Conceptual Understanding, the Framework

Conceptual Framework

Developing conceptual understanding of mathematics has long been a goal of the National Council of Teachers of Mathematics (Smith, Bill, & Raith, 2018), the National Research Council (National Mathematics Advisory Panel, 2008), and the National Council of Supervisors of Mathematics (National NCSM Conference, 2018). This goal has not always reached classroom instruction. It is not that teachers don’t believe the importance, but changing instructional practice is difficult.

Changing classroom practice takes time and necessitates support. Of equal importance is for other key stakeholders (i.e. parents and community) to understand the importance of the shifts in instruction, and support rather than undermine their implementation.

Learning progressions have developed as a result of research into how students learn that has taken place over the last two decades. A learning progression is a set of ‘stones’ that students use to further their learning. These ‘stones’ create a visual or conceptual map providing students a progression of expectations (or path) within a domain.

The first learning progression in mathematics that students are faced with is the Number and Operations- Base Ten and it is an important, one might say critical progression. By the time students get through the progression they need to be fluent with basic computation facts.

Part of the problem we face is that people want to ‘jump’ to the end of the progression and focus on speed and accuracy. Unfortunately, this misguided attempt to help students become better puts up barriers to their development (see above example). The new mathematics standards emphasize the development of strategies (make 10, near doubles, etc.) over use of standard algorithms initially. The goal is to make sure students have the foundations they need to be successful long-term, not just to find the answer to the problem in front of them.

When teachers take the time to help students develop strategy use and better understand number relations, then students can apply those same skills and thought processes to new settings. For instance, when Lucas turned 9 + 6 into (9 + 1) and (6 – 1) or 10 + 5 in his head, he was using the ‘Make 10’ strategy to get the answer. It may not look as easy as just memorizing the fact, but it sets Lucas up to solve larger problems quickly and fluently. For instance, 398 + 25 uses the extension of the ‘barrowing or Make 10’ strategy [(398 + 2) and (25 – 2) or 400 + 23] to solve.

Remember Gavin? He is still counting on from 398 which is now not as efficient. Keira remember had memorized the easy problem. Has she memorized this problem as well? Likely not. What does she do? Does she apply a strategy like Lucas or count on like Gavin?

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