Tracing Student Failure in Algebra to the Upper Elementary Grades

In this thoughtful Teachers College Record article, Thomas Good, Marcy Wood, Darrell Sabers, Amy Olson, and Crystal Kalinec-Craig (University of Arizona), Alyson Leah Lavigne (Roosevelt University), and Huaping Sun (American Board of Anesthesiology) seek to explain why so many American students have difficulty with algebra. Traditionally this course was seen as a way of weeding out “untalented” students. In 1936, an anonymous writer bemoaned this separate-the-sheep-from-the-goats role: “If there is a heaven for school subjects, algebra will never go there. It is the one subject in the curriculum that has kept students from finishing high school, from developing their own interests…” In 2008, Kilpatrick and Izsak echoed this sentiment: “If anything has been constant, school algebra has too often been seen as a source of difficulty and failure – a gauntlet to be run rather than territory to be claimed.”

There is now a broad consensus that algebra is a gateway to success in high school and college, and more students are taking it at an earlier age (15 percent of eighth graders in 1986, 30 percent in 2009). But the failure rate is still high, causing greater and greater concern. The U.S. doesn’t have enough successful students in STEM subjects to fill the technical jobs in our 21^st-century economy, leaving us dependent on recruiting talent from other countries. Algebra, say the authors, is at the center of this problem – our Achilles heel. 

Good, Wood, Sabers, Olson, Kalinec-Craig, Lavigne, and Sun believe the failure of so many students in algebra classes goes back to students not mastering rational numbers in the upper-elementary grades. Why are rational numbers (fractions, decimals, and percents) so difficult? Because they “require entirely new ways of thinking about number relationships,” say the authors. Unfortunately, many teachers “push ahead” with mathematics procedures without developing students’ conceptual understanding. Here are some of the things that confuse students:

• The fact that a number can assume many forms and the same word can mean different things – for example, one half can mean ½, .5, or 3 out of 6;
• The relationship of the numerator and denominator – for example, many students don’t understand why 1/56 is larger than 1/75;
• Rate of change;
• Putting decimals in order of size and generating sequential lists of decimals;
• Over-generalizing the rule for multiplying decimals;
• Understanding the invert-and-multiply rule (some students conclude that it’s impossible to divide fractions since what they do is really multiplication);
• The misconception that multiplying always produces a larger answer and division always produces a smaller answer.

Students who have these confusions and misunderstandings won’t understand how fractions, decimals, and percents are related and how these numbers can be used to understand the world around them.

Why is understanding rational numbers (especially fractions) so important to success in algebra? Because, say the authors, “moving into fractions represents the first opportunity for students to engage in the kinds of activity and reasoning that serve as foundations for algebra” – for example, understanding proportional reasoning and ratios, going beyond the additive reasoning associated with whole numbers, and developing multiplicative reasoning (as when we reduce a picture to 2/3 of its original size). Fractions involve abstract reasoning and generalizations “that define algebra,” say the authors. “As students become familiar and fluent with fraction computation, they can begin to generalize patterns of behavior with fractions, noticing how, for example, equivalent fractions can be generated by multiplying the numerator and denominator of any fraction by the same number. As students realize these general rules with fractions, they can start to represent numbers with variable symbols, making sense of how a/b represents any fraction. This combination of generalization and symbolic notation sets the foundation for algebra…” 

If all this is true, say Good, Wood, Sabers, Olson, Kalinec-Craig, Lavigne, and Sun, the key is teaching rational numbers more effectively in the upper-elementary grades. So they designed a PD program for one large district’s grade 3-5 teachers to see if it would produce better student achievement. The seven 90-minute workshops were designed to fine-tune teachers’ approach to rational numbers, emphasizing ways to build students’ skills and conceptual understanding and help them concretely see and understand the math they were studying. Here are the guiding questions of the program:

• What are fractions, decimals, and percents?
• How do we compute with them?
• How do we solve problems with them?
• Can we relate them to what we do in and out of school?
• How do we order and see better in our daily lives with rational numbers?
• How do fractions, decimals, and percents go together, and why do we use all three?
• What is the language of rational numbers?

The researchers designed the program in collaboration with local teachers, focusing on what seemed to be the most common difficulties students have with rational numbers. 

Two things the researchers noticed as they worked with teachers was that they received state standardized test results too late in the year to use them to improve instruction, and when they did get test results, the data were spread over so many mathematical topics that teachers didn’t get a coherent picture of students’ conceptual and procedural knowledge of fractions, decimals, and percents. So the researchers administered a diagnostic test to students early in the year, gave teachers printouts on which specific items students got wrong, highlighting patterns of incorrect responses and misconceptions, and led team discussions on how these problems could be overcome in classrooms. 

What was the impact of the program? Pre- and post-test data showed students in these classrooms made significant gains in their mastery of rational numbers – and that was true at all three grades and across SES lines. Their only worry was that higher-SES students made greater gains than lower-SES students, resulting in a slight widening of the achievement gap. But the overall picture was very positive: within a single school year, in a large district, students made significant progress in rational numbers, paving the way (the authors believe) for future success in algebra (that study has yet to be conducted).

Good, Wood, Sabers, Olson, Kalinec-Craig, Lavigne, and Sun close with a humble statement about what might have caused these gains. The nine workshops were undoubtedly important, but three other factors may have played a part: (a) providing teachers with detailed, helpful pre-test information on students’ proficiency in rational numbers; (b) focusing special attention and additional classroom time on rational numbers; and (c) involving teachers in the planning and execution of the workshops, which signaled respect and improved credibility and teacher buy-in.

“Strengthening Grade 3-5 Students’ Foundational Knowledge of Rational Numbers” by Thomas Good, Marcy Wood, Darrell Sabers, Amy Olson, Alyson Leah Lavigne, Huaping Sun, and Crystal Kalinec-Craig in Teachers College Record, June 2013 (Vol. 115 # 7, p. 1-45),

https://library.villanova.edu/Find/Summon/Record?id=FETCH-proquest_...

From the Marshall Memo #494