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Mathematics Tasks that Promote Conceptual Understanding

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  Mathematical tasks or exercises are common in many mathematics classes. Our classrooms are loaded with exercises and drills. Students are usually given a lot of exercises thinking that students will learn when they repeatedly do the same exercise. However, teachers need to look at mathematical exercises in terms of what is available for the learner to notice (Marton, Runesson & Tsui, 2003). This is done  by asking progressively and systematically “what  changes and what stays the same” (Watson & Mason, 2006). According to Simon and Tzur (2004)  a well-designed sequence of tasks invites learners to reflect on the effect of their actions so that they recognize key relationships. It also pointed out that mathematics is learned by becoming familiar with tasks that manifest mathematical ideas and by constructing generalizations from tasks. However, Watson and Mason (2006) caution that learning does not necessarily take place solely through learners observin...
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Teaching the (Elusive) Algebra

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“If there is a heaven for school subjects, algebra will never go there. It is the one subject in the curriculum that has kept children from finishing high school, from developing their special interests and from enjoying much of their home study work. It has caused more family rows, more tears, more headaches, and more sleepless nights than any other school subject.” (NCTM yearbook, 2008, p. 3) Four Conceptions About Algebra There are many conceptions about algebra in the literature. According to Usiskin (1988), there are four fundamental conceptions of algebra. First Conception : Algebra is considered as a generalized arithmetic. In this conception, a variable is considered as a pattern generalizer. The key instructions for students in this conception are “translate and generalize”. For example, the use of the four fundamental operations in arithmetic when used can be generalized. For example, understanding the arithmetic expression “5 – 2” as “5 take away 2″ is not a generalization b...
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Letters in Algebra:Misconceptions and Errors

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Students often have difficulty with algebra because of misconceptions in various areas. It is important that teachers are aware of the misconceptions and errors so these could be corrected. When these are corrected, the students seem to grasp the concepts more clearly. One area is the meaning of letters More often, especially for Grade 7 pupils, they completely ignore the presence of letters. For example, if 5 is added to x + 7, the answer is 12 Sometimes students are confused when between letters used as units of measure and as variables. An example is the use of  m . Some students could not distinguish the 5  m  and Some students treat letters as objects.                  Question:  Shirts cost  p  pesos each and pants cost  q  pesos a pair. If I buy 3 shirts and 2 pairs of pants, what does 3 p  + 5 q  represent?                  Student’s answer:...
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