stanford.edu/; accessed 2021-01-23.
Valmari, A. and Hella, L. (2017). The logics taught and
used at high schools are not the same. In Karhum
¨
aki,
J., Matiyasevich, Y., and Saarela, A., editors, Pro-
ceedings of the Fourth Russian Finnish Symposium on
Discrete Mathematics, volume 26 of TUCS Lecture
Notes, pages 172–186, Turku, Finland. Turku Centre
for Computer Science.
Valmari, A. and Rantala, J. (2019). Arithmetic, logic, syn-
tax and MathCheck. In Lane, H., Zvacek, S., and
Uhomoibhi, J., editors, Proceedings of the 11th Inter-
national Conference on Computer Supported Educa-
tion, CSEDU 2019, Heraklion, Crete, Greece, May 2-
4, 2019, Volume 2., pages 292–299, Set
´
ubal, Portugal.
SciTePress.
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