What is the difference between teaching for knowledge and teaching for understanding? Isn’t that a great question? In his book Creating Cultures of Thinking, author Ron Ritchhart tackles that issue head-on (pg 47).
Understanding requires knowledge, but goes beyond it. Understanding depends on richly integrated and connected knowledge. This means that understanding goes beyond merely possessing a set of skills or a collection of facts in isolation; rather, understanding requires that our knowledge be woven together in a way that connects one idea to another.
Sadly, in mathematics, we’ve been teaching fractions without understanding for years. Take comparing fractions. The go-to method for comparing fractions with unlike denominators is to find a common denominator. That’s a great strategy, but completely unnecessary in many cases. Worse, many educators even skip that strategy in favor of short-cuts like cross multiplying. In this series of articles, I will share five different strategies for comparing fractions.
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A quick look at the progression of the standards for comparing fractions shows that both the CCSS and the Texas TEKS are now more aligned with teaching for understanding and recognize that students should employ multiple methods for comparing fractions:
Notice that the foundation for comparing fractions is laid in 2nd grade, which is when students learn the meaning of the denominator–more fractional parts (a larger denominator) means smaller parts. In 3rd grade students use that understanding to begin comparing fractions with the first two strategies–comparing fractions with like numerators and fractions with like denominators.
Comparing Fractions with Like Denominators
When fractions have the same denominator, e.g., 3/6 and 5/6, it’s kind of like comparing apples to apples. Would you rather have 3 apples or 5 apples? In other words, the size of the pieces is the same (sixths), so of course you want more. Please note, however, that even at this most basic level, students should be justifying their conclusions, not just showing the comparison with a symbol (<, =, or >). It is this justification process that takes the learning from knowledge to understanding and develops a student’s fraction sense. Students should also be experiencing fractions using manipulatives, such as fraction tiles. Fractions are an incredibly abstract concept, so every single lesson should incorporate either concrete (hands-on) or representational (pictorial) learning for students to truly develop conceptual understanding.
Comparing Fractions with Like Numerators
This method requires a true understanding of the denominator. When fractions have the same numerator, e.g., 3/4 and 3/12, it is the size of the pieces (the denominator) that drives the comparison. Fourths are much larger than twelfths, and since I’m getting 3 pieces with each fraction, I would rather have 3 fourths than 3 twelfths. Without concrete experiences, that’s a hard concept for children, because all they see is that 12 is bigger than 4. Do not rush through this strategy! It is a fundamental understanding about fractions. Of course, anytime you can turn practice into a game, students will be more engaged. I created a little Memory card game students can use to practice comparing fractions with like numerators. As with any game, be sure to model the math talk you expect to hear when students are playing. Encourage students to use fraction tiles when playing so they can visualize the fractions. Gradually, remove that support to allow the students to reason without the visual.
Grab your freebie here! Head over now for Part 2: Fractions That Are One Unit Fraction Away from the Whole.