One of the great things about coaching at a school where you previously taught is you get to see former students and talk with them about their new year. Recently, a former student approached me and said, “You know the difference between this year and last year is you taught us the why and how. This year, my teacher only teaches us how, not why and it confuses me because I don’t understand.” This was powerful because he was a student who resisted my student-centered approach. He hated doing small groups and often asked if he could just do work out of the textbook.

Within days of this encounter, I read an article from SEDL Insight titled, “Teaching Mathematics Conceptually“. While sitting and reading the article, I couldn’t help but think out loud. My thoughts can be categorized into 3 sections; “Aha!”, “Yeessss!” and “Amen!”

__Aha!__

- To say you are “reducing” a fraction creates misconceptions, the fraction is not smaller. Use simplify instead.
- Mathematics is a precise discipline and teachers don’t realize the slightest deviation from true mathematic language can make the content incorrect.
- Exponents indicate how many times the base appears as a factor, not number of times a number is multiplied.
- -(-4) should be interpreted as the opposite of -4.

__Yeesss!__

- Algorithms and shortcuts are not bad, the key to using them to help rather than hinder understanding lies in the sequence of events.
- Relying only on algorithms and procedures and focusing on shortcuts result in teaching efficiency, not mathematics.
- As a result of naked math, teachers and students easily lose sight of the meaning of numbers and numerals.

**Amen!**

- Memorization of rules and mastery of computation are not the same as true knowledge of mathematical concepts and ideas.
- Shortcuts become the means to get answers with the unfortunate result of bypassing conceptual understanding. This makes it difficult for students to understand more advance or complex mathematics.
- By not taking shortcuts and insisting on illustrating the full process (cra), it’s possible to make explicit connections among concepts.