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In South Africa, very little mathematical research is done in well-resourced schools, which are generally located in middle-class and wealthy suburbs. But there is research to suggest that ninth-grade students in these schools are four years ahead of their counterparts in schools located in low-income urban and rural areas.
The latter schools do not charge fees and get a higher budget allocation per pupil from the government because they are in low-income communities.
However, other research on high achievers writing the National Benchmark Tests, a national college entrance test written by high school seniors, has shown that these high achievers struggle with seemingly basic ideas. in math, like percent and inequalities. Many of these students come from one-fifth of the best-resourced public schools and high-fee independent schools. So, taken together, these results present a troubling picture regarding math achievement.
but this is not new. The results of the International Study of Trends in Mathematics and Science have told a similar story for almost 20 years.
The Wits Maths Connect Secondary project, a research and development project at the University of the Witwatersrand, has been working with teachers in low-performing free schools in Gauteng province since 2010. We were curious to compare the performance of students taught by teachers . who had participated in our program with the performance of ninth grade students in paying schools, to see if there were similarities in overall performance. We identified four wealthiest high-performing schools in Johannesburg that agreed to participate in the research.
In late September and early October 2018, we administered a test to grades 9 that covered number operations, algebra, and functions, and covered content in grades 7 through 9. Not surprisingly, students from the wealthiest schools outperformed students in the poorest schools, even though the latter group had made statistically significant gains since they wrote the same test in February 2018.
But there were similarities in the questions that the students found easier and more difficult in the two groups.
The overall performance of the test was disappointing for both groups. A median score of 54.8% for the wealthiest group indicates that even toward the end of ninth grade, there are many students in high-performing schools who still struggle with basic algebra.
A more detailed error analysis of approximately 90 of these better-off students, conducted in 2019, focused on their errors in solving three linear equations to find out their difficulties with algebra.
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While most students were able to correctly solve an equation like 4p-2 = 10, about 25% of students used flawed approaches to solve equations with letters on both sides, for example 4p-2 = 10 + p, a skill which must be mastered in Grade 8.
Error analysis showed that students’ errors were due more to difficulties with general algebra than to performing the standard procedure for solving equations.
Interestingly, 44% of the errors involved negatives or subtraction in some way, and nearly half of these negative errors involved the wrong simplification of two like terms to a single term, for example -3k + k = -4k and 3k- k = 3.
The “minus symbol” (-) is a major source of difficulty for seventh graders because it has two possible meanings when one begins to learn about negative numbers. For example, in the expression -3k-4, the “minus” before 3 indicates the sign (negative), while the “minus” before 4 represents the operation (subtraction).
Transcendence
This study has four very practical implications for the teaching of eighth and ninth grade mathematics in all schools.
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Students must develop a vision of equivalence of the equal sign so that they can correctly answer questions such as 5 + 6 = _ -2 (that is, 5 adds 6 is the same as something subtract 2). If they only see the equal sign as a symbol before the answer (for example, 5 + 6 = 11), they will still have a hard time solving equations.
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Students need to understand when the minus symbol means “subtract” and when it means “negative.” This requires teachers to pay attention to the addition, subtraction, multiplication, and division of negative numbers throughout the year.
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Students need time to become familiar with algebra and practice how to do it. Our evidence suggests that some students in high-achieving schools are still struggling with this toward the end of grade 9. Teachers should pay special attention to expressions that involve subtraction and negatives, such as 4a- (5-3a).
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Teachers should appreciate the cognitive shifts required to solve equations with letters on both sides. Therefore, time must be allocated specifically to deal with examples such as 4p + 5 = 6p-3. It will also help to include equations with more than two terms on each side, such as 4-2p + 3 = 3p + 1-p.
Read more: COVID-19 outage could be an opportunity to lay a stronger foundation for school math in South Africa
Curriculum designers must recognize that some subjects are more difficult to learn than others and therefore require more time and continuous attention throughout the year. Whole numbers and introductory algebra are possibly the most difficult topics to teach and learn in the middle school math curriculum. Students should practice algebra and work with negative numbers every week, not just in small compartmentalized chunks as stipulated in the annual teaching plans.
Poor decisions about curriculum design affect all teachers and students, but especially students in low-income schools and teachers who lack adequate knowledge of mathematical content. In times of crisis, such as the COVID-19 pandemic, the negative impact is magnified and will not be quickly overcome.
Craig Pournara receives funding from the National Research Foundation and the First Rand Foundation. He is affiliated with Rays of Hope.
By Craig Pournara, Associate Professor of Mathematics Education, University of the Witwatersrand
