Teaching knowledge and difficulties of In-field and Out-of-field Junior High School mathematics teachers in algebra

Abstract This paper sought to explore the Knowledge of Algebra for Teaching and the algebra difficulties of In-field and Out-of-field Junior High School mathematics teachers using the expanded KAT framework. The study employed the descriptive survey design and involved the participation of 374 mathematics teachers using an achievement test instrument. The study projected School Algebra Knowledge as the prevailing knowledge domain and revealed that the two categories of Junior High School mathematics teachers possess knowledge that falls below average for five (5) out of the seven (7) algebra knowledge domains and also for the overall Knowledge of Algebra for Teaching. However, In-field mathematics teachers showed higher knowledge as compared to Out-of-field mathematics teachers in six (6) out of the seven (7) algebra knowledge domains as well as the overall Knowledge of Algebra for Teaching. Also, the research revealed that mathematics teachers with 5 or more years of teaching experience have higher Knowledge of Algebra for Teaching as compared to those with below 5 years of teaching experience. The study finally identified eight major algebra difficulties among Junior High School mathematics teachers. The findings of the study have implications for teacher preparation, policy and practice.

This paper provides insights into the deficiencies in mathematics teachers' Knowledge of Algebra for Teaching in Ghana.Mathematics teachers' repertoire of knowledge in algebra influences learners' achievement in mathematics.This is due to the fact that, it is out of the knowledge teachers possess that they teach and almost impossible for teachers to effectively impart knowledge they do not possess to learners.In view of this, the algebra teaching knowledge of mathematics teachers should be monitored and improved to positively affect students' algebra knowledge, hence their general performance in mathematics.The study projected aspects of Junior High School mathematics teachers' knowledge that need to be improved through effective Professional Development Training and Community of Practice.

Introduction
Algebra functions as the bedrock (National Mathematics Advisory Panel, 2008;U.S. National Research Council, 2001;Wilmot, 2008) and language of mathematics.The numerous applications of algebra in mathematics project the critical and significant roles of algebra in learners' success and advancement in mathematics (Matthews & Farmer, 2008;Star et al., 2015;Wang & Goldschmidt, 2003;Wilmot et al., 2018), making it almost impossible for learners with weak foundation in algebra to perform well in mathematics.In view of this, learners need strong foundation in algebra to excel and progress in mathematics (Makonye & Stepwell, 2016;Moses, 2001;Osei, 2020;Osei & Kubi, 2022).
Researches (Chinnappan, 2010;Sultan & Artzt, 2011) have projected algebra as a particular strand of mathematics that poses several difficulties to teachers.According to Pincheira and Alsina (2021), studies have elucidated several deficiencies in the knowledge of algebra among mathematics teachers.These difficulties include; inability to recognise the different mathematical meanings of the equal sign, especially its equivalence property (Ferreira et al., 2017;Trivilin & Ribeiro, 2015), difficulty in handling algebra tasks that require the representation of unknown quantities with variables (Bair & Rich, 2011), struggles to understand and interpret algebraic symbols, graphical representations and solving algebraic task (Strand & Mills, 2014), issues of generality and symbol language usage (Chick, 2009), deficiencies and less attention to the development of algebraic thinking which leads to the handling of algebraic tasks mostly by trial and error (DiBernardo et al., 2017) and difficulties in handling tasks involving generalisations and connecting different representations (Noviyanti & Suryadi, 2019;Oliveira et al., 2021;Zapatera Llinares & Callejo de la Vega, 2018).The existence of these difficulties among mathematics teachers increases their struggles in providing assistance to learners with difficulties in algebra (Bush & Karp, 2013;Kieran, 2018), thereby reducing the potency of mathematics teachers' ability to offer effective and reliable interventions for remedying learners' difficulties in algebra.
Numerous studies (Baumert et al., 2010;Rowan et al., 1997) have projected the mathematics teacher as a key factor that influences learners' understanding during mathematics instructions.Mathematics teachers' repertoire of knowledge and their difficulties in algebra have potentials of influencing learners' achievement in mathematics (Campbell et al., 2014;Mohr-Schroeder et al., 2017).This is due to the fact that, teachers teach out of the knowledge they have, and it is almost impossible for them to effectively impart to learners, knowledge outside the knowledge they possess.Mathematics teachers with difficulties in the contents they teach are likely to transmit these difficulties to learners during the instructional process (Baumert et al., 2010;Campbell et al., 2014;Eisenberg, 1977;Hanushek, 1972;Shulman & Quinlan, 1996).In laying a strong algebra foundation for learners, mathematics teachers need to purposefully detect and address learners' difficulties in algebra (Fumador & Agyei, 2018), which is almost impossible if teachers themselves struggle in similar areas.Additionally, Mathematics teachers with difficulties in the algebra content they teach, may have challenges in creating effective teaching strategies in addressing learners' challenges in algebra for enhancement in mathematics achievement.Researches (Ball et al., 2008;Hurrell, 2013) have shown that the knowledge of mathematics teachers about the content they teach is an essential indicator of their competency and efficiency.
Studies (Boyd et al., 2010;Harris & Sass, 2011;Kraft & Papay, 2014;Ladd & Sorensen, 2017;Staiger & Rockoff, 2010;Wiswall, 2013;Wolters & Daugherty, 2007) have shown that the knowledge and effectiveness of teachers improve as their years of teaching experience increases, especially at the early stages, according to Podolsky et al. (2019).Koedel and Betts (2007) argue that, even beyond teachers' first decade on field, they continue to improve in knowledge and effectiveness.Darling-Harmmond (2000) emphasized that teachers' years of teaching experience is an indicative variable for their expertise and efficiency.Literature (Bodenhausen, 1988;Klecker, 2002;Rosenholtz, 1986) has revealed that teachers who have more years of teaching experience possess in-depth knowledge about the contents they teach and are effective than those with fewer years of teaching experiences.On this basis, the knowledge of algebra for teaching of Junior High School mathematics teachers is expected to improve as their years of teaching experience increase.This is due to the fact that teachers develop their competency and efficiency through classroom experiences (Klecker, 2002;Rosenholtz, 1986).
In Ghana, mathematics is taught by both In-field and Out-of-field teachers at the Junior High School level.According to McConney and Price (2009), supported by Hobbs (2013), In-field mathematics teachers refer to teachers who possess either major or minor teaching qualification in the area of mathematics education.In the Ghanaian context, In-field mathematics teachers refer to teachers who have experienced professional training in the area of mathematics education from accredited teacher training institutions and have received teaching license from the National Teaching Council (NTC) of Ghana.This group of mathematics teachers consists of teachers who hold either a diploma in Basic Education (with a specialisation in Mathematics) or a Bachelor of Education degree (with a specialisation in Mathematics or Basic Education with a Mathematics option).These teachers are primarily employed by the Ministry of Education through the Ghana Education Service (GES) to teach mathematics in public (Government) Junior High Schools throughout Ghana.Typically, each teacher specializing in this field is responsible for instructing three separate classes (JHS 1, 2, and 3) within a specific school, focusing specifically on the teaching of mathematics.On the other hand, Out-of-field mathematics teachers are teachers without professional qualification (either major or minor) in the area of mathematics education.In Ghana, mathematics teachers within this category may be qualified teachers in other areas other than mathematics education or possess certificates that fall below the minimum teaching qualification of mathematics teachers at the Junior High School level (for instance, Senior High School graduates with the West African Senior School Certificate Examination (WASSCE) certificates).Interestingly, Out-of-field Junior High School mathematics teachers are mostly mathematics teachers of private (non-government) Junior High Schools in Ghana.The engagement of Out-of-field Junior High School mathematics teachers is primarily the responsibility of the management of the private school in question.In view of this, different non-government schools have varied standards for engaging individuals as mathematics teachers for their respective schools.This situation raises concerns about the professionalism of the teaching of mathematics (Buabeng et al., 2020), considering the absence of specified standards for mathematics teachers' engagement in non-government schools, which may lead to the employment of unqualified individuals as mathematics teachers in Ghana.In view of this, the study aimed at exploring the algebra teaching knowledge and difficulties of In-field and Out-of-field Junior High School mathematics teachers in Ghana to ascertain if they possess the knowledge it takes to effectively teach algebra to enhance learners' mathematics achievement.

Theoretical framework
The purpose of the study was to explore In-field and Out-of-field Junior High School mathematics teachers' Knowledge of Algebra for Teaching and their difficulties in algebra.In view of this, the expanded Knowledge of Algebra for Teaching (KAT) framework (Wilmot, 2016), which theorises the comprehensive knowledge of algebra for teaching in seven knowledge domains was employed as the theoretical lens in our exploration of mathematics teachers' algebra knowledge.Studies (McCrory et al., 2012;Wilmot, 2016;Wilmot et al., 2018) have revealed that the foundational teacher knowledge for teaching algebra comprises: School Algebra Knowledge (SAK), Advanced Algebra Knowledge (AAK) and Algebra Teaching Knowledge (ATK), alongside their four advanced algebra knowledge types, namely; Profound Knowledge of Algebra (PKA), School Algebra Teaching Knowledge (SATK), Advanced Algebra Teaching Knowledge (AATK) and Pedagogical Content Knowledge in Algebra (PCKA), which emanate from the interplay among the foundational algebra knowledge domains.In practice, mathematics teachers' ability to connectively employ and operate within these knowledge domains during mathematics instructions are essential for the effective transmission of algebra contents to learners.In this study, the Knowledge of Algebra for Teaching (KAT) of In-field and Out-of-field Junior High School mathematics teachers were measured based on all the seven knowledge domains within the expanded KAT framework (see Figure 1).
The utilisation of the expanded Knowledge of Algebra for Teaching (KAT) framework was to ensure a domain specific measure of mathematics teachers' Knowledge of Algebra for Teaching through the interplay among the seven requisite teacher knowledge domains as postulated in the expanded KAT framework, for effective transmission of algebra contents to learners at the Junior High School level.These knowledge domains are discussed in relation to this study.

School Algebra Knowledge (SAK)
School Algebra Knowledge has been defined by studies (Reckase et al., 2015;Wilmot, 2016) as the algebra contents in the intended curriculum.According to the National Council of Teachers of Mathematics (2000), the principles and standards for school mathematics form the foundation for defining the scope of school algebra knowledge.Stein, Kaufman, Sherman and Hillen (2011) assert that, different curricula may define the contents of school algebra differently.These variations are observed in different contexts based on several factors including; the nature, level of operation and objectives of school mathematics.In the Ghanaian Junior High School context, algebra contents which fall within the school algebra knowledge domain are found in mathematics syllabus, textbooks, workbooks and pamphlets.In this study, mathematics teachers' school algebra knowledge was measured based on the algebra contents stipulated within the Junior High School mathematics curriculum in Ghana.

Advanced Algebra Knowledge (AAK)
The Advanced Algebra Knowledge (AAK) domain include other mathematical knowledge in specific college level mathematics (McCrory et al., 2012;Reckase et al., 2015;Wilmot, 2016) and the application of school algebra in other areas of mathematics (Osei & Kubi, 2022).According to the Conference Board of the Mathematical Sciences (2012), Reckase et al. (2015) and McCrory et al. (2012), the scope of Advanced Algebra Knowledge extends beyond School Algebra Knowledge and provides mathematics teachers with broader and deeper viewpoints on different mathematical ideas and their applications.Also, mathematics teachers' expertise regarding alternate definitions, generalizations and extensions of well-known theorems in mathematics promote advanced mathematical viewpoints (Usiskin et al., 2003).Wilmot (2019) asserts that topics such as operations on numbers, sets of numbers and algebraic expressions incorporates advanced algebra knowledge for algebra applications in these areas.In this research, the advanced algebra knowledge of mathematics teachers was measured based on the applications and extensions of school algebra in other areas of mathematics within the Junior High School mathematics curriculum of Ghana.

Algebra Teaching Knowledge (ATK)
According to Reckase et al. (2015) and McCrory et al. (2012), Algebra Teaching Knowledge is defined as the pedagogical expertise unique to the teaching of algebra that may not be covered in advanced mathematics courses.The Algebra Teaching Knowledge encompasses mathematics teachers' ability to examine what makes a certain algebra idea difficult to understand (Reckase et al., 2015), identify learners' errors (Osei & Kubi, 2022), connect mathematical errors to precise misunderstandings, and deal with these challenging circumstances (Donkor, 2021;McCrory et al., 2012;Yarkwah, 2017).In the context of this study, ATK of Junior High School mathematics teachers was measured based on their ability to identify and connect learners' errors to specific misconceptions and propose appropriate workable pathways.

Profound Knowledge of School Algebra (PKSA)
The intersection of SAK and AAK leads to the development of Profound Knowledge of School Algebra (Donkor, 2021;Osei, 2020;Wilmot, 2016).According to Donkor (2021) and Wilmot et al. (2018), mathematics teachers who operate within this knowledge domain demonstrate thorough command over school algebra and are more likely to explain algebra concepts clearly to learners.Mathematics teachers who possess profound knowledge of school algebra demonstrate alternate definitions, extensions, generalizations of well-known theorems (Donkor, 2021;Wilmot, 2016) and are able to apply school algebra in different contexts (Osei & Kubi, 2022).In this study, items that combine the features of school algebra and advanced algebra were the basis for measuring the Profound Knowledge of School Algebra.

Advanced Algebra Teaching Knowledge (AATK)
The AATK emanates from the intersection of AAK and ATK (Donkor, 2021;Osei, 2020;Wilmot, 2016).According to Donkor (2021), Entsie (2021) and Wilmot et al. (2018), mathematics teachers who possess this proficiency do not only exhibit mastery over advanced algebra, instead they are equipped on how to communicate advanced algebra contents effectively for understanding.This propels application of learned concepts and problem solving.Additionally, the Advanced Algebra Teaching Knowledge equips mathematics instructors to make meaningful connections between different ideas and engage in appropriate trimming and breakdown within and among advanced algebra topics (Donkor, 2021;Entsie, 2021;Wilmot, 2016).In the context of this research, Junior High School mathematics teachers' ability to handle items that combine the features of advanced algebra and teaching knowledge was the basis for measuring Advanced Algebra Teaching Knowledge.

School Algebra Teaching Knowledge (SATK)
SATK emanates from the intersection of SAK and ATK (Donkor, 2021;Osei, 2020;Wilmot, 2016).According to Entsie (2021) and Wilmot (2016), mathematics teachers who operate within this knowledge domain have good knowledge of the trajectory of school algebra.School Algebra Teaching Knowledge enables mathematics teachers to connect different algebra-related concepts, solve problems while preserving some degree of dependability, and unravel complexity to widen the range of the content they teach.The utilization of the SATK enables mathematics teachers to teach school algebra in a flexible way to improve understanding of different groups of learners (Donkor, 2021;Entsie, 2021).In the study, mathematics teachers' performance on algebra items that possess the features of school algebra and algebra teaching knowledge was the basis for measuring School Algebra Teaching Knowledge.

Pedagogical Content Knowledge in Algebra (PCKA)
PCKA is the only trio algebra knowledge emanating from the complex interconnections among the three foundational algebra knowledge domains (Donkor, 2021;Osei, 2020).In Wilmot (2016), the PCKA is described as a form of Pedagogical Content Knowledge (Shulman, 1986) in a domain specific area.Donkor (2021) asserts that mathematics teachers who possess this expertise are capable of handling higher order tasks in algebra and are able to combine numerous teaching techniques to make complex algebraic concepts understandable to learners.PCKA, according to Osei and Kubi (2022) guarantees an effective and flexible transmission of algebra contents for deeper cognitive engagements among learners.In our study, Junior High School mathematics teachers' performance on items that integrate the features of the three foundational algebra knowledge domains was the basis for measuring the Pedagogical Content Knowledge in Algebra.

Research design and research questions
The descriptive survey design was adopted to explore the teaching knowledge and difficulties of In-field and Out-of-field Junior High School mathematics teachers in algebra.According to Fox and Bayat (2007), this approach is appropriate for a comprehensive description of specific characteristics and situations of interest.In meeting the overarching objectives of the study, the following specific research questions were addressed: (1) What Knowledge of Algebra for Teaching domains prevail among In-field and Out-of-field Junior High School mathematics teachers?
(2) What differences are observed in the Knowledge of Algebra for Teaching of In-field and Outof-field Junior High School mathematics teachers as their years of teaching experiences increase?
(3) What difficulties are exhibited in the algebra knowledge of In-field and Out-of-field Junior High School mathematics teachers?

Sample and participants
The study employed the participation of Junior High School mathematics teachers within the Southern part of the Ahafo region of Ghana.The population of the study was 742, constituting 509 In-field and 233 Out-of-field Junior High School mathematics teachers.The two categories of mathematics teachers were considered as strata based on the differences in their backgrounds and experiences for a proportional selection of 226 In-field and 148 Out-of-field mathematics teachers as recommended by Krejcie and Morgan (1970).The proportional selection of participants from each stratum was done randomly using computer-generated numbers to minimize biases in the sample selection.Analyses of the demographic details of participants revealed a male dominance of 197 (87.2%) and 136 (91.9%) mathematics teachers within the In-field and Out-of-field mathematics teachers respectively.

Instrument
In line with the study's objective of exploring the algebra knowledge teaching of Junior High School mathematics teachers based on the seven knowledge domains within the expanded Knowledge of Algebra for Teaching (KAT) Framework, and the various difficulties exhibited as they handle algebra, the research adapted the teacher-made achievement test instruments of Black (2007), Osei and Kubi (2022) and Donkor (2021) for data collection.The instrument had two major sections, these were Personal information section and Test item section.The personal information section sought for demographic information such as respondents' sex, professional status and years of teaching experience.The test item section had sixty (60) questions covering the seven knowledge domains of the expanded KAT framework.Specifically, the instrument had thirteen (13) items on SAK, ten (10) on AAK, nine (9) on ATK, seven (7) on PKSA, eight (8) on AATK, seven (7) on SATK and six (6) on PCKA.The test re-test reliability of 0.79 on 62 Junior High School mathematics teachers was considered acceptable according to Nunnally and Bernstein (1994) and Vaske (2008).On this basis, the instrument was considered reliable for data collection.Table 1 presents sample item for each knowledge domain as employed in the study.

Data collection procedure
An introductory meeting was arranged between the first author and the participating Junior High School mathematics teachers in seven proximal centres within the study area.In each of these meetings, issues regarding the purpose and importance of the study, duration for responding to the items of the achievement test and anonymity of all responses were discussed.Participants' consents were sought and data was collected a week after the completion of the various consent forms.The authors supported by 13 field assistants administered the achievement test instrument within two weeks.
Step 4: 9n ¼ 180 Step 5: 9n 9 = 180 9 Step 6: n ¼ 20 Which of the statements about Gertrude's solution is true?(A) Gertrude's solution is correct (B) Gertrude made a mistake at step 1 (C) Gertrude made a mistake at step 2 (D) Gertrude made a mistake at step 3 PKSA Find the value of x in the equation

Data analyses
A preliminary data analysis on respondents' background characteristics was carried out to ascertain the various demographic compositions within the Junior High School mathematics teachers who participated in the study.The research questions were answered using the achievement test scores and excerpts (snapshots) from the responses of participants.Specifically, the study utilised descriptive statistics, independent samples t-test and Analysis of variance in providing answers to research questions one and two.Descriptive statistics and excerpts (snapshot) of some identified difficulties were used in answering the research question three.Results were presented using tables and charts for meaningful interpretations.

In-field and Out-of-field Junior High School mathematics teachers' Knowledge of Algebra for Teaching
The first research question sought to find out the prevailing algebra teaching knowledge domains and the differences that exist in the Knowledge of Algebra for Teaching of In-field and Out-of-field Junior High School mathematics teachers based on the expanded KAT framework.In addressing this research question, the achievement test scores across the seven knowledge domains (each marked out of 20) and the overall knowledge; a combination of the scores from each knowledge domain (marked out of 140) of the two categories of Junior High School mathematics teachers were utilised.Independent samples t-test at 5% level of significance was employed in comparing the mean scores of In-field and Out-of-field mathematics teachers across the various KAT domains.The descriptive statistics, t-tests and corresponding effect sizes are presented in Table 2.
In reference to Table 2, School Algebra Knowledge (SAK) domain emerged as the prevailing algebra knowledge domain for both In-field (M = 11.26,SD = 3.263) and Out-of-field (M = 11.06,SD = 2.736) Junior High School mathematics teachers.Interestingly, In-field mathematics teachers had ATK (M = 11.06,SD = 3.791) as their second predominant algebra knowledge domain whiles that of the Out-of-field mathematics teachers was AAK (M = 7.91, SD = 1.685).Surprisingly, PCKA which combines the three foundational knowledge domains was the least among In-field (M = 6.69,SD = 3.010) and Out-of-field (M = 5.11, SD = 2.388) Junior High School mathematics teachers.These results seem to suggest that the two categories of mathematics teachers demonstrated mastery over the curriculum contents they teach in the classroom (SAK) as compared to the applications of these contents and the contents of advanced algebra courses (AAK) and, the pedagogy for transmitting algebra contents (ATK).It was however not surprising that ATK which takes care of the pedagogical issues within the scope of algebra was reported as the second-best knowledge domain among In-field mathematics teachers.This is because, the pedagogical issues are projected throughout their professional training as mathematics teachers.Also, it is apparent that In-field Junior High school mathematics teachers possess higher knowledge across six of the seven domains of the Knowledge of Algebra for Teaching (AAK, ATK, PKSA, SATK, AATK and PCKA) as compared to the Out-of-field Junior High school mathematics teachers.These differences are more pronounced in ATK (d = 1.08, p = 0.000), AAK (d = 0.92, p = 0.000) and AATK (d = 0.83, p = 0.000), yielding large effect sizes.In addition, significant differences were also observed in PKSA (d = 0.31, p = 0.004), PCKA (d = 0.58, p = 0.000) and SATK (d = 0.36, p = 0.001) with small, moderate and small effect sizes respectively.Although the overall KAT averages (In-field = 64.20 and Out-of-field = 53.03)out of a total of 140 marks do not show high Knowledge of Algebra for Teaching among Junior High School mathematics teachers, the difference (d = 1.08, p = 0.000) between the two categories of mathematics teachers is significant with a large effect size in favor of In-field Junior High school mathematics teachers.
Even though all the algebra knowledge domains except SAK fall below average for the two groups of mathematics teachers, it is obvious that In-field mathematics teachers possess higher knowledge for teaching algebra as compared to Out-of-field Junior High School mathematics teachers.This is encouraging and projects the relevance of professional training for developing competent mathematics teachers for Junior High Schools.

Differences in the Knowledge of Algebra for Teaching of In-field and Out-of-field Junior High School mathematics teachers based on years of teaching experience
The second research question sought to find out the differences in the Knowledge of Algebra for Teaching of In-field and Out-of-field Junior High School mathematics teachers as their years of teaching experiences increase.In addressing this research question, the overall Knowledge of Algebra for Teaching (marked out of 140) of In-field and Out-of-field mathematics teachers were categorized into three years of teaching experience, namely, Below 5 years, 5-10 years and Above 10 years.Analysis of variance at 5% level of significance was employed for mean score comparisons across the years of teaching experience.Table 3 displays the descriptive statistics of Junior High School mathematics teachers Knowledge of Algebra for Teaching based on years of teaching experience.
A cursory look at Table 3 seems to suggest an upward trend in the mean scores as years of teaching experience increases for the two groups of Junior High School mathematics teachers.Thus, within each group of Junior High School mathematics teachers, teachers with above 10 years of teaching experience obtained the highest means; In-field (M = 67.50,SD = 10.899),Out-of-field (M = 58.91,SD = 8.758) and All mathematics teachers (M = 64.46,SD = 10.954).This supposed upward trend is displayed in Figure 2.
In reference to Figure 2, there appears to be a sharp upward difference in the Knowledge of Algebra for Teaching between teachers with teaching experience below 5 years and those with 5-10 years as compared to the difference between teachers with 5-10 years and above 10 years for In-field and Out-of-field Junior High School mathematics teachers.To ascertain the significance of the differences observed in the means scores, a one-way analysis of variance was employed.Table 4 presents the results of the ANOVA tests and their corresponding post hoc tests based on years of teaching experience.
The results seem to project a sharp positive difference in the Knowledge of Algebra for Teaching within the first decade of Junior High School mathematics teachers' experiences on the field of work.Surprisingly, this improvement in teacher knowledge is not pronounced after the first 10 years of their teaching experiences.This situation is discouraging and does not propel the continual growth expected among mathematics teachers for enhancement in algebra instructions.

Difficulties of In-field and Out-of-field Junior High School mathematics teachers in algebra
The third research question sought to identify the difficulties of Junior High School mathematics teachers in algebra.In addressing this research question, the written scripts of the mathematics teachers from the achievement test instrument were assessed for the identification of the algebra difficulties of In-field and Out-of-field Junior High School mathematics teachers.
The analysis of the written responses of Junior High School mathematics teachers to the algebra items employed in the achievement test instrument revealed eight (8) dominant difficulties.Table 5 presents the various identified algebra difficulties and their corresponding breakdown based on the two categories of mathematics teachers.
According to Table 5, Out-of-field mathematics teachers dominated in terms of percentage in all the identified algebra difficulties except the difficulty of factorising difference of two squares which recorded the highest number (115) and percentage (30.75%)for the two groups of Junior High school mathematics teachers.It is apparent that few (21) Junior High School mathematics teachers representing 5.61% had difficulty regarding the correct application of the order of operations when handling algebra items.Snapshot samples of the various identified difficulties are presented below.

Factorisation of difference of two squares
Factorisation of difference of two squares was identified as a difficulty among Junior High School mathematics teachers.Presented below (See Excerpts 1 and 2) are examples of how two mathematics teachers struggled with two items within the scope of difference of two squares.
In relation to Excerpt 1, the response of In-field mathematics teacher 122 to item 50 of the achievement test looked promising at step 1, where x 2 À 25y 2 was broken down as x 2 À 5 2 y 2 .However, the difficulty was exhibited at step 3 (circled yellow) where x 2 À 5 2 y 2 was simplified as In the case of the response of Out-of-field Junior High School mathematics teacher 89 to item 45 of the achievement test (see Excerpt 2), the difficulty was exhibited at step 3 where a wrong cancellation of the factor   9 resulted in an incorrect simplification of 9ð 1 9 y 2 À w 2 Þ as y 2 À w 2 .The analysis reveals some sort of deficiencies in teacher knowledge regarding factorising difference of two squares which falls within the scope of advanced algebra courses taken by In-field mathematics teachers during their professional training.

More than one term bracket expansions
The two categories of Junior High School mathematics teachers exhibited difficulties regarding the appropriate application of the distributive property in the context of expanding brackets that contain more than one term.See for instance Excerpts 3 and 4 presented below.Excerpt 3 and 4 provide an understanding of how two Junior High School mathematics teachers exhibited difficuty in handling more than one term bracket expansions.The two snapshot samples look similar in terms of how the mathematics teachers handled items of this kind.It is observed that they both teachers failed to multiply through correctly where a bracket contains more than one term.For instance in excerpt 3, 2a was multiplied by and failure to multiply À 1 by x 2 as seen in step 1.This particular difficulty was unexpected, looking at the nature of these expressions and the fact that they fall within the scope of the algebra contents these mathematics teachers teach at the Juinior High School level.

More than one term exponential expansions
Expansion of exponential expressions, specifically more than one term exponential expansion was observed as a challenge among the two groups of Junior High School mathematics teachers.See for instance Excerpts 5 and 6 shown below.
Excerpts 5 and 6 exhibit Junior High School mathematics teachers' difficulty in handling more than one term exponential expansions.It is apparent that the two mathematics teachers incorrectly distributed the exponent (2) to each of the terms within the bracket.This resulted in an incorrect expansion of 2a . This is quite alarming, it was expected that these mathematics teachers demonstrate exceptional control over expressions of this kind for effective transmission of similar alegbra contents within the Junior High School mathematics curriculum to enhance learners achivement in mathematics.

Cancellation of common factors
Inappropriate cancellation of factors was observed among the two categories of Junior High School mathematics teachers.Excerpts 7 and 8 presented below throw light on the challenges of the mathematics teachers regarding the appropriate cancellation of common factors.
Misapplications of the principle of cancelling common factors for easy simplification of expressions were observed in the responses some Junior High School mathematics teachers (see excerpts 7 and 8).These incorrect cancellations led to the wrong simplification of

Equal sign usage
Among the difficulties exhibited in the responses of In-field and Out-of-field Junior High School mathematics teachers was the inappropriate use of the equal sign especially in equations.Excerpts 9 and 10 provide a clearer understanding regarding the improper use of the equal sign among Junior High School mathematics teachers.
The multiple utilisation of the equal sign in single equations as shown in Excerpts 9 and 10 reveals a deficiency in Junior High School mathematics teachers' knowledge regarding the equivalence property of the equal sign.It is apparent in excerpts 9 and 10 that mathematics teachers perceive the equal sign as a symbol that precedes solutions or answers neglecting its equivalence and balancing property.

Application of the laws of indices
Inappropriate application of the laws of indices was also observed among in-field and Out-of-field Junior High School mathematics teachers.The two snapshots (see Excerpt 11 and 12) displayed below exhibit some sort of misapplications of indices rules in the responses of the mathematics teachers.
In excerpt 11, the respondent (In-field mathematics teacher 206) misapplied the indices rule of equating exponents in cases of equal base for the two sides of an equation.This rule which would have worked perfectly if 4 2xÀ 1 ¼ 1 4 ð Þ 2 as seen in step 1 was further simplified as 4 2xÀ 1 ¼ 4 ð Þ À 2 was directly misapplied to 4 2xÀ 1 ¼ 1 4 ð Þ 2 resulting in an incorrect value of x as seen in excerpt 11.In the case of excerpt 12, the Out-of-field Junior High School mathematics teacher misapplied the bracket power rule of indices; This analysis clearly reveals misapplications of some valid indices rules which can be transmitted to learners during mathematics instructions, hence demands attention.

Order of operations
Incorrect order of operations were also observed in the responses of Junior High School mathematics teachers to some of the items of the achivement test instruemnt.Excerpts 13 and 14 provide evidence and understanding of how two mathematics teachers had difficuty handling expressions containing multiple operations (Addition, subtraction, division, multiplication and brackets).
In Excerpt 13, the In-field mathematics teacher correctly changed the mixed fraction 1 1 2 to 3 2 as seen in step 1. Afterwards, the respondent removed the bracket by multiplying the terms within the bracket 3 2 , 3 4 and 1 4 by 1 2 , without following the approprite order of operation.Even though the expression obtained in step 2 is inaccurate 3 4 + 3 8 � 3 4 , the respondent simplified the first two terms 3 4 + 3 8 and afterwards divided the result by the last term 3 4 , instead of dividing the second term 3 8 by the last term 3 4 for a subequent addition to the first term 3 4 .In the case of the Out-of-field mathematics teacher (see excerpt 14), the difficuty was observed where in the expression 2 3 � 27 4 � 8 15 À 10 3 , the last two terms 8 15 À 10 3 were simplified before of the other operations (division and multiplication).Mathematics teachers' dificulties regarding the correct application of order of operations demand attention.This is due to the fact that, they are expected to project the relevenace of adhering to the principles regarding order of operations for expressions that combine multiple arithmetic operations to learners during instructions.

Algebraic representation of word problems
The two categories of Junior High School mathematics teachers exhibited difficulty in transforming word problems into equations.Excerpts 15 and 16 present such evidences for a clearer understanding of how two mathematics teachers incorrectly represented some word problems mathematically.
For excerpt 15, the respondent subtracted the product of as expected in item 44.Interestingly, excerpts 16 also reveals how a mathematics teacher incorrectly represented "X's age 5 years ago is equal to 5 ⁄₄ of Y's age 2 years ago" as These incorrect algebraic representations of word problems in the responses of the mathematics teachers exhibit their difficulty regarding correct representations of word problems in algebraic forms.

Discussion
This study aimed at exploring the Knowledge of Algebra for Teaching of In-field and Out-of-field Junior High School mathematics teachers based on the seven knowledge domains within the expanded KAT framework (SAK, AAK, ATK, PKSA, SATK, AATK and PCKA), and the various algebra difficulties exhibited as these mathematics teachers handle algebra items.The results of the study projected SAK as the prevailing knowledge domain for the two categories of Junior High School mathematics teachers.It was however discovered that ATK and AAK were the second-best knowledge domains for Infield and Out-of-field mathematics teachers, respectively.Even though mathematics teachers' knowledge within each of the seven algebra knowledge domains and the overall Knowledge of Algebra for Teaching fell below average, except ATK for In-field and SAK for the two groups of mathematics teachers, the study showed that apart from SAK, Infield mathematics teachers possess higher knowledge in all the sub-knowledge domains and the overall Knowledge of Algebra for Teaching as compared to their counterparts who are Out-of-field mathematics teachers.Although the results on the prevalence of SAK over the other subknowledge domains and the demonstration of higher knowledge by Infield mathematics teachers as compared to Out-of-field in the overall Knowledge of Algebra for Teaching are consistent with several studies (Donkor, 2021;Entsie, 2021;Osei, 2020;Osei & Kubi, 2022), the breakdown based on the seven sub-knowledge domains deviate from Donkor (2021), where she found out that Infield mathematics teachers did not possess higher knowledge than Out-of-field mathematics teachers in Advanced Algebra Knowledge (AAK), School Algebra Teaching Knowledge (SATK) and Advanced Algebra Teaching Knowledge (AATK).
The result also demonstrates differences in the Knowledge of Algebra for Teaching (KAT) as years of teaching experiences increase for In-field, Out-of-field and All Junior High School mathematics teachers.These findings are consistent with numerous studies (Harris & Sass, 2011;Kraft & Papay, 2014;Ladd & Sorensen, 2017;Staiger & Rockoff, 2010;Wiswall, 2013;Yarkwah, 2017), as they argue for substantial differences in teacher knowledge as years of teaching experiences increase.The study further revealed that, the observed differences in the Knowledge of Algebra for Teaching are not pronounced among Junior High School mathematics teachers with Above 10 years of teaching experience as compared with those with 5-10 years of teaching experience as indicated in Podolsky et al. (2019) and inconsistent with Koedel and Betts (2007) where they argue that even beyond teachers' first decade on field, teachers continue to improve in knowledge and effectiveness.
The study identified eight algebra difficulties among In-field and Out-of-field Junior High School mathematics teachers.These findings align with researches such as Sultan and Artzt (2011), Chinnappan (2010) and Pincheira and Alsina (2021), which projected algebra as a particular strand of mathematics that poses several difficulties to mathematics teachers.Mathematics teachers' difficulty regarding the understanding and utilization of the equal sign as identified in this study sits with studies such as Trivilin and Ribeiro (2015), Ferreira et al. (2017) as they asserted that mathematics teachers are unable to recognise the different mathematical meanings of the equal sign, especially its equivalence property.In addition, the study's finding on mathematics teachers' difficulty regarding algebraic representation of word problems especially using variables to represent unknown quantities sit with Bair and Rich's (2011) assertion that mathematics teachers have challenges in handling algebra tasks that require the representation of unknown quantities with variables.The identified difficulties such as; more than one term bracket expansions, more than one term exponential expansions, Cancellation of common factors, Application of the laws of indices, Order of operations and factorization of difference of two squares buttress Strand and Mills (2014) claim that mathematics teachers have several challenges when solving algebraic tasks.More importantly, the existence of these difficulties among mathematics teachers increases their struggles in providing assistance to learners with difficulties in algebra (Bush & Karp, 2013;Kieran, 2018), thereby reducing the potency of mathematics teachers' ability to offer effective and reliable interventions for remedying learners' difficulties in algebra.

Conclusions, limitation and recommendations
The study revealed SAK as the prevailing knowledge domain among the two categories of Junior High School mathematics teachers.It was further discovered that the two groups of mathematics teachers possess algebra knowledge that fall below average for five out of the seven knowledge domains (AAK, PKSA, SATK, AATK and PCKA) and also for the overall Knowledge of Algebra for Teaching, however, In-field mathematics teachers possess relatively higher knowledge as compared to Out-of-field mathematics teachers in six out of the seven knowledge domains (AAK, ATK, PKSA, SATK, AATK and PCKA) as well as the overall Knowledge of Algebra for Teaching.The study further showed that the Knowledge of Algebra for Teaching (KAT) differs as years of teaching experiences increase for In-field, Out-of-field and All Junior High School mathematics teachers, however, these differences are not pronounced among Junior High School mathematics teachers with above 10 years of teaching experience as compared with those with 5-10 years of teaching experience, but between those with 5-10 years and below 5 years of teaching experience.The study finally identified eight major algebra difficulties among Junior High School mathematics teachers namely; factorization of difference of two squares, more than one term bracket expansions, more than one term exponential expansions, Cancellation of common factors, Equal sign usage, Application of the laws of indices, Order of operations and Algebraic representation of word problems.
This research had some limitations.The utilsation of Junior High School mathematics teachers within the southern part of the Ahafo region in Ghana is a limitation for the generalisation of the findings for the entire country.Additionally, placing sole reliance on an achievement test instrument to evaluate the competence of mathematics teachers in teaching algebra, within a constrained timeframe and controlled setting, presents constraints in capturing a comprehensive grasp of the teachers' overall knowledge of Algebra for teaching.It would have been considerably advantageous if the authors had incorporated teacher observations to gauge additional crucial pedagogical issues (such as classroom management, utilisation of teaching and learning materials, and more) that significantly contribute to successful mathematics lessons, thereby enriching the collected data.
Notwithstanding these limitations, the study provides insights into the Knowledge of Algebra for Teaching of Junior High School mathematics teachers for teacher preparation, policy and practice decisions in Ghana and similar contexts.The deficits and difficulties revealed in the Knowledge of Algebra for Teaching of Junior High School mathematics teachers have critical implication on teacher preparation.Specifically, teacher training institutions should aim at developing pre-service mathematics teachers' mastery in the advanced algebra knowledge domains, especially, PCKA which combines the three foundational algebra knowledge domains for effective transmission of algebra contents to learners at the Junior High School level.For example, when preparing prospective mathematics teachers, teacher training institutions should ensure that future Junior High School teachers cultivate the ability to establish connections between various errors and their corresponding misconceptions in algebra.Additionally, they should ensure that trained teachers possess the proficiency to illustrate the practical application of the algebraic concepts they teach within real-world contexts throughout the training period.
In addition, Policy makers should make clear regulations on the engagement of Out-of-field mathematics teachers in private (non-government) Junior High Schools in Ghana, since Out-offield mathematics teachers' Knowledge of Algebra for Teaching falls below In-field mathematics teachers who are usually engaged in public (government) schools.In this respect, the various district offices of the Ghana Education Service should provide teacher engagement criteria for nongovernment Junior High Schools, which may include an acceptable performance on an aptitude test and further training to improve the knowledge of the Out-of-field teachers engaged for the teaching of mathematics.Again, Out-of-field mathematics teachers could be assigned mentors (preferably, experienced In-field mathematics teachers) to enable them receive assistance in areas of difficulty, especially, the pedagogical components of their Knowledge of Algebra for Teaching.Also, the identified algebra difficulties have critical implications on Junior High School mathematics teachers' potency to offer effective and reliable interventions for addressing learners' difficulties in algebra.In view of this, the study recommends that school administrators (specifically School heads) organise professional development trainings that aim at remedying identified algebra difficulties and also equipping mathematics teachers with innovative and practical approaches of providing assistance to learners with difficulties in algebra.These training programmes should utilise effective learning approaches such as Teacher Design Teams (TDTs) for collaboration among teachers and should be evaluated to ascertain the realisation of their intended goals.
Finally, teachers are also entreated to engage in communities of practice.In this case, teachers within catchment areas should be admonished to come together or network to share knowledge and experiences regarding algebra difficulties.Within the community of practice, the teachers should engage in ongoing interactions, discussions, and collaborative activities that will enhance their teaching skills, pedagogical approaches, and subject knowledge.This will provide an opportunity for teachers to learn from one another, exchange best practice, and collectively address challenges and issues in their field.

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Excerpt 8 as displayed above.The inappropriate cancellations observed in the responses of Junior High School mathematics teachers project the deficiencies in teacher knowledge regarding the correct application of the principle of cancelling common factors during simplifications of algebraic expressions.