A problem-centered approach for making sense of counting, numbers, addition, and subtraction


About the lessons

Counting and Numbers

These conceptual experiences have been designed for children, ages 2-5, to help them Make Sense Of Counting And Numbers. With over 300 5-minute lessons, there are enough for children to successfully prepare for school math.

The lessons provide the structure, imagery, and challenge for them to more deeply understand counting and numbers. Five-frames and a focus on partitions of numbers together with animations provide that structure and challenge.

Children will have many opportunities to practice counting and each of the common counting errors is addressed. Additionally, concepts involving one-to-one matching, the order you count doesn’t affect the number, matching sets, comparing sets, comparing sets indirectly, how moving dots (joining and separating) affect the number, partitions of numbers, one and two more, one and two less, and introductions to addition and subtraction are covered.

For each of the numbers 3 through 5, children are challenged to start with one number and make another, find hidden parts, make sense of the equals sign with addition and subtraction, solve word problems, recognize when a word problem has too much or too little information, examine number patterns, and partition numbers.

Making Sense Of Addition And Subtraction

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.

Enjoy as the children develop new ways of thinking and confidence in their abilities to make sense of addition and subtraction.

Sample Videos

Sample of Explore Problems

Sample of Reasoning Strategy Problems

Sample of Assessments & Reports


Parent Account


Add up to 4 students

Online math videos, lessons, and assessments from the Pre-K section Counting and Numbers

Online math videos, lessons, and assessments from the K-5 section Making Sense of Addition and Subtraction

Student performance tracking and reports

School Account


25 teacher accounts

Each teacher can add up to 50 students

Online math videos, lessons, and assessments from the Pre-K section Counting and Numbers

Online math videos, lessons, and assessments from the K-5 section Making Sense of Addition and Subtraction

Student performance tracking and reports


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Ed Rathmell


Ed Rathmell has spent the past 40 years working at the 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 several 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


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.