The importance of developing conceptual understanding in mathematics

All three students solved this problem successfully. However, only Lucas shows evidence of using number relations to support his automaticity with basic facts.

Written By rodaniel

On April 17, 2019

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A series of blog posts designed to help all stakeholders understand the importance of changing how we teach and learn mathematics. The series also provides a comprehensive vision for mathematics instruction around the Number and Operations- Base Ten learning progression.

Setting the Scene

Imagine for a second this setting in a first grade classroom:

Make Ten strategy

Gavin, Keira, and Lucas each solve 9 + 6 differently. Gavin looks at his fingers and counts on from 9, saying, “nine, ten…fifteen.” Kiera has memorized the answer of 15. Lucas sees (in his head) two tens frames and imagines moving one ‘dot’ over, to re-imagine the expression to be 10 + 5.

The ‘make 10’ strategy he is using sets him up to understand and apply these strategies to a variety of settings in the future. Lucas is on a trajectory to be a proficient mathematics student (National Mathematics Advisory Panel, 2008).

Gavin’s approach (counting on) works and students have to go through this stage before they can progress. If Gavin does not learn other, more sophisticated strategies, he may be stuck as a ‘count on’ student his entire life. There are numerous anecdotal stories of students who in their college algebra classes still use the ‘count on’ strategy when combining like terms (9x + 6x). Continuing to use this inefficient approach limits opportunity in higher mathematics settings as more of his working memory will be engaged in solving this simple (Vasilyeva, Laski, & Shen, 2015).

The final remaining student in our example (Keira), has the problem memorized. Knowing the fact from memory is automatic, but the teacher must figure out if she has just memorized the fact, or understands the number relationships. Memorized facts are often forgotten and then children resort to counting.

It is important to note that a timed, written test focused solely on finding the correct answer would make no distinction among these three children.

Research into how students best learn these strategies is very clear, too early of a focus on speed and accuracy limits development. Yes, it was how we learned mathematics, but IT IS A FLAWED APPROACH! If we want students to learn new or different ways of thinking, we have to give them time to develop their understanding (Baroody et al., 2016).

Link to Part 1- The Setting

Link to Part 2- The Conceptual Framework

Link to Part 3- Instructional Shifts