The Top 10 Most Important Concepts in Middle-School Math

This is my list, compiled from 6 years of classroom teaching. It changes periodically, but I welcome comments. I work with this list repetitively throughout the year. Contrast these concepts with a more specific skill, such as learning to use a stem and leaf plot. If the stem and leaf plot can be authentically embedded into a project, we might work with the concept extensively. More often than not, I teach about 12 to 15 concepts in-depth, and the wide array of specific concepts one day a week in a catch-up tutoring class, or during "test-training" months. Students might see some exact concept only 20 to 40 minutes during the course of a year. Yes, students do miss something by not having a longer period to work with these ideas, but because the concepts we are learning at deep, meaningful level are authentically embedded in context, student engagement remains high and the long-term outcome is positive. Here they are, my top 10 suggestion areas of study that lead to middle-school math success.

Students should have ample and repetitive practice:

1. Using the Number Line. I feel the number line is the building block that forms the foundation of our mathematical thinking. As a base 10 system (decimal), students ought to know that our math comes from 10 fingers, 10 toes and is very old evolutionarily. It's more important, in my opinion, to spend time building a deep understanding of the number line rather than moving to more advanced topics if you notice students without these fundamental understandings.

2. Converting Fractions to Decimals to Percents. Of all the computations I hold as a standard, the ablility for a student to convert fractions such as 3/5 to decimals (.60) to percents (60%) is the core concept I define as giving students the most flexibility within the broader array of math.

3. Estimating. Estimating may be the most important tool for students to learn to approach mathematics in a way that makes sense. Of all the tools in estimation, I recommend that the bulk of the practice of estimation be in a mental format. Too often we are focused on students computing in paper and pencil format. I refer to mental estimation as the light saber of tools in mathematics, referring to its incredible importance and power.

4. Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic rules. Sure, it's a standard that could mean a lot, but I don't worry about choosing one particular method of approaching this concept. It's much, much more important to spend a significant amount using a tool in an authentic manner and gain mastery than it is to abandon deeper for the sake of "going on." Sure, there is a  balance here, but collecting real data and playing with it graphically will lead to a greater sense of purpose.

5. Visualizing and solving problems involving surface area and volume. This is such a specific type of content, but like the number line, I feel it forms the cornerstone of geometry. I can't prove it with any numberical data, but I believe surface area and volume create a type of thinking crucial to how students learn mathematically.

6. Converting from one unit to another. When I teach this concept, I usually have students develop their own units of measure and create a challenge for other students.

7. Formulating questions, designing studies, and collecting data.

8. Using and applying scales, ratios and proportions to quantitative relationships.

9. Enhanced practicing of mental math and solving problems in various methods.

10. Multiplication and Division Tables!

 Sources:  Angela Flicker and the National Council of Teachers of Mathematics and my own years of teaching.

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