Editor’s note: This is the seventh piece 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.” The previous chapters can be found here:
Ch 7: A Gnarly Problem, Critical Thinking, and Authentic Struggle
My meetings with my parole office/mentor Diane occurred once a week in the early morning at a local coffee house a few blocks from my previous school. At one particular meeting she showed me her notes from an observation she had made of a lesson I gave my seventh grade class. Her notes were typical “hunting for problems” comments, such as my not noticing a particular boy who was unfocused, or another student who was talking, and so on.
“Any comments?” she asked.
I resisted the urge to say that it sounded like she was hunting for problems to enter on her online checklist. I talked instead about the lesson itself. It had been about taking a situation like “A bowling alley charges $5 for shoes and $3 per game bowled” and writing an algebraic expression for the cost of x games. (5 + 3x). Knowing that Diane wanted to see me extend JUMP’s scaffolded approach to more “gnarly” problems, I told her about a problem I gave the seventh graders on one of the Warm-Up questions the day after the lesson she observed.
They were to write an expression representing the cost for n hours, if a babysitter charges a flat fee of $10 and $15 per hour, but with the first hour free. I was met with the usual questions of “How do you do this?”
In my highly scientific approach I helped the first person who asked “How do you do this?” which happened to be Kyle. He was a talkative boy who was quite good at problems when he put his mind to it.
“How many hours does the babysitter charge for 6 hours of work if the first hour is free?” I asked.
“Five,” he answered.
“Right: 6 -1 = 5. So for five hours of work what does he charge?”
“Five minus one,” he said. I gave him a few more numbers and then asked “How much for n hours work?”
“Oh! n-1,” he said. He was then able to see it was 10 + 15(n-1), though he and others needed help with the parentheses.
“Yes, that’s a good problem,” Diane said. “But it wasn’t really critical thinking.”
“Why not?” I asked.
“You led them there.”
I said nothing, hoping for an awkward silence and got my wish.
“There’s nothing wrong with what you did,” she said. “But true critical thinking would involve them struggling to come up with a solution.”
I recognized this immediately as the “struggle is good” philosophy which holds that if students aren’t struggling they aren’t learning. There are nuances to this philosophy including “productive struggle”, “desirable difficulties” and “students should be able to use prior knowledge in new situations without scaffolding because otherwise it is inauthentic.” I’ve read variations of this thinking in books that I’ve thrown across the room.
“Let me give you a problem that I want you to solve,” I said. “Two cars head towards each other on the same highway. One car starts from the north heading south, at 80 mph. The other car starts from the south heading north at 70 mph. They meet somewhere on the highway. How far apart are they one hour before they meet?”
She took a gulp of coffee and tried to smile.
“You do not need to know the distance they are apart to solve it”, I said.
She looked perplexed and gave me a look I see on my students’ faces when they ask “How do you do this?”
“Tell me this,” I said. “How far does a car going 80 mph travel in one hour?”
“Eighty miles,” she said.
I drew a line on a napkin and marked a point near the middle with an X. “Where was the 80 mph car 1 hour before he got here?”
“Well, that would be 80 miles north of that point.”
” What about the 70 mph car?”
“Uh, 70 miles south of the point?”
“Good. Can you put that together somehow?”
She suddenly saw it. “Oh, I see! They’re 150 miles apart one hour before they meet.”
“Good work. Now let me ask you something. I gave you some hints. Would you say that you used those hints in thinking about the problem and coming up with a solution?”
She smiled knowingly. “Ah, I see. Critical thinking.”
“So would you say that what you did qualifies as critical thinking?” She agreed.
“Then why would you say that what I did with the baby sitting question did not qualify as critical thinking.”
“I’ll have to think about what I mean by critical thinking,” she said. “I think applying an algorithm repeatedly does not entail critical thinking.”
“Even if it leads to a conclusion? And in essence that was what I had you do when you think about it. And you put it together like my students did. Why would you not call that critical thinking? In your mind is there no difference between thinking and critical thinking?”
“I guess I might have to look up the definition of critical thinking.”
“I’ll send you a definition tonight by email,” I said. “My concern is this. If your goal is to look for examples of critical thinking in my classes using the definition you’ve presented, you will probably never see critical thinking in my classes. I use worked examples and scaffolding and problems that ramp up. That’s how I teach. You’ll see this more in my algebra class, and I hope you observe one of those.”
She said she definitely would. I thanked her for having the discussion with me. “I felt it was important that we understand the language we’re speaking and what I’m about.”
“Yes,” she said. The conversation then shifted to lighter topics. I felt a bit bad for putting her on the spot with my math problem. But then again, her struggle with critical thinking was productive if not authentic.
Fewer and fewer colleges require SAT scores for admission and more and more parents and others are calling for the reduction or elimination of “standardized” tests. Interestingly, there is little call for “no mandated K-12 tests” at all. One might expect that call given the complaints against Common Core-aligned tests and the number of misleading references to what Finland has done.
According to many education writers in this country, there are no tests in Finnish schools, at least no “mandated standardized tests.” That phrase was carefully hammered out by Smithsonian Magazine to exclude the many no- or low-stakes “norm-referenced” tests (like the Iowa Test of Basic Skills, or ITBS) that have been given for decades across this country especially in the elementary grades to help school administrators to understand where their students’ achievement fell under a “normal curve” of distributing test scores.
Yet, a prominent Finnish educator tells us that Finnish teachers regularly test their upper-grade students. As Finnish educator, Pasi Sahlberg, noted (p. 25), teachers assess student achievement in the upper secondary school at the end of each six to seven-week period, or five or six times per subject per school year. There are lots of tests in Finnish schools, it seems, but mainly teacher-made tests (not state-wide tests) of what they have taught. There are also “matriculation” tests at the end of high school (as the Smithsonian article admits)—for students who want to go to a Finnish university. They are in fact voluntary; only students who want to go on to university take them. Indeed, there are lots of tests for Finnish students, just not where American students are heavily tested (in the elementary and middle grades) and not constructed by a testing company.
Why should Americans now be even more interested in the topic of testing than ever before? Mainly because there seems to be a groundswell developing for “performance” tests in place of “standardized” tests. And they are called “assessments” perhaps to make parents and teachers think they are not those dreaded tests mandated by state boards of education for grades 3-8 and beyond as part of the Every Student Succeeds Act (ESSA). Who wouldn’t want a test that “accurately measures one or more specific course standards”? And is also “complex, authentic, process and/or product-oriented, and open-ended.” Edutopia’s writer, Patricia Hilliard, doesn’t tell us in her 2015 blog “Performance-Based Assessment: Reviewing the Basics” whether it also brushes our hair and shines our shoes at the same time.
It’s as if our problem was simply the type of test that states have been giving, not what is tested nor the cost or amount of time teachers and students spend on them. It doesn’t take much browsing on-line to discover that two states have already found out there were deep problems with those tests, too: Vermont and Kentucky.
An old government publication (1993) warned readers about some of the problems with portfolios: ”Users need to pay close attention to technical and equity issues to ensure that the assessments are fair to all students.” It turns out that portfolios are not good for high stakes assessment—for a range of important reasons. In a nutshell, they are costly, time-consuming, and unreliable. Quoting one of the researchers/evaluators in the Vermont initiative, it indicates: “The Vermont experience demonstrates the need to set realistic expectations for the short-term success of performance-assessment programs and to acknowledge the large costs of these programs.” Koretz et al state elsewhere in their own blog that the researchers “found the reliability of the scoring by teachers to be very low in both subjects… Disagreement among scorers alone accounts for much of the variance in scores and therefore invalidates any comparisons of scores.”
Koretz and his colleagues emphasized the lack of quality data in another government publication. And as noted in a 2018 blog by Daisy Christodoulou, a former English teacher in several London high schools, validity and reliability are the two central qualities needed in a test.
We learned even more from a book chapter by education professor George K. Cunningham on the “failed accountability system” in Kentucky. One of Cunningham’s most astute observations is the following:
Historically, the purpose of instruction in this country has been increasing student academic achievement. This is not the purpose of progressive education, which prefers to be judged by standards other than student academic performance. The Kentucky reform presents a paradox, a system structured to require increasing levels of academic performance while supporting a set of instructional methods that are hostile to the idea of increased academic performance (pp. 264-65).
That is still the dilemma today—skills-oriented standards assessed by “standardized” tests that require, for the sake of a reliable assessment, some multiple-choice questions.
Cunningham also warned, in the conclusion to his long chapter on Kentucky, about using performance assessments for large-scale assessment (p. 288). “The Performance Events were expensive and presented many logistical headaches.” In addition, he noted:
The biggest problem with using performance assessments in a standards-based accountability system, other than poor reliability, is the impossibility of equating forms longitudinally from year to year or horizontally with other forms of assessment. In Kentucky, because of the amount of time required, each student participated in only one performance assessment task. As a result, items could never be reused from year to year because of the likelihood that students would remember the tasks and their responses. This made equating almost impossible.
Further details on the problems of equating Performance Events may be found in a technical review in January 1998 by James Catterall and four others for the Commonwealth of Kentucky Legislative Research Commission. Also informative is a 1995 analysis of Kentucky’s tests by Ronald Hambleton et al. It is a scanned document and can be made searchable with Adobe Acrobat Professional.
A slightly optimistic account of what could be learned from the attempt to use writing and mathematics portfolios for assessment can be found in a recent blog by education analyst Richard Innes at Kentucky’s Bluegrass Institute.
For more articles on the costs and benefits of student testing, see the following:
- Phelps, R. P. (2002, February). Estimating the costs and benefits of educational testing programs. Briefings on Educational Research, Education Consumers Clearinghouse, 2(2).
- Phelps, R. P. (2000, Winter). Estimating the cost of systemwide student testing in the United States. Journal of Education Finance, 25(3) 343–380.
- Phelps, R. P., et al. (1993). Student testing: Current extent and expenditures, with cost estimates for a national examination. GAO/PEMD-93-8, U.S. General Accounting Office, U.S. Congress.
Changing to highly subjective “performance-based assessments” removes any urgent need for content-based questions. That was why the agreed-upon planning documents for teacher licensure tests in Massachusetts (which were required by the Massachusetts Education Reform Act of 1993) specified more multiple-choice questions on content than essay questions in their format (they all included both) and, for their construction, revision, and approval, required content experts as well as practicing teachers with that license, together with education school faculty who taught methods courses (pedagogy) for that license. With the help of the president of the National Evaluation Systems (NES, the state’s licensure test developer) and others in the company, the state was able to get more content experts involved in the test approval process. What Pearson, a co-owner of these tests, has done since its purchase of NES is unknown.
For example, it is known that for the Foundations of Reading (90), a licensure test for most prospective teacher of young children (in programs for elementary, early childhood, and special education teachers), Common Core’s beginning reading standards were added to the test description, as were examples for assessing the state’s added standards to the original NES Practice Test. It is not known if changes were made to the licensure test itself (used by about 6 other states) or to other Common Core-aligned licensure tests or test preparation materials, e.g., for mathematics. Even if Common Core’s standards are eliminated (as in Florida in 2019 by a governor’s Executive Order), their influence remains in some of the pre-Common Core licensure tests developed in the Bay State—tests that contributed to academically stronger teachers for the state.
It is time for the Bay State’s own legislature to do some prolonged investigations of the costs and benefits of “performance-based assessments” before agreeing to their possibility in Massachusetts and to arguments that may be made by FairTest or others who are eager to eliminate “standardized” testing.
Editor’s note: This is the sixth piece 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.” The previous chapters can be found here:
Ch 6 The Prospect of a Horrible PD, a Horrible Meeting, and an Unlikely Collaboration
Many schools require teachers to attend some kind of professional development and St. Stevens was no exception. Fortunately it was rather benign even though it was the whole day and was about technology in the classroom. But other than that it was fine.
In my previous school, the principal (and Superintendent) “asked” me and James, the other math teacher to attend six all day professional development (PD) sessions over the course of the school year. The PD, held by the County Office of Education, was to be a forum for “collaboration” among math teachers in the county.
While I don’t mind collaborating with teachers, I don’t like the collaboration to be prescribed. And certainly not for six times. I was also leery of James who hardly gave me the time of day and was passive aggressively hostile. As I told Diane during one of our more productive mentor sessions, “I dislike the idea of going to the sessions with him more than I dislike the idea of the PD itself.” She advised me that most complaints from teachers were about problems with other teachers. “And passive aggressive types are the worst,” she added. It was a valuable piece of advice.
As it turned out, the PD was cancelled. James and I were the only two people in the county who had signed up. Our delight was rather short-lived, however. The moderator met with our principal and suggested having a series of two hour meetings with us at school during the early part of the day when we weren’t teaching. Neither James nor I were too thrilled at the idea of collaborating with each other.
We all met one time. The moderator, a middle aged woman who talked in the cheery tones of a facilitator began describing how she loved math while in school but was just “following the rules and getting an answer”. Later when she taught math, she found she couldn’t explain to students the underlying concepts. Which led her to say, that the Common Core standards were all about “understanding”, and teachers had better teach for understanding because as she explained, “California’s Common Core-aligned tests are not about ‘answer getting’ anymore!” Students had to explain their answers and the tests evaluate whether students are able to solve problems in more than one way.
She went on almost breathlessly: “Students can get full credit on problems where they have to provide explanations—even if they get the numerical answer wrong.”
James and I said nothing.
“Provided the reasoning and process are correct, of course,” she added. “Explaining answers is tough for students and for this reason there is a need for discourse in the classroom and ‘rich tasks’ ”.
My years in education school had taught me the skill of keeping my mouth shut appropriately but at this point I couldn’t contain myself and asked “Could you define what a ‘rich task’ is?”
Her answer was extraordinary in its eloquence at saying absolutely nothing: “It’s a problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”
I had had some experience with rich problems so I knew exactly the type of problem to which she was referring; problems like “A rectangle with a perimeter of 20 inches has what dimensions?” or something similar.
At this point James could take it no longer. He said that meeting for two hours for five sessions was superfluous if it was just the two of us. “I teach three different math classes plus doing the I.T. for the school and don’t have time to delve into alternative approaches other than to follow the script and curriculum as laid out in the book.”
The two of us must have seemed like a rich problem. “Books are just tools,” she proclaimed. “They may be strong in one area but weak in another. Traditional textbooks tend to be lacking in opportunities for conceptual understanding and are old school in their approach.”
She sensed that both of us were more than willing to let her dig her own grave here. “Though there’s nothing wrong with old school,” she quickly added.
As tempting as it was. I saw no need to tell her that I used a 1962 textbook by Dolciani for my algebra class.
She asked if we relied on our textbook for a “script”, meaning scope and sequence “Do you read just one textbook?” she asked me.
“I read lots of textbooks,” I said. She looked surprised.
“He’s also written books,” James said. I was surprised that he knew about them, but I had slipped “Math Education in the US” surreptitiously in the bookcase in the teacher’s lounge so maybe he had read it.
“How nice!” she said and feigned an interest by asking what they were about. I gave a “rich” answer. “Math education,” I said.
“Wonderful!” she said.
I then tried to summarize our thoughts. “Neither of us teaches in a vacuum,” I said. “I read lots of textbooks and talk to lots of teachers.” Since James had played up my authorship, I decided to return the favor.
“And James has a lot more experience than I do so he isn’t exactly ignorant about how to teach math. I really don’t think that this two-hour collaboration is going to add much more.”
I realized this opened me up to her protesting that perhaps I could benefit from his experience so I needed to head that off. “Besides, I’m getting mixed messages,” I said. “On the one hand I’m told by the administration that I’m doing great, and I hear from parents that I’m doing great. But then I’m told that I must attend this PD. Is there something about my teaching that’s lacking? What is this about?”
She assured us that there’s nothing lacking in our teaching and that she’s sure we are both fantastic teachers. “What is it then? Is this about test scores? They think this will raise test scores?”
She had no answer for this except something that I can’t remember. She saw the handwriting on the wall and said “No use beating a dead horse” and said she would talk to the administration about it. And that was the end of our PD.
I decided the next time I met with Diane, I would tell her about the success of James’ and my “collaboration.”
Editor’s note: This is the fifth piece 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, Nonpartisan Education Review, 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.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3, Chapter 4.
Ch 5. The Rituals of School, an Unusual Communion, and the Vast Wasteland of Math 8
Each day at St. Stevens starts out with the entire school of 200 students, plus teachers, gathered around the flagpole to say one or two prayers, and the Pledge of Allegiance. I enjoy the ritual, particularly seeing everyone—first graders through eighth—cross themselves in unison. Before I started at St. Stevens, a friend asked me if knew how to cross myself. “Yes,” I said. “But at this point it’s procedural; I think the understanding will come later.”
My days are built on a set of procedures resulting in an ever-changing understanding of where I am. After “flag”, came a quick walk to my classroom, walking fast to stay ahead of the rapidly dispersing horde of students. I call my classroom The Batcave, partly because my classroom is out of the way and cavelike. I have come to love the room and wouldn’t change it for the world.
It appears to have been a storage closet in a former life and is right next door to the gym. It has a door to the gym which sometimes gets bumped by stray basketballs and other objects. Then there is the music that is played during exercises or dance which usually elicits conversation among my students about the songs being played. I squeezed eight desks in the room to accommodate the students in Math 7 and 8.
The Math 8 class was segmented from the rest of the eighth graders who were in the eighth grade algebra class. During the first week, a rather stubborn and outspoken student, Lou, stated what the rest of the class was feeling. “It’s obvious we’re not too good at math which is why they put us in this class.”
I had heard something similar at my previous school from my seventh graders. In neither case did I respond by talking about “growth mindset”. It was the first time I had taught Math 8 and I was rapidly discovering that the course was a vast wasteland of disparate topics that did little or nothing to prepare them for algebra in the ninth grade.
After my Math 7 and 8 classes, came my algebra class—held in Katherine’s classroom since there were sixteen students. Like most of the algebra classes I’ve taught, this one was full of energetic and motivated students. They were also quite noisy and extremely competitive. Unlike the Math 8 class, they had both confidence and curiosity. The difference between the two classes was never more obvious than when I showed a magic trick to both classes.
It was on a day in which school was dismissed at noon (another cherished tradition in which one’s best plans and schedules written over the summer start to resemble a game of Battleship—a day you thought you’d have for a complicated lesson turns out to be a short one). I had performed this trick many times over the years, starting when I was a sub.
Given the following five cards, someone picks a number from 1 through 31, writes the number on the board and erases it after everyone has seen it so everyone but me knows the number.
I then ask what cards the person sees their number on. I immediately tell them the number. The trick is based on the binary number system. One student in the algebra class who is interested in computers knew how it was done so he kept quiet.
I embellished the presentation for the algebra class by having them help me construct a table of binary numbers from 1 through 31, and then transfer the information onto the five mini whiteboards to make the five magic cards.
“You can fill this table out by looking at the patterns,” I said, realizing that any onlooker who happened to poke their head in to my class would think “Oh, good, Mr. Garelick is teaching them that math is about patterns!”—a characterization that I dislike for reasons I won’t get into here.
I started filling out the first four rows; once I got to the fifth row, they started to see the pattern.
“Oh, it just keeps repeating itself: 01, 10, 11, and 100,” a boy said.
“What do the numbers mean?” someone else asked.
“They correspond to the numbers on the left.”
I tried to explain, showing that you’re adding powers of 2 just as in base 10 the number 11 is (1 x 10) + (1 x 1). Some students understood, but most didn’t.
I then said “Just keep filling out the table. It doesn’t matter right now whether you understand what the binary numbers mean.” If the same onlooker who liked my comment about patterns was looking in again, the reaction would probably be “Oh wait; he’s having them ‘do’ math without ‘knowing’ math”. Or some equivalent bromide.
Once all 31 rows were filled, I had five students transfer the numbers to five mini whiteboards. I then proceeded to do my magic trick. The first time I got the number the entire class shouted.
“Do it again!” I did, and got it right again. Each time I revealed their number they were now screaming. “He’s a wizard!” a boy named Sam shouted out.
I finally revealed the trick: “I look at the first number on each card you told me contained your number and then added them up.” I showed them that for the number seven, the first numbers on those cards are 1, 2 and 4, which sum to seven.
There was a collective “Oh, that’s how!” Their excitement was in stark contrast to the Math 8 class who, although curious and amused, took it in stride as just one more thing that didn’t concern them. I had sets of magic cards that I printed up and asked if anyone wanted them. No one in my Math 8 class had wanted them, but the algebra class immediately surrounded me, some with hands cupped as if receiving communion.
During the prayer before dismissing for lunch (“Bless us, Oh Lord, and these thy gifts…”) I realized I had to do something to fill in the vast wasteland of the Math 8 course. I wondered if perhaps I could sneak in more algebra. And for an extra challenge, doing so without the ritual of “growth mindset”.
I was recently reading a section of the book “The Age of Surveillance Capitalism: The Fight for a Human Future at the New Frontier of Power” by Shoshana Zuboff. This is an eye opener and an excellent book about surveillance capitalism.
Many of the things being addressed in this book are things I see related to or can relate to what I observe happening in our country today, especially in education. Chapter Six, Hijacked: The Division of Learning in Society asks and addresses three questions: Who Knows? Who Decides? Who Decides Who Decides? While the chapter addresses these questions at more length, below are two quotes from the book. The first briefly explains the questions and the second provides an extremely brief answer to each question.
“The first question is “Who knows?” This is a question about the distribution of knowledge and whether one is included or excluded from the opportunity to learn. The second question is “Who decides?” This is a question about authority: which people, institutions, or processes determine who is included in learning, what they are able to learn, and how they are able to act on their knowledge. What is the legitimate basis of that authority? The third question is “Who decides who decides?” This is a question about power. What is the source of power that undergirds the authority to share or withhold knowledge?”
“As things currently stand, it is the surveillance capitalist corporations that know. It is the market form that decides. It is the competitive struggle among surveillance capitalists that decides who decides.”
To me, this is scary to think that we are not only heading in this direction but that we are well on our way. I would say we are already there except things related to technology are always evolving.
Think about this in terms of our education system during three periods of time: 1) prior to the recent education reform era, 2) during the recent education reform era, and 3) the surveillance capitalism present and future. Shifts have taken place in the transition from one time period to the next. The answers to the questions Who Knows? Who Decides? Who Decides Who Decides? with regard to our education system in this country have shifted from parents/local community to state/federal government to surveillance capitalist corporations. These shifts, like their corresponding time periods, as simply stated here does not capture or adequately convey the complexity. The shift to surveillance capitalist corporations driving our education system is well underway, or has already taken place and continues to evolve.
Below is a table that in a simple but incomplete way shows Who Knows? Who Decides? Who Decides Who Decides? for each of the three time periods.
Question Prior to Ed Reform Era Recent Education Reform Era Surveillance Capitalism Future Who Knows? Educators and subject matter experts
state/federal government surveillance
Who Decides? Educators and subject matter experts
market Who Decides Who Decides? parents/local community state/federal government competitive struggle among surveillance capitalists
Different people may place different entities in the various boxes in the table. There are definitely more players involved than just those mentioned. It is possible that others dubbed as “experts” may be included with state/federal government. Such “experts” may be more driven by an agenda or ideology than by any real expertise based on evidence, factual data, or true knowledge.
Foundations and the influential wealthy seem to go hand in hand with the surveillance capitalist corporations in deciding who decides.
These shifts have taken place gradually over time. Has it happened so gradually that most parents and local communities have yet to realize their rights/responsibilities have been usurped? Are parents and local communities okay with this? Have they willingly turned those rights/responsibilities over to the surveillance capitalist corporations? If not, what can be done to restore those rights/responsibilities back to parents and local communities?
What is Surveillance Capitalism?
Editor’s note: This is the fourth piece 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, Nonpartisan Education Review, 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.” The previous chapters can be found here: Chapter 1 , Chapter 2 , Chapter 3.
4. The Parole Officer’s Check List, the Dielectic of Competition, and Gnarly Problems
Upon starting at St. Stevens, I had completed my two years compulsory Teacher Induction Program (TIP) under two different parole officers, otherwise known as mentors. The TIP process consisted of discussions, observations, and the mentors/parole officers having to fill out an online checklist of items which would serve as the final authentication of having gone through the induction.
My first mentor (who I will call Ellen) told me that she would be making suggestions and giving me ideas, but I was under no obligation to follow any of them. This was good because she had no shortage of dubious ideas and quickly learned that I was going to do things my way. At one point early in our meetings, having determined that I didn’t assign group work and rarely had activities, Ellen asked “What are you going to do about Common Core, which requires activities and group work in teaching math?”
“The Common Core standards do not prescribe such pedagogy,” I said and pointed out that the website for Common Core states clearly that the standards do not mandate pedagogical approaches. I expected an argument, but instead she quickly moved on to other business. The mentors are used to teachers fresh out of ed school who are in their twenties and believe whatever they’ve been told about how to teach and what Common Core requires. In any event that was one response she did not expect that remained unrecorded on the online checklist.
My second mentor, who I will call Diane, was assigned during the second year of my induction. Like Ellen, Diane also had teaching experience—second grade mostly—and was now in charge of the mentor/parole program for the county in which I teach. Like Ellen, she would make suggestions that I could either follow or ignore. She would occasionally evince educational group-think that passed as sound advice.
Our first meeting took place in my classroom. “Tell me about your classes,” she said.
I had two that year: a seventh grade math class (non-accelerated), and eighth grade algebra—the latter made up of students I had taught the previous year.
“My seventh grade math class had a rough year last year so they’re coming in with an ‘I can’t do math’ attitude right at the start,” I said.
“That’s never a good thing,” she said.
“And on top of that, they have significant deficits. Like not knowing their multiplication facts.”
Her eyes widened. “Really? How can that be?”
Actually, it can be and is in many schools across the US. I wondered how on earth she could not know this, being in education as long as she had.
“So how are you addressing that?” she asked.
“I’ll let you in on a secret,” I said. She looked intrigued.
“I’ve been giving them timed multiplication quizzes every day to start off the class. My principal told me that timed quizzes stress students, but these kids love the competition, plus I show them how their scores are increasing.”
“Of course!” she said. “Kids love to compete.” I was heartened at this for two reasons: she wasn’t against memorizing multiplication facts, and she appeared to be going against the educationist dialectic of “competition is bad”. But then she added, “Of course, it isn’t good to do that in the first and second grades because it can stress kids out, but it’s perfect for seventh graders.” A few minutes later when I told her I posted the top three scores on quizzes or tests, the dialectic clicked in.
“Are you sure that’s a good thing to do? Some of the students who didn’t make the top scores might feel left out,” she said.
“They ask me who got the top scores, and they don’t seem upset when I post them, so I’m assuming it’s OK,” I said. She had no answer to that and looked around the room. “I like the way you’ve set up your classroom.”
Diane liked various quotes I had tacked up on my walls; random things uttered by my students that I felt worthy of posting. Like “I never get used to math; it’s always changing.” and “Variables don’t make sense and make sense at the same time”.
I kept the ones from my previous year’s classes on the wall, as well as those from my current students. “Seeing quotes from previous classes gives students a sense of legacy and tradition,” I said. I remember being intrigued when my seventh grade English teacher would show us examples of work done by her previous classes, and we would see the names of students from years past—some were brothers and sisters of my classmates.
I pointed out one quote from a student named Jimmy in my current seventh grade class. It had emerged from a dialogue I had with him on subtracting negative numbers:
Me: You lose 5 yards on a play. You have to make a first down. How many yards do you have to run?
Jimmy: Couldn’t you just punt it?
Jimmy had had a particularly rough time in math the previous year and had very little confidence. The first time we had a quiz I had made sure that the students would do well, giving them lots of preparation. Jimmy did do well: 97%. When I was handing back the quizzes he kept saying “I know I failed it.” When he saw his quiz he was silent and then asked if anyone in my class last year had failed.
“No,” I said.
“Do you think it’s possible that I’ll pass this class?” he asked.
“Yes, it’s entirely possible,” I said.
I told Diane about this. “Fantastic that he got 97%. How did that happen?”
I explained how I was using JUMP this year and how it breaks things down into manageable chunks of information that students could master. Given where they were coming from I felt that building up their confidence was very much needed.
“Are you planning to go beyond just mastery and give them some gnarly problems?”
I was tempted to ask her to define “gnarly problems”, though I’m fairly sure it had something to do with “If they can’t apply prior knowledge to new problems they haven’t seen before, there is no understanding”. But my answer to her was “Of course”. JUMP does in fact provide “extension” problems in the teacher’s manual. I made a note to self which when roughly translated was something like: “Come up with something.” Given the deck I had been handed with this class, I had other things on my mind besides giving the class gnarly problems, however it was being defined.
Editor’s note: This is the third piece 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.” Chapters 1 and 2 can be found here and here.
3. Understanding, and Outliers in a Sea of Outliers
During my second week of school at St. Stevens the principal, Marianne, called me in to her office to tell me some good news. “I just want to let you know that we heard from Mary’s mother and that Mary said she is really happy in your class; she says that “Mr. Garelick really wants us to understand.”
I was glad to hear that Mary’s mother was pleased, and while I haven’t taught for very long, I knew enough not to believe I was any kind of miracle worker—particularly during the first few weeks at school when everything is new and has the cast of a halo over it. At my previous school during back-to-school night one year before, a parent of one of my students in my seventh grade class said something similar. “My son said that this is the first time in any math class that he actually understood the math.”
In both cases, it didn’t hurt that the word “understand” was used in conjunction with my teaching, although the word has different meaning for me than what others in education think it means. I want students to be able to do the math. That’s pretty much what students mean when they say they understand. It isn’t something I obsess over.
Mary was one of two girls in my eighth grade math class (Math 8), who had to come in twice a week for intervention help for half an hour before classes began. The other student was Valerie who had been classified as special needs since the lower grades. They were both very animated girls; Mary was very outgoing and friendly with me. Valerie was more guarded. In her world of Smart Phone, songs, reality TV, I felt she viewed me as frightfully out of touch with what was really important. Math was certainly not on her list.
My Math 8 class was similar in some ways to my last year’s seventh grade class. My prior school, like St. Stevens, had two seventh grade math classes—one accelerated, the other not. I taught the non-accelerated group who considered themselves “the dumb class”. Their doubts were compounded by their last year’s math teacher who was not popular with parents or students, and was finally let go by the school. My Math 8 class similarly knew they didn’t make the cut for the eighth grade algebra class (which I was teaching). Their previous math teacher was similarly unpopular—and also let go by the school.
In a school as small as St. Stevens, there weren’t enough students to form a remedial class by itself. As a result, in the midst of a class in which the students already doubted their abilities, Mary and Valerie felt they were outliers. I worked with them as best as I could. I called on them infrequently in the main class, and focused on them during my intervention time.
At first, I tried to get them up to speed with what the rest of the class was doing. During one of my sessions with them, I went over one-step equations. I asked them to solve the equation 6x = 12. I had reached the point where neither one was trying to subtract the 6 from 6x to isolate x. But while Marie understood that 6x meant 6 multiplied by x , Valerie could not see that; nor could she see that solving it meant undoing the multiplication by division.
“How do we solve it?” I asked.
“You put the 6 underneath both sides,” Valerie said.
Putting one number underneath another meant divide to Valerie which is as procedural-minded as you can get. If ever pressed to justify my acceptance of her level of understanding to well-meaning doing-math-is-not-knowing-math types I could say that her method at least incorporated the concept that a fraction means division. No one ever asked, but just to make sure, I said “And what do we call the operation when we put another number ‘underneath’ another?”
She thought a moment.
Mary whispered in Valerie’s ear: “Division”.
“Oh; it’s division,” Valerie said.
Over the next few weeks, I worked with the two girls privately while trying to keep them on track in the Math 8 class. I realized that their deficits were so significant that to hold them to the standards of Math 8 would result in failure. Katherine, the assistant principal agreed with me, and said to focus instead on filling in the gaps and to base their grade on their mastery of those.
I leveled with them one morning when they came in for their intervention.
“The Math 8 class must be extremely painful for you,” I said.
Valerie for the first time let down her guard. “I just don’t understand what’s going on.”
I found her statement true in a number of ways. One was just the fact that she admitted it. But more, it brought home the issue of understanding. Of course she didn’t understand—she barely had the procedural and factual tools that would allow her even the lowest level of understanding of what we were doing.
While there are those who would say “Of course they didn’t understand; traditional math has beat it out of them,” such thinking is so misguided that, in the words of someone whose name I can’t remember: “it isn’t even wrong”. In the case of Valerie and Mary, they needed more than the two half-hour interventions every week. They needed someone who specialized in working with what is known as dyscalculia. But qualifying as a special needs student doesn’t guarantee the student will get the kind of help to deal with a disability.
They had finished what I gave them to do early that day, and Mary being in a celebratory mood said the two were going to make a drawing for me. They giggled while they drew a rather strange looking bird laying eggs and eating blueberries among other odd things going on and presented it to me. I put it on the wall where it remained for the entire school year. Sometimes other students would ask what the drawing was, and they would explain it, excitedly. The excitement was partly due to them having drawn it, and partly due to my keeping it on the wall for all to see.
Issues related to and surrounding the Common Core State Standards (CCSS) are controversial and “toxic” (as Mike Huckabee put it) for many people both in and outside of education, including decision-makers. Rather than truly replacing the CCSS, some states have simply rebranded them. As a result, “College and Career Readiness Standards” and setting “higher” national standards are viewed as euphemisms for the CCSS. Rebranding has taken many forms, from simply changing the name to having committees review the standards, make minor, unsubstantial changes, add some front material, and possibly reformat their presentation.
For those familiar with pre-CCSS state math standards and who can compare them with the Common Core State Standards for Mathematics (CCSS-M), it can be seen the CCSS-M are uniquely written. Once familiar with this uniqueness, a person can usually determine if CCSS-M standards have been used as a base or model for a standards revision or rewrite.
Two states, Alabama and Florida, have been making noise about getting rid of the Common Core State Standards. Some headline terms used include repeal, end, ditch, eliminate, and scrap. As time goes on, more states will consider changing their standards. It will be interesting to see how they go about it and what the resulting product (set of standards) looks like.
Here are some possible scenarios of what states might do as they consider changing their CCSS-M standards. These are listed from worst to best case
- Adopt the Common Core State Standards as they are
- Rebrand the CCSS-M in name only
- Rebrand CCSS-M in name with minor changes*
- Rewrite standards using CCSS-M as the model**
- Rewrite standards using another state’s weak pre-CCSS standards as a model
- Rewrite standards using an A rated set of pre-CCSS standards as a model
- Adopt an A rated set of pre-CCSS standards (IN, CA, or even the unrated WEMS)
*changes some states made, even minor ones, significantly weakened their standards
**this results in standards that are basically CCSS with phrases that have been rewritten
I would recommend states work to avoid paths 1 though 5 and if possible and only accept paths 6 or 7.
Some states have expended a lot of resources on rebrands or rewrites that have resulted in adopting a set of standards that in essence are the CCSS (or worse). It doesn’t appear that any state completing a rebrand or rewrite has done anything that actually improved the CCSS.
One strategy that has been used in a few states is to have a survey set up for the public to provide specific input on the current standards, often standard by standard. This strategy will mostly result in a set a standards that closely resembles or is the same as the current standards. And if the current standards are the Common Core or a rebrand a brand makeover results. This strategy fits with path 4 where the standards are rewritten using the CCSS as a model.
Do states that make noise about the CCSS want to repeal, revise, replace, rebrand, or update their standards? Do they really want to have a better set of standards? Or do they just want to make noise having people think they are doing something that will result in a better set of standards when the real result will be little to no change or something worse?
Editor’s note: This is the second piece 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.” The first piece can be found here.
An espresso-based job interview, a 1962 algebra book, and procedures vs understanding
In the remaining two weeks at my previous school, I applied for the few math teaching positions that were advertised. I had the typical non-responses except for one—a high school that specialized in problem-based learning. I had applied there out of desperation never expecting a response. I received an email saying they were interested in interviewing me. Despite my skills at making my teaching appear to be what people wanted to see, I knew that this one required too much suspension of disbelief on both sides of the aisle.
I cancelled the interview saying something along the lines of having to wash my hair that day. Shortly after that, I received another email from St. Stevens, a K-8 Catholic school needing a part-time math teacher. A fellow math teacher from another school had put in a good word for me. Like the school where I had been, this school was also part of a small community and had about two hundred students.
A few days later I was at the school for a 2 PM interview. I tend to get a bit logy in the afternoon so I thought I’d have an espresso prior to coming in. The principal, Marianne and assistant principal, Katherine, interviewed me and asked the usual questions: What does a typical lesson look like, what are my expectations and so on. My inner voice tried to keep me from extended caffeinated responses. I emphasized how I leave time for students to start on homework in class, do the “I do, we do, you do” technique, and in my controlled ramblings managed to get across that I am, by and large, traditional.
But when the assistant principal asked me what my approach was in teaching algebra the espresso kicked in big time and my inner voice was having a hard time keeping up. I said that I taught using a 1962 algebra textbook by Dolciani.
Inner voice: “You shouldn’t have said that.”
“I bought about fifteen of them over the internet when they were selling for one cent a piece about four years ago. So I was basically paying for shipping. But now the prices increased because of Amazon’s supply and demand algorithm, so they’re selling for about $60 a copy last time I looked. Which tells me a lot of people are buying them.”
“Please shut up.”
I talked about how I liked the sequence, structure and explanations of Dolciani’s book much better than the official textbook. As it turned out, so did the students despite the increased amount of word problems—and the word problems were yet another plus for using the book.
“But did you cover what was in the Common Core standards?” Marianne asked. I assured them that I did, supplementing with topics that weren’t in Dolciani, like exponential growth and decay.
I hastened to add that I did not spend inordinate amounts of time on exponential growth and decay functions.
“Did you cover exponents at all?” Katherine asked. “Because students generally are weak on those.”
“You mean like products and quotients of powers? Oh yeah, big time. The Dolciani book is very big on those.” I was about to repeat that I didn’t spend time on exponential growth and decay, but the caffeine was mercifully wearing off.
They did not seem perturbed by any of my ramblings. Then again, it was during the last week of school and I imagine that they were so exhausted that they were probably amenable to anything I said.
They asked about my classroom management techniques. In any interview or evaluation process, one has to have some weakness to talk about and I freely admitted that classroom management is not my strong suit. I mentioned that my seventh grade math class had behavior problems even though there were a total of 10 students in the class.
“How did you handle the problems?” Marianne asked.
“I had a warning system; two warnings and they got a detention. I wasn’t too faithful in carrying that out though.”
“Why was that?”
“When I gave a detention, the two main troublemakers were really good at carrying on about it and crying.”
“I hated giving detentions. I always got talk-back, like ‘But I wasn’t talking’. And then the crying. Which they did with all the teachers, I found out.”
“So how did you deal with them most of the time, then? What did you do?”
I explained how this group had large deficits in math skills and most of the boys in the group had given up at believing they could learn math.
“I used an alternative textbook, which my school let me use: JUMP Math. It was developed in Canada and broke concepts down into very small incremental steps. It scaffolds problems down to incremental procedures and builds on those.”
I went on about how procedures can lead to understanding and you can teach understanding until the cows come home, but most students are going to grab onto the procedures.
“Did it work?” Katherine asked.
“Well, let’s just say that it would have been even worse had I not tried to fill in their deficits.”
They had no response and then the usual niceties ensued and the interview was over.
They called my references, as well as the principal of my school as I found out the next day. “Marianne called me about you. She sounded excited,” the principal told me.
“What did she ask?”
“She wanted to know more about the seventh grade class; she was curious about their behavior.”
“Oh,” I said. “I thought she might.”
“I told her they were really a tough bunch of students but you handled them well.”
“Did she ask about the algebra books I used for my algebra class? Or about procedures versus understanding?”
“No; just about the seventh grade class,” she said. “She sounded positive.”
And a few days later I was offered the job at St. Stevens. “I think you’ll like it there,” the principal told me.
I hoped so. There are always doubts about starting any new job, particularly in teaching. I had given them fair warning in the interview about how I taught. I hoped I would be allowed autonomy, but for the most part, I was glad they brought me in out of the rain.
The research is mixed. Some reviews (of the many studies on retention or promotion) have shown little benefit for retention in grade 3 compared with social promotion to grade 4. For example, a 2017 review looked at the results of Florida’s “A+ plan,” begun by Jeb Bush when he was governor. Little benefit has been found for the retained students.
Other reviews have shown that retention in the primary grades correlated with dropping out in high school. Nevertheless, some students who don’t pass a grade 3 reading test (for promotion to grade 4) do go on to grade 4 after submitting a portfolio, taking an alternative assessment, and/or attending a special summer school session and thereby qualifying for promotion to grade 4, even though they still may not do grade 4 work adequately. The quality of the work these low achievers do may depend on whether upper elementary teachers can group them for skills work in their self-contained classrooms. School or state policies may forbid grouping practices in the teaching of reading, especially in elementary school.
Unfortunately, neither promotion nor retention has solved the problem of low reading achievement, it seems. Earlier intervention than grade 3 is now often recommended. For example, see a 2008 review. However, it is not clear if earlier intervention has helped poor readers in grades K, 1, and 2 to pass a high-stakes grade 3 reading test or graduate from high school at a more frequent rate compared with a similar group of poor readers without early intervention or help in school. For example, see this recent review. As we all know, there are many low achievers in elementary reading classes to this day.
Indeed, because of a growing number of state laws requiring retention (probably in desperation), many third graders in this country’s schools today will not be promoted to fourth grade. For example, we are told that thousands may be held back in Mississippi. That newspaper article from Mississippi refrains from pointing a finger at anyone—the students or their teachers or parents. However, many education policymakers seem to fault, implicitly at least, elementary classroom teachers for the failure of many kids to learn how to read by grade 3.
So-called “retention” studies also seem to assume teachers or school policy makers are to blame when researchers find few long-term differences between low-achieving third graders who repeated grade 3 and similar low-achieving third graders who were promoted to grade 4. Researchers as well as journalists, nevertheless, are reluctant to criticize struggling students or their teachers.
But when a large group of kids in a state have not learned beginning reading skills by the end of grade 3 (remember, they’ve been in school for at least 4 years), it is fair to ask if the problem may lie with their teachers’ training programs, not their teachers. Few parents or other readers would guess that teachers’ training programs may be the source of the problem because the studies on retention in grade 3 rarely provide information about the beginning reading program these students have had or the preparation program their teachers had. Their focus is on students’ achievement in school after grade 3.
Why are large numbers of students who don’t pass a grade 3 test of beginning reading apt to be an indictment of a state’s preparation programs for primary grade teachers? Because, as I learned in Massachusetts, most elementary teachers have not been trained to use effective, research-based strategies. How do we know this? I learned this by examining licensure tests for prospective teachers of young children before helping to develop one in the Bay State. Most licensure tests of beginning reading knowledge for prospective teachers of young children, I discovered, do not assess or assess adequately the major elements of research-based knowledge of beginning reading as set forth in the National Reading Panel’s report of 2000.
The components of effective beginning reading programs and strategies one would expect researchers to look for, or professional development providers to provide, are well-known and listed here. But, alas, they are not apt to be found in many primary classrooms. Teachers teach the way that they are taught to teach in their training programs.
Since around the 1960s, teacher training programs in the U.S. have tended to promote guessing from context (often called Whole Language) as the primary strategy. Even if some decoding is taught (as in many misnamed “Balanced Literacy” programs), kids are not taught the purpose for an alphabet. Nor are they taught systematically how to decode the alphabetic symbols used for beginning reading in English (the symbols for the sounds made in words read by children in the short stories created for beginning readers). Yet, somehow, teacher preparation programs have escaped the (often implicit) fault-finding that their own students—prospective teachers—have not. For reasons that are not clear yet, low achievement in K-12 students is perceived by education policy makers, researchers, and many others as the fault of their teachers. At least, that is who the framers of the Race to the Top grant competition in 2011 decided should be held accountable for low K-12 student scores on federal or state-mandated achievement tests. Indeed, sometimes as much as 50 percent of a teacher’s evaluation is based on her students’ test scores.
Strangely, while K-12 teachers, under current education policies, are held accountable to varying degrees for the low scores of their K-12 students, faculty in teacher preparation programs are NOT held accountable for the failure of their own students (the prospective teachers they recruited and prepared) to teach K-12 reading well enough so that the racial and ethnic “gaps” between low-achieving K-12 students’ average reading scores and the average reading scores of higher-achieving K-12 students have narrowed. What is worse, many education policy makers seem to believe today that the chief reason low-achieving readers are low-achieving is because their teachers, principals, or communities are bigoted and have discriminated against them.
One might think that the requirement to pass a well-constructed licensure test in beginning reading skills for all prospective teachers of young children would ensure that all young children in our schools have adequately trained teachers. But only a few states (Massachusetts, Arkansas, Connecticut, Mississippi, New Hampshire, North Carolina, Ohio, and Wisconsin) today seem to use a well-constructed licensure test of beginning reading skills for prospective special education, early childhood, and/or elementary teachers, as I showed in my own research on the content of licensure tests for special education teachers.
Moreover, it turns out that in some states there are differences in pass rates for “white” and black prospective teachers on their required licensure tests in reading, leading some policymakers and researchers to imply that racial or ethnic differences in pass rates for prospective teachers also means discriminatory tests (licensure tests).
For example, in Wisconsin: “According to Department of Public Instruction (DPI) records, two-thirds of people who took the Foundations of Reading Test (FoRT) between 2013 and 2016 [a test developed in the Bay State in 2000] passed on the first try. Including those who took it two or more times, 85% passed. Pass rates were better for white test-takers than for minority test-takers, which led to concerns that the test keeps a disproportionate number of minority potential-teachers out of classrooms. Department of Public Instruction officials say many who have not passed FoRT would be good teachers and passing FoRT isn’t the only sign someone will be a good teacher.”
A major part of the problem with the thinking expressed by education policy makers at the Wisconsin DPI is the idea that raw scores on licensure tests predict teacher effectiveness. They weren’t intended to do so at the inception of teacher licensing, and still are not. Passing a licensure test in most if not all professions means only that the test-taker has adequate entry-level knowledge for the profession. It is assumed that those who don’t pass the test don’t get a license. While adequate entry-level subject knowledge is needed for effectiveness in any profession and is necessary. But it is not sufficient. For prospective teachers of young children, student teaching experiences are expected to indicate to supervising personnel whether the test-taker is apt to become an effective teacher. In other words, NOT passing a well-constructed licensure test of beginning reading skills is a sign that the test-taker is UNLIKELY to become a good or effective teacher of beginning reading.
It is therefore not surprising that thousands of children across the country fail a reading test at the end of grade 3 when the basic problem may well be that they have not been taught how to read by their teacher because she has not been trained in her preparation program to use research-based knowledge of beginning reading, or tested for this knowledge on her licensure tests. This may well be the case in Florida today despite almost two decades of the A+ Plan.