This is the final part of our article that seeks to address the problem of mathematics learning in Ghanaian schools. The first part discusses the historic factor that negatively affected mathematics learning in Ghana and the culture of mathematics teaching that has contributed to the mathematics underachievement of Ghanaian students. This final part focuses on language of instruction, the culture of mathematics learning, some recommendations and our conclusion.

Language of Instruction

In accordance with Ghana?s government policy teachers, administrators, and students are compulsorily required to use English as their sole medium of communication in school. This has two major implications. First, this implies that Ghanaian indigenous languages are banned in schools either as a means of instruction or communication between students, teachers, administrators and parents. The official sanctions for this prohibition are left to the discretion of school administrators to determine. Second, the tradition of using Ghanaian indigenous languages for instruction in primary 1-3 has been discontinued. The philosophy of using students? mother tongue in their first three years of schooling was based on the idea that using the mother tongue of students for instruction in the early years of schooling increases the students? self-esteem and self-worth. The philosophy was grounded in the empirical evidence that for students to learn and identify themselves with the learning processes students? indigenous language and culture should form an integral part of the curricula. Nonetheless, the government has rationalized its language policy on the grounds that it would help to establish a socio-linguistic environment in which students would be able to practice English speaking and eventually attain the acceptable level of competency required for academic functioning. Others have argued that in a multi-linguistic society such as that of Ghana the adoption of a neutral language is more likely to lead to national cohesion and unity. Yet other writers have argued that because of the internationalization of the English language, Ghanaians would be better off to be instructed in English in school rather than in our local languages. While we disagree absolutely with any of the philosophical arguments relating to the government?s language policy, all that we can say at the moment is that the government?s language policy has exacerbated the discontinuity between school culture and home culture. It has also justified British colonial educational policy that aimed at making English language the dominant means of communication in our schools and society. Furthermore, it devalues our culture by relegating our languages to the periphery of our society.

Regardless of the government?s language policy, the glaring reality is that Ghana is not an English-speaking country and would never be one. Instead, English is a second- language to an overwhelming majority of Ghanaian students. Research has demonstrated that second-English language speakers experience difficulties in learning mathematics in English and this may have little to do with difficulties in processing mathematical ideas. This suggests that English language deficit will lead to a mathematics deficit if taught in English. Our observation is that the problem with these groups of students has much to do with the language used for processing mathematical ideas rather than the processing of mathematical ideas per se. From our experiences, all L2 students (those whose mother tongue is not English) ? whether Ghanaians, Native Americans, Mexicans or Inuit ---experience common problems with mathematics learning in English. These L2 students, first, have to process mathematical ideas in their mother tongue and translate them into English. Alternatively, some of them process mathematical ideas first in English and translate them into their mother tongue to ensure that they make sense and then they translate them back into English. This double processing characteristic of L2 students constitutes a huge problem in mathematics learning for these students. Unlike other disciplines, in learning mathematics even at an early stage one has to know not only the facts but also to use the facts to argue a case. In secondary school, for example, if a student is required to factor a trinomial expression she has to understand the algorithm for factoring that expression and apply the algorithm to the specific case that requires factoring. This becomes more difficult where students are required to provide solutions to application questions. For application questions requires not only an understanding of the specific algorithm, but also the ability to understand the context of the question, organize the relevant information from the question, decide what strategies or algorithms would help to solve the question, and after the proposed solution determine whether the solution really answers the question.

However, this is different from a situation, for example, in learning history, where one has to write to account for factors that led to the fall of the Mali Empire. Students simply regurgitate the factors as transmitted to them by their teachers or textbooks.

It follows that L2 students may experience difficulties in translating English words into mathematical symbols. Some researchers have investigated the relation between language and mathematics ability in data obtained from Hispanic and Native American students. Among other things, they found that the students have difficulties translating words into mathematical symbols. In translating the statement? there are twice as many students as desk? more than 95 percent of the students wrote D=2D, instead of S=2D as their answer. This may make it increasingly difficult for L2 students to engage in mathematical problem solving that has been dabbed the richest aspect of mathematics learning at the primary and secondary school level. It is with word problems (we prefer the term application questions) that students learn the skill of extracting relevant information, organizing the information, selecting appropriate strategies to solve the problem, and testing to determine whether the problem has been solved. Owing to the fact that application questions are heavily language-based, most elementary and secondary mathematics teachers in Ghana avoid focusing their instruction on them. Nor do teachers assign to their students open-ended application questions or investigative problems because that would entail using English language to write a report and argue a case.

We regard the government?s language policy as detrimental to the mathematical proficiency of primary school students. From our observation and professional experience, we found that lower elementary students could acquire proficient numeracy skills in their own languages. For example, children could perform counting better in their mother tongue than they could in English. Edu baako (eleven), edu mmienu (twelve), edu mmiensa (thirteen), and so on in Akan language of Ghana is much easier for a child learning counting to remember than it if she/he were to learn it in English. The reason is that it is ten (edu) and something. And as soon as the child learns 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 (baako, mmienu, mmiensa, enan, enum, ensia, enson, enwotwe, enkron, and edu), the child is more likely to have no problems learning the other counting. For that matter a child?s brains would be able to recognize, store, and grasp the meaning and retrieve edu baako than eleven. This is another reason we disagree with the government?s language policy that bans the use of local languages in schools.

In concluding this section of the paper, we would like to reject the belief that children who speak languages outside of the Mathematico-Technological (MT) culture group (Europe, the United States, and Japan) would not be able to construct mathematical meanings and relationships from their mother-tongues. Other researchers also suggest that to understand the component of MT culture that provides contexts for mathematics learning, children from other cultures should master the socio-cultural characteristics of that cultural group. Such an idea seems to suggests that the MT cultural group were the original inventors of mathematics and that Chinese, Babylonians, and Egyptians (the three great civilizations) that laid down the fundamentals of mathematics could not learn mathematics until they were immersed in the MT cultural beliefs, institutions, and sentiments. It also implies that though mathematics is socially constructed and validated, its principles have no universal applications except within the cultural realms of the MT. We know as a fact that language is a human invention and that developing an appropriate vocabulary in Ghanaian languages for mathematics teaching and learning in the first six years of elementary should not pose any difficulties. Lastly, we wonder whether the Republic of Korea and Taiwan speak languages that belong to the MT cultural group. As far as we know these two countries used their own languages for instruction in their schools, colleges, and universities. In the Third International Mathematics and Science examination in 1999 for eighth graders, the Republic of Korea scored an average of 587 in mathematics and 549 in science. The score for Taiwan were 585 and 569 respectively. The United States scored 502 in mathematics and 515 in science. England?s achievement were 495 and 538; New Zealand achieved 491 and 510; Japan 579 and 550; Italy 479 and 493. In fact, England and the United States, which may be considered the most integral members of the MT cultural group, achieved low marks in mathematics and science in contradistinction to other countries that may not be considered members of that group.

National Culture of Mathematics Learning

The culture of mathematics learning in Ghana?s educational institutions is the direct consequence of both mathematics teaching culture and students? perception about the nature of mathematics and its importance in formal education. How do Ghanaian students learn mathematics? What beliefs, attitudes, or assumptions influence the way Ghanaians learn mathematics in schools, colleges, and universities? To answer this question, we should look critically at the culture of teaching mathematics in Ghana, for as we have already alluded to mathematics teaching culture shapes significantly mathematics learning culture. The two are inseparable. We summarize below the culture of mathematics learning in Ghana. We would like to emphasize that the culture of mathematics learning in Ghanaian educational institutions is a generalization and do not reflect individual variations or individual subculture. We also would like to add that this culture of mathematics learning is identified through our lived experiences (Fredua had both his elementary and secondary schooling in Ghana, whereas Ahia had all his schooling in Ghana except his doctoral studies) and series of observations of Ghanaian students learning mathematics both in Ghana and also in North America.

1 Students learn mathematics by listening to their teacher and copying from the chalkboard rather than asking questions for clarifications and justification, discussing, and negotiating meanings and conjectures. Consequently, students learn mathematics as a body of objective facts rather than a product of human invention.

2.Students hardly read their mathematics textbooks or other mathematics texts books. Where students read the prescribed mathematics textbooks, they read them like the way they read novels or newspapers.

3. Students could go to the library to read newspapers or novels, not mathematics. Mathematics is learned only in the mathematics classrooms or for examinations, quizzes, or tests.

4. Students could form a small study group outside of their classroom to do home work assignments or prepare for an examination or tests, but not for discussing mathematical concepts that were taught to them in the classrooms.

5. Students learn mathematics by regurgitating facts, theorems or formulas instead of probing for meaning and understanding of mathematical concepts. That is to say, students hardly ask the logic or philosophy underlying those mathematical principles, facts, or formulas.

6. Students accept whatever the teacher teaches them. The teacher is the sole authority of mathematical knowledge in the classroom, while the students are mere receptors of mathematical facts, principles, formulas, and theorems. Thus, if the teacher makes any mistakes the students would also make the same mistakes as the teacher made.

7. Most students do mathematics assignments and exercises not as a way of learning mathematics, but as a way of ?disposing off? those assignments to please the teacher. This implies that mathematics assignments are not construed as an instrument for learning mathematics.

8. Students go to mathematics classes with the object to calculate ?something?. Therefore, if the classes do not involve calculations they do not think that they are learning mathematics. So students learn mathematics with the goal to attain computational fluency, not conceptual understanding or meaning. For a conceptual understanding requires students to think critically and act flexibly with what they know. Students are fond of asking, ? How do you calculate that?? instead of asking? why do you calculate it in that way??

9. Students learn mathematics with the aim to pass a test or examination. After passing the test or examination mathematics is no longer of importance to the students.

10. Students have internalized the false belief that mathematics learning requires an innate ability or the ?brains of an elephant?.

11. It is generally believe that only science-oriented students must learn and master mathematical principles, not so-called arts or business students. Alternatively, most people (including some mathematics teachers) believe that art or business students require a pass in mathematics in their final examinations. Though people believe that artisans or technicians must learn mathematics, they not believe that they have to master as much mathematics as science students (those who want to study engineering, medicine, architecture, computering, electronics, etc).

Recommendations

We conclude this paper with recommendations to promote effective mathematics education in Ghanaian educational institutions. These recommendations, which are mostly teaching strategies, policies, and motivational modalities, would require a change in government?s language policy, parental attitude, students? perceptions, and a radical transformation of teacher education programs in teacher training colleges. Mr. Anamuah Mensa, the Vice Chancellor of the University of Education at Winneba, made a profound statement during a national forum under the caption Teacher Education in Ghana at Crossroad: New Perspective in Focus. We feel that it is worthy of repetition here:? Any teacher education system that fails to reflect changes in society would be producing teachers for the past and not the present or the future?(Ghanaweb General News May 13, 2004). As applied to mathematics education, we need changes in teacher education in order to make mathematics accessible to a majority of Ghanaian students and to establish a strong national mathematics culture for industrialization.

1. The government?s policy on language must be changed to allow mathematics teachers in both elementary and secondary schools to use a language of instruction most suitable to the needs of the students. There is in fact sufficient evidence to support the statement that Chinese and Japanese students who learned mathematics in their own languages do very well when enrolled in schools in North America. For in general (these students), it is easier to search for the words to express an understanding of a concept or idea, than it is to search for the understanding of a concept or idea to express in words.

2. Mathematics instruction should shift from the transmission model to the transaction model, in which the teacher allows the class to engage in critical discussion of mathematical ideas and the classroom is designated one of the real places to learn. We regard the Japanese teaching method called ?whole class instruction? as one of the most effective methods of teaching mathematics. In this method the teacher assigns a mathematical question or problem to the class. When the class is done each student presents his or her solution or answer to the class. Individual students have the opportunity to respond by agreeing or disagreeing with the solution or answer. Indeed, from our experiences allowing students to debate or talk about mathematical ideas helps to reduce considerably epistemological anxieties in mathematics learning. Again, understanding of mathematical concept is also enhanced by having students work in groups to solve a mathematical problem or question. Our society is said to be communal yet grouping in the mathematics classroom for learning is hardly utilized. Why?

3. We suggest that in order for mathematics to make sense to students, the knowledge that students bring to the classroom must be the starting point for teaching any mathematical concept. This implies that mathematics teachers should do culturally responsive teaching. This pedagogy recognizes the importance of including students? cultural references in all aspects of teaching. For example, home and in-class assignments should focus on issues or concepts that apply to students? community.

4. Students should be allowed to write their own mathematical questions or problems and find answers to them. Alternatively, students could share their questions or problems and present the solutions to the class and get feedbacks from their fellow students. In fact, at the beginning students may have the tendency to write simple questions or problems. However, as their understanding of the concept deepens, they are more likely to write complicated questions. This would facilitate an understanding of the concept involved.

5. Every secondary school should establish a mathematics-learning centre, where students could consult for homework assignments, examination or test preparation or exploration of mathematical concepts. The centre should have different mathematics books from basic to advanced on arithmetic, algebra, trigonometry, statistics, geometry, calculus, combinatorics, mathematical games, graphing calculators, calculators, computers (if possible) and assorted manipulatives for learning mathematics.

6. Secondary schools, the University of Education, University of Cape Coast, and Ghana Education services (GES) should offer periodic seminars and workshops for students, parents, teachers, and school administrators designed to promote positive attitudes toward mathematics.

7. Secondary and elementary schools often organize science fairs to display scientific inventions and discoveries of their students. Mathematics fairs could also be organized to exhibit the different uses of mathematics, the different branches of mathematics, careers that require knowledge of mathematics, mathematics games, geometric shapes, mathematical problems and puzzles that students have written, and problem-solving strategies students have discovered. The mathematics-learning centre could also include a display of mathematicians of African descent both in the continent and in the Diaspora.

8. Mathematics should be taught through a contextual situation. This does not suggest that abstract mathematical concepts are of no value. For example, if the teacher wants to teach the quadratic equation, he/she should be able to model a situation or two in our culture in which quadratic equation arises. Such practical situations could provide a solid referent which student would fall on to make sense of the concept of quadratic equation.

9. Mathematics teachers should discount the practice whereby students give answers to mathematical questions or problems without explaining the logic behind it. Similarly, mathematics teachers must provide justifications for any algorithms they teach their students. Simple justifications in words, diagrams, pictures or drawings would serve this purpose.

10. Again, we consider the Japanese model of mentor suitable for Ghanaian educational organizations. In the Japanese mentor model, a new mathematics teacher is assigned to an experienced mathematics teacher. The two meet from time to time to discuss issues relating to teaching and learning events that the new teacher had encountered. The experienced teacher also visits the classroom of the mentored, observes his or her teaching, and after that discusses some of his/her observations with the mentored. There are several good things about this model. The mentored has an accessible resource to draw on when needed. It additionally provides on-job learning experience to the mentored and the mentor an opportunity to utilize his/her accumulated experiences, skills, and knowledge in mathematics teaching and learning.

11. Another interesting Japanese model is that of team-based organization of teaching. In this case, teachers who teach the same grade meet periodically to discuss issues of teaching and learning mathematics, lesson plan preparations, assessment strategies, and instructional delivery modalities. This is in line with the Japanese philosophy that an effective learning always occurs in a social context, not in an individual context. This would find a perfect fit in our culture that is basically communal.

12. Assessment of mathematical knowledge, understanding, and skills must change. Presently, most teachers use tests and examinations as their instruments for assessing students? knowledge of mathematical concepts they have been taught. We suggest that a variety of assessment instruments should be used, including interviews, oral presentations, oral and written quizzes, journal writing, portfolio, class and home assignments, projects.

13. As we have stated, the main purpose of teaching mathematics is to cause learning just as the main object of material production is consumption. The classroom is a legitimate location for learning. In order to enhance mathematics learning and eliminate epistemological anxieties, students should be made comfortable to ask questions or make comments without any threats, fears, or intimidations from either the teacher or peers. Students cannot learn anything effectively when they are in a state of fear.

14. Mathematics teachers without teacher education, regardless of educational background, should complete a course in pedagogy organized by the Ministry of Education. This would help those teachers to understand theoretical conceptualizations of teaching and practical models of teaching methodology. Such knowledge would definitely be of an immense use in enhancing the teaching of mathematics in our educational institutions.

15. The Ministry of Education in conjunction with the Ghana Education Services, University of Cape Coast, and University College of Education should promote continual professional activities in mathematics education for elementary and secondary school teachers. The mathematics professional activities should ensure improvement towards excellence in mathematics teaching and learning. The professional activities could take the form of specific mathematics education courses offered during school vacation. Mathematics teachers who complete the courses should be awarded a certificate of completion. This certificate should be taken into consideration in determining pay scale and promotion.

16. Ghana Education Services should design series of advertising campaigns?radio, television, newspapers, and posters ?designed to dispel mathematics phobia in teachers, students, parents, school administrators, and society in general.

17. The Ministry of Education in conjunction with the Ghana Education Services should institute a National Presidential Award for mathematics teachers, one from each region, who have contributed immensely to mathematics education in their schools.

18. Three annual national mathematics competitions should be established and administered to primary 6, JSS 2, and SSS 2 students. Schools with the best average should be awarded prizes in cash or kind. We propose that these national mathematics competitions be named ?Alottey National Mathematics Competition ?, in honour of professor Alottey, a prominent Ghanaian mathematician and physicist.

19. The mathematics curricula of elementary schools and junior secondary schools must be enriched to include problem-solving, introduction to algebra and combinatorics (mathematics of counting). Research has shown that elementary students do better in mathematics in secondary school and take up mathematics studies at the university level when they are introduced early to algebra.

20. The teaching of mathematics and the assessment of mathematical knowledge should be weighted towards conceptual understanding in relation to computational fluency and algorithmic memorization. If conceptual understanding is emphasized, students could be provided formula sheets during tests or examinations and test questions would include more communication of mathematical processes or reasoning.

Conclusion

We have analyzed factors that we think are the causes of mathematics phobia in Ghanaian educational institutions. We have also made recommendations to reduce, if not eliminate, the incidence of mathematics phobia in Ghanaian students. However, we would like to clarify one important point. The problem of mathematics learning in Ghanaian schools is not that teachers do not teach mathematics well. We appreciate the excellent work and the professional commitments of many mathematics teachers in elementary, secondary, colleges, and universities. The crust of the problem is that Ghanaian students are not getting better at mathematics as the quote at the top of the first page of part 1 succinctly points out. This is why we have recommended an action plan for the solution of the problem. From our experience, Ghanaians are usually good at talking about their problems and making headlines in the national dailies. Nevertheless, for some strange reasons, we think that the problem will go away after talking about it! For instance, when the British government realized that mathematics learning is becoming increasingly a problem in its schools it formulated and implemented National Numeracy Strategy. Accordingly, to solve our national mathematics underachievement or mathematics phobia, we need a concrete, feasible plan with an enthusiasm to implement the plan and the passion to ensure that the plan is working.

We know for a fact that mathematics phobic behaviour would not go away in our educational system unless a plan is put in place to help mathematics teachers to improve their teaching practices, particularly in elementary school where generalist teachers are required to teach all subjects including mathematics. The generalist teachers may have their own mathematics phobia and this may be transmitted inadvertently to students. As one elementary school teacher once remarked: ? I have never liked mathematics, so it is not surprising that I find it very difficult to understand fractions; how would you expect me to teach it to my students?? Solving the mathematics phobia problem too requires a committed government that would take effective steps to change general societal attitude toward mathematics. The problem of math phobia or mathematics underachievement in Ghana is neither religious nor biological; it is a combination of psychological and sociological problem and should be viewed through that prism.

We find it destructive to students? morale and self-esteem the government?s language policy that mandates English language as the only means of instruction in Ghana?s schools. Our perspective is that mathematics teachers who speak the indigenous language(s) of their students must be allowed to instruct in that language. This would enhance students? understanding of mathematical concepts they are being taught. In fact, an understanding of mathematical concepts could be obtained in any language other than English. Nevertheless, some Ghanaians are likely to argue that mathematics instruction in Ghanaian indigenous languages would eventually lead to lower educational standards. They would support this statement on the grounds that students would be unable to build an appropriate mathematical vocabulary in English for further studies. Our experiences as mathematics educators are contrary to that assertion. Our position is that understanding and meaning must be the primary goal of mathematics education at any level. And this understanding and meaning could be conveniently acquired in a non-English language (Chinese, Indians etc.), so we regard it shortsighted and neo-colonialist any policy that mandates instruction only in English in Ghana. Our experience in providing mathematics instruction to Asian immigrant children in metropolitan Toronto (who hardly speak one word of English) is that the children search for meaning of English words that describe specific concepts, not the meanings of the concepts in English. These students already have an understanding of the concepts in their own languages. This is one of the most important reasons that Asian students obtain the highest mathematics achievements in North American schools. As a matter of fact, this Asian case contrasts sharply with a Ghanaian immigrant child who has neither an understanding of mathematical concepts nor the meaning of the English words that describe the concepts. In this case whose education is more ?transportable? to the Western world? Those who argue that mathematics instruction in non-English languages would devalue our educational standards are indirectly saying that mathematics education in English is the only acceptable standard to them. If this is the case, we do not have anything else to say. It is because the argument is cast in an ideological term rather than logical or experiential terms. Nevertheless, we would like to say to the apostles of British colonialism that Indira Gandhi once stated that India education is for the benefit of India not for Britain and that she did not care whether the British found it up to their standard or not. We are more concerned with elevating the mathematical achievements of Ghanaian students in Ghana and we are absolutely unconcerned whether the Brits or Yankees approve our methods, suggestions, or recommendations.

In this paper we have not added any gender dimension to our analysis. This is not because we think gender issues are unimportant to the analysis of mathematics learning problems in Ghana?s schools. Our experience in Ghana?s schools teaches us that girls are more likely to underperform in mathematics than boys. And men are more likely to be mathematics teachers than women. We think that girls? underperformance in mathematics has more to do with the military-like nature of mathematics instruction in our schools, where individual students? voices are stifled, the collective class seized with constant fears, and mathematics taught as if it has nothing to do with the lives of the students. Consequently, some of our recommendations indirectly deal with gender issues in mathematics learning in Ghana?s schools. Perhaps a separate paper devoted to analyzing the impact of gender on mathematics learning in Ghana would be in order. We may take up this project in future.