Editor’s Note: This is Chapter 16 in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder” by Barry Garelick, a second-career math teacher in California. He has written articles on math education that have appeared in The Atlantic, Education Next, Education News and AMS Notices. He is also the author of three books on math education. Says Mr. Garelick: “At its completion, this series will be published in book form by John Catt Educational, Ltd. Please read at your own risk.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 , Chapter 4 , Chapter 5, Chapter 6, Chapter 7, Chapter 8, Chapter 9, Chapter 10, Chapter 11, Chapter 12, Chapter 13 and Chapter 14 and Chapter 15.
Chapter 16 Instructional shifts, Formative Assessments, and Taking Matters into My Own Hands
Whenever the word “shifts” appears in an article about education, it is highly likely that what you’re reading is blather, claptrap, drivel, garbage and idiocy. (Sorry for all the adjectives; I was trying to avoid saying the word “crap”.) Even more so, if the article talks about formative and summative assessments. While formative assessment is a valid concept, its meaning may vary depending on who you talk to or what article you happen to read.
For example, formative assessment may be defined as evaluating how someone is learning material while summative assessments evaluate how much someone has learned. So says one expert. Another says summative assessments can be used formatively, by using the results to guide approaches in subsequent courses.
These hermaphroditic definitions have provided me much cover in my quest to appear aligned with whatever shiny new thing happens to be in vogue. My first parole officer, Ellen, got me started down the path of formative assessment. Although she had no shortage of suggestions for things I would never consider doing, there was one that I thought I’d try. “Have you ever let students use their notes for a quiz or test?”
I liked the idea and during my first year at Cypress, I allowed my classes to use notes for quizzes, but not tests. I felt that this would reinforce the idea of the value of notes. The problem was that some students’ organizational skills were lacking—resulting in this typical conversation:
Student: How do you do this problem?
Me: Look in your notes.
Student: I can’t find it.
Me: (Drawing a diagram on a mini-white board.) How would you find the time each of the cars are driving?
Student: I don’t know.
Me: (Writing “Distance = Rate x Time” underneath the diagram)
Student: Oh!
Such incidents led me to provide help to students in a direct manner rather than the “read my mind” approach that entails asking vague questions that serve to frustrate rather than elucidate. Sometimes I would partially work out the equation for a particular problem. Other times I would use an example of a similar problem. Expanding from a worked example to solve similar problems demands critical thought, and does exactly what math reformers pretend that unguided discovery does.
I continued this approach with my Math 7 class during my second year at Cypress. I was intent on bolstering the confidence of my students who had suffered the previous year and were convinced they could not do math. I was making headway with them using JUMP, and I could see that getting decent test scores had positive results. But as we got into more complex topics, they were having difficulty and asking for help.
I knew that there was a potential that such approach could quickly blossom into grade inflation and an artificial sense of achievement. So I justified my giving them help by telling myself that their difficulties helped guide my instruction. But I knew there were limits.
“It’s hard for me to not give help when I see they’re on the wrong track,” I told Diane during one of our sessions.
“Yes!” she said. “They have to learn from mistakes.”
Fearing a foray into Jo Boaler’s money-making “mistakes make your brain grow” motif, I rapidly changed the subject and tried out a new idea. “I’ve been thinking of giving students a choice when they ask how to do a problem, or whether it’s correct. If I answer, it will cost them points deducted from their score. I need to wean them from this dependence on my help.”
“Brilliant!” she said, took a sip of coffee and said again “That’s brilliant!”. And so I tried it. For the most part it worked. Jimmy asked if a problem were correct and I said it would cost 5 points for me to answer. “Never mind,” he said. For those students clearly lost I would not deduct points. Over time it became a judgment call—do they really need help or hand-holding?
I continued this technique and have used it at St. Stevens. It has evolved so that I will offer help as needed, but at a certain point in the school year, I will announce my policy of deducting points for certain questions.
If there are many questions in the course of a test or quiz, I find myself falling back on one of the many definitions of formative assessments, telling myself I’m using the results to guide future instruction.
My algebra class at St. Stevens was a case in point. The class was a mix of students, most of whom were able to stay afloat and do well on tests and quizzes. But there were others who perhaps should not have been placed in the algebra class who struggled and were falling behind. I would offer hints and help for those who were clearly lost. Some students would ask for help, some would not. And for those that did, they would also attempt problems on their own.
And then there was Lucy. Despite the one victory in which she was motivated enough to find a method for factoring more complex trinomials, she once again settled into her usual mode of angrily putting down answers that she thought made a kind of sense. In fact, I found that she had forgotten how to factor trinomials. She rarely asked for help during tests. I gave it to her anyway.
In keeping with summative sometimes being formative, I advised her parents that it would be best if she repeated algebra 1 in ninth grade. Lucy and her parents were receptive to this. There was one other student for whom I made the same recommendation and it was accepted, no question. Both went on to get A’s in algebra their freshmen year.
My interpretation of formative and summative assessments may not be what others think it is. Also, well-intentioned learning scientists may view me as not providing students with enough “retrieval practice”, “interleaving” and “spaced repetition”. I’ll let you look those terms up on your own. (I assure you I do all those things.) In the end, it all boils down to what used to be called “teaching”.
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