The effect of great math teaching

Many, many years ago I walked into the classroom of a new teacher during her prep time and found this on her chalkboard…

five to the zero power

Doing my best of embody the famous quotation of G.H. Hardy, “A mathematician, like a painter or a poet, is a maker of patterns.”, I created a table of some powers of 5. Then we filled it out gradually.

table of fives with missing

Finally, we looked for patterns in the table. Looking at the middle column, my colleague said, “See? The fives are gradually disappearing. There will be no fives for 5 to the power of 0, so the answer is zero.”

“That seems to make sense, but let’s look at the third column. What is happening as we move down the column?”, I asked.

“It divides by 5”, replied. I filled in the table only to hear her gasp in amazement. “Oh my gosh! It’s ONE!”, she almost screamed.

table of fives with ones

“But what about the middle column?”, she wondered out loud to herself.

“You were half right”, I said. “There are no fives…but the answer is one, not zero.”

“Because the pattern says so?” she asked. And before I could answer, she said “Because the pattern says so!”

I am reminded of this story from close to 30 years ago by this wonderful article on the impact of teacher effectiveness on student achievement. If my colleague was unintentionally teaching her students misinformation with exponents, where else might she have been misinforming or half-informing her students. What impact might this have on her students?

Now…I’m NOT bagging on my colleague. It is not her fault that no one taught her this simple “proof” of the Zero Exponent Rule. I’m honored to have taught her a little something. Indeed, she became a great teacher, only the tiniest part because of me. For that moment, however, I performed the role of mathematics instructional coach, helping her with both mathematics AND the instructional strategies that might be used to get the mathematics across to the students. As a result, she took a huge step towards being a high performing teacher.

Why does this matter? In this research progress report, we learn that there is a huge difference in student academic achievement depending on whether the students experiences three consecutive years of high-performing teachers versus three consecutive years of average-performing or low-performing teachers.

high high high versus low low low

In comparing 5th grade math achievement after experiencing three consecutive years with high-performing teachers versus three consecutive years with low-performing teachers, the report states,  “With an even start, the difference in these two extreme sequences resulted in a range of mean student percentiles in grade five of 52 to 54 points!!”


Unfortunately, the effect of student achievement after experiencing a low-performing for a single year can still be measured even after multiple years of having high-performing teachers. It is essential that no student is subjected to the instructional strategies of low-performing teachers. Or at least, site administrators should insure that students who experience a year with a low-performing teacher should then experience multiple years of high-performing teachers.


Since it is so obvious that the quality of the math instruction is of paramount importance, we teacher leaders and math leaders need to be vociferous advocates for meaningful mathematics coaching. Occasional sit-and-get professional development is not good enough. We need the kind of ongoing training and coaching that will bring about real change in the classroom.

Our students deserve it.