BECOME A MEMBER! Sign up for TIE services now and start your international school career

GORDON ELDRIDGE: LESSONS IN LEARNING

Improving Student Skills and Results in Algebra

By Gordon Eldridge, TIE Columnist
11-Feb-14


Stronger skills in mathematics, and in algebra in particular, have been correlated with both university and career success (Vogel, 2008). If this is the case, it is imperative that we provide our students with every possible chance to develop strong skills in this area.
Algebra is, however, often an area where many of our students do not succeed to the degree that we might hope. The National Mathematics Advisory Panel (2008) found a sharp decline in mathematics achievement when students begin studying algebra.
Possible reasons for this include: (a) the far deeper level of abstract reasoning and problem-solving required in algebra as compared with the arithmetic students have previously studied, (b) the need to learn a completely new language of mathematical symbols, and (c) the need to recognize and understand the structure of mathematics embedded in algebra.
These three challenges combined can be a formidable barrier for some students, so evidence that particular instructional strategies may help us support students in overcoming these difficulties would be welcome. Researchers from the University of Louisville, Kentucky conducted a meta-analysis of 82 different studies involving 22,424 students in the area of instructional interventions in the teaching of algebra, with a view to determining the kind of instructional strategies which best support student learning.
What were the results of the analysis?
• The researchers determined five categories of instructional intervention in algebra: instructional strategies, manipulatives, technology tools, technology-based curriculum, and nontechnology curriculum.
• Each category showed a positive effect in students’ learning of algebra which was statistically significant.
• The duration of the treatment did not make any difference in effectiveness, suggesting that even instruction over short periods of time may still be beneficial.
• There was also no difference found between whether the treatment occurred across a whole department, a whole school, or in a single classroom, suggesting that interventions by individual teachers can make a difference.
• Perhaps the most important finding of all was to do with the purposes strategies were used for, rather than the strategies themselves. Strategies which were specifically used to foster conceptual understanding produced effect sizes more than twice as large as strategies used to foster procedural mastery.
What do these advantages mean for us in the classroom?
Unfortunately, the process of collapsing the multitude of strategies investigated in the original studies into the categories listed above means that it is difficult to determine precisely what strategies a teacher could implement in his or her classroom in order to improve student learning in algebra. It is interesting to note, however, that all of the treatments produced positive results!
Some research indicates that the very process of implementing a new methodology in the classroom can cause teachers to pay closer attention to how students are responding to particular strategies, and that this in turn can lead to strategy refinements which facilitate learning. This may partially explain the positive effects of all strategies investigated here.
The potentially richest piece of information for us as teachers to consider is the strong indication that focusing strategies on developing conceptual understanding, rather than just procedural mastery, is hugely beneficial to student success. The difference between the two is illustrated by the student who may be able to fluently follow procedures for plotting points on a graph from an equation, for example, but may still be unable to extract any meaning from the graphical representation.
Further, students who only have procedural knowledge can often get lost when confronted with an unfamiliar situation. Their lack of conceptual understanding often means they are unable to determine which procedures would be appropriate in this unfamiliar context and why.
What does it mean to teach for conceptual understanding?
Hiebert and Grouws (2007) describe classrooms focused on conceptual understanding in mathematics as: (a) focusing explicitly on making connections between facts, procedures and ideas, and (b) allowing students to struggle with important mathematical concepts. In this model, procedures are learned through the process of connecting key concepts.
The two key hurdles in implementing a pedagogy focused on developing conceptual understanding in algebra may be (a) defining precisely what the concepts of algebra are and what is it that students need to understand about these concepts, and (b) developing assessments that allow us to determine the degree to which students have understood the key concepts.
In terms of assessment, the researchers who conducted this meta-analysis give an example of how a question might be re-focused to get a window into a student’s conceptual understanding. The question “Use the quadratic formula to solve x² + 6x + 8 = 0” (p. 377) can be solved using the formula without a student necessarily having any understanding of the meaning of quadratic functions. Refocusing the question to “ask students to graph y = x² + 6x + 8 and explain how the x intercepts of the graph are related to the factors of the equation” (p. 377) requires students to make connections between the factors of the function, the roots of the function, and the x intercepts of the function.
It is this kind of demonstration, of the ability to make connections, that allows us to see student understanding. Defining the fundamental concepts of algebra is a more difficult task, however, and the researchers note that an agreed definition of algebra “remains elusive” (p. 373). The National Council of Teachers of Mathematics recently defined algebra as “... a way of thinking and a set of concepts and skills that enable students to generalize, model, and analyze mathematical situations” (2008).
The researchers note that concepts related to variability are central the subject, and they also note that students are often confused by the concept of equivalence and its meaning within algebraic contexts. Research needs to be undertaken to elucidate this emerging set of key algebraic concepts.
As teachers it seems essential that we focus our teaching and assessing on conceptual understanding of algebraic concepts, and on the connections between concepts, algorithms, representations etc. as suggested by Hiebert and Grouws (2007). The potential advantages of this are noted by Skemp (2006), and include: (a) improved ability to transfer learning to unfamiliar contexts, (b) reduced need to memorize rules, (c) enhanced motivation to learn mathematics, and (d) increased probability that students will become independent learners in mathematics and possibly in other subjects.
The study
Rakes, C., Valentine, J., McGatha, M. and Ronau, R. (2010). “Methods of Instructional Improvement in Algebra: A Systematic Review and Meta-Analysis.” Review of Educational Research 80 3, pp. 372-400.
Other references
Hiebert, J. and Grouws, D. (2007). “The Effects of Classroom Mathematics Teaching on Students’ Learning.” In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 371-404). Reston, VA: National Council of Teachers of Mathematics.
Skemp, R. (2006). “Relational Understanding and Instrumental Understanding.” Mathematics Teaching in the Middle School 12, pp. 88-95.
National Council of Teachers of Mathematics (2008). “Algebra: What, When, and for Whom.” Retrieved from http://www.nctm.org/uploadedFiles/About_NCTM/Position_Statements/Algebra%20final%2092908.pdf.
National Mathematics Advisory Panel (2008). “Foundations for Success: The Final Report of the National Mathematics Advisory Panel.” Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.
Vogel, C. (2008). “Algebra: Changing the Equation.” District Administration 44, pp. 34-40.




Please fill out the form below if you would like to post a comment on this article:








Comments

There are currently no comments posted. Please post one via the form above.

MORE FROM

GORDON ELDRIDGE: LESSONS IN LEARNING

What Are the Elements of an Effective Global Citizenship Curriculum?
By Gordon Eldridge, TIE Columnist
Mar 2021

Designing Curriculum for Global Citizenship
By Gordon Eldridge, TIE Columnist
Dec 2020