SHORTENED TITLE 2
Error Analysis Case Studies
Error analysis in mathematics is a well-researched set of procedures designed to help
teachers identify errors (both common and uncommon) in student problem-solving and to aid in
the design of lesson plans addressing the specific type of error(s) identified. Through careful
examination of the patterns of errors students exhibit, teachers can correct misconceptions and/or
remediate skill deficits more effectively. The general method of error analysis consists of
collecting data on a sample set of math problems, identifying any error patterns, recognizing the
reasons for those errors, and making data-based decisions on how to intervene.
Errors in mathematics generally fall into one of three categories and are related to either
student misunderstanding or a lack of knowledge. Patterns of errors can be factual, procedural, or
conceptual in nature. Factual and procedural errors are generally the result of gaps in knowledge,
while conceptual errors tend to follow from misinterpretations or imprecise understanding of the
principle(s) of the problem at hand. Procedural errors represent the largest observable set in most
student populations, although students with disabilities may be prone to careless errors. Careless
errors or errors that are very sporadic and/or do not follow a definable pattern often do not need
more than a simple prompt from the teacher to correct.
Case Studies
Level A, Case 1 – Dalton
Dalton is making a very consistent procedural error, when multiplying with decimals.
Pattern analysis indicates that his specific error borders on conceptual, as he seems to believe
that the number of placeholders in the result equals the highest number of placeholders in each
set (and not the sum of all digit-bearing placeholders). Each result had the decimal in the wrong
place, but the remainder of the facts, concepts, and procedures were mastered. Other examples of
SHORTENED TITLE 3
possible procedural errors in these types of problems include (1) forgetting to add additional
placeholders when beginning a new row in the computational steps and (2) adding the two
decimal sets versus multiplying, then adding the individual rows in the computational step.
A visual strategy, specifically modelling how to calculate the number of placeholders by
underlining all digits to the right of the decimal, may be the best approach. The student would
then be shown to count the number of underlined digits and transfer the same number of digits to
the right of the decimal in their answer. The strategy would help sharpen the focus on the
placeholders by providing a visual prompt.
Level A, Case 2 – Madison
Madison’s primary errors are factual, as she does not know the vocabulary word
“quarter”. Pattern analysis indicates that she demonstrates mastery when digital representations
of time (even “quarters” represented as :15) are provided but did not provide accurate responses
on all items requesting the “quarter to” to “quarter past” positions. Other examples of potential
factual errors include not knowing that each space between digits on the analog clock represents
5 minutes, not being clear on which hand is assigned minutes and which is assigned hours, and
which direction the clock hands are moving when judging which hour is the last that the hour
hand passed.
Building vocabulary relevant to analog clocks (specifically targeting “quarter” and
“half”) would likely prove beneficial for Madison. The lesson(s) could involve games such as “I
have, who has?” in this effort. Everyone in the class could be involved. The game can be adapted
several ways, but for Madison a descriptor such as “quarter past 2” would be on the bottom of
her playing card and a clock face with a different time would be on the top. Each student would
have different cards with different descriptors and clock faces. The teacher reads the first prompt
SHORTENED TITLE 4
and whichever student has the corresponding time on the clock on their card, gets to ask who in
the class has the new time on the bottom of their card. Students view and compare cards after
each play. New cards are drawn and the round occurs again. The strategy would reinforce the
vocabulary and help Madison (and other students) match the analog clock face to the terms.
Level B, Case 2 – Elias
Elias is falling prey to a common procedural error whereby he is placing the “carried”
digit (the “1” in sums greater than 9) directly to the left of the “ones” column in the answer
rather than aligning it above the “tens” column. Pattern analysis indicates that he does this every
time he is required to represent numbers in the sum line larger than 9. Other likely examples of
procedural errors in these types of problems include omitting the “carried” one entirely and not
keeping true to the columns in their answer.
Using a visual strategy such as a practice sheet with problems of increasing difficulty
may help this student remember where the “carried” number is placed. The early practice sheets
would need to clearly indicate where the “carried” one goes, possibly by placing a colored box
above the digits in the appropriate column and slowly fading the box on future worksheets. The
strategy would help provide a visual prompt to aid in Elias’ memory of this key step in the full
algorithm.
Level C, Case 1 – Wyatt
Wyatt’s procedural error borders on conceptual, in that he understands that both the
numerator and denominator are multiplied to attain the new fraction; however, he consistently
forgets this when the denominators are equal. Pattern analysis indicates that when he encounters
problems that have the same denominator, he will multiply the numerators but treat the
denominators as “common” as if he were adding the fractions. Other examples of procedural
SHORTENED TITLE 5
errors observed on these types of problems include cross-multiplying and adding the
numerators/denominators rather than multiplying. The latter of the possible examples could
easily be a careless error as well, although that is not as likely if the error is constant.
A visual strategy to help Wyatt could be as simple as having him rewrite the problems with two
multiplication signs (one for the numerator, the other the denominator) or he could insert them in
the original problem in a different color. This would provide him with a visual prompt which
could easily be faded after he has mastered the procedure.
Conclusion
Error analysis provides a set of strategies to help teachers identify patterns of errors in
mathematics and to assign prescriptive interventions across students. While errors can be factual,
procedural, conceptual, or simply careless, the greatest number of errors tend to be procedural.
This was true for the majority of the case studies herein. While conceptual and procedural errors
may be difficult to differentiate, conceptual errors may be far less likely than careless ones
(especially if they are sporadic rather than consistent). Collecting and analyzing samples of
student work can help teachers concentrate instruction on specific student needs more efficiently
and effectively.
