Dr. Cotter Invited to Speak at the University of Cambridge!
RightStart Math’s author Dr. Joan A. Cotter, and her daughter Kathleen Cotter Clayton, were invited to speak at the 16th International Conference of Mathematics Education for the Future Project in August 2022 at King’s College, Cambridge University, UK. Dr. Cotter discussed why appropriate place value instruction positively influences students’ understanding of elementary math. She explained how reducing the focus on counting and increasing the use of transparent number naming greatly improves number sense. Kathleen presented how using the linear fraction model and strategies greatly improves the student’s understanding and application of fractions. You can find a summary of Dr. Cotter’s presentation here, and the PowerPoint presentation of Kathleen’s talk here. The conference proceedings and all the abstracts from the conference can be found at this link. If you are interested in reading Dr. Cotter’s and Kathleen’s abstracts, you can find them on pages 117 and 111. Dr. Cotter and Kathleen presented and stayed at King’s College, Cambridge University, UK, which was founded in 1209. Here is a picture of the courtyard at King’s College. It was an incredible honor for Dr. Cotter and Kathleen to be invited to such a noble institution. They were thankful for the opportunity to present some of the reasons why RightStart Math makes a positive impact on how students learn math. Dr. Cotter and Kathleen were privileged to meet, share ideas with, and learn from attendees from Poland, Turkey, Ireland, Greece, South Africa, Croatia, and many other countries. What an amazing trip! Contact us today for information on how RightStart Math can improve your elementary and middle school students’ math education.
Fractions, by Dr. Cotter
Many people have the idea that fractions are incomprehensive and unpredictable. In fact, cartoons often express the fear of them as a source of amusement. However, when they are understood, they are amazing. They are also necessary for everyday life. For example, telling time, counting money, and cooking utilize fractions. History of Fractions Historically, fractions were only considered as part of a whole and could never be greater than one. The term fraction comes from the Latin word frangere, which means “to break.” In the 1600s, the concept of fractions expanded. Mathematicians began using fractions as dividing two whole numbers. For example, one-third became thought of as 1 divided by 3. This new idea expanded fractions to include what we now call improper fractions. For example, 3/3 and 4/3 are now legitimate fractions because 3 could be divided by 3, and 4 could be divided by 3, resulting in a number equal to or greater than 1. Not everyone welcomed this new view. So, those fractions that followed the traditional understanding of being less than a whole were named proper fractions. The new fractions, those that were not less than a whole, were called improper fractions. Unfortunately, we still use the traditional idea of fractions as being part of a whole or as a part of a larger amount. In fact, the dictionary defines fractions as ‘a tiny part.’ If you look up fractions in a Thesaurus, you will see synonyms such as “fragment” or “snippet.” However, this is a limited view and can prevent or reduce the student’s understanding of them. Fractions as Division If fractions are now considered division, why do we still use the terms numerator and denominator instead of dividend and divisor? These terms were used in the 1540s, where numerator referred to a counting process. The denominator referred to the parts being counted. This process created useless memory work. When I teach fractions, I delay introducing these terms for as long as possible. When teaching math to children, the main goal is to help the child see the relationship between concepts. Therefore, we need to teach fractions as division as early as possible because they need to understand the link between fractions and division. Circular Fraction Model A common model used today, and one of the earliest models used for teaching fractions, is the circle. Many teachers use the “pizza model” to engage their students and help understanding. However, this model has serious drawbacks. Many would agree that it is easy to see 1/2, 1/3, and even 1/4 of a circle. But, it is significantly more challenging to see or visualize 1/7, 1/8, or 1/9. There are other drawbacks when using the circular fraction model. How do you compare fractions? For example, how do you help the student 4/5 with 5/6 when using the circular fraction model? Can they cut out the pie pieces to compare the size of the shapes? What do you do with fractions that are greater than 1? Linear Fraction Model The Linear Fraction Model or fraction strips is a more powerful and versatile model to use when teaching fractions. The linear fraction model includes a series of ten rectangles that are 8 to 10 inches long each. The top strip has ‘1’ written in the center of it. The second strip is broken in half, with 1/2 written in the center of each segment. This pattern continues for the remaining strips to the tenths. These strips are placed in order, one right under the other, to form a fraction chart. There are several variations of this model. Some fraction charts have different colors for each fraction family. While this may seem helpful for instructions, the colors actually create a distraction, and the size of the pieces is not as quickly seen. For this model to be effective, all the strips should be the same color. Another variation of the fraction chart is removing the lesser-used fractions and adding additional ones. For example, some of these charts remove the 7ths and the 9ths and then add the 12ths. Not only are we telling our students that if we do not like the numbers, we don’t use them, but we also keep them from seeing interesting patterns within the fraction chart. When looking at a complete fraction chart, the student can see interesting arc patterns (hyperbolas). When you introduce the fraction chart to your students, have them begin by assembling the strips like a puzzle while having the full fraction chart available as a reference. Even young children enjoy building the fraction puzzle again and again. Your students can do another fun activity by creating the fraction stairs. By the way, they can see half of a hyperbola when they do so! A Story About Mike Several years back, I tutored a third-grade student named Mike. He was moving out of the area and needed help in math. To help Mike learn the foundation of fractions, I started by having him assemble the fraction chart like a puzzle. When he completed that task, I asked him, “Which is more 4 or 5?” He answered, ‘Five.” Then, I asked him, “But which is more, 1/4 or 1/5?” After viewing the chart, he stated, “1/4.” Once I saw that he quickly responded, I continued to ask him similar questions, such as “Which is more 2 or 3?” and “Which is more “1/2 or 1/3?” Then I expanded on the topic by asking him different questions, such as “How many thirds is needed to make a whole?” After viewing the chart, he said, “3.” Then, I asked how many fourths, fifths, etc. And he was able to answer the questions. After I explained to Mike how 3/4 was three 1/4s, Mike showed me where 3/4s was on the fraction chart. We spent some more time working through similar fraction associations. At the end of a 45-minute tutoring session, I told Mike that he had done a great job but that he still had not yet learned everything there was to know
Celebrating You & All You Do
We, at RightStart Math, want to pause and say, “Thank you!” We truly appreciate the hard work, dedication, and heart you bring to your students. You invest time, creativity, and care into helping every learner grow—not just academically, but in confidence and perseverance as well. We Celebrate You Today, we celebrate you—the lesson planners, the encouragers, the problem-solvers, and the champions for your students. Your work matters, and it makes a lasting difference. “Teachers plant seeds of knowledge that grow forever.” – author unknown We Appreciate You Every day, you show up with patience, creativity, and an unwavering commitment to your students. The work you do matters deeply—and it often goes unseen. “The influence of a good teacher can never be erased.” – author unknown Thank you for your perseverance, your care, and your belief in every child’s ability to learn and succeed. We are grateful to partner with you in this important work. If we can help you in any way, do not hesitate to reach out! Contact Us Product Spotlight: RightStart Math Card Holders Small tools can make a big difference—especially in busy classrooms. RightStart Math Place Value and Base-10 Picture Card Holders help reduce clutter and distractions while actively supporting visual learning and hands-on engagement. Place Value Card Holder The RightStart™ Math Place Value Card Holder provides a clear, structured way to display Place Value Cards during instruction. By keeping cards organized and visible, it helps students see how numbers are built, supporting stronger understanding of place value and the base-ten number system. Base-Ten Card Holder The RightStart™ Math Base-Ten Card Holder keeps Base-Ten Cards neatly organized and easy to use. This simple tool supports hands-on learning and helps students make sense of number relationships through a clear, visual representation. Whether you’re working with a full class or a small group, these card holders help keep the focus where it belongs—on thinking, strategy, and understanding. Once again, thank you for the care, dedication, and heart you bring to your students each day. We are honored to support you and grateful to walk alongside you in your teaching journey. Your RightStart Math partners, Teresa, Rachel, Maren and Kathleen Filed Under: Newsletters
