Unlocking chemistry calculation proficiency: uncovering student struggles and flipped classroom benefits

2.1 Chemistry calculations as a critical topic

Chemistry calculations are integral to every laboratory course and increasingly emphasised within the chemistry curriculum as students progress. Despite its importance, this topic is frequently cited as a contributing factor to the low popularity of chemistry. Several studies have addressed the challenges associated with chemistry calculations, ^{1 , 11 , 12 , 13 , 14} leading to attempts to enhance student performance in this area. ^{2 , 8 , 11}

Chemistry calculations are widely regarded – and explicitly referred to – as critical topics in the subject. ^{7 , 9 , 13} However, students often perceive them as unimportant, overly difficult, and demotivating. ^{6 , 11 , 15} Teachers, despite recognising these perceptions, frequently allocate a significant number of lessons to chemistry calculations, which can further diminish students’ attitudes toward the topic. ⁶ Moreover, research suggests that teachers’ efforts may have limited impact on improving students’ proficiency in chemistry calculations. ^{9 , 14 , 16}

2.2 Students’ problems with chemistry calculation tasks

One of the key factors discussed in the literature is students’ mathematical skills. Leopold and Edgar ¹² argued that poor performance in chemistry calculations is largely a result of students’ inadequate mathematical abilities. Similarly, a study by Larsson and Palmgren ¹⁷ highlighted that teachers frequently emphasised the necessity of strong mathematics skills for success in chemistry calculations. However, this view is not universally supported. Other research. ^{10 , 13} has instead identified a range of arithmetic problems faced by students. Tabinas et al. ¹⁴ attributed these difficulties to poor problem-solving skills, while additional studies noted students’ lack of science-specific chemistry reading skills. ¹⁰

Two studies ^{9 , 13} evaluated the role of general mathematics skills but reported conflicting results. While Scott’s findings pointed to gaps in mathematics proficiency, Rusek et al. found no significant influence of mathematics skills on students’ performance in chemistry calculations.

The issue appears to extend beyond general mathematics skills. Two consecutive studies ^{9 , 18} identified a critical problem in students’ inability to transfer their general skills to chemistry-specific problems. This phenomenon was further clarified by Tóthová and Rusek, ¹⁹ who demonstrated that students often perceive typical chemistry tasks as exercises requiring the application of memorised facts, rather than as problems solvable through reasoning.

These studies also revealed a tendency among students to rely on prefabricated formulas, sometimes using illogical or inappropriate variations of calculation relations, which they applied to standard calculation types without evidence of logical reasoning. This suggests that mastering chemistry calculations often involves learning a prescribed set of arbitrary steps rather than developing the ability to apply knowledge and concepts meaningfully to chemistry problems.

2.3 Chemistry calculations in Czech national curriculum

In the context of this study, which focuses on pre-service chemistry teachers, two types of upper-secondary curricula, i.e. those students entering chemistry-oriented university program went through, were considered: grammar school and chemistry-oriented vocational school. The curricula for these types of schools are quite vague regarding chemistry calculations, stating only that “students perform simple chemical calculations…” For this reason, given the known influence of textbooks on school curricula, ²⁰ the extent of topics covered in Czech chemistry textbooks for upper-secondary schools ²¹ provides the best approximation of the chemistry calculations students are likely to be exposed to.

Although there is consensus among textbooks regarding basic chemical calculations (e.g., mass fraction and molar concentration), there are significant differences in the extent and depth of other topics, such as calculations from chemical equations and pH. The textbooks’ content varies not only in their approach to explanation and presentation of alternative solution methods but also in the number and type of model and practice problems. Some topics that are typically more challenging for students are not adequately represented in the textbooks. ²¹ This, together with the natural variation in students’ abilities, is a probable cause of the differences in students’ initial chemistry calculation test results.

2.4 Flipped classroom principle

As synthesised by Wright and Park, ²² (p. 96) from multiple sources, the flipped-classroom principle can be best described as “a student-centred pedagogical model that maximises face-to-face time with students by moving teacher-dominated lectures outside of class via digital technologies; in-class time is then organised for students to collaborate with peers, deeply engage with content, and take ownership of their learning.” With the constructivist teaching-learning approach gaining increasing popularity, the flipped-classroom principle has attracted concentrated research attention for many years. ^{22 , 23 , 24 , 25} The growing body of research on this topic has been systematically analysed in a recent review by Bancroft et al. ²⁶

The flipped classroom has been applied effectively in chemistry education. ^{27 , 28 , 29 , 30} In addition to numerous instructional videos available on platforms such as Khan Academy, the 2015 ConfChem Conference on Flipped Classrooms highlighted its continued relevance in the field. The flipped-classroom approach has been shown to be effective at both high school ^{25 , 31} and university levels. ³² Its application has been evaluated in organic chemistry, ²⁷ general chemistry, ²⁹ and more recently, in problem-solving within a chemistry course. ³³

Despite these promising findings, Bancroft et al. ²⁶ concluded that identifying robust, generalisable trends remains challenging, and further research is required to fully understand its impact. In light of this, the present study focuses on employing the flipped-classroom principle to address the complex topic of chemistry calculations, particularly during the pandemic-enforced transition to fully online teaching.

4.1 Research sample

This study utilised a convenience sample, comprising first-year bachelor’s pre-service teacher students enrolled in the Chemistry Calculations course at the Faculty of Education, Charles University, Prague, Czechia. General information about the students was collected, their initial chemistry calculation skills were assessed, and the study materials, practice tasks, and instructional approach were controlled. These logistical considerations made it technically unfeasible to include students from other institutions.

The experimental group consisted of the class of 2020/2021, while the control group comprised the class of 2021/2022. A total of 31 students participated in the experimental group, with 24 completing the post-test, and 36 students formed the control group.

Although the national curriculum does not mandate a fixed sequence for topics, chemistry calculations are typically introduced in the first year of study. Both groups – affected by the COVID-19 lockdowns – received instruction in chemistry calculations under comparable conditions. The primary distinction lay in the mode of delivery: the experimental group experienced the flipped-classroom, online method, while the control group was taught using a traditional, in-person approach. However, the preparation for the final upper-secondary school exam, the last formal exposure to the subject, was similarly influenced by pandemic-related conditions, ensuring comparable prior learning experiences.

Since the course instructor was also the study’s author, the course’s scope, materials, and practice tasks were kept consistent between the in-person (control) and flipped-classroom (experimental) formats. Pre-test results, derived from the same set of tasks, confirmed that the groups were comparable in terms of initial chemistry calculation skill levels, enabling the implementation of the pedagogical experiment.

From the experimental group, 12 students were selected for interviews, based on their pre-test performance, to represent three subgroups: successful, partially successful, and unsuccessful, with four students in each category. The interviews, conducted at the end of the course, provided qualitative insights into their experiences and progress.

Students’ initial performance in chemistry calculations was generally low, except for tasks involving mass fraction and molar concentration calculations. Success was categorised as follows: students scoring 7 or more points (out of 10) were deemed successful; those scoring 4 to 6 points were partially successful; and those with three points or fewer were classified as unsuccessful. This scoring system was applied consistently to both the pre-test and post-test results.

4.2 Research design

This study focused on evaluating students’ outcomes following a course in Chemistry Calculations. The same methodology was applied over two consecutive years, employing an identical set of tasks (tests) for both cohorts. In the first year, the course was delivered using the flipped-classroom model and conducted entirely online. In contrast, the following year, the course was taught using a traditional approach, combining lectures with seminars, and conducted in person.

4.3 Research tools

Initially, students completed an entrance-level test consisting of 10 chemistry calculation tasks. ⁹ The test was designed to assess students’ knowledge and skills across fundamental calculation types, including mass fraction, pH, molar concentration, dilution problems, and chemical equation calculations. The content validity of the test was established by six experts, each holding a Ph.D. in chemistry education. Following the methodology of Fach et al., ¹¹ each calculation type was represented by a word problem and a task formulated solely using symbols and numerical values ⁹ (see Supplementary File 1).

Additionally, each task included two supplementary items: a five-point scale for evaluating students’ confidence judgment (CJ) with the prompt, “To what extent are you confident in the correctness of your result?” and a yes/no question asking, “Do you know such a task from your upper-secondary studies?”

The post-test contained the same items as the pre-test, with minor changes to variables such as volumes of reactants or compounds to ensure that the operations required to solve the problems remained consistent. It should be noted that the difficulty level of the calculations was aligned with the upper-secondary school curriculum.

4.4 Data analysis

The entrance-level test score was calculated as the number of points the students received. In the paper, the test score is utilised in the form of percentages calculated as: the number of points achieved/total number of points in the test x 100. The results were used to identify problematic calculation types as well as to sort the students into groups according to their achievement in the test.

The results were transcribed and analysed, focusing on students’ correct responses, perceived mastery of each task, familiarity with similar tasks, and the steps taken to solve each problem (e.g., calculation of the amount of substance, use of the rule of three, balancing a chemical equation). This approach facilitated a deeper analysis of students’ progress.

Drawing on experience from previous research, ⁹ the analysis traced the thought processes (algorithms) employed, students’ conceptual knowledge, application of appropriate formulae, and the accuracy of mathematical operations used in their calculations. To compare the test results, IBM SPSS 26 software was used. As the Shapiro-Wilk test (p < 0.05) showed the data were not normally distributed, non-parametric tests were used. Non-parametric McNemar’s test was used to compare the students’ results in calculation or assignment types, with φ as a corresponding effect-size test. ³⁸ To evaluate students’ overall success in specific calculation types, a chi-squared test was used. Wilcoxon’s test, with r as the effect size, assessed shifts in pre-post-test confidence judgments (CJ) for calculation types and assignment types. The Kruskal-Wallis test evaluated differences in CJ across calculation types. The Mann–Whitney U test compared pre-test and post-test results across two study years, with r as the effect size. The students’ confidence was evaluated using standard variables introduced e.g. by Stankov and Crawford: ³⁹

1. Mean confidence (CF) – an overall value of the students’ confidence judgement, also general confidence,

2. Mean students’ confidence when correct answer (CFC) or a wrong answer (CFW) was provided,

3. Mean confidence accuracy quotient (CAQ = (CFC − CFW)/SD) which provides gives the standardised value of confidence discrimination indicating whether the students are aware of what they know and what they do not know. ^{39 , 40}

The interviews were conducted in Czech. They were transcribed verbatim and coded using open coding by one researcher with a second researcher (author of this paper) checking of the codes independently. ⁴¹ As the agreement was almost total, a consensus by agreement was reached in the individual cases. For this paper’s purposes, concrete statements were translated in English.

4.5 Description of the chemistry calculations course

The instruction for the experimental group was designed based on the flipped-classroom principle, aiming to maximise face-to-face interaction by shifting teacher-dominated lectures outside of class through digital technologies. This approach focused on addressing the challenges students encountered after initially engaging with the material independently, ²² (p. 96). The course was conducted entirely online during the Covid-19 lockdown.

The first two lessons were dedicated to introducing the course structure and assessing students’ initial chemistry calculation skills. The third session covered two well-mastered topics identified in the pre-test: mass fraction and molar concentration. Subsequent lessons adhered to the flipped-classroom model, with one exception – a traditionally structured control lesson delivered during the second half of the course.

All synchronous meetings were 45 min long, with each topic allocated an equal amount of time, except for pH calculations, which were divided into two sessions. Students were assigned topics for independent study, utilising resources such as book chapters, online materials, lecture notes, and videos. They were encouraged to use any additional resources they found beneficial. The independent study materials were accompanied by calculation tasks for practice and homework.

During flipped-classroom sessions, students began by addressing clarifying questions and reviewing homework tasks. This was followed by a discussion of practice calculation results and the completion of more challenging tasks. Each lesson concluded with the assignment of homework tasks and the topic for the next session.

The control lesson lasted 90 min and provided additional support, offering students extended opportunities to ask questions and interact with the teacher and peers. This lesson aimed to assess whether students could follow the course syllabus independently or preferred a more traditional instructional approach. The teacher presented the topic and theoretical concepts, demonstrated illustrative tasks, assigned calculation problems, reviewed results and procedures, answered questions, and assigned additional homework tasks.

The control group, taught in the regular in-person format, received identical instruction to the control lesson. The course topics included Balancing Chemical Formulas, Calculations from Chemical Formulas (Stoichiometric Problems), Introduction to pH, Advanced pH Calculations, and the Equation of State.

5.1 Students’ initial skills in chemistry calculations – RQ1

The students’ initial chemistry calculation level was assessed from three perspectives: overall achievement, confidence judgment (CJ), and familiarity with the tasks.

The students’ overall achievement results are shown in Figure 1, highlighting significant differences (p < 0.001, η ² = 0.4) indicate statistically significant differences with a large effect-size in their mastery of various calculation types. The students were overall successful on the mass fraction and molar concentration tasks and overall unsuccessful on the three other chemistry calculation tasks. The differences between the students’ performance on the mass fraction and molar concentration tasks (p = 0.715, r = 0.055), as well as the other three calculation types (p = 0.4223, r = 0.0044) was insignificant. On the contrary, the differences between the students’ results on the calculation tasks the students succeeded in and failed in showed a statistically significant difference (p < 0.001, r > 0.5) with a large effect size.

Figure 1:

Students’ initial level calculations skills. *Note: w – mass fraction calculations, pH – pH calculations, c – molar concentration calculations.

An overwhelming majority of the students reported familiarity with the two most successful calculation types (both over 90 %). However, despite also being familiar with pH calculations (approximately 85 %), calculations from chemical equations (around 65 %), and dilution problems (75 %), most students struggled with these tasks.

Given that the Chi-squared test revealed no significant differences between assignments presented as word problems and those using only symbols (p > 0.05, φ values showed only a small effect), the results for students across these types of tasks were combined. The findings are presented in Figure 2.

Figure 2:

Overall students’ results from chemistry calculations pre-test. Note: Form. – assignment using formulas and symbols, WP – word problems; w – mass fraction calculations, pH – pH calculations, c – molar concentration calculations.

Despite the overall findings, eight of the 12 interviewed students reported finding assignments using formulas and symbols easier, explaining that “they are not forced to think” about the assignment, as the physical quantities are directly provided. Three students stated that such tasks are faster and simpler to complete, while one noted that the task format often prompts the correct definitional relationship required for the calculation. Additionally, one student attributed their preference to familiarity with this type of assignment from upper-secondary school. Another student highlighted the previously mentioned greater difficulty of word problems: “When it is assigned as a word problem, I can get lost in it or forget something.”

These results underscore variations in student performance across calculation tasks and highlight the uneven impact of instruction on these topics, despite all tasks being part of the upper-secondary curriculum. The analysis of students’ solutions identified specific issues requiring targeted interventions.

The concept of mass fraction was well mastered by most students. However, common errors among those who failed included neglecting to sum the mass of the solute and solvent or confusing variables by misplacing numerators and denominators. Only one student failed to attempt one task, and another did not provide a result.

Similarly, in molar concentration tasks, most students applied the correct relationships for calculations. Unsuccessful attempts were attributed to the use of incorrect formulae or errors in constructing cross-multiplication relations for word problems. Unit conversion was a minor issue, with only one student failing to convert units correctly.

In contrast, success rates for other types of calculations were significantly lower, though this was not consistently reflected in students’ confidence judgment (CJ) values. Dilution problems proved particularly challenging, with only slightly over 20 % of students using the correct mathematical relationship. For symbol-based tasks, 21 % of students left the task incomplete, and 15 % did not attempt it. For word problems, 55 % did not finish, and 25 % did not start. Mathematical errors were prevalent, affecting 45 % of students for symbol-based tasks and 40 % for word problems.

Within pH calculations, success rates were the second lowest. Only 39 % of students correctly calculated concentration from a pH value, indicating widespread misunderstandings of logarithms. Similarly, 45 % successfully calculated pH from pOH, while approximately 36 % made mathematical errors. Discrepancies between low results and relatively high CJ values were linked to the mechanistic application of the “pH = −log[c]” formula, often without recognising complexities such as the diprotic nature of sulfuric acid. A substantial number of students did not attempt these tasks (33 % for symbols, 27 % for word problems).

For calculations from chemical equations – the third least successful task type – most students either did not attempt the task or failed to complete it. Common issues included difficulty balancing equations and, in one instance, failure to account for a surplus reactant.

These findings offer critical insights for refining the content and instructional strategies of the Chemistry Calculations course, ensuring a more effective focus on problematic areas.

Students’ results strongly correlated with their CJ values. Pearson’s correlation showed a strong (r = 0.503) and statistically significant (p = 0.006) correlation. When students were divided into three groups according to their overall results, their CJ strongly (r = 0.914) and significantly (p < 0.001) correlated with their test results. Nevertheless, the lowest-scoring group still reported a medium CJ (Mdn = 3), indicating overestimation.

For example, in pH calculations, students’ CJ was considerably low (Mdn = 2) for the task with a lower success rate (acid’s concentration calculation when pH was provided) and quite high (Mdn = 4) when pH was the unknown – the success rate was higher as this task resembled more common problems. Conversely, students assessed their performance on dilution problems in the middle of the CJ scale (Mdn = 3), estimating average performance despite poor results.

Interviews revealed that students assessed their chemistry calculation skills after upper-secondary school as “4” on a scale of 1–5 (1 being the highest rating); six students rated themselves as “4,” two rated themselves “3,” and four rated themselves “2.” Since the interview sample included equal numbers of high-achieving and low-achieving students, the results clearly indicate low confidence in their skills, which is corroborated by their performance outcomes.

The national curriculum provides limited guidance regarding the scope of chemistry calculations, and only a few studies ^{21 , 42 , 43} have assessed this area at the curriculum level, including an analysis of chemistry textbook content. ²¹ To better understand students’ performance, their familiarity with the assigned tasks was also investigated.

Overall results revealed that 42 % of students did not recognise problems presented using symbols, while 28 % failed to recognise these in the form of word problems. This lack of familiarity was also evident in interviews, where eight out of 12 students reported that dilution problems were not covered in their upper-secondary curriculum.

The second least familiar task type was calculations from chemical reactions, with 33 % of students failing to recognise word problems and 17 % failing to identify symbol-based tasks. Surprisingly, only two students mentioned not performing these calculations in upper-secondary chemistry, a trend also observed for pH calculations. In the test, 20 % of students reported unfamiliarity with pH calculations involving symbols (e.g., converting pOH to pH), and 14 % were unfamiliar with word problems requiring pH calculations for diprotic acids.

Interviews further revealed that of the 12 selected students, only four had prior exposure to all calculation types. Three students stated that their instruction focused primarily on mass fraction tasks, while the remainder indicated that chemistry calculations were covered only superficially. One respondent highlighted a phenomenon likely relevant to others: “I have the feeling I learnt the calculation from scratch when I came to university.”

The students’ confidence judgements (CJ) are presented in Table 1, with mean confidence values (CF) calculated on a 1–5 scale. The Calibration for Confidence Judgement (CFC) values indicate that successful students’ below-average confidence on pH tasks suggests potential confusion. CFC values near “3” reflect limited confidence among successful students, while overconfident responses were evident in tasks involving word problems (w-wp), pH word problems (pH-wp), and symbol-based tasks (c-s). Low CJ values in other tasks suggest that students accurately gauged their performance.

The Calibration Accuracy (CAQ) values show that students generally demonstrated good calibration across most tasks. Values near “1” indicate that students’ CJ closely aligned with their performance. However, CAQ values >1 in three cases indicate overconfidence, while higher negative values in three other tasks reflect underconfidence.

This analysis underscores the variability in students’ confidence levels and familiarity with task types, suggesting targeted interventions are needed to improve both knowledge and self-assessment accuracy.

5.2 The effect of the flipped chemistry calculation course – RQ2

5.2.1 Students’ performance in the post-test – RQ2

The impact of the flipped course on students’ performance was evaluated using a mixed-methods approach, incorporating pre-test and post-test comparisons as well as interviews. The overall differences between the pre- and post-test scores are presented in Figure 3. The course demonstrated a positive effect, with significant improvements observed across all calculation types. The sole exception was mass fraction calculations, where students had already achieved high scores in the pre-test.

Figure 3:

Post-test scores according to the assignment type.

Despite this exception, the effect-size values indicated a large effect for all improvements, suggesting that one semester of such instruction is sufficient to raise students’ proficiency to a satisfactory level.

However, while the trend was positive, the results were not optimal, particularly given that the tasks were designed with a basic level of difficulty. A detailed analysis of the post-test outcomes was therefore undertaken to examine potential intervening factors and identify areas requiring further improvement.

Over 75 % of the students performed well on all tasks except the stoichiometry problem involving calculations from chemical equations assigned with symbols. The success rate for this task remained comparable to the pre-test, indicating that despite its apparent simplicity, students found it particularly challenging. This difficulty primarily stemmed from the need to balance the equation correctly and apply the appropriate stoichiometric coefficients, which many students failed to accomplish.

In mass fraction calculations, students did not achieve a 100 % success rate solely due to incorrect rounding (three students) or errors in substituting values into the mathematical formula. Nonetheless, the procedural understanding of this task was evident among all participants. Significant improvement was observed in pH calculations, with only four students failing to recognise the diprotic acid and two struggling with the conversion from pOH to pH. Only one student left the task unanswered, while the remainder solved it correctly.

The course’s effect was limited for two specific types of calculations. In dilution problems, unsuccessful students either did not attempt the task or made decimal errors. These students also tended not to report their confidence, resulting in data gaps. For calculations from chemical equations, while improvement was evident in one task, the overall success rate remained low (40 %). This was attributed to the nature of the task rather than its format. Eight students did not attempt the task, and among those who were unsuccessful, one applied an incorrect procedure, while another failed to account for a compound reacting in surplus.

Regarding students’ confidence judgments (CJ), no statistically significant differences were observed, primarily due to a lower response rate for this item in the test. Many students left this item unanswered. An analysis of the omitted responses revealed that a student’s likelihood of completing the confidence item was not influenced by their success on the task. Instead, it appeared more closely related to their overall approach to addressing third-tier items across the test.

Table 2 shows the shift in experimental group students CJ in the post-test. The mean confidence values rose in all cases showing the students reported higher self-confidence.

Table 2:

Comparison of students’ confidence judgement values in pre and post-test (N = 31).

	Mass fraction		pH		Molar concentration		Problems		Calculations from chem. equations
	Pre	Post	Pre	Post	Pre	Post	Pre	Post	Pre	Post
CF	3.85	4.25	2.85	3.08	3.38	4.06	2.35	3.45	2.70	2.91
N	15	10	10	9	17	10	10	10	13	9
SD	0.81	1.05	1.15	3.31	1.09	1.01	1.37	0.89	1.25	1.16
CFC	3.92	4.47	2.13	3.31	3.46	4.06	3.67	3.46	3.96	2.91
CFW	4.13	3.00	3.29	1.50	3.50	NA	1.92	NA	2.30	1.33
CAQ	−0.25	1.40	−1.01	0.55	−0.03	NA	1.27	NA	1.33	1.35

The specific values presented in Table 3 indicate a significant increase in students’ confidence judgments across nearly all tasks. This finding was corroborated by the interviews, where all 12 participants expressed feeling more confident about their performance compared to the pre-test.

Table 3:

Differences in students’ confidence between pre and post-test (N = 31).

Chemistry calculation tasksa	w-s	w-wp	pH-s	pH-wp	c-s	c-wp	dill-s	dill-wp	eq-s	eq-wp
p-value	<0.001	<0.001	0.057	0.04	0.19	<0.001	0.83	0.076	0.013	<0.001
Effect-sireb (r)	1.272	0.868	0.448	0.67	0.262	1.037	0.408	0.419	0.527	1.043

1. ^aThe tasks are labelled with an abbreviation according to the type of calculation and with the symbol s for tasks (concentration – c, mass fraction – w, calculations from chemical equations eq, dilution problems – dp and pH for pH calculations), assigned using symbols and wp using word problems. ^bThe values over 0,3 suggest medium effect-size, the values over 0.5 suggest large effect-size.

Despite this improvement, the Calibration Accuracy Questionnaire (CAQ) values for tasks where students provided responses suggest a shift from initial overconfidence to underconfidence. This shift reflects the students’ improved results, indicating that while their confidence increased, they remained cautious in evaluating their performance.

5.2.2 Comparison of the classical versus flipped-classroom instruction in chemistry calculations course

To compare the effectiveness of the Chemistry calculations course based on the flipped-classroom principle (experimental group) with that of a traditionally structured course (control group), the pre-test results of both groups were analysed first. These results are illustrated in Figure 4. The data indicate differences in students’ performance on pH calculations and molar concentration tasks, with both showing a moderate effect. From this perspective, the two groups can be considered comparable, although the experimental group demonstrated slightly better performance in these two calculation types.

Figure 4:

Control and experimental groups comparison in pre-test.

The analysis of both groups’ results indicated that the two groups were comparable. A detailed examination of specific calculation tasks provided further insights.

As shown in Figure 5, statistical significance tests revealed a notable difference in student outcomes only for dilution problems. For mass fraction tasks, the effect size was small, likely due to relatively strong pre-test performance. For pH calculations and molar concentration tasks, a small to medium effect size favoured the experimental group. Conversely, a small to medium effect size favouring the control group was observed for dilution problems. The performance difference for tasks involving calculations from chemical equations was negligible.

Figure 5:

Comparison of experimental and control groups’ post-tests.

Overall, the findings suggest that instruction based on the flipped-classroom principle produces comparable or superior outcomes to traditional teaching methods, with the exception of dilution problems.

6.1 Chemistry calculations as a critical aspect of chemistry education

Although the upper-secondary chemistry curriculum does not specify the extent of instruction for chemical calculations in detail, analyses of school curricula ^{42 , 43 , 44} and textbooks ²¹ highlight a generally accepted foundational level. However, the results of this study, alongside previous research, ^{9 , 10} indicate that this foundation is not being adequately achieved by students.

Given that the tasks assessed only basic calculations, this study corroborates earlier findings, ^{9 , 10} demonstrating that even students who choose chemistry for their university studies only partially achieve the expected outcomes in chemistry calculations by the end of their upper-secondary education. Similar concerns about students’ chemistry calculation skills have been raised in other countries, including Poland, ⁸ Germany, ¹¹ the UK, ¹³ and Kenya. ⁴⁵ This points to a shared instructional approach across nations that leads to similar, suboptimal results. However, to substantiate this hypothesis, the test would need to be administered to students in a broader international context.

The deficiencies in these basic skills are particularly striking given that chemistry calculations are included in school-leaving exams. ⁴² Students opting for university chemistry programmes are expected to have reviewed and demonstrated these skills to an exam committee approximately six months before the calculation test. The lack of proficiency, especially with calculations from chemical equations, may stem from insufficient experimental activities in upper-secondary school chemistry. ¹⁸ This highlights the need for a more accurate mapping of this topic to align expectations between upper-secondary and university educators.

Some students reported unfamiliarity with specific calculation types during both the tests and interviews, which was largely influenced by the assignment format. The only exception was dilution problems, which appear to be rarely addressed in school curricula. ⁴⁴ The test tasks in this study were selected based on textbook content ²¹ and aligned with the upper-secondary chemistry curriculum, as confirmed through interviews. While the sample size was too small for definitive conclusions, it was evident that students were more unaccustomed to word-problem formats than to the calculation types themselves. This finding supports the conclusions of Rychtera et al., ⁷ who noted that teachers often adopt a mechanistic approach, focusing on basic tasks and limiting the curriculum to essentials such as nomenclature and simple equation balancing. By doing so, they fail to teach the deeper conceptual meaning of chemical equations and their associated calculations in a meaningful context.

The consequences of such a restricted approach were recently demonstrated in a study by Hamerská et al., ⁴⁶ which found that balancing chemical equations is often treated as a “school chemistry skill,” rather than a transferable competency that students can apply to broader problem-solving contexts. This underscores the need for instructional practices that prioritise conceptual understanding over rote learning.

6.2 Chemistry concepts knowledge

As evidenced by the students’ success rates, the concepts of mass fraction and molar concentration are well understood by most students even at the start of their university studies. This recurring finding ^{9 , 10} can be attributed to at least two factors: students’ conceptual understanding and their mathematical skills. For instance, in mass fraction calculations, common errors include failing to add the mass of the solute to the mass of the solvent. Similarly, a frequent cause of failure in pH calculations is misunderstanding the concept of diprotic acids, with students often directly inputting the acid’s concentration into a calculator using the formula −log[c], without grasping what the value represents.

Addressing these issues requires more nuanced testing to distinguish between conceptual deficiencies and mathematical difficulties. In the original test, multiple steps overlapped, increasing cognitive load and obscuring critical problem-solving points. Since not all students provided complete solutions, identifying specific causes of failure was only possible for a limited number. Future assessments could incorporate qualitative methods, such as think-aloud protocols, or employ eye-tracking technology to provide insights into the entire problem-solving process.

Despite observed improvements in other calculation types, persistent challenges remain. First, the test results demonstrate that many students perceive pH calculations merely as entering the negative logarithm of concentration into a calculator. The underlying concepts, such as the relationship between [H₃O⁺] ions and an acid’s concentration, as well as the mathematical meaning of logarithms, are often not fully grasped. These fundamental ideas need to be explicitly and thoroughly explained in introductory courses.

Second, although chemical formulas are considered fundamental ⁷ and important ⁶ by teachers, the meaning of stoichiometric coefficients remains unclear to many students. The inability of most tested upper-secondary chemistry graduates to balance even a simple equation and apply it to calculations points to significant gaps in high school chemistry education. While these skills improved during the Chemistry Calculations course, the symbolic representation of chemical reactions often lacks meaning for students. Greater emphasis on sub-microscopic representations, for instance, through the use of interactive applets or physical models, may help bridge this gap. ^{47 , 48}

6.3 The assignment type’s role

The original expectation that students would favour assignments using symbols was not supported by their calculation results. The insignificant difference in performance, consistent with findings from previous studies, ^{9 , 18} suggests that this factor does not significantly influence overall scores. This outcome may be attributed to students’ greater familiarity with the word problem format, commonly used in upper-secondary chemistry textbooks. ^{21 , 44}

However, interviews revealed that students perceive assignments using symbols as easier. Despite this perception, their preference did not consistently translate into better performance, indicating that other factors, such as conceptual understanding and problem-solving strategies, may play a more critical role.

6.4 The role of mathematics skills

The results also highlighted significant weaknesses in students’ ability to integrate and apply mathematical concepts (e.g., fractions, logarithms, exponential calculations) in chemistry tasks. While students demonstrated proficiency in calculating percentages and performing basic mathematical operations, their approach often reflected a mechanistic process without deeper understanding. This may explain the notably higher success rates for mass fraction and molar concentration tasks. These tasks, introduced in lower-secondary chemistry education, are familiar to students and require only elementary mathematical operations, which they have already mastered. In contrast, other calculation types involve more advanced mathematics, likely contributing to students’ difficulties.

This finding aligns with Leopold and Edgar’s ¹² observation of students’ insufficient mathematical fluency but suggests this is only part of the issue. In this study, challenges such as misunderstanding digits’ place values, difficulties with unit conversions, minor computational errors, and problems converting logarithmic to exponential forms further compounded poor performance. These issues underline the need to strengthen the integration of mathematics within chemistry education.

Students frequently exhibit a mechanistic application of mathematical operations without proper conceptual comprehension. ^{11 , 49} Moreover, the assessment of chemistry skills in schools often neglects the connection between these skills and the types of problems encountered in the field, potentially influencing students’ performance. ^{50 , 51} Addressing this gap could improve students’ ability to approach complex chemistry calculations more effectively.

6.5 The effect of flipped-classroom principle on chemistry calculations course

The findings of this study demonstrate a positive effect of the Chemistry Calculations course on students’ ability to solve chemistry calculation tasks. Considering that students devoted approximately one semester, or about 50 h (including direct instruction and homework), to this topic, the observed improvement aligns with expectations. Importantly, progress was evident even in calculation types where students initially struggled.

However, the fact that students did not achieve maximum scores in the post-test, which comprised only basic tasks, is noteworthy. This shortfall was most pronounced in pH calculations and calculations from chemical equations, indicating that certain areas remain particularly challenging. These findings suggest the need for increased focus on specific problematic areas of chemical calculations. Mapping students’ errors and developing targeted instructional strategies to address these challenges will be critical. Previous studies, ^{11 , 52 , 53} provide useful frameworks for improving the design and practice of these types of calculations.

6.6 Students’ confidence in their ability to perform chemistry calculations

The analysis of confidence judgment (CJ) and calibration accuracy quotient (CAQ) values indicated that students generally demonstrated good calibration across most tasks. However, instances of both overconfidence and underconfidence in specific tasks highlight the need for targeted interventions to help students refine their self-assessment skills.

The confidence-if-correct (CFC) values in both the pre-test and post-test suggest a reluctance among students to assign the highest confidence ratings, consistent with findings by Putica. ⁵⁴ Post-course, the CFC values exceeded the confidence-if-wrong (CFW) values in four out of five calculation types (data for one type was insufficient). This aligns with Lundeberg et al.’s ⁴⁰ observation that students tend to exhibit higher confidence when their answers are correct than when they are incorrect.

A significant increase in students’ overall confidence was observed, driven by two factors: increased confidence among students who solved tasks correctly and a reduction in confidence among those who did not. This shift indicates an improvement in students’ self-awareness and self-assessment, further supported by the observed enhancement in confidence discrimination.

Nevertheless, these findings should be interpreted cautiously due to the relatively low response rate on the confidence judgment scale. This limitation underscores the need for additional data to draw more robust conclusions.

6.7 Implications for introductory courses

Since chemistry is fundamentally grounded in research and the verification of hypotheses in the laboratory, chemistry education must incorporate experimental activities. ⁵⁵ These activities inherently necessitate the mastery of chemical calculations. The results presented in this study, consistent with previous research, indicate that chemical calculations pose significant challenges for students.

The findings, in agreement with others ^{46 , 56} provide a compelling impetus for re-evaluating the approach to introductory chemistry courses. Despite being at the university level, it is evident that chemistry calculation courses must begin with fundamental concepts, as many students struggle even with basic tasks. A lack of foundational skills in chemical calculations from upper-secondary education could jeopardise students’ academic success, increase the difficulty of their chemistry studies, and potentially contribute to university dropouts. This underscores the need to provide students with comprehensive study materials and tasks designed to address these gaps, even if they are below the traditionally expected university level. Such an approach is critical not only for flipped-classroom courses but for all instructional formats, as effective study resources are a cornerstone of student success. Furthermore, the findings highlight several specific issues and suggest practical implications, including:

1. Students’ problems with certain chemistry calculation types

Most students who choose chemistry as their field of study after completing upper-secondary school struggle with fundamental tasks, such as balancing a simple chemical equation and calculating the theoretical yield from a given mass. Similarly, other practical types of chemical calculations, including mixing and diluting solutions or calculating pH, have also proven challenging.

The time allocated to chemical calculations in chemistry education ⁵⁷ appears to be insufficiently effective. One possible explanation is the weak connection between chemical calculations and practical activities, likely stemming from the overall lack of experimental work in chemistry classes. ¹⁸ Additionally, what was once a practical tool and an integral part of scientific process skills has, over time, become a purely theoretical topic, similar to balancing equations. ⁴⁶ While students may achieve the required outcomes during school-based assessments, the retention of these skills is often low, as highlighted by prior research. ⁹

To address these shortcomings, educators should consider revising the curriculum to include more targeted and differentiated instruction on these challenging areas. Emphasising the integration of chemical calculations with experimental activities can provide context and relevance, fostering deeper conceptual understanding. Furthermore, incorporating varied practice opportunities and diverse task formats, coupled with detailed and timely feedback, could enhance students’ mastery of these concepts.

1. The role of mathematics

The results of this study, consistent with the findings of Scott ¹³ and Rusek et al., ⁹ did not indicate a direct correlation between insufficient mathematical skills and students’ difficulties with chemical calculations. However, a detailed analysis reveals that tasks in which students achieved higher success rates were predominantly based on basic arithmetic operations. In contrast, more complex problems, such as dilution calculations and tasks involving chemical equations, require the manipulation of expressions, while pH calculations demand an understanding of logarithmic functions. This observation aligns with Leopold and Edgar’s ¹² conclusions.

To ascertain whether these challenges stem from inadequate knowledge of related chemical concepts or insufficient mathematical skills, future research should focus on individual types of chemical calculations. This could involve presenting students with a series of differently structured problems and integrating questions that specifically assess their grasp of fundamental concepts. Such an approach would help distinguish the relative contributions of conceptual and mathematical deficiencies.

The recurring errors identified in arithmetic, logarithmic functions, and unit conversions underscore the need for enhanced support in these areas. Educators must ensure that students are not only proficient in mathematical operations but also comprehend the underlying principles and their application in chemical contexts. Collaborative initiatives between chemistry and mathematics departments could facilitate the development of interdisciplinary teaching resources and methods, providing students with a cohesive understanding of these interconnected subjects.

1. The assignment type

The effect of task assignment style was found to be inconclusive, aligning with previous research in mathematics. ^{58 , 59} Consistent with Johnstone’s ⁶⁰ and later Hartman et al.’s ⁶¹ theories, the requirement to perform multiple successive steps emerged as a significant challenge for students. While the motivational context provided in word problems did not appear to influence students’ performance, it raises the question of whether such contexts are worthwhile in tasks aimed primarily at developing skills other than reading.

Conversely, the application of chemistry calculations to real-world scenarios demonstrates their practical relevance and underscores why students should master them. Eliminating meaningful contexts risks reducing these calculations to artificial exercises, similar to the often-criticised focus on equation balancing as an isolated skill. ⁴⁶

Future research should explore the impact of broader contextual framing on a series of tasks within the same calculation type, incorporating a graduated level of reading complexity. Such studies could better assess the influence of contextual and linguistic factors on students’ performance, offering insights into how to design tasks that effectively integrate both conceptual understanding and practical skills.

1. The flipped-classroom approach

Although the literature ^{23 , 26} frequently highlights the advantages of the flipped-classroom principle, its suitability for a challenging topic such as chemical calculations remained uncertain. The positive impact of the flipped classroom model on students’ performance and confidence in chemistry calculations, as demonstrated in this study, indicates that this approach can be at least as effective as traditional methods in improving students’ problem-solving abilities. In some respects, it may even prove more efficient. ^{62 , 63}

Future research should expand beyond traditional assessments of teaching effectiveness to include an exploration of the level of support perceived by students in both traditional and flipped classroom settings. Moreover, it should evaluate the adequacy of available self-study materials and, crucially, the role of perceived self-efficacy as a result of mastering chemical calculations.

1. Students’ confidence judgement

The significant improvement in students’ confidence and self-assessment accuracy, as indicated by the CAQ values, highlights the importance of fostering metacognitive skills in students. Additionally, the observed correlation between students’ results and their confidence judgements. ^{49 , 64} further underscores the relevance of promoting accurate self-assessment as a key component of effective learning. Educators should continue to integrate self-assessment and reflective practices into the learning process. Equipping students with tools to accurately evaluate their understanding and pinpoint areas requiring improvement can enhance learning outcomes. This objective can be achieved by providing students with access to a diverse range of progressively graded calculation problems. Practical experience suggests the necessity of encouraging students to engage with a substantial number of calculation exercises rather than relying solely on their initiative. Furthermore, students should be actively encouraged to seek assistance from course instructors when difficulties arise. ⁶⁵