Exploring Mathematical Values Held by High School Mathematics Teachers

doi:10.21203/rs.3.rs-4893810/v1

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Exploring Mathematical Values Held by High School Mathematics Teachers

https://doi.org/10.21203/rs.3.rs-4893810/v1

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This current study explored mathematical values espoused by senior high school (SHS) mathematics teachers in mathematics teaching and learning in a municipality in Ghana. In achieving this aim, we employed an explanatory sequential mixed methods design. In the quantitative study, a census was employed to include all 53 SHS mathematics teachers in a municipality. For the qualitative phase, purposive sampling was used to select six SHS mathematics teachers recruited to complete a ‘What I Find Important (WIFI) in my mathematics learning’ questionnaire in the quantitative phase to participate in interviews. The quantitative data were analyzed using inferential statistics (principal components analysis [PCA] with Oblimin rotation and one-way multivariate analysis of variance [MANOVA]). On the other hand, the qualitative data was analyzed using a thematic analysis. Among Bishop’s six-value cluster, the findings revealed that SHS mathematics teachers preferred mathematical values of rationalism, progress, and control in mathematics teaching. Notwithstanding, there was no statistically significant disparity in what SHS mathematics teachers value across class levels. Implications for practice and future research are reported.

mathematical values

values in mathematics education

mathematics teachers

principal components analysis

sequential explanatory mixed methods

Mathematics has played a major role in many advancements in every human effort sphere. Mathematical knowledge, for instance, offers a highly efficient method for developing cognitive discipline. It fosters the development of logical reasoning skills that are beneficial for learning other academic disciplines, such as science and social sciences. Despite the significance of mathematics as a school subject, it is widely acknowledged that numerous students develop negative attitudes towards it. Some students dislike the subject, others feel inept, and still others perceive it as irrelevant to their lives. According to Seah et al. (2016), such unfavorable perceptions of mathematics by many students are not a result of mathematics’ inherent qualities. This probably arose from designing and implementing a mathematics curriculum that included values, skills, and methods but did not explicitly address the values of mathematics education. Lovat and Clement (2008) assert that “values education [should] be at the heart of all pedagogical and curricular ventures and that any educational regime that sets out to exclude a values dimension in learning will be weakening its potential effects on all learning and student wellbeing, including academic learning” (p.13). In light of this, educators are urged to instill values consistent with universal principles in the learning and teaching of mathematics, either directly or indirectly (Mahmud et al., 2020).

Value is key to improving students’ mathematical knowledge and performance (Movshovitz-Hadar & Edri, 2013). The nature and forms of a teacher’s instructional decisions are usually impacted to varying degrees by the teacher’s internalized sets of values (Bishop, 2008a). According to Bishop (2008a), values significantly impact the interests, preferences, and actions of instructors and students in mathematics. Therefore, values can be useful for promoting cognitive and emotional development in mathematics education. Many educational systems anticipate that the many academic subjects, including mathematics, will instill ethical and moral values in its young citizens. For instance, the Ghanaian basic education system promotes the imbuing of values as one of the essential learning domains that should serve as the foundation for instruction and assessment in mathematics education (National Council for Curriculum and Assessment [NaCCA], 2019). Holistically, this is aimed at helping individuals develop a mathematical mindset.

It is believed that mathematics instruction should place equal emphasis on cognitive and affective dimensions, including attitude and values (Tan-işman & Kirez, 2017). This is because teachers who teach mathematics concepts by emphasizing cognitive development and leaving out key affective components such as values would produce students who could not appreciate and apply the mathematical concepts they learn to their daily lives. To curb this situation, there is a need to foster students' complete, balanced, and value-driven growth by developing the affective dimensions of mathematics education.

The inclusion of values in mathematics education is a relatively new phenomenon. Research on values in mathematics education emerged in the 1980s, focusing on integrating them into the cultural aspects of mathematical education (Bishop, 2004). Values might have varying interpretations for individuals based on the circumstances in which they find themselves (Kyeremeh & Dorwu, 2023). Values in mathematics education are qualities that individuals consider relevant and valuable. These values provide individuals with the motivation and drive to continue with their chosen path in mathematics learning and teaching (Seah, 2016; Seah & Andersson, 2015). In mathematics education, it is believed that values play a role in influencing the cognitive capacities and emotional states of teachers, aligning them with the process of teaching mathematics concepts (Seah et al., 2017a).

Values in mathematics teaching and learning have been classified by different academics using various methods (Bishop, 1988, 2008b; Seah, 2016). Bishop (1996) categorized educational values in mathematics education into mathematical, mathematics educational, and general educational values. Mathematical values are “values that have developed as the knowledge of mathematics has developed inside ‘Westernized’ societies” (Bishop, 2008a, p. 83). Examples of these mathematical values include control and openness. The mathematics educational values are directly linked to the teaching methodology, such as through practice and group projects. These valuing occur within the framework of actions undertaken to enhance the teaching and learning of mathematics. Also, general educational values encompass values considered valuable in school education and are imparted through many academic disciplines, such as mathematics.

Studies (e.g., Bishop, 2008a; Mahmud et al., 2020) have found that mathematics teachers widely believe that mathematics is the most value-free subject taught in schools, of which SHS mathematics teachers in Ghana is no exception. This is often reinforced by the idea that there is a solitary right answer and the best way to find it. According to FitzSimons et al. (2001), these lasting impressions help explain the alienation from this subject of many students and teachers. These misconceptions by teachers deprive many students of seeing the relevance of mathematics to their real lives and, as a result, develop positive attitudes toward learning mathematics.

There is little research (e.g., Aktas et al., 2019; David & Abass, 2023; Kyeremeh & Dorwu, 2023) about mathematics teachers’ awareness and preferences of mathematical values in mathematics teaching. Given this, this current study explored mathematical values espoused by senior high school (SHS) mathematics teachers in mathematics teaching in a municipality in the Ahafo Region of Ghana. This study, therefore, was guided by the research question, “What mathematical values are espoused by SHS mathematics teachers?” Based on the research question, we sought to test the null hypothesis, “There is no statistically significant difference in the mathematical values espoused by SHS mathematics across class levels.” This study’s findings are anticipated to help create awareness among mathematics educators and scholars about the value-ladenness of mathematics and provide some pedagogical strategies for values’ enactment in the classroom. Also, the study’s findings would serve as a valuable foundation for developing evidence based on Ghanaian teachers’ mathematical values and their contribution to the literature on values in mathematics education.

Values in Mathematics Education Held by Mathematics Teachers

The Values and Mathematics Project (VAMP) in Australia examined the planned and implemented values, the legislation governing these values in teaching, and the enhancement of mathematics instruction through values education for teachers. During their group talks, teachers were prompted to deliberate on subjects about values in their weekly journal entries. FitzSimons et al. (2000) disclosed that mathematics teachers had limited awareness of the values of teaching mathematics.

In Ghana, Kyeremeh and Dorwu (2023) conducted a cross-sectional survey to investigate the mathematical values espoused by pre-service mathematics teachers in mathematics education. One of the discoveries was that pre-service mathematics teachers highly value understanding, connections, exploration, and fluency in mathematics teaching and learning. The study's findings showed a statistically significant variation in the mathematical values espoused by pre-service mathematics teachers based on their class levels. These mathematical values, fluency, exploration and connection espoused by pre-service mathematics teachers relate more to what Bishop (2008a, 2008b) describes as control, progress and empiricism values of mathematics education, respectively.

In a similar context, David and Abass (2023) conducted a survey to investigate the mathematical values of 66 SHS mathematics teachers. It was found that SHS mathematics teachers prioritized authority, engagement, strategy, and communication in students' mathematics learning. The mathematical value of authority reflects the value of control, whereas the mathematical values of engagement and communication reflect openness in mathematics education, as described by Bishop (2008a, 2008b).

In a survey conducted by Bishop (2008b), the author examined the perspectives of Australian primary and secondary mathematics teachers regarding their preferences for mathematical values. The study also investigated how these values are implemented across six value clusters in mathematics classrooms. Through the sociological analysis, it was discovered that mathematics teachers prefer the values of control and progress when teaching mathematics. Teachers prioritize empiricism over rationalism in mathematics at the primary level, while both values hold significance. This approach differs from the previous research findings. At the primary level, mathematical activity mostly relies on empirical evidence. The ideological factor among secondary instructors is characterized by their ideas, with mathematics inclined towards rationalism, in contrast to the primary teachers. In terms of sociological and sentimental aspects, secondary teachers largely concur with their primary counterparts in advocating for openness, as an illustrative example.

The results also indicated that secondary teachers mostly choose rationalism as their value preference in mathematics teaching and learning. Similarly, empiricism was listed as the top choice regarding the elementary teachers' desired mathematical values. This implies that the things that secondary and elementary mathematics teachers consider important in mathematics differ. The study by Bishop et al. (2006) revealed similar findings when secondary mathematics teachers rated rationalism as a top value in mathematics education. Contrary, primary mathematics teachers favored empiricism over rationalism for mathematics, though both are important, and this contrasts with the findings above. The reason could be that a significant portion of mathematical tasks at the primary school level rely on empirical evidence.

In a study conducted by Dede (2012), the author examined the perspectives of Turkish and German mathematics teachers regarding the significance of mathematics and delved into the underlying mathematical ideals that shape their viewpoints. This study utilized data from the "Values in Mathematics Teaching in Turkey and Germany [VMTG]" project, which had a duration of two years. The VMTG project conducts a comparative analysis of the values held by mathematics teachers and their pupils in Germany and Turkey. German mathematics teachers prioritize mathematical values such as rationalism, progress, openness, control, and mystery. In contrast, Turkish mathematics professors prioritized the mathematical significance of rationalism, objectism, progress, control, and other related concepts. The findings indicated that the mathematical values of German and Turkish mathematics teachers exhibited both parallels and differences.

In their study, Zhang and Seah (2015) examined the fundamental principles of successful mathematics instruction as seen by four secondary mathematics instructors in mainland China. Teachers highly appreciate common qualities of good teaching, such as incorporating enjoyable activities, utilizing various instructional approaches, promoting student engagement, and providing illustrative examples. The preference for employing several ways, such as using various approaches or techniques, demonstrates a recognition and appreciation for the importance of progress. Haciomeroglu’s (2020) study in Turkey examined the values in mathematics education demonstrated by pre-service mathematics teachers. Pre-service mathematics instructors were found to prioritize values in mathematical instruction, such as cognition and concretization. The findings indicated that the program's duration did not significantly impact the mathematics education values of student teachers.

Given this literature review on values in mathematics education advocated by mathematics teachers, it is evident that there are available studies in this area of values research. However, there appear to be limited studies on mathematical values espoused by mathematics teachers. This study, therefore, fills this gap in the literature.

Theoretical Framework

Investigating SHS mathematics teachers’ mathematical values and teaching practices is novel. In order to have some understanding of the mathematical values advocated by SHS mathematics teachers and how these values are promoted through teaching, using a preexisting theoretical framework is crucial while examining the values under investigation. In this direction, we used Bishop's (1988) six-values cluster model, which was established through a review of texts on the actions of mathematicians in Western history and culture. This model organizes six value clusters into three pairs complementary to each other and associated with the three aspects of ideology, sentiment/attitude, and sociology. Bishop (1988) summarized six value clusters as follows:

“The particular societal developments which have given rise to mathematics have also ensured that it is a product of various values: values which have been recognized to be of significance in those societies. Mathematics, as a cultural phenomenon, only makes sense if those values are also made explicit. Bishop has described them as complementary pairs, where rationalism and objectism are the twin ideologies of mathematics, those of control and progress are the attitudinal values which drive mathematical development, and, sociologically, the values of openness and mystery are those related to potential ownership of or distance from mathematical knowledge and the relationship between the people who generate that knowledge and others” (Bishop, 1988, p.82).

Research Design

We utilized explanatory sequential mixed approaches to accomplish this study's objective. We employed this design to garner qualitative data to provide a more comprehensive explanation of the initial quantitative results.

Selection of Participants

In the quantitative study, a census was employed to include all 53 SHS mathematics teachers within a municipality in the Ahafo Region of Ghana. An effort was made to involve all 53 SHS mathematics teachers in the quantitative study due to the relatively small population of mathematics teachers. One of the greatest advantages of using the census frame is that all population members have the same opportunity to participate in the study and are more capable of yielding representative results (Cohen et al., 2018). Among these selected participants, 48 offered to respond to the questionnaire. Purposive sampling was used to select six SHS mathematics teachers recruited to complete the questionnaires in the quantitative phase to participate in interviews. Constantinou et al. (2017) found data saturation with five interviews in a qualitative study; therefore, choosing six participants is deemed appropriate.

Instruments

We employed questionnaires and interviews in this study to gather the quantitative and qualitative data, respectively. In the first phase of this study, we used the ‘What I Find Important (WIFI) in my mathematics learning’ questionnaire to gather quantitative data on SHS mathematics teachers’ preferred mathematical values in mathematics teaching and learning. For this study, the WIFI questionnaire was modified for use. The original WIFI instrument designed for students alone comprised four sections A, B, C and D (Seah, 2016). However, only 64 Likert-type items in Section A of the original WIFI questionnaire were adapted for the quantitative phase. The characteristics of these items allowed for the investigation of SHS mathematics teachers' mathematical values. In order to make it possible for teachers to respond to Section A items, certain items, such as Teacher asking us questions (item 35), Teacher helping me individually (item 41), and Feedback from my teacher (item 44) were modified as Teacher asking students questions (item 35), Helping students individually (item 41), and Providing feedback to students (item 44) respectively.

In the qualitative phase, the study employed semi-structured interviews to explain the key findings on mathematical values espoused by SHS mathematics teachers in mathematics teaching from the quantitative phase. The semi-structured interview guide was designed based on the findings obtained from the quantitative study questionnaire. The interview protocol consisted of five items.

Ensuring Validity and Reliability

In ensuring quality measures for data-gathering methods in the quantitative phase of the study, validity and reliability were established. In order to determine the face and content validity of the questionnaire, we presented the preliminary version of the questionnaire to two experts in the field of study to provide their expert assessment regarding the correctness and suitability of the instrument and its items and their relevance to the study purpose. Based on experts' comments, some of the items, such as Teacher helping me individually (item 41) and Feedback from my teacher (item 44), were revised as Helping students individually (item 41) and Providing feedback to students (item 44), respectively.

In order to ensure the reliability of the questionnaire, a pilot test was conducted among 20 SHS mathematics teachers from a different municipality in the Ahafo Region of Ghana using the questionnaire. This sample is believed to possess characteristics similar to those of the population regarding available resources, conditions of service, and competencies. In the context of this study, we conducted internal consistency reliability on the data from the pilot test to measure the extent to which the items clustered initially within the questionnaire. The study assessed the internal consistency of the questionnaire by computing the Cronbach Alpha coefficient. This test yielded a Cronbach’s Alpha reliability coefficient of 0.79, suggesting the instrument's acceptability (Taber, 2018).

Trustworthiness of the Qualitative Study

In the qualitative phase, we ensured trustworthiness by establishing credibility, dependability, transferability, and confirmability. In establishing the credibility of the qualitative findings, we cross-checked with the participants to verify their experiences during the interview to ensure an accurate interview representation. In order to guarantee dependability, the study procedures were made replicable through meticulous documentation. Confirmability was ensured by keeping an 'audit trail' that provides a detailed record of the decisions and procedures outlined. During the audit trail procedure, impartial and discerning reviewers were tasked with assessing the methodologies employed for data collection. To achieve transferability, we provided a comprehensive account of the context, participants, location and a clear data analysis procedure.

Data Collection Procedure

In this study, quantitative and qualitative data were gathered in two phases. The first phase gathered quantitative data using a close-ended questionnaire. Before data collection, we applied to the Humanities and Social Sciences Research Ethics Committee of Kwame Nkrumah University of Science and Technology for ethical approval. Once permission was granted by the Ethics Committee, we wrote a letter with the attachment of ethics approval to the Municipal Directorate of Ghana Education Service (G.E.S) for permission to conduct the study in the public SHSs within the municipality. The obtained ethics and the Directorate’s approval letters were used to seek permission from the Headmasters of public SHSs in the municipality to carry out the study in their SHSs. We invited the sampled teachers to familiarize them with the study and discuss any potential apprehension they may have.

We established a strong rapport to gain the participants' trust and willingness to respond to the questionnaire and participate in future interviews without any fear. We also emphasized the confidentiality of the data and the results. Each participant who volunteered to participate was given an informed consent form to read and sign. Afterwards, a discussion was initiated, and agreement was reached concerning the date for administering the close-ended questionnaire. One volunteer from each of the three SHSs was engaged to assist in distributing the questionnaire and their retrieval. The questionnaires distributed to participants were collected the same day after their completion. This helped to ensure timely response and increase the retention rate of the questionnaires distributed. The close-ended questionnaire for the quantitative phase was administered over four weeks.

In the qualitative study phase, data was collected through face-to-face interviews. During this step, a visit was made to obtain the agreement of potential participants and familiarize them with the study through direct interaction. Regarding this matter, we provided the interviewers the option to choose the location where they would feel at ease and secure throughout the interview. This was implemented to guarantee a shared comprehension between the interviewer and the interviewees within the research environment. During the interview sessions, we actively listened to the participants and posed questions that enabled me to extract further information on many topics. The data was documented using field notes, brief written notes, and written summaries of interviews. Participant interviews lasted approximately 50 minutes. The interviews were carried out a fortnight after the quantitative survey to elucidate certain assertions made by participants in light of the study's findings.

Data Analysis

The study data was analyzed using a combination of quantitative and qualitative methodologies. Data from the questionnaire were analyzed using inferential statistics (principal components analysis [PCA] with Oblimin rotation and one-way multivariate analysis of variance [MANOVA]). These analyses were done using SPSS version 23. Specifically, the data analysis method used PCA to explore SHS mathematics teachers’ mathematical value preferences in response to research question one. To test the hypothesis, one-way MANOVA was used to investigate whether a statistically significant disparity exists in SHS mathematics teachers’ mathematical values across class levels. Using one-way MANOVA allowed for comparing two or more groups in terms of their means on more than one dependent variable (Pallant, 2016).

The qualitative data collected from the interviews was analyzed using a thematic analysis. We began by reading the transcripts thoroughly to familiarize myself with the data and understand the patterns (Braun & Clarke, 2021). The coding step commenced after my thorough reading and familiarization with the data. We then created a comprehensive profile for every participant and organized them into categories based on their grade levels. The identified codes were compared with the data extracts that exhibited the codes. We initiated the process of examining the data for recurring patterns or topics. The subsequent stage of the analytical process involved establishing and designation the themes, which commenced when we formulated the topics for ultimate enhancement. In order to accomplish this, we reviewed the compiled data for each theme and systematically arranged the themes to ensure internal coherence with the participants' narratives and study goals. The final phase of writing the study report commenced after we formulated thoroughly elaborated themes.

What mathematical values are espoused by SHS mathematics teachers?

To answer this research question, we subjected the 64 items on the WIFI Scale to PCA with Oblimin rotation to determine the smallest possible number of factors needed to effectively represent the relationships between the variables in the data. Prior to the PCA, the suitability of the data for factor analysis was assessed. Table 1 displays two tests: the Kaiser-Meyer-Olkin Measure (KMO) of Sampling Adequacy and Bartlett's Test of Sphericity, which determines whether the correlation matrix is suitable for factor analysis.

Table 1

KMO and Bartlett's Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy.		.652
Bartlett's Test of Sphericity	Approx. Chi-Square	2474.941
	Df	741
	sig.	.000

From Table 1, KMO recorded a value of 0.886, which was above .6. This suggests that conducting factor analysis is realistic and has merit (Howard, 2016). Bartlett's test of Sphericity value was statistically significant (p = 0.00), so factor analysis is appropriate.

Furthermore, a significant number of coefficients with values of 0.3 or higher were identified upon examination of the correlation matrix. Utilizing Kaiser's criterion and scree tests established a determination regarding the optimal number of elements to keep. Only components with an eigenvalue larger than or equal to 1 were considered when applying Kaiser's criterion. For this purpose, only the initial 13 components with eigenvalues greater than 1 (21.6, 6.3, 4.8, 4.1, 3.3, 2.5, 2.4, 2.2, 2.1, 1.7, 1.4, 1.3, and 1.0) were considered in evaluating the number of components that meet this condition. These 13 components accounted for a total of 85.7% of the variance.

Upon careful examination of the scree plot, it became evident that there was a clear break in the pattern after the second component. This indicates that the initial two components account for a significantly greater proportion of the variability than the remaining components. We selected two components for additional investigation with Catell's (1966) scree test. The parallel analysis results, which showed that only two components had eigenvalues higher than the threshold values for a randomly generated data matrix of the same size (64 variables × 48 respondents), further supported these conclusions.

We employed a method of investigation, testing various variables until a viable solution was identified (Tabachnick & Fidell, 2013). Consequently, Oblimin rotation was executed, specifically imposing a two-factor solution. Table 2 displays the proportion of total variance explained for each factor derived from the two-factor solution.

Table 2

The total variance explained for each factor resulting from the two-factor solution (Before Deletion)
Component	Explained Variance	% Variance	Cumulative %
1	19.143	45.579	45.579
2	5.477	13.041	58.620

From Table 2, the two-component solution explained 58.6% of the variance, with components 1 and 2 contributing 45.6% and 13%, respectively. The results demonstrate linear correlations among the different components, which offer a contextual framework for conducting factor analysis. Following a two-factor rotation, we noticed that 18 things had poor compatibility with the other items in their respective components, as indicated by the communalities result. We have taken into account Pallant's (2016) assertion that in order to achieve a favorable result and ensure accurate interpretation, the factor loadings must be above a threshold of 0.3. As a result, the 18 items with communality scores below 0.3 were removed from the WIFI scale. Furthermore, after implementing the two-factor rotation, certain variables with significant loadings were selected as the representation for each component. However, we could not obtain a simple structure where every variable loaded substantially on only one component. We found nine cross-loading items from the pattern matrix output, which were also deleted. Pallant (2016) suggests eliminating variables that do not load exclusively on one component in order to achieve a more optimal solution.

Table 3 displays the percentage of total variation explained for each factor arising from the two-factor solution after removing these items.

Table 3

The total variance explained for each factor resulting from the two-factor solution (After Deletion of 27 items)
Component	Explained Variance	% Variance	Cumulative %
1	18.099	48.915	48.915
2	5.240	14.163	63.078
Extraction Method: Principal Component Analysis

As evident in Table 3, after some items were deleted, the two-component solution explained an overall 63.08% of the variance, with components 1 and 2 contributing 48.92% and 14.16%, respectively. The results demonstrate a clear correlation between the various factors, which validates factor analysis. Comparatively, the total variation explained for each of the four components has grown. This supports Pallant's (2016) claim that eliminating items with low communality values generally increases the total explained variance. The PCA with Oblimin rotation was repeated with the remaining 37 items after the deletion of 27 total items to obtain a simple structure (Tabachnick & Fidell, 2013).

Oblimin rotation was performed to help interpret these two components. Table 4 illustrates that the rotated solution confirmed the presence of a straightforward structure, where all variables had significant loadings on a single component, and the two components displayed a range of strong loadings.

Table 4

Pattern and Structure matrix for PCA with Oblimin rotation of two-factor solution of mathematical value preferences
Items	Pattern Matrix		Structure Matrix		Communalities
	Component		Component
	1	2	1	2
3. “Small-group discussions”	.909	.005	.908	− .166	.824
34. “Outdoor mathematics activities”	.906	.021	.902	− .151	.814
23. “Learning mathematics with the computer”	.905	− .003	.905	− .174	.820
1. “Investigation”	.888	− .107	.908	− .275	.836
50. “Getting the right answer”	.885	− .068	.898	− .236	.811
38. “Giving students a formula to use”	.871	− .073	.884	− .238	.787
48. “Using concrete materials to help students understand mathematics”	.854	− .067	.867	− .229	.755
61. “Stories about mathematicians”	.842	.052	.832	− .107	.695
27. “Being lucky at getting the correct answer”	.823	− .052	.833	− .208	.696
35. “Teacher asking students questions”	.817	.011	.815	− .143	.664
46. “Students asking questions”	.785	− .116	.807	− .265	.664
29. “Students making up their own mathematics questions”	.774	.003	.773	− .143	.598
42. “Working out the mathematics by myself”	.772	.043	.763	− .103	.585
52. “Hands-on activities”	.764	.145	.737	.001	.563
21. “Students posing mathematics problems”	.760	.100	.741	− .044	.559
20. “Mathematics puzzles”	.759	.049	.749	− .094	.564
30. “Alternative solutions”	.746	.044	.738	− .097	.546
17. “Stories about mathematics”	.732	.058	.721	− .080	.523
44. “Providing feedback to students”	.726	.103	.707	− .034	.510
5. “Explaining concepts to students”	.701	− .080	.716	− .212	.518
18. “Stories about recent developments in mathematics”	.694	.005	.693	− .126	.481
32. “Using mathematical words”	.693	− .118	.716	− .250	.526
51. “Learning through mistakes”	.691	.016	.688	− .115	.474
54. “Understanding concepts/processes”	.687	.193	.650	.063	.459
60. “Mystery of mathematics”	.677	.044	.669	− .084	.449
10. “Relating mathematics to other subjects in school”	.672	.024	.668	− .103	.446
39. “Looking out for mathematics in real life”	.625	− .065	.637	− .183	.410
36. “Practicing with lots of questions”	.588	− .172	.620	− .283	.413
11. “Appreciating the beauty of mathematics”	.568	− .024	.573	− .132	.328
28. “Knowing the time-table”	.550	− .089	.567	− .193	.329
16. “Looking for different possible answers”	.532	− .091	.549	− .192	.309
24. “Learning mathematics with the internet”	.027	.987	− .159	.982	.964
19. “Explaining solutions to the students”	.001	.976	− .184	.976	.953
45. “Students giving feedback to colleagues”	− .002	.968	− .185	.968	.937
49. “Examples to help students understand”	.015	.949	− .164	.946	.896
31. “Verifying theorems/hypothesis”	− .081	.941	− .259	.957	.922
40. “Explaining where rules/formulae came from”	− .035	.835	− .193	.842	.710
Note. Major loadings for each item are bolded.

Based on the results presented in Table 4, the items that strongly correlate with each component were labeled using similar explanatory criteria. The first component (C1) was labelled understanding, with 31 items accounting for 48.9% of the overall variance. Some of the items that loaded on C1 include, Explaining concepts to students (item 5), Understanding concepts/processes (item 54), Using concrete materials to help students understand mathematics (item 48), Hands-on activities (item 52), Teacher asking students questions (item 35), Students asking questions (item 46), Relating mathematics to other subjects in school (item 10), Looking out for mathematics in real life (item 39). However, a few items, such as Learning mathematics with the computer (item 23), Mystery of mathematics (item 60), and Being lucky at getting the correct answer (item 27), which were loaded onto C1, do not seem to reflect understanding. This suggests that SHS mathematics teachers value understanding, which relates to the value of Rationalism in mathematics teaching and learning.

The second component (C2) was labelled authority, with six items accounting for 14.2% of the overall variance. Some of the items that loaded on C2 include Explaining where rules/formulae came from (item 40), Verifying theorems/hypotheses (item 31), Explaining solutions to the students (item 19), and Examples to help students understand (item 49). However, a few items, such as Learning mathematics with the internet (item 24) and Students giving feedback to colleagues (item 45), which were loaded onto C2, do not seem to reflect authority. These results suggest that SHS mathematics teachers prefer mathematical values, understanding, and authority in mathematics teaching, which relates to the values of rationalism and control in mathematics education, respectively.

There is no statistically significant difference in the mathematical values espoused by SHS mathematics teachers across class levels.

A one-way MANOVA was performed to investigate whether a statistically significant disparity exists in SHS mathematics teachers’ preferences for mathematical values across class levels. The two components, understanding and authority, were the dependent variables. The independent variable in this study was the class level.

An initial assessment was conducted to evaluate the assumptions of normality, linearity, univariate and multivariate outliers, homogeneity of variance-covariance matrices, and multicollinearity. However, no major violations of these assumptions were detected. For instance, the test of the assumption of homogeneity of variance-covariance matrices was performed, and the result is presented in Table 5.

Table 5

Box's test of equality of covariance matrices
Box's M	6.073
F	.945
df1	6
df2	30117.552
Sig.	.461

From the Box’s test of equality of covariance matrices results in Table 5, the p-value (.46) was not significant. Therefore, we have not violated the assumption of homogeneity of variance-covariance matrices. Also, the check for sample size adequacy was performed, and the result is presented in Table 6

Table 6

Descriptive statistics of the SHS mathematics teachers
	At what level do you teach?	Mean	Std. Deviation	N
Authority	Basic 10	19.47	6.681	17
	Basic 11	18.61	7.625	18
	Basic 12	14.31	7.825	13
	Total	17.75	7.516	48
Understanding	Basic 10	105.88	26.949	17
	Basic 11	105.28	26.797	18
	Basic 12	108.85	37.984	13
	Total	106.46	29.624	48

As shown in Table 6, more cases exist in each cell than the number of dependent variables. Given this, any violations of normality that may exist will not matter too much (Pallant, 2016).

Table 7 provides multivariate tests of significance to assess whether there are statistically significant differences between the groups on a linear combination of the dependent variables.

Table 7

Multivariate tests of SHS mathematics teachers’ mathematical values in across class levels
Effect		Value	F	Hypothesis df	Error df	Sig.	Partial Eta Squared
Intercept	Pillai's Trace	.959	518.125	2.000	44.000	.000	.959
	Wilks' Lambda	.041	518.125	2.000	44.000	.000	.959
	Hotelling's Trace	23.551	518.125	2.000	44.000	.000	.959
	Roy's Largest Root	23.551	518.125	2.000	44.000	.000	.959
Class Level	Pillai's Trace	.083	.968	4.000	90.000	.429	.041
	Wilks' Lambda	.918	.968	4.000	88.000	.429	.042
	Hotelling's Trace	.090	.966	4.000	86.000	.430	.043
	Roy's Largest Root	.090	2.015	2.000	45.000	.145	.082

Table 7 indicates no statistically significant variation in the pooled dependent variables based on the class level, F (4, 88) = .97, p = .43; Wilks’ Lambda = 0.92; partial eta squared = 0.04. Therefore, we deduced that there is no significant disparity in SHS mathematics teachers’ preferences for mathematical values across class levels.

In gaining insight into what comes to mind when SHS mathematics teachers think about values in mathematics education, interviews were conducted among six (MT1, MT2, MT3, MT4, MT5, and MT6) in the qualitative phase of the study. From the interview, SHS mathematics teachers explained that values in mathematics education are considered important in teaching and learning mathematics. One of the participants stated: “When we [teachers] say values in mathematics education, I think it is about how we [teachers] regard mathematics. Whether it is important or not” (MT2–3/11/2023). Another also commented: “For me, it is about the importance of learning mathematics in school. What we [teachers] appreciate when it comes to the learning of the subject [mathematics]” (MT5–12/11/2023).

Conversely, some SHS mathematics teachers explained values as the moral aspect of mathematics education. This is captured in the following excerpt: “I will say it is the moral aspect of the mathematics we learn in school. That is to say that, values are what we consider to be right in learning of mathematics” (MT4–10/11/2023). MT3 also mentioned, “In my view, values in mathematics education have to do with what we consider to be acceptable in the learning of mathematics” (MT3–3/11/2023).

From the quotes above, it is evident that SHS mathematics teachers hold different views about the meaning of values used in mathematics education. Some understand values to mean what is considered worthwhile or important in the teaching and learning of mathematics. Others also understand values to mean what is considered right or wrong regarding mathematics teaching and learning.

Interviews were conducted among selected participants to gain insight into SHS mathematics teachers’ mathematical values. From the participants' accounts, it was revealed that SHS mathematics teachers value constant practice of the use of formulas in teaching and learning of mathematics. MT5 recounted that: “As it stands, you cannot learn the subject [mathematics] without the use of formulas. Students will need them [formulas] to solve problems in exams” (MT5–12/11/2023). Another participant remarked: “…for me, mathematics is all about learning how to use formulas and rules to solve problems. In order to solve problems, you will have to find a formula that will work and apply it” (MT3–3/11/2023). MT1 also added that: “You see if you consider the nature of mathematics, you will realize that in order to learn the subject [mathematics] and learn it well, you have to do constant practice” (MT1–2/11/2023).

From the quotes above, SHS mathematics teachers' preference for the constant practice of the use of formulas in mathematics education relates to the value of Control. This suggests that SHS mathematics teachers value Control in mathematics teaching and learning.

From the interviews, it was also revealed that SHS mathematics teachers prefer using different ways/methods in solving mathematical problems. This is captured in the excerpts below:

In mathematics, there are different ways of solving a particular problem. So, you have to learn all these methods so that I can impact on my students. For example, we can use 2πr or πd we to find the area of a circle of circle (MT4–10/11/2023).

…when it comes to the learning of mathematics and you want to find a solution to a problem, there are at times you may find more than one approach. So there are so many ways, you just have to use the one [approach] you are comfortable with it (MT5–12/11/2023).

Based on the above quotes, SHS mathematics teachers’ preference for using different ways/methods in solving mathematical problems relates to the value of Progress. This suggests that SHS mathematics teachers prefer value Progress in mathematics teaching and learning.

Contrary to mathematical values, SHS mathematics teachers preferred perseverance and creativity related to mathematics educational values. In the interview, MT3 stated, “Alright, I know mathematics brings creativity. If you check the curriculum, they have stated clearly that we should help build creativity among students” (MT3–3/11/2023).

MT6 also commented that:

The learning of mathematics is such that you have to persevere. You try a problem at first and you don’t get, you have to try again. You will find some students getting answer wrong at first trial and stop there (MT6–20/11/2023).

The above quotes revealed SHS mathematics teachers’ lack of understanding of what constitutes mathematical values. SHS mathematics teachers’ preferences for perseverance and creativity are constituents of mathematics educational values embedded in the mathematics curriculum due to other values (Bishop, 2008a).

The study’s findings showed that SHS mathematics teachers prefer the mathematical value of understanding and authority in mathematics education, which relates to the value of rationalism and control in mathematics teaching and learning. The study also discovered that SHS mathematics teachers prefer using different ways/methods in solving mathematical problems which relates to the value of Progress in mathematics teaching and learning. These findings largely concur with what Kyeremeh and Dorwu (2023) found in their cross-sectional survey among pre-service mathematics teachers in a College of Education in Ghana. Pre-service mathematics teachers preferred the mathematical values, fluency, exploration and connection in mathematics teaching, which relate more to what Bishop (2008a, 2008b) describes as control, progress and empiricism values of mathematics education, respectively. In another study, Dede (2012) observed that German mathematics teachers prioritize mathematical values such as rationalism, progress, openness, control, and mystery. According to the literature, values play a crucial role in the affective contexts of mathematics classrooms and significantly influence teachers’ instructional decisions in mathematics (Bishop, 2008a).

This current study also found no statistically significant disparity in SHS mathematics teachers’ preferences for mathematical values across class levels. This suggests that SHS mathematics teachers' mathematical values in mathematics education were consistent as they taught across class levels. These findings contradict the findings from previous studies (e.g., Bishop 2008a; Bishop et al., 2006; Kyeremeh & Dorwu, 2023). For example, Kyeremeh and Dorwu (2023) found a statistically significant variation in the mathematical values espoused by pre-service mathematics teachers based on their class levels. Similarly, the study by Bishop et al. (2006) revealed that secondary mathematics teachers rate rationalism as a top value in mathematics education, with primary mathematics teachers favoring empiricism over rationalism in mathematics education. The reason could be that at the primary school level, much mathematical work is empirical in nature. The lack of disparities in the mathematical values espoused by SHS mathematics teachers indicates that the class level at which they teach does not influence, to some degree, their mathematical values system.

An argument is also made for giving more attention to specific broad educational values such as critical thinking, problem-solving, and creativity when considering the categories of values that may be present in mathematics courses. In mathematics education research, the values construct has the potential to go beyond students' comprehension and performance in mathematics. It can also contribute to developing a value-based mathematics education, which is crucial for thriving in the Fourth Industrial Revolution. Mathews (2001) asserts that value is a mediator in learning practices. This perspective on value considers the aspects of mathematics learning that students and teachers deem significant as a factor that influences the teaching and learning of mathematics. Values form the fundamental essence of civilization. Values are inherent to human nature and have a deliberate or spontaneous impact on individuals' behaviors, decisions, and choices.

This current study has presented findings on SHS mathematics teachers’ mathematical values and teaching practices in a municipality in the Ahafo Region of Ghana. There was contention among SHS mathematics teachers about the meaning of values as used in mathematics education. Among the SHS mathematics teachers, there was a contention about the meaning of values used in mathematics education. The SHS mathematics teachers preferred mathematical values of rationalism, progress, and control in mathematics teaching. In the case of Progress, SHS mathematics teachers preferred using different ways/methods of solving mathematical problems in their teaching of mathematics. Notwithstanding, there was no statistically significant difference in SHS mathematics teachers’ preferences for mathematical values across class levels. This suggests that SHS mathematics teachers' mathematical values in mathematics teaching and learning were consistent as they taught across class levels. To enhance comprehension of values and their significance in mathematics education at the primary and secondary levels, it is crucial to research to determine the values that basic school mathematics teachers prioritize in their teaching. This research should also examine how these values align with the values held by students and those promoted in the mathematics curricula.

Ethical Statement

This study received approval from the Humanities and Social Sciences Research Ethics Committee for using human subjects with approval number: HuSSREC/AP/105/VOL.1.

Conflict of Interest Statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Exploring Mathematical Values Held by High School Mathematics Teachers

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Abstract

Introduction

Values in Mathematics Education Held by Mathematics Teachers

Theoretical Framework

Methodology

Research Design

Selection of Participants

Instruments

Ensuring Validity and Reliability

Trustworthiness of the Qualitative Study

Data Collection Procedure

Data Analysis

Results/Findings

What mathematical values are espoused by SHS mathematics teachers?

Discussion

Conclusions

Declarations

Ethical Statement

Conflict of Interest Statement

References

Additional Declarations

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