By Mary Leonard, Math Department Chair, and the A-S Math Department
Mathematics is, in part, about recognizing patterns and making predictions. These patterns are all around us. Inspiring our boys to key into their beauty at a young age can instill in them a sense of wonder and curiosity about their world. Giving them experiences and encouraging them to harness their ideas, evaluate them, and grow them is the basis of mathematics. Call it logic or intuition, but either way we want them to connect their thinking with what their experience proves true, and clearly express their thinking, both orally and in writing.
Unlocking the language of algebra is essential to building a strong foundation in numeracy skills, and one of the concepts that lays the groundwork for deep understanding in advanced mathematics is that of fractions. Fractions are the building blocks to patterns in music, nature, engineering, life sciences, art, finance, physics, and sports—to name just a few. Understanding their properties, such as how to add, subtract, multiply and divide, is key to unlocking the principles of algebra, geometry, and calculus.
Our earliest learners use blocks to explore shapes and their attributes. In the First Grade, the boys develop fractional thinking using blocks to compare sizes and to differentiate between a whole and part of a whole. They move on to trying to cover shapes with other shapes using congruent pattern blocks, which tangibly represent parts of a whole. As they progress through the early grades, they continue to use manipulatives but also draw pictures to express different parts of a whole. Giving them context is the basis of the K-4 curriculum. It helps the boys move from whole-number theory to the realm of rational numbers, as they see that there are values less than one, or there are values that lie somewhere in between two whole numbers on the number line. Sophy Joseph, Math Learning Specialist, observes that much of the learning is done through students sharing their ideas and their thinking with one another.
In Grade Two, for example, students learn the difference between money, which is part of 100, and clock time, which is part of 60. Alice Heminway, a second-grade teacher, states, “In the Time and Money unit, we study the analog clock and how it can be split into halves; we use the language of half past. We then look at how the clock can be split into four equal parts and introduce the term quarters. We use the language of quarter to and quarter past. Some students are able to distinguish between a quarter of a dollar and a quarter of an hour. Then they have these big “Aha” moments where they say, ‘Oh! That’s why that coin is called a quarter.’”
By the time our boys reach the Fourth Grade, they see division problems as fractions and recognize remainders as both fractions and decimals, thus using mixed numbers as well as decimals to express quantities and fluidly convert between them. As they grow in their reasoning abilities, they start to see decimal numbers as part of a whole and can express the difference between decimal place value as powers of ten and that of a denominator of a fraction. Crucial in their understanding is that the denominator of the fraction depicts the number of equally sized pieces the whole is divided into. To build on the second-grade time unit, for example, as students progress in their thinking they are able to compute with time and differentiate between the notation of part of an hour expressed as a decimal and part of an hour expressed in minutes out of 60, rather than 10 or 100, so they see that a time-lapse of 3.20 hours is not equivalent to 3 hours and 20 minutes.
Grade Five is the year when we would like to see students solidify fraction operations. There are many types of fraction problems liberally furnished throughout the curriculum to encourage the students to see how the composition of something—from musical notes to recipes to measuring— requires them to develop a facility with parts of a whole. Some examples include:
• Putting musical notes together as ¼ + ¼ + ¼ + ¼ to equal a full measure in common time, then substituting notes with different powers of two in the denominator
• Following recipes to explore doubling and halving
• Measuring polygonal dimensions to the nearest eighth or sixteenth of an inch and calculating area and perimeter
Exposure to fractions in these various platforms helps drive home the “why” behind the procedures for fraction operations. Rather than memorizing the procedures, we would like to see students detect patterns and come to realize through their own experience why combining fractions necessitates a common denominator and what it means to multiply and divide fractions. We would like them to intuitively understand, for example, that multiplying two proper fractions together yields a smaller product than either of the two proper factors, something different from what they initially experienced when multiplying whole numbers.
By the time a student is in the Sixth Grade, he is working with fractions as ratios and rates so that when he makes the transition to Pre-Algebra, he uses proportional thinking to solve problems with similarity. “Proficiency with fractions in all their different forms and applications is one of the boys' first introductions to abstraction, and it’s the secret sauce to successfully mastering algebra. Sixth-grade mathematics offers boys the opportunity to see the interconnectedness between fractions as ratios, decimals, and percents. They learn that there are many ways to use their knowledge of fractions to tackle problems,” says Sixth Grade Math Teacher Annette Perez.
As students transition to Pre-Algebra and Algebra I in the Seventh and Eighth Grades, they generalize as algebraic fractions are introduced and the concept of negative fractions brings to life more abstract concepts. Math Teacher Robin Keats says of the seventh-grade Pre-Algebra course, “In Seventh Grade, students use their new understanding of negative fractions and deepening understanding of variables to explore the new realms of probability and statistics.” (see probability example below) Suppose a student has a strong foundation with fraction concepts when the numerator and denominator are numerical. In that case, substituting variables will not throw them off course, and they can extend their thinking to apply their understanding to multivariate expressions and equations.
As students move onto secondary school mathematics, the theme of rate of change is a central concept in advanced algebra and coordinate geometry, which relies on a solid understanding of fractions. Students graph functions to see how they change over time. They must differentiate between a constant rate of change (linear) to rates of change that are exponential or parabolic. (see graphs below that show rates of change) And, being able to graph functions, interpret the data, and predict their behavior is a building block of the sciences.
Our Science & Engineering and STEAM classes extend the boys’ work with numeracy since ratios and rates are a central concept that is key to their understanding properties of physics and chemistry. Throughout the curriculum, our boys have many opportunities to put into practice the lessons learned in math class. Connecting those lessons to their broader experience will help deepen their number sense so that they think critically and correctly interpret their experiences. Sparking imagination and curiosity to bring about positive change is what it means to educate a boy at Allen-Stevenson.
