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September 4, 2017

Often we are in a rush to get quick results in our classrooms. Unfortunately, quick results in our classrooms (and in life) are rarely lasting.  In order for learning to be lasting, replicable, and generative meaning students can take one idea and connect it to others; it must be connected to what they already know and understand. Students may know that addition and subtraction are related, but do they use 98 + ___ = 101, to solve 101 98 = ____ ?

One way to support lasting learning in your classroom is by linking concepts to procedures. Consider the best way to learn how to play tennis.

Would it be by listening to somebody explain how to play the game, or watching videos of others playing tennis, or by getting a racket, getting on the courts, and playing tennis yourself?

We can anticipate errors will repeatedly be made when learning anything new. There will be confusion about the rules. There will be frustration and fatigue. There will also be improvement over time. These are natural occurrences in any area of learning. Yet, we try to remove these natural occurrences from the classroom environment. If we can come to understand that challenges are a natural part of learning, we can better anticipate and support our students’ needs when they occur.

In order to link concepts and procedures, use physical representations and allow for approximations. Instead of telling or showing our students mathematical concepts and procedures, let’s allow them the time to engage in them independently of teacher directions. Let students count collections (efficiently and inefficiently), let them draw pictures and cross them out, let them use a variety of tools and reflect on their helpfulness, give them real world problems to make sense of and solve. Then, we can sit back and expect moments of brilliance and moments of struggle to occur.

Approximation and exploration does not have to last an extended amount of time. For some students, it may take a day, for others it may take a week. However, I would much rather spend a week letting students gain true understanding versus fighting against years of frustration because they don’t realize that subtraction can be used to solve an addition problem or 3 x 7 could be the score from a football game.

 Once students show a basic understanding of the problem or concept, extend their thinking by linking the concept to a procedure. For example, if students drew a picture of 100 blocks in 5 rows, we can ask them, “What would it look like if we used ten frames to organize our work or what would it look like if we used an equation to match our work?”   This is a small question that yields huge results. We are asking kids to be both reflective and generative by asking them to link their beginning and informal ideas to more sophisticated and abstract representations. By allowing the space for approximation, exploration, and elaboration, we push students to make connections between concepts and procedures that are lasting.   Spend a little extra time allowing your students to understand concepts. Then, link those concepts to procedures simply by asking, “What would it look like if…?” Allow students to share their correct and incorrect or partially correct ideas. Yes, you want to listen to and represent all contributions, even the incorrect or partially correct ideas. If we do not examine all ideas, students may hold onto false ones.         Look at the following students work. Notice how the student goes from drawing 20 blocks, to representing the 20 objects with the numeral 20, and using it in a repeated addition equation, to writing out a two- step equation using multiplication. For this particular student, this process occurred over two days, some students will need a little more time to practice and approximate. By doing this, students can see the natural link between concepts and procedures.

For help with planning for meaningful instruction that helps students make connections between mathematical properties, concepts and procedures go to www.TeachingOneMoore.org