Tagged with ' Math'

"Would you rather have half of one chocolate bar or a quarter of a different chocolate bar? Most popular answer: It depends on the size of the chocolate bars! As teachers know, the relative size of fractions depends on how the whole is defined. Authors Jim Callahan and Marilynn Varricchio address these common problems with fractions in their new book Fractions Made Easy (Didax, 2016). Drawing on material from the book, we will focus on how visual models can be used to support a solid conceptual understanding of fractions in third grade.

"With current math standards' emphasis on number lines as a visual math model, number lines are being used in nearly all primary grades classrooms for learning such things as counting and early operations. However, research has shown that number lines are conceptually too difficult for young children to understand and instead we should be using number paths, at least until second grade (Fuson, et. al., 2009). A number path is a visual model for counting, addition, subtraction and more. Experts say that the number path is superior to the number line as a visual model for early math learning. We caught up with Educator Margaret McGinty to learn why.

When I was teaching in the high school, we taught a unit on rational expressions and equations. In simple terms a rational expression is a fraction that has numbers and variables in the numerator, denominator, or both. Because rational expressions behave a lot like fractions, I usually started this unit with a day or two of review of fractions to help students build confidence with this foundational concept. Every year, I was surprised how many students struggled with fraction concepts, and it was clear to me that we needed to do more to build their conceptual understanding in the early grades. Generally, we are doing better with this, using more and different models to help students really understand the relationships between the part and the whole and also between fractions. Number lines help build conceptual understanding of fraction relationships and area models are useful tools for both relationships and operations. Another tool that helps students build an understanding of both fraction relationships and operations are interlocking fraction circles. The short video below explains how these circles support students’ understanding of fractions.

As I travel around the country working with teachers, I frequently hear, “My students just don’t like fractions.” Teachers are right, fractions are confusing. Let’s explore the foundational understandings and possible misconceptions that students may go through as they are learning about fractions. As articulated in the progression on “Number and Operations-Fractions, 3-5,” fractional understanding begins in Grades 1 and 2 as students partition shapes. This is certainly a logical beginning as students have had experiences identifying when their share is smaller than someone else’s!  In Grade 3, students begin considering breaking a whole into equal parts.  Students work with wholes that are varying shapes such as rectangles or circles  and the focus is placed on equal parts.  The emphasis in Grade 3 is on unit fractions (i.e., fractions having 1 as the numerator). Just as our whole numbers are composed by combining 1s, fractions can be similarly constructed by combining unit fractions. For example, ¾ = ¼ + ¼ + ¼.

When I was working in the school district office, we spent significant time putting together a plan that would meet the needs of a range of learners. Where there were almost an endless number of resources from which to choose for Reading, there were very few for Math. We looked at many options but struggled to put together anything as comprehensive as what we could offer for Reading. It was during this struggle that I was introduced to the work of Kathy Richardson, who is one of the leading math educators in the country. As I studied Kathy’s work, I came to understand that the very foundational concepts of number—counting, for example—were much more complex than I had given them credit for. As I became familiar with the Critical Learning Phases that Kathy identified, I realized just how important it is for students to build their conceptual understanding of number relationships.