Pearson Algebra 1, Geometry And Algebra 2 Commo...
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To get beyond metaphor, what I hope you might have a mental image of for a fraction, is some numerator and denominator pair with a line segment in between. If we say "fraction" in general these may be any values, but if we qualify it as a "rational fraction" we signify having integer values for each; possibly with variables standing for integer values, e.g. a/b, as we are often given as a model in algebra texts. 1 I am hoping you've pictured this because it is "rational fractions" that I want to discuss most in this paper, though I commonly say "fraction" in much of the text below.
I see fractions as an ideal vehicle to help my students attain more mastery of a ubiquitous topic from their primary skills, while at the same time using this mastery to find engagement with and access to algebra and advanced secondary studies. I will approach this by first considering some difficulties that my students have had with notation, language, and the way in which they have studied fractions in their primary school experience. Then I will state objectives and present strategies, with specific examples of assignments that I will use and other teachers may find helpful.
As a last thought on language, it is important that students simply understand "what" they are studying. In his book Innumeracy, mathematician John Allen Paulos introduces what he considers some of the worst misconceptions about mathematics, the second of which is that mathematics is a completely hierarchical subject in which one topic or skill set follows another without commingling. 5 Students' general notion that they are learning "normal math" for years, and then suddenly are doing things like "algebra," works against their comprehension of the grand scheme in which they are working. Students should learn the terminology to appropriately consider their "math" studies.
According to Howe, "We will do well to regularly point out to students the methods we employ that are part of arithmetic and where and how they transfer to skills we would appropriately term algebra." 6 This could be very helpful to students in finding the nuances within the reiterated skills of managing fractions, because students might better understand a step forward is not an abandonment of previous skill sets.
Liping Ma, an international education researcher, claims that students use among three approaches to developing arithmetic skills: counting, memorization, and, what Ma calls, "extrapolation," by which she means a development of reasoning around properties of numbers (which are actually informal applications of both arithmetic and algebra), which is reliant upon student rigor, but neither upon the exhaustiveness of counting nor the inflexibility of memorized facts.
When we have difficulty justifying extrapolation, over algorithmic approaches, to administrators or skeptical colleagues, we do have some very supportive statements via the NCTM: "National Council of Teachers of Mathematics asserted in their Curriculum and Evaluation Standards (1989) that proportional reasoning is of such great importance that it merits whatever time and effort must be expanded to assure its careful development." 7 In other words, learning how to manage fractions is worth the time. I also hope to show in my strategies below that it can be coordinated in coursework so that it needn't "take time away from" learning algebra skills.
We should also expect lasting rewards for this investment of time. The perspective of solving the known from the unknown, i.e. using an open-ended problem solving approach based on what we know with confidence v. unqualified algorithmic exertions, initiates a lasting attitude toward learning and mathematics. 8 If this type of attitude toward the notation and communication of rational numbers can be established through work with fractions, it can only be a positive reinforcement of a similar attitude toward all algebra work that involves them.
It would be nice to leave it at that, but, of cou