Thinking With Numbers

Ed Rathmell has spent the past 40 years working athe University of Northern Iowa to help prospective teachers learn to teach mathematics. Throughout those years he also collaborated with primary grade teachers to learn how to help children make sense of basic addition and subtraction facts, practical everyday word problems, mental computation, and to develop deep understandings that lead to number sense

Ed has also been active in the National Council of Teachers of Mathematics serving on serveral editorial boards and on the writing team for the original NCTM Standards. His efforts, as an officer, speaker, and through planning and conducting professional development in collaboration with the Iowa Department of Education earned him a life-time achievement award from the Iowa Council for Teachers of Mathematics.

Larry Osthus is an independent mathematics consultant who has worked with several schools and educational nonprofit organizations. From 2003 through the present he worked for Thinking With Numbers in various capacities. In 2011-2013 Larry provided professional development for Iowa Adult Education teachers and Iowa Department of Corrections teachers through his program called The Iowa Numeracy Academy. In 2005-2007 he participated in a document review and curriculum alignment audit of four large New York City school districts and in a curriculum alignment audit in Elkhart, IN. He has been a keynote speaker for CASIO at the 2007 National Council of Supervisors of Mathematics Annual Conference. Larry has made several presentations at both the state (Iowa) and national level on using CAS with under-performing students and the resulting impact on student attitude toward mathematics. He has authored two online courses focused on the use of formative assessment in the mathematics classroom. Prior to working as an independent consultant he served as a mathematics consultant for Heartland Area Education Agency in Johnston, IA. Larry earned his M.A. degree in Mathematics from the University of South Dakota. He has several years of both high school and middle school teaching experience. His present interests include using technology to aid student learning of mathematics, using Computer Algebra Systems (CAS) with under performing students, and using formative assessment in the mathematics classroom.

These are conceptual lessons, not “naked” numbers, designed to provide the structure and imagery for students to gradually begin to answer questions about addition and subtraction without counting from one each time. The ten frames and a part-part-whole diagrams provide that structure. Helping children make sense of answering “How many?” questions without counting is a major step towards helping them use more mature thinking while adding and subtracting. These lessons will provide kindergarten children a great opportunity for success with mathematics in the primary grades and help older children make mathematics more than just magic for those who happen to know the secrets.

The lessons also will help children conceptually make sense of equality. A delayed introduction to the equals sign allows children to make sense of operations independently. This approach eliminates many of the misconceptions that children develop about the equals sign and promotes a much deeper understanding of equality.

The lessons are not intended for master of the facts. In fact, do not “drill” children on answers to basic facts. Premature memorization interferes with understanding. That mastery will be developed in lessons that will come later, when the children have the conceptual understanding to benefit from them.

While using these lessons, be sure to read to the children and wait for the directions box in the bottom right hand corner of the screen. Give the children an opportunity to answer and discuss the questions without telling them how to do it, or telling them the answer. Initially these lessons may take longer than five minutes, but as the children improve their abilities to reason and explain, five minutes should be enough time to complete each lesson. By waiting for the children to solve and explain the questions, you will have the opportunity to listen and learn how they are thinking. Enjoy as the children develop new ways of thinking and confidence in their abilities to make sense of addition and subtraction.