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Mathematics educators have recently highlighted the need for “low floor, high ceiling” tasks that lead students into rich areas of inquiry. The “Four 4’s” problem is a fairly well-known example cited by Jo Boaler and others, but such challenges are hard to develop. It is often difficult to find an exercise, game, or a puzzle that is instantly accessible at a basic level, yet also leads to the exploration of higher-order thinking and deeper mathematical insights. The PEMDice game is designed to scratch this itch. Very simple in concept and, when played in its elementary form, it is ultimately as complex and as challenging as anyone cares to make it.Read More
Every spring, millions of people turn their attention to what is, mathematically speaking, a tree diagram. Some think about it only when their favorite team is playing, while others are completely immersed in the annual College Basketball tradition known as “March Madness.” This frenzied tournament provides multiple opportunities to engage students in math, although we sometimes focus so narrowly on probability and statistics that students miss out on other opportunities to learn. In the spirit of the season, we’d like to share some ideas for teachers of all levels to bring March Mathness to the classroom. You can use the link below to access the activity sheets to use in your classroom. A printable tournament bracket is available here.Read More
Early in my publishing career, I was working with an author on an updated book series. In the course of our discussion, he mentioned the Rekenrek, a tool for building early numeracy. This was, at the time, a very new tool in the United States even though its use could be tracked back for many years in the Netherlands. I was intrigued, but not quite sure what we could do with this tool that would help teachers enhance their instructional practice. Fast forward several years, and the Rekenrek is far more familiar. Many educators are embracing this tool with varying degrees of success. To make this implementation a little easier, Don Balka has developed two activity books for the Rekenrek: Working with the Rekenrek (with Ruth Harbin Miles) and Working with the 100-Bead Rekenrek. These books provide both teacher support and engaging student activities to help us all use the Rekenrek more effectively. If you’re looking for some ideas to get started with the Rekenrek, here are two new ideas based on these resources.Read More
Here is a simple story problem. Rose has 5 pennies. Eva has 9 pennies. How many more pennies does Eva have than Rose? I have posed this problem to many, many children. More than a few of them have answered, “Eva has 9. You just told me that.” Those children didn’t understand the question. It is not simply that they got the wrong answer. It is not simply that they made a minor mistake. Those children really didn’t understand what I was asking. Here is another problem. Trixie has 3 baskets. There are 4 cherries in each basket. How many cherries does Trixie have altogether? I often ask children to draw pictures to represent story problems. For this problem, I have seen children draw 3 baskets, draw 4 cherries in each basket, and then miscount the cherries – maybe they count 11, or maybe 13. Of course, 11 is not right. And 13 is not right. But compare those children who miscount with those children who start out by drawing 4 cherries and 3 baskets. Miscounting is one thing – everyone makes minor mistakes. But children who draw 4 cherries and 3 baskets don’t understand the question. Those children need help.Read More

It’s hard to believe that 2018 is here, and we’re quickly approaching the 100th day of school. When I first started working with elementary school teachers, the concept of a 100 days celebration was foreign to me—it wasn’t something that we did in high school. Over time, I’ve come to appreciate this tradition and the mathematical opportunities it brings. In honor of 100 days of learning this school year, here are a few ideas for your 100th day activities. Try them out and let us know what you think!

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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.

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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
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