The following is the philosophical motivation for the use of "for every" statements in the classroom. Making the slight adjustment, which I propose in the following paper, to the language you use in talking with your students about proportions and ratios can make all the difference for their understanding, especially with students who are still concrete thinkers.
This paper was presented at ChemEd 2011 at Western Michigan University in July of 2011. The "For Every" Speak handout, which contains robust examples of use and explanations, from the full presentation that was given at ChemEd 2011 can be found on the ChemEd page of my website.
According to Piaget, proportional reasoning is an essential cognitive function that indicates the formal operational stage of cognitive development (Santrock, 2008). The ability to think in an abstract fashion, as is associated with the formal operational stage of cognition, is necessary to the successful study of chemistry (Lythcott, 1990). Beyond the concrete properties of matter and the directly observable features of chemical change, the study of chemistry includes abstract (and sometimes nebulous) concepts. Some of these topics are quantitative in nature, such as dimensional analysis, stoichiometry, and equilibrium, requiring sound mathematical talent to successfully master them. Often, students struggle with the chemistry when their math ability, especially their proportional reasoning, is underdeveloped (Lythcott, 1990). Students can become easily intimidated and discouraged by a struggle with chemistry that is in reality a fledgling ability with mathematical reasoning. If teachers of chemistry do not address the disparity with math ability in students' chemistry performance, a struggle with the content for many students will be inevitable and meaningful learning will not take place in the chemistry classroom (Krajcik & Haney, 1987).
Arons (1996) describes that proportional reasoning is a fundamental skill that students need in order to be successful in any science discipline. Demonstrating successful proportional reasoning ability is a telltale sign of good scientific thinking. To better assist all students, and not merely 'luck out' with teaching the mathematically strong students in the chemistry classroom, physical meaning must be explicitly connected to quantities and the relationships between quantities must be explicated (Hestenes, 1987). Building the conceptual understanding behind chemical quantities will allow students to relate those quantities in all situations and reason through quantitative problem solving. By treating the quantitative aspects of the study of chemistry with concrete pedagogical approaches, students can more easily attain the concepts behind the principles, reason their way to a solution, and transfer their learning to a novel situation (Darling-Hammond, Low, Rossbach, & Nelson, 2003).
When you honestly consider the majority of students in a typical chemistry classroom, they are within the early years of teenage development. According to Vygotsky, to most successfully work within a student's zone of proximal development during this age, teachers' instructional methods must appeal to the concrete operational thinking that is characteristic of this developmental period (Darling-Hammond, et. al., 2003). Merely teaching chemistry in a way that relies upon algorithms and strong mathematical abstract reasoning appeals only to students who are further cognitively developed. This method will not be effective; furthermore, it can serve as a discouragement to students whose present stage of cognition is mismatched to the instructional methodology. The language teachers use to communicate physical meanings of quantities and relationships can either promote algorithmic memorization or support useful learning.
Avoiding the algorithmic pitfall, that will surely lose a number of students in the teaching process, is possible. This can be achieved by making a very small adjustment to the language used to talk about chemical quantities, proportions, and relationships (Cramer & Post, 1993). Utilizing traditional language such as, "100 centimeters is equal to one meter," can give students a sense of the magnitude of quantities of length or the relationship between their magnitudes; however, this is of little use to students when they go to think about the physical meaning of the quantities or determine the number of centimeters in 13.4m (Lesh, Post, & Behr, 1988). Without an algorithm for using that equality, students have no reliable means to arrive at an answer for the number of centimeters in 13.4m. There is no mathematical operation or procedure embedded in the language used to describe the relationship; therefore, it is effectively and functionally useless to employ this language as a primary means of talking about quantities, relationships, or solutions to problems.
To illustrate the impact that language can have on the way we conceptualize mathematical quantities or operations, consider the difference between English and Chinese Language. Take the example of how we refer to the fraction 3/5. In English, we say, "three-fifths," which tells nothing about the relationship between the numerator and denominator. In fact, many students become used to referring to 3/5 as, "three over five." This is neither an accurate interpretation of the fraction nor a useful one. What does "over" mean as a mathematical operation? Ineffective language where 3/5 means "three-fifths," reduces fractional relationships and ratios to tacit facts not functional knowledge. In contrast, consider the Chinese language for the fraction 3/5, which would translate, "for every five parts, take three." This interpretation of the fraction uses functional language that contains a concrete conception of the relationship represented by the fraction (Gladwell, 2008). Knowing that 3/5 means, "for every five parts, take three," can lead to a cognitive process of determining how many equivalent parts would you take in a group of ten parts. One could proportionally reason that since ten are twice five that twice as many as three would be the answer. Even if students struggle with the multiplication operation, they could still reason their way to the corresponding solution steps using "for every" language, because they could prove what the answer would be based on reason.
Gladwell continues to posit that the power of embedding instructions into language empowers thinking about mathematics. The second most powerful example, according to his argument, describes how the counting system in China is very intuitive. In English, we would say the number '13' as "thirteen," but in Chinese they would say "ten-three." This pattern continues in Chinese, as the number '25' is called "two tens-five;" in English, we say, "twenty-five." This subtle difference entails having to think about more or less information when doing calculations. In English, to add '13 + 25' we have to know what thirteen and twenty mean, and then we combine them algorithmically. In Chinese, they base it on the number of 'tens' and the rest is just the 0-9 counting numbers. So in Chinese, saying '13+25' contains the instructions for adding it: "ten-three and two tens-five." This easily allows the Chinese student to calculate the answer "three tens-eight."
Though it is a consequence of language differences that students become used to thinking about fractions, ratios, proportions, and calculations differently, in English we have the option of using functional language to describe quantities. Using "for every" statement language to describe quantities and relationships contains cognitive instructions for thinking one's way through solving a problem. Algorithmic methods are typically what teachers rely upon to help students solve proportional reasoning problems, because that's what they were shown by their teachers before and it "made sense" to them, e.g., predict the theoretical yield of a certain chemical reaction based on a starting masses of the given reactants and show your work. The typical approach used is the cross-cancelling table method. This is the one with a tic-tac-toe board where quantities and units are matched up so that the units cancel and give rise to the desired unit. The issue with an algorithmic approach is two-fold: first, solving the problem using an algorithm does not necessarily entail or demonstrate understanding of the relationships between the quantities being compared; second, if an algorithm is taught in lieu of the physical meaning behind the quantities in the algorithm (the shortcut first instead of the circuitous route) then the algorithm takes on no meaning. Some students can pick up on algorithms and arrive at the right answer fairly efficiently Algorithms can be utilized, but not until after a student truly understands and demonstrates proficiency with the concept and their reasoning through a problem (Darling-Hammond, et. al., 2003). In order to successfully use algorithms, they cannot just be shown to students for memorizing the procedural steps; instead, the thinking that underlies the algorithm, which is the proportional reasoning, must be made transparent to students through a cognitive apprenticeship (Collins, Brown & Holum, 1991) that includes scaffolding, coaching, and independent practice with the steps prior to implementation of the algorithm.
Consider the following question: "What is a ratio?" A fundamental answer explains that a ratio a comparison of two quantities. Most frequently, students are introduced to and exposed to ratios in a pure sense, using numbers without physical meanings, often in a mathematics course. This leads to the disconnection of ratios from the physical meanings of the quantities they relate as well as a tendency to utilize poor representations of the ratio (Karplus, Pulos, & Stage, 1983). As students gravitate toward, "three over five" language to represent 3/5, the ratios become decontextualized even before they get to a science class (Krajcik & Haney, 1987), let alone a chemistry class. Since many students enter the study of chemistry still at a concrete operational stage of cognition, it is important that proportional reasoning is utilized in a concrete way (Lawson, & Renner, 1975). The "for every" statement language can effectively accomplish maintaining a connection between the physical meanings of ratio quantities and the relationships between them; furthermore, "for every" statement language provides a reliable means of problem solving and reasoning that can easily transfer to novel contexts. By employing "for every" statement language in the chemistry classroom, teachers can reach students at a concrete operational level, creating a springboard for developing proportional reasoning skills (Cramer, & Post, 1993) and making quantitative problem solving less intimating. Students who have fewer struggles with the mathematics involved in chemistry are more likely to approach the content with confidence and be successful with it. Using "for every" statements is a way to elucidate how to think about the quantities to students while preserving the physical meanings behind quantities. This small change to the way we refer to ratios and have students think about ratios will yield large returns of success in the long run from dimensional analysis to stoichiometry and beyond.
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