I have had this debate many times with my friend and former colleague, Ginny. I think that we both agree that you need to change practices before core beliefs. To change a core belief, you need firsthand evidence, and you will only get that firsthand evidence if you try something different and get unexpected results.

One of the milestones in Algebra I is learning how to find the equation of a line given two points. The summative assessment for the seventh-grade functions unit we developed in Holliston included this problem:

Find the slope, y-intercept, and equation for the line that passes through (-9, 2) and (6, -4).

Start by finding the slope:

Then substitute the slope and one of the points into the slope-interecept form of the equation for a line to find the y-intercept:

Finally, use the slope and y-intercept to write an equation for the line in slope-intercept form:

Most teachers firmly believe that finding the equation of a line given two points is hard. They regularly see high-performing eighth-grade students struggling with it. Learning it requires a student to memorize a number of steps, and many students are either unwilling or unable to do that.

I, on the other hand, believe that this process is fairly intuitive and accessible to most sixth-graders. You may have noticed that the summative assessment for the sixth-grade functions unit we developed in Holliston contained this problem:

Find the rule to predict the area of the colony on Day x.

Day | Area (cm2) |

0 | |

5 | 32 |

9 | 52 |

13 | 72 |

17 | 92 |

Okay, we are giving the students four points on the line here, but that doesn't change the nature of the problem.

Start by finding the rate of growth. The bacteria colony is growing at a rate of 20 square centimeters every four days, which is 5 square centimeters per day.

Then find the area of the colony on Day 0. To do that, we work backwards. From Day 0 to Day 5, the colony grew for five days. At 5 square centimeters a day, it grew 25 square centimeters. So the area of the colony on Day 0 must have been 7 square centimeters. (A student also could have counted backwards by 5, finding the area of the colony on Days 4, 3, 2, 1, and then 0.)

Finally, write a rule to predict the area of the colony on Day x.

If the colony starts with an area of 7 square centimeters and grows at a rate of 5 square centimeters per day, then on Day x, the colony will have an area of 5x + 7.

Within a week, we had close to 100% of all sixth-graders doing this type of problem, including special education students in our substantially-separate math class. More importantly, they viewed this problem as trivial and did not feel as though they needed to memorize anything to figure out how to solve it.

By convincing the sixth-grade math teachers to try a new curriculum with new instructional practices, we were able to generate firsthand evidence that caused some of them to reconsider their core beliefs about the difficulty of tasks and what students can and cannot do. It was a small shift, but it was enough for me to build on and introduce even more new practices. Presenting this evidence at the professional day in January to the rest of the department caused a stir, but it did not really move any of the seventh- or eighth-grade math teachers. Because feedback is so noisy in educational systems, it is very easy to dismiss secondhand evidence.

My overall strategy in Holliston was to build working relationships with the math teachers and to encourage a few individual teachers to try some new practices. Because teachers would be experimenting with new practices without a change in core beliefs, none of those new practices could be that radical. However, the new practices had to be powerful enough to generate short-term evidence that would register above the noise. If the firsthand evidence was too powerful to ignore, then a small shift in core beliefs would occur and I could then suggest a few slightly more advanced new practices. This would lead to another small shift in core beliefs and more advanced new practices. I would essentially be tacking into the wind. As evidence mounted and core beliefs shifted, more teachers would take ownership in the work and, together, we'd be able to convince more teachers to take that small first step.

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