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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.

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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.

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Every year around this time, my family is getting ready for back-to-school night. Now that I have children in high school, junior high, and elementary school, it’s always fun to see how this event is handled at the different levels. When I was teaching high school, we were very structured, with parents moving from class to class as though on a regular schedule; I think we had each group for ten minutes, just long enough to quickly review the syllabus and policies and send them off to the next class. Regardless of the structure of back-to-school night at your school, there are a few things you can do to make the evening more engaging for students and their parents. A good place to start is having some manipulatives you’ll be using during the year out for parents to handle. Things like Unifix Cubes, Pattern Blocks, Ten-Frame Floor Mats and Fraction Tiles are always good choices.

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A few weeks ago, we shared a new tool for Unifix cubes, the corner cube. This week I wanted to introduce you to something else that’s new in the world of Unifix: the Jumbo Unifix cube. My three-year-old recently got into my box of Unifix cubes and was having a good time building with them, until I handed him the jumbo cubes. His eyes lit up and he immediately went to work with the larger cubes. Bigger than our traditional Unifix cubes, these are perfect for smaller hands in preschool and kindergarten classrooms. They are also popular with special education teachers and occupational therapists for students who are still developing their fine motor skills.Read More
I taught an Algebra 1 or Algebra 2 class every year I was teaching, and I was always looking for ways to make the content engaging for the students. I firmly subscribe to John Van de Walle’s notion that drill and practice are two very different things, and sought opportunities for the students to have meaningful practice with the concepts they were learning. As a result, I avoided the lengthy problem sets and worksheets that are prolific in high school math classes, opting instead for problems, explorations, and games that encouraged thinking and discussion.Read More
Traditional dominoes have a variety of uses in the classroom. A simple internet search for “math domino games” yields thousands of ideas for using these tools to build number concepts. But eventually, the novelty wears off and they become more of a toy than a learning tool. But the matching aspect of dominoes lets us expand this teaching tool to domains beyond number.Read More

The first time I ever saw a teacher using Unifix cubes in the classroom, it was not in an elementary school. I was coaching a middle school teacher who was introducing the concepts of mean, median, and mode to her students. They were using the cubes to “graph” the data and then find these measures of central tendency. Since then, I’ve seen Unifix cubes in classrooms at every grade level, used in a variety of ways to teach number sense, data, measurement, patterns, and an array of topics. They were never as useful a tool for Geometry… until now.

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"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.

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"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.

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Kathy Richardson, one of the leading authorities on elementary math, has spent years educating children in the early grades. Her recent book, How Children Learn Number Concepts, is a wonderful introduction to the Critical Learning Phases that elementary children move through as they are developing a sense of number. This concise book is full of information to guide teachers of all levels as they help students through the phases.Read More
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