^{a}

^{b}

Student struggles with fractions are well documented, and due to fractions’ importance to later mathematics achievement, identification of the errors students make when solving fraction problems is an area of interest for both researchers and teachers. Within this study, we examine data on student fraction problem errors in pre- and post-quizzes in a digital mathematics environment. Students (n = 1,431) were grouped by prevalence of error types using latent class analysis. Three different classes of error profiles were identified in the pre-quiz data. A latent transition analysis was then used to determine if class membership and class structure changed from pre- to post-quiz. In both pre- and post-quiz, there was a class of students who appeared to be guessing and a class of students who performed well. One class structure was consistent with the idea that early fraction learners rely heavily on whole number principles. Identification of co-occurrence of and changes to fraction errors has implications for curricular design and pedagogical decisions, especially in light of movements toward personalized learning systems.

Educators have long been concerned with students’ struggles with fractions (

With fractions critical to later mathematics achievement, it is important to understand some of the reasons students struggle with fractions. One way to do this is through understanding the common errors students make when solving fraction and other mathematics problems (

Prior research has offered a number of reasons why students struggle with fractions: fractions are multifaceted (e.g.,

Fractions are thought to have five subconstructs: part-whole, ratio, quotient, operator, and measure—also known as magnitude (

Sub-construct | Definition ( |
Example with 2/3 |
---|---|---|

Part-whole | an object is partitioned and the fraction compares the number of parts to the total number of partitions | two out of three slices of pizza |

Ratio | comparison of two different quantities | two slices of cheese pizza and three slices of pepperoni |

Operator | function applied to number, object, or set | two-thirds of a pizza, regardless of the number slices |

Quotient | division | splitting two pizzas between three people |

Measure/ magnitude | does not fit easily into the pizza example because it represents where a fraction is placed on a number line (or how big the fraction is) |

Although many children may understand fractions as a part-whole, this concept alone is not enough for complete understanding of fractions (

The algorithms used for addition, subtraction, multiplication, and division are also more complex for fractions than for natural numbers. When adding or subtracting natural numbers, one can simply combine the digits of the same place value and regroup if necessary. However, when adding or subtracting fractions, one must first find the least common denominator, transform the fractions so they will have the same common denominator, then add/subtract the numerators.

Procedures | Natural Numbers | Fractions | Fraction Arithmetic |
---|---|---|---|

Form | Takes the form ab | Takes the form a/b | |

Ordering | Ordering number depends on comparing similar place values | Ordering depends on the relationship between numerator and denominator | |

Quantities | Discrete | Discrete and continuous | |

Addition/Subtraction | Combining digits of same place value | Use equivalent fractions with common denominator then combine numerators | a/b + c/d = da/db + bc/db = (da+bc)/db |

Multiplication | Multiplication makes number bigger | Multiplication/division makes number bigger or smaller depending on the fractions | a/b × c/d = ac/bd |

Division | Division makes number smaller | a/b ÷ c/d = a/b × d/c = ad/bc |

The typical order of mathematics instruction may compound student difficulties in understanding fraction complexity. In most curricula, students learn natural numbers and then fractions (e.g.,

Each of the aforementioned challenges to understanding fractions highlight the difficulties students may have in gaining conceptual knowledge of fractions. Conceptual and procedural knowledge have a bidirectional relationship, but are separate entities that contribute to fraction success (

It is important to understand both

There is an extant body of work identifying student errors in fraction problems and examining why students might make these errors (e.g.,

Using data from within the digital mathematics software ST Math, we identify common fraction errors, examine co-occurrence of errors to define student error profiles, and investigate the changes in profiles from before to after instruction. Specifically, we ask:

What errors are made when students solve fraction problems within ST Math?

How can students be grouped into classes by the errors they tend to make together? How do student characteristics differ between these classes?

How do these classes change from pre-quiz to post-quiz?

We expect that certain errors will be more prevalent in the pre-quiz due to a naïve understanding of fractions, such as the ratio error or whole number ordering (

ST Math, created by MIND Research Institute (MIND), is an interactive instructional software for computers and tablets that is based on theory that suggests that the ability to visualize mathematics concepts leads to better conceptual knowledge and performance (see

Within ST Math, students progress through a number of objectives focused on specific math concepts. The ST Math content follows a hierarchical pattern: objective, sub-objective, game, level, puzzle. Within the objective/sub-objective, there are a variety of games that use the same imagery and design throughout their levels. Each of these games contains between one and 10 levels that increase in difficulty. Within each level, students complete interactive puzzles, which are the delivery method for the mathematics content. For each level, the student has between one and three lives—if they answer more puzzles incorrectly than they have lives, they are removed from the level and can chose to replay it or replay a previously-passed level. Before students begin the objective, they must complete a five-question multiple choice pre-quiz on that objective’s content. After demonstrating mastery of the content within an objective by successfully completing all levels within, the student then completes a five-question multiple choice post-quiz that mirrors the pre-quiz, question-for-question in topic, but uses different specific examples and numbers. The pre- and post-quizzes have either three or four answer choices for each question (see

The third-grade curriculum for the 2015-2016 Florida version of ST Math had 23 required objectives and eight optional objectives. These objectives covered content such as place values, addition and subtraction, multiplication and division, and fractions. The three objectives covering fraction content were part of the required curriculum and began approximately 65% of the way through the curriculum.

Participants were third graders from a school district in central Florida participating in an NSF-funded project relating gameplay within ST Math to student achievement and motivation. Students within the district played ST Math as part of their normal instruction during the 2015-2016 school year. The sample (

Variable | Percent of Sample |
Percent of Total |
---|---|---|

54 | 52 | |

10 | 17* | |

46 | 58* | |

11 | 13* | |

19 | 13* | |

Race | ||

Asian | 6 | 4* |

Black | 13 | 21* |

Hispanic | 16 | 18* |

White | 60 | 53* |

Other | 5 | 4 |

*

As implemented in Florida during the 2015-2016 school year, the ST Math curriculum aligned with the Florida Standards, which introduced fractions in the third grade (

Quiz | Question Types | Example Question |
---|---|---|

match a fraction figure to a written fraction match whole numbers to their equivalent fractions (e.g., 3 = 3/1) |
||

match a point on a number line to a fraction match three fractions to points on the number line match a fraction to a point on the number line (possible points labeled with letters) |
||

match a fraction figure to a written fraction match three fractions to points on the number line identify the correct statement of magnitude comparisons
identify the correct fraction sentence select the appropriate fraction to complete the number sentence |

Data were the answer choices on the pre- and post-quizzes on fraction objectives. Only incorrect answer choices were included in the analysis of errors. Overall, there were 86 possible incorrect answer choices for the six quizzes (pre- and post-quizzes for three objectives). Error coding was completed in three key steps. First, answer choices were qualitatively coded using a priori and a posteriori codes. The a priori codes were taken from previous research on fraction errors (e.g.,

To make the error codes usable data, several steps were followed in the statistics analysis software

the student did not make the error;

the student made the error between 0% and 25% of the time (0 < x ≤ 0.25);

the student made the error between 25% and 50% of the time (0.25 < x ≤ 0.50);

the student made the error between 50% and 75% of the time (0.50 < x ≤ 0.75);

the student made the error between 75% and 100% of the time (0.75 < x ≤ 1).

Error proportions were used in analyses instead of number of errors made to control for the number of errors students made overall. Using number of errors would bias results by heavily weighting on student performance and we would lose the relative frequency of each error.

Latent class analysis (LCA) is a mixture modeling method used with categorical variables. It attributes the relationship between variables to an unobserved, latent variable (

To determine if class membership changed between time points, such as between a pre- and post-quiz, latent transition analysis (LTA) was used (

as a statistical model in which (i) latent categorical constructs are defined at two or more time points, (ii) parameters are included that assess initial status and transition probabilities from time

LTA regresses the class variable at the second time point onto the first time point to determine the likelihood of the classes remaining the same. LTA was used to determine if classes remained the same between pre- and post-quiz. The

Logistic regressions were used to determine how students differed between the classes, as was done with pre-quiz classes. Additionally, logistic regressions were used to predict the movement from pre- to post-quiz class. All logistic regressions were clustered at the teacher level to account for classroom similarities.

Three of the 10 error codes were developed a priori—illogical sizing/spacing (

The complement error occurred when the student chose the complementary fraction to the correct answer choice. For example, if the question asked what fraction represents the shaded part of a shape and the answer was 2/3, the complement would be 1/3. On average, students made this error 11% of the time in pre-quizzes and 9% of the time in post-quizzes. About 29% of students made the complement error in pre-quizzes and 23% in the post-quizzes.

The filler error encompassed answer choices that did not appear to follow a known error logic. Consider the example in

This error represented misunderstanding that fraction parts must have equal sizes and consistent spacing (

Question Type | Number of Questions | Errors Possible |
---|---|---|

Matching written fractions to visual models | 8 | complement illogical sizing/spacing incomplete information ratio reciprocal reducing fractions to whole numbers |

Making equivalent fractions | 4 | complement incomplete information reciprocal reducing fractions to whole numbers |

Placing fraction(s) on a number line | 12 | complement illogical sizing/spacing incomplete information reciprocal whole number ordering |

Comparing fractions | 6 | same numerator/denominator ordering whole number ordering |

This error had two different characterizations. The first was when the answer was partially correct but missing a critical part that would make it correct (e.g., if there was a mark placed at one third of a number line but the number line was zero to two, the student might have answered 1/3 when in reality the answer was 2/3). The second was when the answer matched only part of the question. For example, if the student was asked to identify the mixed number 1 3/4, a student making this error would answer only 3/4 or only 1, missing the totality of the original prompt. Students made this error, on average, 19% of the time on pre-quizzes and 8% of the time on post-quizzes. Sixty-seven percent of students made the incomplete information error at least once on pre-quizzes and 35% of students made it at least once on post-quizzes.

Ratio errors were made when the student selected the fraction as the shaded parts over the unshaded parts instead of the shaded parts over all of the parts (

The reciprocal error occurred when the student flipped the numerator and denominator. For example, if the question asked what fraction represents the shaded part of a shape and the answer was 2/3, a student making the reciprocal error would select 3/2. Students made this error 15% of the time on pre-quizzes and 10% of the time on post-quizzes. Roughly half of the students made this error at least once on pre-quizzes (49%) and post-quizzes (45%).

Unlike whole numbers, fractions can be equivalent to both other fractions and to whole numbers (

Example of errors in the quizzes. The answer choices highlighted in each question represents the labeled error. For the filler question, both C and D were coded as filler.

Minimum | Maximum | |||
---|---|---|---|---|

Pre-Quiz | 6.621 | 2.757 | 0 | 13 |

Post-Quiz | 4.135 | 2.542 | 0 | 14 |

Error | Description | Proportion of Times Error is Made |
Max Times Error Was Made | Number of Students Who Made This Error | ||
---|---|---|---|---|---|---|

Mean | Min | Max | ||||

Complement | Complement of correct fraction (e.g., correct: 2/3, incorrect 1/3) | 0.11 | 0.00 | 1.00 | 3 | 421 |

0.09 | 0.00 | 0.67 | 2 | 327 | ||

Filler | Incorrect answer but no obvious error | 0.11 | 0.00 | 0.71 | 4 | 706 |

0.07 | 0.00 | 0.50 | 3 | 843 | ||

Illogical size/spacing | Unequal parts of a shape or illogical spacing on the number line^{a} |
0.09 | 0.00 | 0.75 | 3 | 422 |

0.10 | 0.00 | 0.75 | 3 | 503 | ||

Incomplete information | E.g., leaving out the whole number in a mixed fraction | 0.19 | 0.00 | 1.00 | 4 | 855 |

0.08 | 0.00 | 0.67 | 2 | 338 | ||

Ratio | The fraction is the shaded part over the unshaded parts^{a} |
0.07 | 0.00 | 1.00 | 1 | 99 |

0.02 | 0.00 | 0.67 | 2 | 82 | ||

Reciprocal | Reciprocal of correct fraction (e.g., correct: 2/3, incorrect 3/2) | 0.15 | 0.00 | 0.75 | 3 | 705 |

0.10 | 0.00 | 0.60 | 3 | 639 | ||

Reduction to whole number | Misconception of what fractions are equivalent to whole numbers | 0.17 | 0.00 | 0.80 | 4 | 938 |

0.10 | 0.00 | 0.60 | 3 | 635 | ||

Same numerator/ denominator ordering error | Ordering incorrectly when the fractions had the same numerator or denominator | 0.12 | 0.00 | 0.33 | 1 | 498 |

0.08 | 0.00 | 0.50 | 2 | 399 | ||

Whole number ordering | Ordering only the numerator or denominators^{a} |
0.26 | 0.00 | 0.60 | 6 | 1,282 |

0.12 | 0.00 | 0.57 | 4 | 822 |

^{a}a priori code.

This ordering error was possible only when students were asked to compare fractions with the same numerator or denominator in the comparing fraction questions. Within these questions, the student would only have to compare the numerator or denominator, but still made an error. For example, a student may say 1/3 > 2/3 (same denominator). There was one question in the pre-quiz and two questions in the post-quiz for which this error was possible. Students made the same numerator/denominator ordering error 12% of the time on pre-quizzes and 8% of the time on post-quizzes. Approximately 35% of students made this error on the pre-quizzes and 28% made it on the post-quizzes.

The whole number ordering error was made when the student ordered the fractions based on their numerator or the denominator, without apparent consideration of the relationship between the two (

As noted above, nine types of errors could be made in the third-grade quizzes—complement, filler, illogical size/spacing, incomplete information, ratio, reciprocal, reducing to whole numbers, same numerator/denominator ordering, and whole number ordering. Analyses were run in

A comparison of models with one to five classes is found in

Class | Sample Size Adjusted BIC | BLRT |
Bayes Factor |
---|---|---|---|

1 | 21746.513 | ||

2 | 21493.295 | < 0.001 | > 10 |

3 | 21430.938 | < 0.001 | < 1 |

4 | 21422.483 | < 0.001 | < 1 |

5 | 21454.988 | < 0.001 | < 1 |

Pre-quiz latent class model. Each color represents how often students made the error. The y-axis represents how many students in that class made a specific error a certain percentage of time (e.g., in the Few Errors Class, over 86% of the students did not make the complement error and about 12% made the complement error 25%-50% of the time).

The first class had the widest distribution of errors, meaning that they made the most error types of all of the classes. The majority of the errors were made at least once, with six of the nine errors being made at least 25% of the time. About 30% of the students (

The students in the WNO Class primarily made the whole number ordering error. Although these students also made other errors, the majority of students made the WNO error at least 25% of the time but rarely made other errors this often. Approximately 40% of the students (

Students in this class primarily did not make errors, except for the whole number ordering error. However, this class is distinct from the WNO Class because errors were made so infrequently—over 70% of the students in this class did not make six of the nine errors. Of the sample, 30% (

To determine who was in each class, separate logistic regressions were conducted for each class (0/1 whether the student was a member of that class). When running logistic regressions, an odds-ratio of greater than one indicates an

Variable | Distributed Errors |
WNO Error |
Few Errors |
|||
---|---|---|---|---|---|---|

Odds-Ratio | Z-score | Odds-Ratio | Z-score | Odds-Ratio | Z-score | |

1.24 | 1.65 | 0.80 | -2.00 | 0.90 | -0.70 | |

1.00 | -0.09 | 1.00 | 0.05 | 1.00 | 0.09 | |

0.94*** | -15.38*** | 0.98*** | -7.42*** | 1.12*** | 17.60*** | |

0.87 | -0.60 | 1.13 | 0.62 | 0.90 | -0.41 | |

0.98 | -0.13 | 1.24 | 1.53 | 0.68 | -2.31 | |

1.10 | 0.44 | 1.25 | 1.14 | 0.60 | -1.57 | |

1.35 | 1.60 | 0.49*** | -4.02*** | 1.37 | 1.40 | |

Race | ||||||

Asian | 1.33 | 1.08 | 0.82 | 0.75 | 0.84 | -0.54 |

Black | 1.13 | 0.59 | 1.26 | 1.39 | 0.70 | -1.53 |

Hispanic | 0.93 | -0.35 | 1.02 | 0.13 | 1.09 | 0.38 |

Other | 1.22 | 0.70 | 1.24 | 0.76 | 0.63 | -1.37 |

11.60*** | 1.95 | 0.00*** |

***

For the latent transition analysis, a post-quiz model was constrained to three classes for two reasons. First, the three-class model had the lowest sample size adjusted BIC value (see

Post-quiz latent class model. Each color represents how often students made the error. The y-axis represents how many students in that class made a specific error a certain percentage of the time (e.g., in the Reciprocal Class, over 90% of the students did not make the complement error and about 8% made the complement error 0%-15% of the time).

Students in this class made errors that were largely distributed across the types, as is illustrated by the relatively high amount of blue in the bars (the error was made between 0% and 25% of the time). However, over 80% of the students made the reciprocal error at least once, in stark contrast to the percentage of students who made this error in the other two post-quiz classes. Approximately 30% of students (

Similar to the Distributed Errors Class in the pre-quiz, the post-quiz Distributed Errors Class comprised students who made a variety of errors, but the reducing fraction to whole numbers and whole number ordering errors were made most often. However, students also made other errors more often than in the other classes. Of the sample, 29% (

Like the pre-quiz Few Errors Class, students in the post-quiz Few Errors Class made the least errors. This class does differ from its pre-quiz counterpart by having most errors made less often, especially in regards to the relatively small number of whole number ordering errors. Approximately 40% of students (

As with the pre-quiz, logistic regressions were run to determine which demographic and game-play variables predicted class membership. A student had higher odds (1.09-fold increase) of being placed in the Few Errors Class if they had a higher pre-quiz average. Conversely, higher-performing students were

Variable | Reciprocal Error |
Distributed Errors |
Few Errors |
|||
---|---|---|---|---|---|---|

Odds-Ratio | Z-Score | Odds-Ratio | Z-Score | Odds-Ratio | Z-Score | |

0.99 | -0.05 | 0.90 | -0.77 | 1.00 | 0.04 | |

1.00 | 0.70 | 1.00 | -1.37 | 1.00 | 0.90 | |

0.97*** | -8.63*** | 0.95*** | -12.16*** | 1.09*** | 15.87*** | |

1.04 | 0.20 | 0.87 | -0.65 | 1.06 | 0.24 | |

1.04 | 0.30 | 1.18 | 1.13 | 0.75 | -1.78 | |

1.56 | 2.14 | 1.02 | 0.08 | 0.53 | -2.33 | |

0.48*** | -3.74*** | 0.79 | -1.20 | 2.14*** | 3.43*** | |

Race | ||||||

Asian | 0.92 | -0.28 | 0.91 | -0.33 | 1.15 | 0.48 |

Black | 1.47 | 2.13 | 1.09 | 0.46 | 0.59 | -2.49 |

Hispanic | 0.93 | -0.39 | 1.02 | 0.12 | 1.07 | 0.35 |

Other | 0.91 | -0.31 | 1.92 | 2.23 | 0.54 | -1.90 |

1.60 | 10.47*** | 0.00*** |

***

Students who were in the Few Errors pre-quiz class, the class that was highest-performing at pre-quiz, moved only to the Reciprocal Error and Few Errors post-quiz classes, among which they were almost entirely placed into the post-quiz Few Errors Class (99.8%; only one of the 506 transitioned to the Reciprocal Error Class). Students who were in the pre-quiz Distributed Errors Class, the class that was lowest-performing at pre-quiz, moved to each of the three post-quiz classes. Primarily, these students were placed in the post-quiz Distributed Errors Class (56%), followed by the post-quiz Reciprocal Error Class (27%), with only 16% placed in the post-quiz Few Errors Class. Lastly, the students in the pre-quiz Whole Number Ordering Class, followed a similar transition pattern to those in the pre-quiz Distributed Errors Class. Of the WNO Class, 64% moved to the post-quiz Reciprocal Error Class, 35% moved to the Distributed Errors Class and 6% moved to the Few Errors Class. See

Pre-Quiz Classes | Post-Quiz Classes |
||
---|---|---|---|

Reciprocal Error | Distributed Errors | Few Errors | |

Distributed Errors | 0.274 | 0.561 | 0.163 |

WNO Error | 0.642 | 0.352 | 0.006 |

Few Errors | 0.002 | 0.000 | 0.998 |

Logistic regressions were run to better understand which students transitioned between classes. Because all but one student who started in the pre-quiz Few Errors Class remained in the post-quiz Few Errors Class, logistic regressions were only run on students who started in either the Distributed Errors or WNO Class. Among these six transition possibilities, statistically significant predictors were only identified for movement into the post-quiz Few Errors Class. If a student started in the pre-quiz Distributed Errors Class, the odds of them moving into the post-quiz Few Errors Class increased if they had a higher pre-quiz average or were classified as gifted (1.04-fold and 3.58-fold increase, respectively). Being classified as gifted also increased the odds of being placed in the post-quiz Few Errors Class if students started in the WNO Class (44.86-fold increase). However, being an English Language Learner also increased the odds of students transitioning from the WNO Class to the Few Errors Class (11.30-fold increase). See

This study had three main questions:

What errors are made when students solve fraction problems?

How can students be classed by the errors they tend to make together?

How do these classes change from pre-quiz to post-quiz?

Within this study, we expand upon previous fraction error analysis in both the context and content of our questions and in the identification of our errors. First, breaking from the tradition of using open-response researcher-administered questions (e.g.,

As our second contribution to the nature of error identification, we described new categories of errors. Of the nine errors coded, six of them were a posteriori in that they were not previously described in research on fraction errors—complement, filler, incomplete information, reciprocal, reducing fractions to whole numbers, and same numerator/denominator ordering error. These a posteriori errors were made just as often, if not more, than those assigned a priori codes from prior research. It remains to be seen if these errors would be found outside of the context of ST Math or in non-multiple-choice examinations.

As the multiple-choice nature of the ST Math quiz questions may have influenced the errors that arose, so too may have the visual nature of the questions. The focus within ST Math on visual-spatial instruction is reflected in a number of quiz questions that used visual representations, such as shaded figures or number lines. These questions may elicit different errors than those seen in prior studies, which, other than writing factions based on visual models, relied primarily on questions that were symbolic (e.g., 1/2 + 3/4 = ____;

Three classes of errors were identified using LCA at pre-quiz—Few Errors, Distributed Errors, and Whole Number Ordering Error. Students in the Few Errors Class tended to have the highest pre-quiz averages. The students in this class made few errors, and when they did make errors, tended to make the same error the majority of the time. Although the LCA identified this group as a class, many of the students had little in common regarding the type of errors they made—instead, they were joined together merely by their propensity to make few errors. It is intuitive that high performers would make few errors overall and that these errors may not be consistent among the high performers. There are two possibilities we offer for why these students did not make similar

The Whole Number Ordering Error Class contained less than five percent of students who did not make the whole number ordering error and almost 60% who made it at least half of the time. The high rate of this error—ordering only numerators or denominators without considering the relationship between the two—is likely due to a naïve understanding of fractions and an improper reliance on knowledge of whole numbers and their ordering schema (

Lastly, the largest group of students were placed in the pre-quiz Distributed Errors Class. This class made a wide range of errors but did not make most errors more than 25% of the time. The seemingly random nature of their errors may indicate that most of their errors are due to a general lack of fraction knowledge and unfamiliarity with fraction problems. This conclusion is further supported by students in this class having the lowest pre-quiz averages, indicating a lower level of fraction knowledge. This may indicate that students with little fraction knowledge do not make errors that indicate one type or a few types of conceptual or procedural misunderstandings, but instead, exhibit a pattern that may be more indicative of guessing. It may be that these students would be those that would leave open-ended questions blank, and therefore would not be attributed to a specific error or classified into an error profile on the types of exams frequently used in prior fraction error research.

Both class membership and class structure differed between pre- and post-quizzes. This was expected, because the number of total errors being made decreased as the students learned from ST Math. Two of the three pre-quiz classes remained in the post-quiz—the Distributed Errors Class and the Few Errors Class. Interestingly, the percent of students in the Few Errors Class increased by 10% from pre- to post-quiz. The class also changed slightly in its composition of errors, as depicted in

Distributed Errors and Few Errors classes transition. The left bars represent the pre-quiz class and the right bars represent the post-quiz class.

The most notable change from pre to post is that the proportion of errors increased for the filler and illogical size/spacing categories in both classes and the complement error in the Distributed Errors Class. For the illogical size/spacing error, students who make this error have some aspect of the question correct—they understand that 1/5 is one of five parts (for example) but do not recognize that the parts must be equal, or they correctly order fractions but do not properly space them on the number line. Thus, the students who make this error may have a partially developed understanding of fractions. Alternately, students who selected answer options in line with this error may have been confused at encountering what may be viewed as a

Only movement into the Few Errors class could be predicted by student characteristics. Students who had higher pre-quiz averages and were labeled as gifted were more likely to move from the Distributed Errors Class in the pre-quiz to the Few Errors Class in the post-quiz. Gifted students, in addition to English Language Learners, were also more likely to move from the Whole Number Ordering Class to the Few Errors Class. Movement between the classes from pre- to post-quiz followed a pattern similar to the one found with

As noted, although the multiple-choice nature of the questions allowed us to collect authentic educational data from a large number of students, it also provided a limitation in that it constrained the possible number of errors to those already provided within the software. It also allowed for the number of times an error could be made to vary. For example, the same numerator/denominator ordering error could only be made on one specific type of question—comparing fractions. Similarly, the number of times an error could be made varied between the pre- and post-quiz, preventing a true measure of difference, although this difference was small (differences ranging from zero to two possible errors). Additionally, although we grounded our coding decisions in both prior research of fraction errors and understanding of mathematics and mathematics education, we lacked insight into each student’s thought process as they solved the problems. Future studies may rectify this shortcoming by providing for methods such as cognitive interviewing to complement error coding.

Our results are immediately applicable to the digital platform from within which the data came, ST Math. First, we can work with the platform developers to reduce the number of filler items in ST Math quizzes and provide for a variety of error options with which to identify student misconceptions. These same guiding principles can be used by other test and platform developers to allow for more fine-grained data collection and identification of opportunities to assist student learning. Even after instruction, students demonstrated difficulty with the proper placement of numerator and denominator (reciprocal error). It may be that these are difficult concepts for all students, which would explain the persistence of these errors at post-quiz. Alternately, it could mean that ST Math is not presenting content in a way that teaches this idea as well as it does other fraction concepts. Examination of the curriculum and the puzzles that cover material related to these concepts can help to answer this question. Experimental studies altering this material and examining resulting error patterns can inform future iterations of both ST Math and other elementary fraction curricula.

Our work can also be applied to the realm of personalized or individualized instruction (e.g., learner profiles;

We set out to better understand third grade students’ errors with fractions in a digital learning environment using multiple choice tests of fraction concepts and algorithms. We identified new types of errors, such as incomplete information and reciprocal errors. We then examined the co-occurrence of these errors using Latent Class Analysis, identifying three patterns of student errors at pre-quiz: one that demonstrated a naïve fraction schema, one that contained high performers without consistent errors, and one that included the majority of the students and likely reflected student unfamiliarity with fraction problems. We found that, although some patterns remained the same, such as distributed or few errors, some patterns changed after instruction. Our work contributes to the growing field of research on fractions, adding to the understanding of the complexity of fraction knowledge and offering insights into the particulars of student struggles, insights that can contribute to the design of assessment and instruction, and ultimately, to improved fraction and mathematics achievement.

Error type | Pre-quiz |
Post-quiz |
|||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | |

Pre-quiz | |||||||||||||||||

1. Complement | – | ||||||||||||||||

2. Filler | .06* | – | |||||||||||||||

3. Illogical Size/Spacing | .05 | .15*** | – | ||||||||||||||

4. Incomplete Information | -.06 | -.05* | -.06* | – | |||||||||||||

5. Reciprocal | .10** | .10** | .05* | -.03 | – | ||||||||||||

6. Reduce to Whole Number | -.16*** | .18*** | .12*** | .10** | -.12*** | – | |||||||||||

7. Ratio | .08** | .21*** | .09** | .13*** | -.00 | .07** | – | ||||||||||

8. Same Num/Denom Ordering | .06* | .18*** | .13*** | .11*** | .10** | .11*** | .11*** | – | |||||||||

9. Whole Number Ordering | .03 | .06* | -.08** | .01 | .12*** | .16*** | .02 | .15*** | – | ||||||||

Post-quiz | |||||||||||||||||

10. Complement | .08** | .12*** | .08** | -.00 | .08** | .08** | .06* | .15*** | .14*** | – | |||||||

11. Filler | .04 | .17*** | .12*** | .08** | .11*** | .15*** | .06* | .16*** | .27*** | .08** | – | ||||||

12. Illogical Size/Spacing | .05* | .10** | .09** | -.01 | .04 | .08** | .04 | .08** | .16*** | .08** | .14*** | – | |||||

13. Incomplete Information | .09** | .17*** | .08** | .08** | .09** | .09** | .05* | .10** | .14*** | .08** | .13*** | .01 | – | ||||

14. Reciprocal | .06** | .11*** | .03 | .04 | .22*** | .02 | .07** | .12*** | .13*** | .09** | .08** | .03 | .06* | – | |||

15. Reduce to Whole Number | .07* | .08** | .07** | .03 | -.01 | .21*** | .01 | .07* | .09** | .01 | .08** | .07* | .03 | -.23*** | – | ||

16. Ratio | .03 | -.00 | .03 | .01 | .02 | .02 | .07** | .03 | -.01 | .03 | .08** | .03 | .07 | .06 | .05 | – | |

17. Same Num/Denom Ordering | .07** | .19*** | .11*** | .03 | .13*** | .09** | .13*** | .22*** | .14*** | .15*** | .19*** | .06* | .16*** | .18*** | .07** | .05 | – |

18. Whole Number Ordering | .10** | .13*** | .12*** | .03 | .13*** | .12*** | .10** | .21*** | .27*** | .21*** | .27*** | -.11*** | .14*** | .13*** | .11*** | .06* | .33*** |

Class | Sample Size Adjusted BIC | BLRT |
Bayes Factor |
---|---|---|---|

1 | 20187.229 | ||

2 | 19757.500 | < 0.001 | > 10 |

3 | 19693.396 | < 0.001 | < 1 |

4 | 19714.993 | < 0.001 | < 1 |

5 | 19741.642 | < 0.001 | < 1 |

Error type | Pre-Quiz |
||
---|---|---|---|

Distributed Errors | Whole Number Ordering Error | Few Errors | |

Complement | 0.05 | 0.19 | 0.08 |

Filler | 0.17 | 0.12 | 0.04 |

Illogical size/spacing | 0.15 | 0.07 | 0.05 |

Incomplete information | 0.22 | 0.18 | 0.17 |

Reciprocal | 0.10 | 0.23 | 0.10 |

Reduction to whole number | 0.32 | 0.11 | 0.10 |

Ratio | 0.13 | 0.07 | 0.01 |

Same numerator/ denominator ordering error | 0.19 | 0.14 | 0.04 |

Whole number ordering | 0.29 | 0.31 | 0.18 |

Error type | Post-Quiz |
||

Reciprocal Error | Distributed Errors | Few Errors | |

Complement | 0.13 | 0.12 | 0.03 |

Filler | 0.19 | 0.21 | 0.07 |

Illogical size/spacing | 0.11 | 0.13 | 0.08 |

Incomplete information | 0.13 | 0.12 | 0.02 |

Reciprocal | 0.20 | 0.07 | 0.05 |

Reduction to whole number | 0.00 | 0.22 | 0.09 |

Ratio | 0.03 | 0.03 | 0.01 |

Same numerator/ denominator ordering error | 0.13 | 0.14 | 0.01 |

Whole number ordering | 0.17 | 0.22 | 0.04 |

Variable | Distributed Errors → Reciprocal Error |
Distributed Errors → Distributed Errors |
Distributed Errors → Few Errors |
|||
---|---|---|---|---|---|---|

Odds-Ratio | Z-Score | Odds-Ratio | Z-Score | Odds-Ratio | Z-Score | |

0.94 | -0.27 | 0.95 | -0.23 | 1.23 | 0.70 | |

1.00 | 0.60 | 1.00 | -1.86 | 1.00 | 1.55 | |

0.99 | -1.12 | 0.99 | -1.35 | 1.04*** | 3.10*** | |

0.86 | -0.37 | 0.81 | -0.61 | 1.86 | 1.38 | |

0.56 | -2.24 | 1.46 | 1.76 | 1.12 | 0.39 | |

2.04 | 1.66 | 0.94 | -0.18 | 0.29 | -2.28 | |

0.47 | -2.16 | 0.74 | -1.05 | 3.58*** | 3.58*** | |

Asian | 0.44 | -1.46 | 1.14 | 0.32 | 2.19 | 1.52 |

Black | 1.27 | 0.68 | 1.32 | 0.85 | 0.33 | -2.36 |

Hispanic | 0.92 | -0.23 | 1.01 | 0.03 | 1.08 | 0.19 |

Other | 1.02 | 0.04 | 1.58 | 0.83 | 0.35 | -1.25 |

0.52 | 6.13 | 0.00 |

***

Variable | WNO Error → Reciprocal Error |
WNO Error → Distributed Errors |
WNO Error → Few Errors |
|||
---|---|---|---|---|---|---|

Odds-Ratio | Z-Score | Odds-Ratio | Z-Score | Odds-Ratio | Z-Score | |

1.36 | 1.50 | 0.75 | -1.40 | 0.33 | -0.65 | |

1.00 | 0.21 | 1.00 | -0.19 | 1.00 | -0.73 | |

1.01 | 1.42 | 0.99 | -1.46 | 1.01 | 0.23 | |

1.10 | 0.30 | 0.91 | -0.29 | 1.00 | omitted | |

1.15 | 0.61 | 0.88 | -0.57 | 0.41 | -1.23 | |

1.17 | 0.47 | 0.81 | -0.65 | 11.30*** | 3.49*** | |

0.80 | -0.71 | 1.06 | 0.17 | 44.86*** | 5.79*** | |

Race | ||||||

Asian | 1.01 | 1.30 | 0.52 | -1.22 | 1.00 | omitted |

Black | 1.35 | 1.08 | 0.74 | -1.07 | 1.00 | omitted |

Hispanic | 0.85 | -0.55 | 1.15 | 0.49 | 2.94 | 1.37 |

Other | 0.58 | -1.18 | 1.81 | 1.29 | 1.00 | omitted |

0.63 | 1.56 | 0.00 |

***

Support for this research was provided in part by the National Science Foundation, grant number 1544273 and based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No DGE-1746939.

The authors have declared that no competing interests exist.

This research was completed to satisfy the first authors’ thesis requirements. Substantial sections of that thesis have been reproduced in the current paper and the empirical data was reported in the thesis as well. The thesis is available through North Carolina States University’s Theses and Dissertations Repository at

The authors would like to thank MIND Research Institute and the school district for participating in this study.