Happy October – one of my favorite times of year in education. By now, classroom teachers have routines in place and have gotten to know their students. Schools start to have that familiar hum of learning as we roll through the crisp fall days. Armed with routines and familiarity, teachers work diligently to move student’s forward in their quest for knowledge and understanding.

A recent twitter post about this very work of educators struck me as poignant, “managing instruction with a system of ‘covering’ standards that are too many and too far from students’ prior knowledge is not management but posturing … the key is to respond to students’ thinking, not their so-called ‘needs’ …’need’ names a sled to low expectations”

If the key is to respond to student thinking, the education stance required is to shift towards uncovering how students understand various concepts. Questioning becomes critical here, as we need to ask students to explain, show, prove, and share their thoughts. Classrooms come alive as students work towards deeper levels of understanding with their classmates and teachers pushing for their explanations.  Through this process of challenging student thinking through rich mathematical tasks, students’ needs are met and standards are covered because students are constructing deep understanding.

In 1976, Skemp gave us an interesting clarification about two differing definitions of understanding. “Students who are taught instrumental understanding come to see mathematics as isolated pieces of knowledge. They are expected to remember procedures for each and every concept/skill.  Each new skill requires a new set of procedures.  However, those who are taught relational understanding make connections between and within concepts and skills.  Those with a relational understanding can learn new concepts easier, retain previous concepts, and are able to deviate from formulas/rules given different problems easier because of the connections they have made.” Well worth reading more here from Mark Chubb’s blog about this distinction.

relational understanding - teaching mathematics

Tweet: Teach less for instrumental understanding, more for relational understanding. http://ctt.ec/35Y14+ #teaching #mathematics @looneymath Perhaps that slippery slope of low expectations can be avoided by reflecting on Skemp’s message – teach less for instrumental understanding and more for relational understanding. So, how do educators do this? Well, for starters, we do not do things like teach tricks that are void of understanding. Consider this trick echoed in many classrooms, “When you multiply a number by a ten, just add a zero” What happens to that trick when students multiply 4.3 x 10 and find out it is NOT 4.30! Or consider the damage of teaching key word strategies for solving story problems.  This approach encourages students to become number grabbers and ignore all of the words EXCEPT for those key words that signal an operation. What happens to these students in testing situations where there are no key words? This is another ineffective strategy for developing understanding. Such shallow rules, procedures, and tricks will not lead to success for our students. And, more importantly, it will not lead to understanding. (For more on this try: Nix the Tricks and Teaching Key Words? Forget about it!.)

So, if we know what not to do, then what DO we do? We choose rich tasks like those found at Yummy Math. We ask students for explanation and justifications. We engage learners in productive struggle with activities like those found at Week of Inspirational Math.


My call to action this month is for you to respond to students’ thinking. Keep a journal where you note opportunities of relational understanding in a lesson – a student making a connection, or a task that lends itself to productive struggle, a conversation that reveals deep understanding, etc.  At the end of the month, generate a list of generalizations regarding when and how these opportunities occurred. Share these results with a colleague and brainstorm how to incorporate these into daily instruction.

The reality of 2016 is that educators are accountable to for their grade level standards and closing gaps for students who begin the year performing below grade level expectations. Perhaps we can learn a lesson from Skemp from forty years ago. Making the distinction between instrumental understanding and relational understanding is profound when we think about teaching and what is expected on next generation assessments, as well as what is expected of individuals to be successful in the 21st century.

relational understanding - teaching mathematicsTweet: When we move into #teaching for relational understanding, we navigate towards understanding. http://ctt.ec/F67c_+ #mathematics @looneymath When our teaching moves into teaching for relational understanding, we navigate towards true understanding.

Happy Learning!
Sue Looney, Ed.D.

This Month’s Resource Publications:

Please contact LMC to discuss how we can help you obtain your goals for mathematics education in your district.