This document describes an expert system called the Expert Multiplication Diagnostician (EMD) that diagnoses children's errors in multiplication problems. The EMD system incorporates knowledge of 35 known systematic errors children make in multiplication. It uses a modular approach with 8 interconnected programs to diagnose errors while fitting within the memory constraints of microcomputers available in schools. The system interacts with students in a tutorial manner by providing problems, receiving answers, diagnosing errors, and explaining errors to help students learn.

1. InstructionalScience13 (1984) 79-92
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
79
AN EXPERT SYSTEM FOR DIAGNOSING CHILDREN'S
MULTIPLICATION ERRORS
M. ATTISHA and M. YAZDANI
Computer Science Department, UniversiO,of Exeter, Exeter EX4 4QL, U.K.
ABSTRACT
We describe a microcomputer-based system for diagnosing children's multiplication errors.
The system incorporates the knowledge base of all known systematic errors which children make.
In order to cope with the complexity of designing this computer-assisted learning package, we
use a modular approach. Chaining the modules allows us to fit the overall system on
microcomputers with limited memory capacity available at schools.
Introduction
We have earlier reported the development of a microcomputer-based ex-
pert system dealing with the diagnosis of children's subtraction errors (Attisha
and Yazdani, 1983). Here we describe a similar but more complex system for
multiplication. The greater complexity arises from the difficulty of collecting the
property database (i.e., identifying systematic errors) and of designing a compu-
ter-assisted learning system for this domain. Great care has been taken in
designing the system to make the environment in which a pupil works similar to
his usual one. For example, the notations he is asked to use closely resemble the
formal mathematical notations. But even so, more responses are needed from
the user (pupil) than in our subtraction system. To implement our design, we
adopted a modular approach, introduced here together with the logic of chain-
ing these modules to form the overall system.
Multiplication and Error Analysis
Two main methods of the multiplication operation of a n-digits multipli-
cand by a one-digit multiplier are taught at primary schools (Hargreaves, 1982).
In the first method, a distributive technique in a vertical form is used. It could, to
a certain extent, be easier for the pupil to understand.
0020 4277/84/$03.00 © 1984 Elsevier Science Publishers B.V.

2. 80
e.g. 38 multiplicand
× ~ multiplier
16 first partial product
60 second partial product
76 final answer
The second method (adopted in designing our system) is more advanced, gives a
shorter layout, and mainly involves "telescoping" two of the steps of the distribu-
tive technique into one. The pupil replaces the addition by pencil and paper of
the "1" (tens in"16") to 60 by a mental process. The above example is re-solved in
the traditional layout as follows:
38
× 2
76 product result
1 carry subscript
The pupil is then introduced to two-digit multipliers by using multiples of 10. At
this stage the pupil will not necessarily need to use two steps (for partial product)
when dealing with such problems (for example 24 × 30), but will be able to write
down the product immediately, starting with the zero in the unit column and
then carrying out the multiplication of 24 × 3 (tens) mentally to give 720 as the
final answer. Multiplying by 100 and then 200,300 etc. involves the same pattern
but this time moving two columns to the left, thus leaving the tens and units as
spaces to be filled by zeroes.
The case of carrying out multiplication of any two-digit numbers is seen as
an extension of the second method of multiplication by a single digit in the
traditional long multiplication layout. For this reason, pupils are taught to
multiply by the unit digit of the multiplier first, followed by the tens digit.
34 lead on 34 34
7 naturally to X 27 ratherthan × 27
238 238 680
680 238
918 918
X
However, as it can be seen from the above example, either method leads to the
correct answer, provided the pupils are familiar enough with place value to
realise that the figure "2" represents "2 tens."
From what has been mentioned above, it can be seen that the pupils need to
learn three processes in order to master the multiplication of multi-digit
numbers, each of which has its associated difficulties as far as the pupils are
concerned. Firstly, they should know 100 different multiplication facts in order
to be able to multiply quickly without any possible fact error. Downes and
Paling (1958) have concluded that the main difficulties in the multiplication
process arise from an imperfect knowledge of the primary multiplication facts.

3. 81
Secondly, they must know the multiplication technique; for example, knowing
the procedure of multiplying the multiplicand by each digit of the multiplier. In
the multiplication of large numbers (more than two digits), one common error
the pupils make is forgetting one of the partial products (Cacha, 1972), and
finally, the partial addition when the multiplier consists of more than one digit.
Cox (1975a), in her diagnostic test, realised that the largest number of
systematic errors in arithmetic occurred in multiplication. She also claimed that,
since the multiplication algorithm is so complex, almost every error was unique
for each child and not many identical systematic errors occurred. We believe her
prognosis is too gloomy. A study of the literature identifies 35 systematic errors
(see the Appendix) from empirical diagnostic studies made by Cox and other
researchers (Cox, 1974; Ashlock, 1976; Reisman, 1978).
Since the multiplication process discussed so far depends mainly on the
number of digits in the multiplier(i.e., the existence of partial sums) we therefore
believe that the most convenient classification of systematic errors is as follows:
1. Systematic errors due to one-digit multiplier, for example
307
x 8
406
2. Systematic errors associated with the occurrence of more than one digit in the
multiplier and no partial addition performed. The most common error of this
type occurs when the pupil always multiplies each digit of the multiplicand
(top number) by the digit directly beneath it:
327
x 241
687
The structure of the Expert Multiplication Diagnostician (EMD) from the
programming point of view is different from our subtraction system when
handling the analysis of the incorrect answer and its explanation. The EMD
system consists of eight programs. The interconnection of these programs with
each other and their execution philosophy appears in Fig. 1. The function of
each program and how they interact according to the user response are explained
below.
1. MULT: The initiation of the system is via a MULT program when the
user requests it. The first action to be taken by the system is to display instruc-
tions explaining the aim of the system in order to direct the pupil throughout the
execution of the system. The user then identifies the way of supplying the
problem (multiplicand and multiplier). Alternatively he can accept questions
provided by the system. The numbers (multiplicand and multiplier) are checked
when the pupil decides to provide the system with his own problem. The process
of checking is held in procedure DGTCHECK which involves the validation of
the digits and the size of those numbers. The validation rules accept characters

4. 82
I MRANDOML_
Program J
I
,, MHELP ', V,'~eqp'--~
I Program
---j--" j~;
Q~
J MULT I
Program
no.
tN
I ONEDM10
Program
I CONEDM11 I
Program
Program
Y error
®
(9
_r"~WODM,2
t
7 Program
J PADDM13
] Program
J ~MULTERR
I
"'] Program
1
Fig. 1. Principles of the EMD system.

5. 83
0-9 only, the multiplicand being not more than nine digits and the multiplier not
more than two digits.
The pupil is then asked to solve the problem. The typing in at this stage is
done from right to left. The system will perform the multiplication in a similar
fashion to the traditional long multiplication layout and at each stage the system
will compare the result with the pupil's result. In case of any error the system will
select one (or more) of the programs ONEDM 10, TWODM11 and PADDM 13,
depending on the number of digits in the multiplier and on whether the partial
product (if any) has been found correctly.
2. ONEDM10 and CONEDM11: These programs deal with the systematic
errors (bugs) concerning the product of a n-digit multiplicand and a one-digit
multiplier. Thereare 26 procedural bugs. The programs are called when the
system identifies a bug in the pupil's performance, by either program MULT
(when the problem consists of a one-digit multiplier) or by program TWODM 10
(when there is more than one digit in the multiplier) in order to give a
particular answer (wrong answer). If any of these answers matches exactly the
erroneous answer given by the pupil, an indication will be stored in an array,
which later is used as a code to enable the system to execute the required
remediation statement within the ERRMESS program.
3. TWODM12: This is executed by the system when the pupil's response is
incorrect. TWODM12 deals only with errors associated with a more than
one-digit multiplier with no partial product, i.e.
34
X 52
158
If the program detects such a bug, program TWODM12 will call the program
ERRMESS, otherwise the program MMULTERR is called.
4. PADDM 13: As stated in the previous section, there are some systematic
errors stemming from the fact that incorrect performance of partial addition has
occurred. This type of bug occurs when the multiplication problem consists of a
multiplier of more than one digit. PADDI3 handles all partial addition errors.
There are six error procedures in this program, taking only the necessary
addition bugs (Attisha, 1983) into consideration.
5. MMULTERR: This is responsil~le for diagnosis of any systematic error
accompanying another one (systematic error, non-systematic error or fact er-
ror). It is only called when the system fails to diagnose any consistent error. The
strategy of MMULTERR is achieved by implementing the column-by-column
technique to the particular problem with the erroneous answer. This program
gives priority to any possible systematic error occurrence. However, if the
program fails to detect a systematic error, then the errant behaviour of the pupil's
response will be considered as a fact error (whether multiplication or addition).

6. 84
6. ERRMESS: Each procedural bug has a remediation for the cause of that
bug. All these remediations are held in the program ERRMESS. It is called
whenever the system detects a bug. The purpose of ERRMESS is to explain the
cause(s) of any bug which the system has detected. It selects and displays the
exact location of each error, in terms of columns and row (partial product).
ERRMESS is called by either CONEDMll or PADDM13 only when the
system has found at least one procedural bug producing an answer correspond-
ing to the pupil's erroneous answer. It can also be called by MMULTERR.
7. MRANDOM: This program is optionally executed (on the pupil's re-
quest) to supply a problem (multiplicand and multiplier) to the user by using the
provided random number generation facility. It is only called by MULT. The
pupil is asked to specify the size of the multiplicand (not more than ten digits)
and the multiplier (not more than two digits). Exceeding this limitation will
result in the pupil being asked to try again.
Design of Pupil-Machine Interaction
A crucial stage in any education system design is that of specifyingthe form
of interaction between pupil and computer. This involves deciding exactly what
kind of information will be presented to the pupil and what sort of responses the
pupil will have to make. Depending on the chosen interactive policy, the pupil
should be able to set all the relevant parameters in any combination he wishes. In
our system we aimed to keep the form of input simple and short by minimising
the typing-in by pupils while responding to the system. An insight into some of
pupil-computer interactions is best gained by some examples.
Suppose that after reading the instruction provided by the package, the
pupil decides to ask the system to create a problem for him (multiplicand and
multiplier in a vertical form). The system will ask the user to type in the
parameters for determiningthe size(number of digits) of each number(multipli-
cand and miltiplier) in the problem. The pupil reacts by typing in one non-zero
digit for each number (see Fig. 2 in which a pupil's responses are shown in bold
characters).
*HOW MANY DIGITS DO YOU WANT
FOR THE MULTIPLICAND? 2
AND FOR THE MULTIPLIER? 2
Fig. 2.
The pupil is then asked to solve the problem in the same environment to which he
is used and to indicate whether he has finished entering the partial product, if any
(Fig. 3).

7. 85
87
43
32
*MULTIPLY THESE NUMBERS
Fig. 3.
If the pupil fails to give the correct answer, the system will then analyse the
answer(s) and attempt to diagnose his misconception. The example given in
Fig. 3 shows that the pupil has had difficulty in handling the "carry unit"
sub-procedure. The following explanation will appear to him on the screen as a
remediation of his misconception:
1 -YOU MULTIPLIED THE NUMBER IN THE MULTIPLICAND BY THE NUMBER
DIRECTLY BENEATH IT IN THE MULTIPLIER AND YOU WROTE DOWN THE CAR-
RIED NUMBER, IGNORING THE UNITS NUMBER.
This explanation has been adopted, since we believe that in the first column he
performs 3 × 7 to get 21, and then writes down the "carry" (which is 2 in this case)
thus ignoring the units digit. He then multiplies 4 X 8 to get 3 for the second
column.
When the pupil is told his mistake, he has the choice of making another
attempt. This can easily be done by answering "Y" for the question given in
Fig. 4. The previous problem may be skipped by entering"N" in answer to the
above question. These choices give the pupil a high degree of freedom with
minimal response.
87
× 43
261
3480
3741
** THESE ARE YOUR**
** NUMBERS AGAIN **
* MULTIPLY THESE NUMBERS
FINISH PARTIAL PRODUCT?. N
FINISH PARTIAL PRODUCT? Y
* TYPE THE FINAL ANSWER
- - GOOD... YOUR ANSWER IS NOW CORRECT = =
! IF YOU WANT ANOTHER MULTPROBLEM !
! TYPE"Y" FOR RANDOM NUMBER GEN. !
v OR"G" TO GEN. YOUR OWN NUMBER !
! OTHERWISE TYPE ANY OTHER CHAR. !
Fig. 4.
There is more than one possible way to solve a particular problem in the
multiplication operation. In essence, the pupil might solve the problem directly
without the need of partial product, e.g.,

8. 86
328
X 40
13120
or by using the traditional layout method. This forces us to slightly modify the
environment and allows a slight deviation from the normal method. In other
words, the system needs some indication from the pupil to recognise which
procedure is being used. This indication is in the form of questions and re-
sponses, after any number is typed in. Suppose that the pupil feeds the package
with two numbers for multiplication. After a checking process, the system will
ask the pupil to solve the problem. Figure 5 gives a picture of the pupil-computer
interaction to show how the pupil responds after each typing in.
ENTER THE TOP NUMBER 302
ENTER THE BOTTOM NUMBER 84
X
302 **IF NUMBERS CORRECT **
84 **TYPE Y ELSE TYPE N **Y
*MULTIPLY THESE NUMBERS
1248
24860
26108
FINISH PARTIAL PRODUCT? N
FINISH PARTIAL PRODUCT? Y
*ENTER THE FINAL ANSWER
*YOUR ANSWER IS WRONG
** POSSIBLE CAUSES OF ERROR
Fig. 5.
The system will react to the pupil's response and display the following messages
according to the pupil's responses appearing in Fig. 5 above.
1 - INCORRECTLY MULTIPLES BY ZERO PERFORMING "NXY = N".
2 - ADD CARRIED NUMBER & MULTIPLICAND TOGETHER
BEFORE MULTIPLYING.
3 - YOU MULTIPLIED BY CARRIED DIGIT INSTEAD OF
MULTIPLICAND.
These messages represent the diagnosis of the system for the pupil's erroneous
answer.
A further clarification is provided by the system to give the-exact location of
the error. The remaining responses include a choice on what path to follow after
this problem. This involves finishing the execution, repeating the problem, or
moving to another one. This is illustrated in Fig. 6.

9. ERROR IN ROW [122] RESPECTIVELY
87
T WOULDYOU LIKETO TRY AGAINTYPEY IF YES ?
N
? IF YOU WANTANOTHER PROBLEM,TYPE !
? "Y"TO PROVIDE YOU WITH THE EXERCISE v
? OR "G" TO ENTER YOUR OWN EXERCISE
OTHERWISETYPE ANY OTHER KEYTO END v
Fig. 6.
Concluding Remarks
The microcomputerised system for the diagnosis of multiplication errors
(EMD) has been designed for all the known errors. We have done this according
to our belief that more systematic errors may be discovered in the future and
such a system should be capable of accommodating those future bugs. This will
enhance the diagnostic ability and give an up-to-date database including those
bugs. We have taken great care in designing the system, controlling the execution
time and minimising the storage area taken. Although the system was complex
to design, it now offers structured features, and is simple to follow, maintain and
modify.
The system can determine what bugs, if any, underlie a pupil's misconcep-
tions. We have attempted to make the pupil's working environment as close as
possible to the pupil's usual pencil and paper writing. This has been achieved by
reducing the pupil's interaction with the computer and introducing the most
common method of handling the multiplication process. Also, the system is
designed to be flexible when handling different modes of the process (i.e., dealing
with a multiplication problem of a multiplier of more than one digit) of multipli-
cation.
We believe that some of the systematic errors in addition are not worth
including in our system since the pupils have been introduced to the multiplica-
tion operation after a concentrated effort by the teacher to help them master the
principle of the addition operation. For this reason, the systematic errors arising
from the possibility of adding two numbers, disregarding the column sums, i.e.,
adding all the digits of the question together (46 + 23 = 15) have been ignored by
the system. Such an error will only be met during the early stages of teaching
addition. Therefore we have only included those addition errors which have been
diagnosed by some multiplication operation researchers.

10. 88
The system is not designed to invent new bugs. Discovering new bugs is very
important in order to enrich our database of bugs since it is only by having a
comprehensive database of bugs that we can ascertain the general efficiency of
the system. EMD is, however, capable of accommodating any further extension
with simple modifications as a result of adding new systematic error procedures.
The system gives priority to diagnosing any single error by comparing the
given answer (pupil's erroneous answer) with the answer created by each pro-
cedural bug. If it fails to detect a single error, EMD will examine the problem
digit by digit to detect the occurrence of any concurrent errors.
In most cases a single bug accounts for each error, but there are cases where
bugs interact to produce a single surface error. Brown and Burton (1978) have
shown that, in arithmetic, pupils often have a set of bugs which interact to
produce a non-obvious pattern of error. From the EMD point of view, if these
errors interact within the same column, they cannot be detected. However, they
can be detected if they interact in different columns.
In the process of multiplication for multiple-digit numbers, there may be
two stages to deal with partial products and the addition of these partial
products. For these cases, the system will not carry on to the second stage should
it diagnose an error at the first stage. We have reached the conclusion that there is
no point in allowing the system to proceed with the second stage. Any error
discovered in the second stage may be related to the error in the first stage, which
may eventually confuse the pupil during the remediation process. There is in
addition the undesirable delay.
In fact, the techniques used by our subtraction system (Attisha and Yazda-
ni, 1983) and addition system (Attisha 1983) to generate more than one hypothe-
sis for a particular error have been excluded in the design of the EMD system.
This is because we believe that the system will rarely need it due to the fact that
the errors of multiplication are independent. These simplifications have made it
possible to fit the system on a microcomputer available in schools. The evalua-
tion of the system within a school environment is currently under way in order to
establish the relevance of such systems at a primary school education level.
Appendix
The errors currently covered by the system
Code Description Example
M10-11 When the pupil encounters a multiplication of 302
the form N X 0, he writes N as the answer. X 4
1248

11. 89
M10-12
MI0-13
MI0-14
MIO-15
M10-16
MI0-17
M10-18
MI0-19
M10-20
M10-21
M10-22
The pupil multiplies correctly in the 'ones' column
only, and he writes the remaining digits of the
multiplicand in the answer.
The pupil adds the carried digit to the
multiplicand before multiplying
The pupil multiplies by the carried digit.
When facing a zero in the multiplicand, the pupil
multiplies the carried digit to the multiplier.
The pupil multiplies the carried digit by the
remaining digits of the multiplicand.
The pupil always adds 1 as a carry regardless of
the number required
If a carry results from a multiplication, the pupil
adds 1 instead of adding the actual carry number.
The pupil adds the carried digit and multiplicand
together and the sum is placed in the answer.
The pupil renamed when it was unnecessary.
The pupil forgets to add the carried digit to the
next product.
The pupil places the entire number of the "ones"
column in the answer and does not multiply the
other digits.
313
× 3
319
327
× 6
3662
5217
X 8
43206
406
X 73
1238
30820
32058
76
X 4
144
2429
× 2
5958
514
× 7
3588
536
X 8
573
401
X 4
1704
149
× 40
4660
247
X 4
28

12. 90
M10-23
M10-24
Mll-10
Mll-ll
Mll-12
Mll-13
Mll-14
Mll-15
Mll-16
MIl-17
Mll-18
The pupil carries the wrong number by writing
the carried digit in the answer.
The pupil multiplies the carried digit instead of
adding to the product.
The pupil performs the addition operation instead
of multiplying.
No multiplication is performed. Instead, the pupil
places the multiplicand in the answer.
The pupil places the entire number of the "ones"
column in the answer without renaming, and
copies the remaining digits of the multiplicand in
the answer.
The pupil does not add the carried digit to the
product of the "tens" column.
The pupil does not add the carried digit after
performing the necessary multiplication
The pupil uses the technique of the vertical
multiplication algorithm, but instead of
multiplying, the digits are added.
The pupil multiplies the carried digit by the
multiplier.
The pupil writes the digits of the product from left
to right.
The pupil adds the carried digit to each of the
remaining multiplicand digits and writes the
result as the answer.
724
× 6
4812
29
X 3
127
725
X 3
728
2OO
X 5
200
237
X 4
2328
216
X 6
1266
574
X 8
4062
320
X 4
764
308
X 4
422
712
X 9
8064
536
X 8
978

13. 91
Mll-19
M12-10
Ml2 ll
M12-12
M12 13
Ml2-14
M12-15
M12-16
M13-10
M13-11
The pupil inserts an extra zero incorrectly to the
first partial product.
The pupil multiplies the number in the
multiplicand by the number directly beneath it in
the multiplier.
The pupil multiplies the number in the
multiplicand by the number directly beneath it in
the multiplier and thinks that N × 0 = N
When the multiplier is a multiple of ten, the pupil
does not multiply by zero in the "ones" column.
When multiplying by a multiple of 1O, the pupil
annexes too many zeros to the product.
When there is a zero in the multiplicand, the pupil
ignores that zero and multiplies the remaining
digits of the multiplicand by the number beneath
it in the multiplier.
The pupil multiplies the number in the
multiplicand by the number directly beneath it in
the multiplier and writes down the carry, ignoring
the units digit.
The pupil does not multiply by the "tens" column
of the multiplier.
The pupil multiplies correctly, but instead of
adding the partial product, the smaller is
subtracted from the larger one.
When adding the partial product, the pupil does
not rename but instead he writes both digits in the
answer.
436
× 51
4360
21300
26160
524
× 731
3564
296
× 371
676
53
× 2O
106
247
X 2O
494000
809
× 52
4018
347
X 282
631
346
X 28
2768
53
× 74
212
3710
3502
28
X 17
196
280
3176

14. 92
M13-12
M13-13
When adding the partial product, the pupil does
not add the "carried" number to the next column.
When adding the partial product, the pupil adds
the number to be carried to the same column and
writes the result in the answer.
927
X 73
2781
6489
66571
36
X 42
72
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