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Cate recalled that as a high school student she saw mathematics as the task of memorisingformulas' and reproducing step-by-step algorithms; she claimed she was successful in doingthat. That belief lingered through her collegiate mathematics experience:

Even now, I'm still more comfortable if I know the steps that I'm going to do. It's just, well,it's easier. It's less effort if they give you a procedure and you follow it, it's much easier than creating

your own.

If you can get them to see a direct connection between their life and what's going on in math class,then that provides motivation for increased learning and interest and participation. If the kids justdon't give a rip and don't see how the heck this will ever help them, the average student won't bewilling to participate.

Horace's Compromise (Sizer, 1984) to describe the agreement she hoped to establish withher students: I agree with Sizer in that you have to understand and be open. Students have to understand what the teacher is trying to do and why. I think you have to be open with your studentsand say, "Look, this is what we're going to try to do." And negotiate that, so they feel the ownership,they feel part of what's going on

Another aspect of good lessons mentioned by many of the interviewees was the need tomake the mathematics relevant to students. This most commonly required the teacherto choose examples from the real world and present these to students as a way ofmaintaining their interest. It also involved showing students how the mathematicalconcepts they were learning would be useful to them in later life, though there was asense among the group that this would be more difficult in topics such as algebra whichwere regarded as highly abstract.

In order to expose further some of their thinking on the role of realistic examples, someof the participants were asked about how they might teach an introductory lesson ondifferential calculus and, here too, there were echoes from their own experiences asstudents and as tutors. For some, the need to make learning more meaningful was bornout of dissatisfaction with their own schooling.

I didnt like learning calculus because it was all so theoretical and I wanted to know what wewere trying to achieve, like why do we take the power minus one, and so on. Whats the point? So I thinkits good to say, This is why we do derivatives.

"It think it [mathematics] has to do with the complex relationships between numbers andthe symbols we use to make sense of the world among us. More and more I see maths asa system put in place to help us better [] make sense of the world around us. Mathsallows us to group things, to calculate, to categorize. It's a great way to bring order fromchaos." (Heng-Zi)

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Beliefs about MathematicsToolbox Aspect"My first impression is that math is numbers, quantities, units. In math there is alwaysone right answer. [] Math is about [...] memorizing formulas that yield the rightanswer." (Stephanie)"When first pondering the question "What is mathematics?" I initially thought thatmathematics is about numbers and rules. It is something that you just do and will do wellas long as you follow the rules or principles that were created by some magical manthousands of years ago." (David)System Aspect"Mathematics is the science of pattern and structure. It uses number sense andmathematical concepts to develop a flexible understanding of the world around us."(Nora)"Mathematics is a universal language. It is the study of numbers, proportions,relationships, patterns and sequences. Becoming literate in this language is important in

order to understand space and time; to develop logical thinking and reasoning; []."(Rachel)Utility Aspect"Math is all around us. We live in a quantitative society. On any given day, we may berequired to use math to tell us how far over the speed limit we are driving, how muchmoney we have left in our bank account to pay our loans, how many more universitycredits we need to graduate, how much prozac we need to take to get through the day orsimply how many people in this world matter to us." (Sandee)Process Aspect"For me, math has truly transformed from being a skill or procedure that can be used

merely for efficiency to being imbedded within a process of meaning-making that goeson inside the individual, a construction of understanding that we make up." (Becky)Beliefs about Learning and Teaching MathematicsToolbox Aspect"For me math is a puzzle to figure out. All of the questions or problems we were given inschool had a solution that I just needed to apply a formula or rule to and the answerwould be clear." (Chealsy)System Aspect"To learn mathematics is to learn how numbers are used to represent concepts and matter,as well as show relationships and solve problems." (Diana)

"Learning math means understanding patterns, quantities, shapes, []. To teachmathematics is to teach fundamental number concepts []." (Lorena) Utility Aspect"We also teach mathematics that is related to everyday life, for example our system ofcurrency and how to measure how tall we are." (Jacqueline)"[We teach mathematics] to enable students to function successfully in our world. It issuch

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an integral part of everything in and of our world, the more they know, the more choicesin life they'll have." (Diana)Process Aspect"The other thing that stands out is the difference between formally teaching students, andactually facilitating learning. By being a facilitator of the learning process, we are able tochoose situations, activities and problems for the students to work on either individuallyor in groups, and through this approach, students are able to [] try different ideas, anddevelop strategies." (Robyn)"I think to teach mathematics you need to let the thinking be put in your stude nts hands. You need to give them ownership of ideas and let them feel safe and free within theclassroom." (Michelle).

"Math is a language that helps individuals reason, problem solve, and distinguishrelationships. In order to do these activities, we need an understanding of the basics of thelanguage, such as symbol meaning, number values, number relationships, and basic skill

counting." (Amber)

One of the great features about mathematics is that it has an endless frontier

Cut out partsMost Ghanaian students in their second year at the Junior High School are less likely to

answer correctly, for example, the following algebraic question that appeared on the TIMSS test:The cost, C, of printing greeting cards consists of fixed charge of 100 cents and a charge of6cents for each card printed. Which of these equations can be used to determine the cost of

printing cards? (A). C= (100 + 6n) cents (B). C= (106 + n) cents (C). C= (106n) cents

(D) . C= (600n) cents. The above question does not require solving a linear equation. Instead, itrequires students to set-up an equation using the given information. This suggests that the skill tosolve an equation is as important as the skill to set-up an equation. However, students cannotlearn the skill of setting up an equation unless they are provided appropriate opportunities tolearn problem-solving and problem-posing skills (Fredua-Kwarteng & Ahia, 2004). It istherefore not surprising that Ghanaian students who took part in the TIMSS study could not

perform at competitive levels.