Example
Essay on Integrating Science and Mathematics:
A growing number of predominant scholars in the field of education call for a radical change in math education. Whereas the assumptions and their consequent suggestions may vary, it is generally agreed that math education should provide better understanding of real-life numerical problems. This implies putting bigger emphasis on branches such as probability and statistics and less on calculus.
The rationale behind such claim is rather straightforward: quantitative reasoning is an essential element in rational decision-making, and thus underlies the thinking processes that enhance one’s ability to make sound economic decisions, to understand relationships such as causality and correlation and to ease the understand of more advanced mathematics when needed. Furthermore, the fact that computers have cleared the way to scientists to focus on their endeavors instead of spending enormous time on performing and recording calculations may lead to a change in our perception of numerical skills, namely a shift from the mechanics of math towards more conceptual perception of numerical phenomena.
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Needless to say that the influence of such trend on the work of math teachers is very likely to be far-reaching, and primary school teachers will not be an exception. Following this rationale, I developed a simple but effective unit of study to teach the concept of percentages. In addition to providing an engaging introduction to the topic, the unit has two other targets:
First, it develops basic scientific thinking skills by allowing the student to hypothesize the results of percentage calculations and test their assumptions immediately afterwards. They do so by dividing sums of money, which allow them to understand why and by far they were wrong when they make mistakes. They use currency as a medium of investigation that they can count, touch and know from their daily lives.
Second, the unit aims to demonstrate a practical use for percentage calculation, namely to understand the meaning of %-denominated marketing messages that they see every day. This may give them some basic decision-making skills as young consumers, but more importantly will engage them to try and make such calculations as an extra practice in their everyday lives.
The unit has an unfolding structure. It capitalizes on the student’s prior familiarity with fractions to show how easy it is to translate fractions into simple percentages. When students understand that 100/100 equals one whole unit of something and that 50/100, for example, is exactly half unit, there is a strong reason to believe that accepting % as a simple way to denote a constant denominator of 100. And so, saying ‘half’ is as good as saying 1/2 or 50/100, but with a smart twist – its shorter to write and easier to read.
The next part invites the students to reflect on the role of percentages as a convenient means to manage our possessions. The students are now incredibly reach; each one received $100 in $1 (monopoly) notes, which they can touch and develop a sense of ownership towards it. But what is there treasure built from? Two piles of $50 (50%), four piles of $25 (25%) and so on (thirds are not addressed in this unit in order to avoid dealing with decimals).
An equally important and interesting question is the dollar value of halves, quarters, etc. when there are now only $40. The percentages are the same but now they stand for less money, of course. The variability of the dollar values as opposed to the constancy of the percentages help to convey the message that percentages are directly related to (and dependent on) the value of the ‘underlying asset’ they describe. To emphasize this point from another point of view, the students turn to cooperate with each other (i.e. in groups of two students) in order to explore the dollar value of simple percentages when 100% equals $200. The possibility that $200 might sometimes imply 200% is not discussed in this unit for the sake of simplicity.
Once we ensured understanding of the basic concepts we can take the students into the domain of healthy consumerism. They receive a ‘Weekly Offers’ worksheet, which quotes various prices and offers percentage discounts. The challenge here is to convert percentages into real advantages; we can get more for fewer dollars, but we need to know how much money we save and what is the price after the discount. Students will be asked to make predictions regarding the amount they can save when accepting the offer.
After making their predications they will check them by holding the required amount to pay the original price, and then dividing it and putting away the notes they can keep thanks to the discount they were offered. A computer game, for example, might cost $50 but is offered at a 10% discount this week. In order to know how much money they can save, students will have to take 50 $1 notes, divide them into piles of ten and take one pile out. The price is now $45, they have saved $5, and should write down the conclusion that 10% of $50 is $5, or simply 10%*50=5.
Some struggling students may make false predictions regarding the value of the discount. Such students may benefit from the proposed method to a great extent because it allows them to get a physical perception regarding the extent to which they were wrong. Therefore, they can engage in a process of cognitive accommodation (i.e. gradually improving their accuracy by getting closer and closer into the rationale) through physical feedback. This approach poses an alternative to the rigidity of feedback methods that can only distinguish right and wrong answers.
Percentages are indeed effective means to understand relative values, which are essential element in understanding numerical propositions. The unit proposed here aims to train students to think critically about such propositions and to simplify students’ ability to quickly quantify percentage-denominated offers and claims. It obliges the student to work hard in order to solve rather simple problems and thus leads them into the ability to make fast and accurate evaluations by using mental math instead of traditional, paper-based and quite archaic evaluation approaches.
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