Instructional shifts, Formative Assessments, and Taking Matters into My Own Hands

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 AtlanticEducation NextEducation 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 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 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”.

Professional Development, Memorization, and Dubious Rubrics

Editor’s note: This is Chapter 15 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 AtlanticEducation NextEducation 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. If it is made into a movie I will be played by either Jeff Bridges or Harrison Ford. The part of Ellen will be played by Jamie Lee Curtis; Diane will be played by Helen Mirren.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12 and Chapter 13 and Chapter 14

Ch 15.  Professional Development, Memorization, and Dubious Rubrics

As part of the parole/credentialing process, I was required to have nine hours of Professional Development (PD) for the school year. I didn’t realize it at the time, but a conference I attended prior to my starting at Cypress came in handy when it came time for my first mentor Ellen to fill in the electronic checklist on professional development (PD).

I had attended a conference given at Oxford University, sponsored by a grass-roots organization called researchED, a teacher-led organization dedicated to disseminating information on effective teaching practices backed by scientific research. I had in fact given a presentation at this conference about the state of math education in the U.S., how it got that way, and how it looked like it was going to stay there thanks to Common Core.

I asked Ellen whether I could count my attendance at the researchED conference, given that it occurred in the summer before I started at Cypress. “Of course it does,” she said. “I wish we could count it double, since you presented there.”

She didn’t ask what my presentation was about, nor did I volunteer it. In fact no one at the school ever asked. While I’d like to paint myself as a totally altruistic hero, I have to say I really wish someone had shown even the slightest interest.

“Can you describe a session that you attended?” Ellen asked.

I told her about a session on the role of memory in learning and understanding. She looked at me over her lap top.

“Memorization?” she asked.

“Yes.”

“Memorization is not a good thing,” she said as if she were talking about parents beating their children. “Was this person advocating it?”

 “It was about how memory plays a role in learning.”

“How?”

This wasn’t looking good. “You taught biology, right? Did you need to know a lot of information?”

“Well, yes.”

“Names of organisms, what’s in a cell, and so forth, right? Somehow that gets into your long-term memory doesn’t it?”

She started typing information into her electronic form. “OK, how does this sound?” she asked.  “The session focused on long term memory and its role in understanding.”

“Sounds good,” I said. While she did not appear entirely convinced that this was true, she did look satisfied that it would pass muster by her superiors. I use the same technique. For example, if asked to describe in writing my preferred teaching style, I might say “I use direct and explicit instruction with worked examples to fulfill my intentionality of having students construct their own knowledge.”

“What other PD did you have this year?” she asked.

“This is where it gets a bit difficult,” I said. “I was required to attend a six hour session held here at the school the week before school started.”

“Why is this difficult?”

“Because I really didn’t like it.  It was called ‘How to lesson design like a rock star teacher.’ “

“It was about designing lessons?”

“More or less. I guess. I don’t know. It was six hours of being all over the map, and the guy clearly didn’t like certain things.”

I stopped there. It was hard to know what to say or not say about it. There was the “ice breaker” in which the moderator—a jovial know-it-all who name dropped several constructivist leaders he admired—had us state what our “super power” is? (Why is so much PD steeped with the vocabulary that has teachers being “rock stars” or “super heroes”?)  I noticed that James, the union rep said “sarcasm” which I found interesting. When the leader got to me, I said “Card magic”. Although the moderator has a rejoinder for each person’s response, he didn’t know what to say to mine, so he moved on.

There was the comparison we had to make between various instructional methods, using a scoring rubric based on Creativity, Communication, Collaboration and Critical Thinking—a textbook example of confirmation bias. Creativity was based on whether the method incorporated open ended questions with more than one answer. The moderator showed the first candidate on the screen:

My group agreed that there’s nothing wrong with a math workbook and we gave it high points, but we didn’t exactly follow the rubric either. We saw the need for practice, and felt that not everything has to be open ended or collaborative. Since there are no wrong answers in situations like these, the moderator upon seeing that we gave it a good score exclaimed “Good for you!” and then added “There’s nothing wrong with workbooks, they have their place, but you have to be aware of the potential for creativity.” Which was the edu-reform way of saying: “You really shouldn’t have given workbooks such a high rating.”

I told Ellen none of this given her educational inclinations.

“I can see that a six hour session on lesson design is a bit much,” she said. “But can you think of anything that you got out of it?” I could see she needed something positive in order to fill out her electronic form. 

“Well there was one thing that made sense,” I said. “He was critical of projects like building models of the California missions out of sugar cubes, or making a model of a Navajo village, because it is not teaching anything other than the construction itself.”

“Ah, good,” she said and started typing. “How does this sound? ‘Effective lessons should reflect and reinforce what students are expected to learn about a particular subject.’ ”

“Sounds good,” I said.

I wasn’t being completely honest about this part of the PD.  I neglected to tell her that after making his point about how sugar cube missions had no educational value, he told us what he thought was in fact a good activity. (Wait for it).  “Minecraft!” he said.

For those who don’t know, Minecraft is a video game version of Lego blocks in which players build structures while discovering and extracting raw materials, making tools, and fighting computer-controlled mobs.

I’m not sure what rubric he was using to give Minecraft high marks, but I suspect it had to do with the “potential” for Creativity.  Or words to that effect.

Operating Axioms, the Death March to the Quadratic Formula and an Unimpressed Student Teacher

Editor’s note: After a long hiatus because he teaches during the school year, Mr. Garelick returns once again in presenting the fourteenth chapter in a series called “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”.  Garelick is a second-career math teacher in California.  He has written articles on math education that have appeared in The AtlanticEducation NextEducation 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. Also, to my devoted readers, I decided to name the first school I taught at as Cypress School rather than “my previous school” to reduce confusion and irritation with the author.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3 Chapter 4 , Chapter 5, Chapter 6, Chapter 7Chapter 8Chapter 9Chapter 10,  Chapter 11Chapter 12 and Chapter 13.

Ch 14 Operating Axioms, the Death March to the Quadratic Formula and an Unimpressed Student Teacher

In math, assumptions held to be self-evident are accepted without proof and are called axioms. In teaching, as in many situations in life, one also makes many assumptions. I accept them without proof not because they are self-evident, but because 1) they seem like they could be true, and 2) I lack absolute proof. My axioms change from day to day depending on circumstance and observation but eventually they coalesce into a consistent set. Associated theorems then follow with the proviso that I could be dead wrong.

During my first year at Cypress I was forming many axioms particularly for my algebra class. It was a small class—only 11 students—and included two sets of twin girls. One set determined the tone of the class—they were somewhat dour and projected a “don’t mess with me” vibe. The other twins were very bright, and students clamored to be seated next to or near them whenever I changed the seating. 

The class was unusually quiet and for the first few months I was always in doubt as to where I stood with them. It wasn’t until about March when we hit the chapter on quadratic equations that I felt I was hitting my stride with them.

At the start of the chapter I announced that we would now be continuing on our death march to the quadratic formula. “We’ve learned to solve some types of quadratic equations by factoring, but now were going to look at more complicated cases when we can’t factor,” I said. “By Friday of this week we will learn the quadratic formula.”  I then wrote it on the board: 

I tend to stay away from things like posting “Today’s Objective” since most students ignore them as do I. I find it far more effective to let them know what they’ll be doing in a week’s time, or even a month’s. Plus the looks of horror and disbelief on their faces tell me I have their attention. “By Friday this will not look as ominous to you as it does now,” I said.

And sure enough, by the time Friday came, and after working with solving quadratic equations by completing the square, they were ready for the much easier way of solving equations by the formula. After they felt comfortable with the formula I told them that next week I would show them how the formula is derived.

“What does deriving the formula mean?” one of the dour twins asked.

“It means solving the equation ax2 + bx + c = 0 using the steps of completing the square.”

Silence.

“And the derivation of the quadratic formula will be an extra credit problem worth 10 points.”  

“It must be hard,” the other dour twin said.

“I don’t think it’s hard,” I said.  “If you can complete the square, you can derive the formula.”

“I love completing the square,” one of the bright twins said.

Earlier that week the third grade teacher, Sandra, had asked me if the student teacher she was mentoring in her class could observe one of my lessons.

“She’s interested in teaching middle school math and wants to see a class.”

“She doesn’t want to teach elementary school?” I asked.

“She’s exploring options.”

“Does she know what middle school is like?”

“I think that’s why she wants to observe a class,” Sandra said. Sandra had in fact taught algebra at Cypress a few years before, team teaching with James, the union representative. An opening for a third grade teacher came up and Sandra went for it, apparently preferring it to middle school.

“Fine,” I said. “I’m deriving the quadratic formula next Monday in my algebra class.  Have her come by.”

The student teacher was in her twenties and projected an aura of confidence that comes from a belief that the (forgive me) crap ideas she had been fed in ed school were actually worth following. (I’m assuming this as an axiom and feel fairly confident in doing so.)

I started my lesson that day by pointing to a poster I had made which bore a quote from Rene Descartes: “Each problem that I solved became a rule which served afterwards to solve other problems.” 

“Nowhere is this more evident in Algebra 1 than in the derivation of the quadratic formula,” I said, and proceeded to show the steps. The students knew how to complete the square, having done it as part of last week’s death march. As I worked through the derivation I asked them for the next steps. For the most part, they knew them, though it often took some prodding on my part. I note that this is how I normally teach but I was particularly aware of keeping up a dialogue lest I be judged guilty of too much “teacher talk” as traditional teaching has come to be characterized.

At the end of the class, the student teacher left without a thank you—or anything. Her head was held obnoxiously high. I assume but cannot prove that she thought that all I was doing was promoting memorization and imitation of procedures, but not “deeper understanding”.

I never heard from Sandra on what the student teacher might have thought. And in fact, I noticed that she was no longer as friendly to me as she once had been. I assume (but cannot prove) that I was somehow discredited in her eyes.

 I did ask Sandra how her student teacher was doing, hoping to get some feedback.  “Oh, she’s doing some innovative things in the class room,” she said. What these innovative things were she didn’t say, nor did I ask. I assume with some degree of confidence that it involved group work, collaboration, student-centered, inquiry-based projects and not answering students’ questions.

As it stands, four out of my eleven students got the derivation correct on the test. Two or three more got partial credit for getting halfway through. I hold out belief that at least one person was as fascinated as I was years ago in seeing how a method for solving problems could be turned into a formula.

I have no proof of this of course.