What is the difference between teaching for knowledge and teaching for understanding? Isn’t that a great question? I just started reading the book Creating Cultures of Thinking, and 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.
I have been a long time reader of your blog. Applaud your work and thank you for outstanding posts, ideas, and lessons.
I am super excited to share a new manipulative that has been a game changer in my third and fourth grade classrooms. Frac Track and Frac Track Plus are new products available through EAI Education. Check out this video to see how they work. https://youtu.be/Zib7n9mP6ss
I have used these resources for fractions on a number line, comparing fractions, finding equivalent fractions, adding, subtracting, multiplying, and dividing fractions. I don’t even “teach” students how to change fractions greater than one to mixed numbers. They own it after they use the Frac Track Plus because they see the relationship between the numbers. Decimals are also part of the experience so the fractions come full circle.
I will be attending NCTM in San Francisco and CAMT in San Antonio. If you stop by the EAI Booth, I can give you samples. If you prefer I would be happy to mail them to you!
Would love to get your feedback!
I love this new tool, Dianne! I will be in CAMT in San Antonio, but I’d love to try them out with my students before then. Look for an email from me!
I came across your video and would like to know how to go about getting the Frac Trac and Frac Trac Plus. I work in a Title 1 school and having this “hands on” material would be extremely beneficial for my students.
The FracTracK and Plus are available through EAI Education.
I just wanted to say a quick thanks for what you do – there is no one I trust more on the new math TEKS than you! I just came by because I was thinking to myself “what’s the best way to teach comparing fractions with unlike denominators?” and sure enough, this comparing fractions was the second from the top on your blog. I’m definitely struggling trying to figure out how to get my students (who are significantly below grade level – I’m at an IR campus!) to a true understanding of how to compare fractions — just because “the butterfly” is an easy fix doesn’t mean it’s good teaching.
Thank you for everything that you do.
Kelly in Dallas
Thank you, Kelly!I’m glad the fraction posts were so timely. 🙂
Your articles are very resourceful. I enjoy reading them and at times practising with them.
I would like to ask for your permission to share them, if you do not mind.
Of course you may share, Kevin!!
I am a college student studying math education in the general education classroom this semester. Throughout our classes, my fellow classmates and I discuss similarities and differences between how we were taught and how we are expected to teach our students. You are completely right when you say, ” we’ve been teaching fractions without understanding for years”. My classmates and I have discussed how our foundation of the abstract idea of fractions is very wish washy. Our teachers have said a lot of “just do this” and “just do that”. This led all of us with a scared sense of teaching fractions because we had a lot of relearning to do in order to get to UNDERSTANDING rather than just the knowledge of how to compare fractions. The questions of “why” we do the things we do was a difficult thing to describe. However, with the help of blogs, like yours, and our professor, we are able to reanalyze the concept of fractions and get to where we need to be as future teachers!
Thank you for your article and your help on the idea of making fractions more clear to me and my future students! 🙂
Fractions are incredibly abstract! I’m so happy that you’re hearing better instructional strategies through your coursework. Good luck with your teaching career!
I am reviewing third grade material for the STAAR. In many of the resources, although it is not stated, the students are asked to compare 2 fractions with both unlike denominators and unlike numerators. I know that cross multiplying is not the best method, but the students still struggle to draw accurate comparable number lines or fraction bars. What would you suggest for tackling this problem? We have discussed more or less than 1/2 as a strategy but even that is sometimes not applicable. Please help!
I would question the alignment of those resources, Anna. The TEK clearly states that comparisons are only for fractions with like numerators or denominators. It’s not until 4th grade that students should be using additional comparison strategies. There is a reason for that–focusing on the meaning of the numerator and denominator is key for truly understanding fraction concepts.
3.3(H) compare two fractions having the same numerator or
denominator in problems by reasoning about their sizes and
justifying the conclusion using symbols, words, objects, and
Thank you for responding so quickly! I have used this blog as a resource for awhile and it is incredibly helpful, and reassuring that I have a place to turn when I “the teacher” struggle with the material. I hope I am not the only one to realize that I have lots to learn when it comes to 3rd grade math standards! Thank you again.
I love the crosswalk snippet you have between the CCSS & TEKS. I have been searching for one so that I can use the CCS lessons that I find online. Do you know of a crosswalk that has already been compiled?
No, Valerie, I’m afraid I don’t.
Thank you for such a wonderful post. Having just finishing studying different techniques to teach fractions to children, this really sheds a new light of some techniques. I agree where you talk about how teacher take shortcuts and avoid strategies when working with children, This leads to the teaching without understanding. How is one able to teach a child about math, if they them self do not understand the topic that they are teaching. Using the techniques that you provided, I myself now have better understanding of fractions and am able to better teach my students, better prepared to answer the questions that will arise from them.