I was the math and science curriculum specialist at the Robert Adams Middle School in Holliston, Massachusetts from 2007 to 2010. In the fall of 2008, I worked with the 6th-grade math teachers to develop and implement a new functions unit. In most middle school math programs, students learn a little about slope in pre-algebra. But their first real introduction to functions is in Algebra I, where they are expected to master a number of discrete skills, including:
- Using the slope and y-intercept to graph a line
- Writing the equation for a line in slope-intercept form
- Finding the slope and y-intercept of a line given two points
- Working with linear equations in point-slope form
- Working with linear equations in standard form
All of these skills should sit on top of a single conceptual framework, but because very few students enter Algebra I knowing anything about slope, there really isn't time to develop a conceptual framework and then get students to mastery in the time allotted. Instead, we are forced to ask students to learn each skill in isolation.
In Holliston, about 60% of our students take Algebra I in eighth-grade, and only half of those students do well on the functions chapter test. I would describe functions as a significant pain point for the eighth-grade math teachers; they really wanted their students to do better on this key concept, but they didn't know how to make that happen.
I decided to target the functions pain point in 2008 for five reasons:
- I didn't have a very good working relationship with the eighth-grade math teachers and alleviating this eighth-grade math pain point would be a big step toward establishing one.
- Functions is something that we could do completely in-house in the middle school. We could introduce functions in sixth-grade without relying on any help from the fifth-grade math teachers based in the elementary school, and we could have everything wrapped up and tied in a bow by the end of the eighth-grade, so we wouldn't need the high school math teachers to do anything different to take advantage of what we had done.
- The math teachers all recognized that procedural approaches weren't working and wanted students to have a conceptual understanding of functions.
- If we could help students develop conceptual understanding and double the number of students reaching mastery, then people would notice and applaud the work we were doing. It would also go a long way toward changing some of the core beliefs held by the middle school math teachers themselves.
- The functions pain point isn't something that any one teacher can successfully tackle on his or her own. This would reinforce the fact that I wasn't pointing my finger and saying that individual teachers weren't doing their jobs. It would also highlight the good things that could happen if we set aside our egos and worked together as a department.
I spent about a month working with the sixth-grade math teachers to design a four-week functions unit for sixth-grade. We did this work after school and during their prep periods. As curriculum specialist, I had no say over how they spent their prep periods, so this was a purely volunteer effort. When we finally implemented the unit in November, I spent some time modeling lessons and coaching teachers in their classrooms.
Here are some of the problems we used in the summative assessment we gave at the end of the unit:
3) At the start of an experiment, a bacteria colony has an area of 22 cm2. It grows at a constant rate of 3 cm2 per day. Fill in the table and come up with a rule for predicting the area of this bacteria colony on Day x.
| Day | Area (cm2) |
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 28 | |
| 55 |
4) Bob uses the following rule to predict the area of a bacteria colony on Day x:
area = 21 + 4x. How fast is the bacteria colony growing? What was its area at the start of the experiment? Explain how you can tell by looking at the rule.
5) 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 |
Use the graph to answer problems 7-10 and the bonus:
7) Which bacteria colony is growing at a constant rate? How can you tell just by looking at the graph?
8) When is Bacteria Colony B growing faster than Bacteria Colony A? How can you tell just by looking at the graph?
9) What was the average growth rate per day for Bacteria Colony B between Days 4 and 7?
10) Describe the growth of Bacteria Colony B over time. Explain when it is growing faster or slower, when it is shrinking, and when it is staying the same size.
11) Bonus: Write a rule for the growth of Bacteria A over time.
We spent a professional day in January analyzing the results of the summative assessment as a department. 85% of the students scored a B or higher. The seventh- and eighth-grade math teachers were floored by both how well the students did and the quality of their written responses. We estimated that 75% of the students had a solid conceptual understanding of linear functions and could fluently translate among tables, equations, graphs, and written descriptions as long as functions were presented in a real-world context. 55% of the students completed the bonus correctly even though we had never asked them to write an equation from a graph with a fractional rate of change before, and another 20% understood what they needed to do, but just made a procedural error somewhere along the way.
One of the sixth-grade special education teachers worked closely with us as we developed the sixth-grade functions unit. She used a pared down version of the unit with the students in her substantially-separate math class. (A substantially-separate math class is for students with math learning disabilities so severe that it has been determined that their needs cannot be met in a general math class.) Her students loved the unit and got 75% of the pattern, relationship, and algebra questions on the state test correct when they only got 45% of the questions in the other four strands correct. The state test was administered in May, about five months after she taught the functions unit.
In 2010, I developed the seventh-grade functions unit with the seventh-grade math teachers. By this time, the first group of sixth-graders that had taken our sixth-grade functions unit were now seventh-graders. The seventh-grade math teachers had been excited by how well these students had done on the sixth-grade summative assessment, but they were anxious about how much those students would actually retain. We decided to start the unit with a warm up activity designed to re-activate what the students had learned sixteen months ago.
Although the students said that they had forgotten everything about functions from sixth-grade, the warm up activity demonstrated that the conceptual framework for functions they had developed was still there, and everything came flooding back to them.
Here are some of the problems we used in the summative assessment we gave at the end of the end of the seventh-grade unit:
A plant is growing at a constant rate. Use the table to answer problems 1-5.
| Day | Plant Height (cm) |
| 0 | |
| 20 | 17 |
| 28 | 23 |
| 36 | 29 |
| 44 | 36 |
1) What is the rate of growth of the plant? Explain how you found it.
2) What is the starting height on Day 0 of the plant?
3) Write a rule for the height of the plant (y) after x days.
4) Use the rule to find the day when the height of the plant will be 80 cm. Write the rule and show all steps.
5) Create a Plant Height vs. Time graph.
Use the following rule to answer problems 6-8: w = 50m/3 + 100 where w = gallons of water in a tank and m = minutes.
6) Use the rule to fill in the rest of the table. Show all steps.
| Minute | Water in Tank (gal) |
| 0 | |
| 10 | |
| 15 | |
| 620 | |
| 800 |
11) Graph: y = 3x/4 − 2
13) Write an equation for the given graph.
In problems 14 and 15, graph the two given points, then find the slope, y-intercept, and equation for the line that passes through those two points.
14) (-9, 2) and (6, -4)
15) (-8, 2) and (-5, 10)
16) Explain how you found the slope for the graph in problem 14.
17) Explain how you found the y-intercept for the graph in problem 15.
18) Challenge: Find the point (x, y) where the two lines in problems 10 and 11 would cross if they were on the same graph.
We found that the 75% of the students who had developed a solid conceptual understanding of functions in sixth-grade were able to pick up immediately where they had left off and build on what they had learned. In fact, the seventh-grade math teachers found that they had to do very little teaching in order to get students to generalize from real-world contexts to abstract x's and y's, or to introduce fractional rates of change. However, the 25% of the students who were shaky after sixth-grade got left in the dust.
I left Holliston after the 2009-10 school year, but the plan was to develop a new functions unit in eighth-grade. The 75% of the students with a solid conceptual understanding should have easily been able to extend what they knew to linear equations in standard and point-slope forms. This means that all of the eighth-grade Algebra I students (instead of less than half) should have mastered linear functions before moving on to Algebra II in ninth-grade, saving the high school math teachers 3-4 weeks of re-teaching time. And the students that didn't take Algebra I should have been much better prepared for Algebra I in ninth-grade. All of this should have then opened the door to a district-wide conversation about collaboration and curriculum and instruction.
Within the middle school math department, I was hoping that the success of this conceptual approach to functions would lead to a willingness to work together and to try new instructional strategies. Up until this point, getting teachers to try some new curriculum hadn't been too difficult, but getting them to accept coaching was like pulling teeth. And new curriculum was substantially less effective without the necessary coaching and shifts in practices. My next goal would have been to analyze what we were doing and then to try to increase the percentage of students developing solid conceptual understanding. If you've been reading my blog, then you know that the first step is to design a curriculum that breaks down tasks to intuitive subtasks. I think we did that in the sixth- and seventh-grade functions units. The next step is figuring out how to help students learn how to perform those tasks. We got to 75%, which is excellent for our first try, but now we needed to push that up to 95%+. Doing that would mean challenging even deeper core beliefs and forcing more changes in how we operated as a middle school and district.
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