Welcome! Today is a special day! As part of Math Solution’s Fractions February, I was given the opportunity, and honor, of interviewing Julie McNamara, author of Beyond Pizzas & Pies and Beyond Invert and Multiply. Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense (3-5) helps teachers combat common misconceptions related to teaching and understanding fractions—misconceptions that hinder students’ deep understanding and development of fraction sense. Beyond Invert & Multiply: Making Sense of Fraction Computation (3-6) assists teachers in further strengthening students’ foundational understanding as it applies to fraction computation. Both are MUST have resources! At the end of this interview, enter for your chance to win one of these phenomenal resources. Many thanks go out to Julie McNamara for graciously granting this interview and sharing her expertise and passion.
Why is working with fractions where students often begin to struggle?
One big issue students have is that when working with fractions they have to consider numbers in new ways. When working with whole numbers, a number such as 27 was always 27 – even if it referred to 27 people, 27 dogs, 27 thousand dollars, etc. Now, given a number like 2/7, students have to consider the relationship between the numerator and denominator, and understand the whole. Is the 2/7 referring to the number 2/7 that is somewhere between 0 and ½ , or is it referring to 2/7 of some larger number, such as 2/7 of a class of 28 students? In the latter case, the 2/7 is actually 8 — which can be super confusing!
Also, many of the ways that we approach fractions instruction can be very shallow and give a false sense of students’ understanding. For example, when shown an image like the one below, most students who have had any instruction in fractions will answer ¼.
What fraction of the large square is shaded?
However, when shown an unequally partitioned image like the shown below, many students, even students in middle school, will say 1/3.
What fraction of the large square is shaded?
In your books you talk about fraction sense. What is fraction sense? Why is it so important? What traits does a student with fraction sense possess?
To quote Beyond Pizzas & Pies, “Fraction sense implies a deep and flexible understand of fractions that is not dependent on any one context or type of problem. Fraction sense is tied to common sense: Students with fraction sense can reason about fractions and don’t apply rules and procedures blindly; nor do they give nonsensical answers to problems involving fractions.” In other words, students with fraction sense understand why a fraction such as ¼ is greater than 1/8, even though four is less than eight. They have a sense of the relative magnitude of fractions and can estimate answers and check for reasonableness after performing calculations. They can decompose fractions into other fractions in order to compute mentally. They understand fractions as numbers and know that the properties of operations apply to fractions just as they do to whole numbers.
Can you talk a bit about what it means for students to have a flexible understanding of fractions?
Having a flexible understanding of fractions is very much tied to fractions sense. In situations involving computation, it means that students don’t blindly apply rules and procedures but instead consider the numbers (fractions) they are working with and then chose an appropriate method of proceeding. One of my favorite activities in Beyond Invert & Multiply is called Get to the Whole! This is a strategy that can be successfully used when adding or subtracting fractions. Given two fractions such as 4/5 and 3/5, one student might add them by decomposing the 3/5 into 1/5 + 2/5, add 4/5 + 1/5 to make 1, and then add the remaining 2/5. Another student might decompose the 4/5 into 2/5 + 2/5, add 2/5 + 3/5 to make 1 and then add the remaining 2/5. This strategy can also be used when working with fractions with unlike denominators. Students are often taught that the only way to add two fractions such as 5/6 and ¾ is to create equivalent fractions. While doing so will result in the correct answer, it is also possible to rename only the second fraction to 9/12 and add two of the twelfths to the first fraction to make 1 (since my fraction sense allows me to understand that 2/12 = 1/6) and then add the remaining 7/12. It can also be as simple as understanding that I can change the order to fractions to add them (the commutative property) or group them in different ways (the associative property) to make computation more efficient.
Your first book, Beyond Pizzas & Pies provides educators with the understanding and tools they need to combat fraction dilemmas/misconceptions at the 3-5 level. What suggestions can you give educators at the K-2 level who want to help students develop an early understanding that does not lead to misconceptions?
Providing K-2 students with visual models that they can touch, move, manipulate, and create is a must. Also, building on their intuitive sense of fairness to help them understand the importance of equal shares. Helping them make sense of the protocol for naming and (when appropriate) labeling fractions. When working with materials it is essential that students understand that a label such as ¼ is NOT the name of a block or region of a circle, but is a description of the relationship between the part and the whole.
What tools for teaching and understanding fractions are MUSTS? Why are they MUSTS?
As I wrote before, providing visual models, especially those that are generalizable across contexts are a must. I am hesitant to name specific materials because it is not about the actual items that you use, but about how you use them to encourage students to develop generalizable principles. That said, I really like to use Cuisenaire Rods because they allow for a great deal of flexibility and can be a wonderful support for moving from area models of fractions to linear models – in particular, the number line.
You devote Part IV of Beyond Invert & Multiply to discourse with fractions. Why is fostering student talk so important and what are some ways to get students started?
Through talk we often find out how we are thinking about things and where our gaps in understanding lie. Many misconceptions (or limited understandings) can be present in students’ conceptions but talk helps to bring these ideas to the surface so they can be interrogated and utilized to deepen understanding. In addition, through mathematical discourse, students often present ideas in unique ways that other students can latch on to and build upon.
Presenting open-ended prompts such as the “Which One Doesn’t Belong?” routine or “What do you notice? What do you wonder?” helps students get over their initial fear of being wrong. Also, as teachers we need to create a safe place for students to begin sharing their thinking with others. When students realize that they have important ideas to contribute, and that their teachers and classmates are truly interested in what they think and have to say, they will be much more willing to add to the mathematical conversation.
Are there any current projects or plans for future publications you can share with us?
I am hoping to begin work on a book addressing algebraic reasoning in grades K-8 soon. It will follow a similar format as Beyond Pizzas & Pies and Beyond Invert & Multiply. There has also been some discussion of a book aimed at the primary grade standards that address partitioning and early work with fractions. I would not be one of the main authors but hope to have a contributing role if this manuscript goes forward.
Thank you so much for having me!
It was a pleasure to have you stop by today! Now, it’s your chance to win a free copy of Beyond Pizzas & Pies and Beyond Invert & Multiply. AND here’s a little video teaser for Beyond Invert & Multiply.
Winners will be announced March 1st!
All the best–