FROM INDIVIDUAL DEFICITS TO COMPLEX INSTRUCTION
A written creative work submitted to the faculty of
San Francisco State University
in partial fulfillment of
The Requirements for
Master of Arts
James Richard Sheldon
San Francisco, CA
James Richard Sheldon
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Appendix C, the Ordering Numbers Task, is used with permission by Dr. Marcy Wood, and the CC-BY-SA 3.0 license does not apply to Appendix C.
CERTIFICATION OF APPROVAL
I certify that I have read From Individual Deficits to Complex Instruction by James Richard Sheldon, and that in my opinion this work meets the criteria for approving a written creative work submitted in the partial fulfillment of the requirements for the degree: Master of Arts in Special Education
Dr. Susan Courey
Professor of Special Education
Professor of Secondary Education
The traditional response to encountering students who have profound difficulties in mathematics is to diagnose them with a “learning disability” and to blame their mathematical inabilities on an individual deficit in the brain. Students so diagnosed are then relegated to separate classes and rarely have the opportunity to develop problem-solving skills. In light of the Common Core, however, special education students are going to be responsible for solving rigorous problems and to demonstrate comprehensive proficiency in mathematics. Considering these challenges, this creative work project proposes an alternative vision for a mathematics classroom in which all students participate in meaningful ways in general education and where students of all levels work together on multiple-ability groupworthy tasks. In doing so, it offers an hour and a half workshop to educate both general and special education teachers and offers some critical reflection based on the author’s experiences teaching this workshop at a conference of mathematics educators.
This creative work would not have been possible without the support and encouragement of the numerous faculty with whom I’ve worked with at San Francisco State. In particular, I want to acknowledge the Chair of my creative work committee, Dr. Susan Courey, for her enthusiasm, suggestions, and mentoring throughout this year-long project. I would like to acknowledge Dr. Judith Kysh for spending this year as my faculty mentor for the Predoctoral Scholarship program, for serving as a second reader on this creative work, and for initially introducing me to the idea of complex instruction. I would like to acknowledge Dr. Philip Prinz for serving as my faculty advisor throughout this Master’s program as well as being my University supervisor for my student teaching and for his patient listening and encouragement throughout these past three years. I would like to acknowledge Dr. Phyllis Tappe as well for her continued mentoring and guidance throughout my program.
I would also like to acknowledge some outside faculty who supported me along this project; my extradepartmental advisor Dr. Kai Lundgren-Williams for his critique and feedback on drafts of this project and the preliminary papers leading up to it; my co-conspirator on numerous other projects, Dr. Kai Rands for many long conversations and ideas for this project; and my summer research internship mentor, Dr. Marcy Wood for inviting me to participate in her research and for continued advice and support on my own projects and career path.
I would finally like to acknowledge the help of Steven Macaris in the Special Education office; there have been many, many times that I rushed into the office just before they close with some panicked question about paperwork, bureaucracy, or department processes and he has always been there with a prompt and detailed answer as well as a helpful reality checks.
MOVING FROM THE DEFICIT APPROACH TO COMPLEX INSTRUCTION
If I’m at a party, I generally introduce myself one of two ways; either I’m a special education teacher or a math teacher. If I say I’m a special education teacher, the usual response is something along the lines of “you must be very caring and compassionate,” reflecting the cultural stereotypes of special education as a sort of do-gooder profession that only the most exceptional can ever hope to work in. If I instead opt to say I’m a math teacher, about 90% of the time I get the response “I was never good at math.” For many students, though, it’s not simply a mere dislike of mathematics; there are students that struggle with mathematics to a level that we might call a “disability.” I spoke with a nursing student who had failed Algebra 1 at the community college level 4 times, for whom this was the sole obstacle keeping him from being admitted to the Bachelor of Science program of his choice. I spoke with another student who had wanted to do a Masters in Psychology and switched to Theology after repeatedly failing Statistics; his school refused to let him substitute another class and he was forced to change programs.
There are many different theories about why certain students struggle more than others with mathematics. Some people see it as a deficit that occurs within an individual student’s brain – for example, a “processing disorder” that interferes with working memory or executive functioning, a “language learning disability” that prevents the processing of language, or sometimes even a “mathematics learning disability” that specifically impairs their mathematical abilities.
This medical model of disability focuses on finding a discrete set of symptoms that constitute a “disease” or “disorder” and then position doctors, psychologists, and teachers as the judges of who has a disability. In doing so, they reify these categories (make the abstract categories concrete and permanent) and make a list of symptoms or criteria into an actual “thing” that exists. In other words, A student “has” a learning disability rather than merely displaying some (perhaps undesirable) cognitive patterns. The medical model of disability also presupposes the existence of these “disabilities.” For example, one study that I reviewed (Mishna et al, 2011) used a half-dozen graduate students to screen all the high school students in their study to make sure that they all met the criteria for a “learning disability” using IQ and achievement tests; any student that didn’t meet this criteria was excluded from this study even though they were enrolled in a class for students with “learning disabilities”. The authors, therefore, missed out on the opportunity to actually interrogate the categories, and instead presupposed them.
In a way that seems rather hegemonic, research in special education tends to perpetuate this medical model in its focus on whether a particular intervention works for “students with LD” instead of actually testing the categories themselves. When researchers have examined these categories (see for example, Gina Borgioli, 2008, p. 135), they find that there is almost no evidence that students with LD differ from other underachieving students. The category of LD itself comes from a very problematic history in terms of race and imperialism; Beth Harry and Janette Klingner (2006, p. 5) cite Sleeter’s 1986 work in which she “argued that LD came into being as a result of the 1960s concern with making the United States more competitive in the era of Sputnik.” As the push to intensify math and science instruction, and to upgrade American education for global competitiveness began to leave more and more students behind, “White middle-class families whose children were not proving competitive in that ethos sought the LD label as an alternative to the more generalized, more stigmatizing label of mental retardation” (Harry and Klingner, p. 5). The label of LD was constructed to exclude “environmental factors” and to require high IQ scores on racially and culturally biased tests in order to qualify
Disability studies as a field, however, redirects our attention towards the societal barriers constructing the experience of disability. Thus, disability studies proposes, we should locate the problem less in the individual student’s brain and more in a complex mix of social, cognitive, and behavioral causes; therefore the problem lies more in the interplay between the individual and society than in the individual person alone. Disability studies suggests that the traditional model of looking for a “cure” for a disability is faulty and what really needs “curing” is the societal barriers that block certain people from full participation (see, for example, Garland-Thomson, 2010 and Edwards, 2010).
Similarly, scholars in the growing field of complex instruction (for example, Cohen 1994a and 1994b, and Featherstone et al. 2011) propose a key factor in determining academic achievement is not individual potential or individual deficits. Rather, drawing upon extensive sociological literature, they point to the role of status within the classroom and within a school; certain students are ascribed greater mathematical status and thus their mathematical ideas are validated by the teacher and classmates while other students (who have less status) have their ideas discounted or often utterly ignored. Complex instruction is a term for a specific set of ideas for how teachers can address status in a classroom through careful instructional design and pedagogical strategies. This is done within a framework of delegating classroom authority to small groups, with the teacher focusing more on status interventions in the classroom rather than providing direct instruction (Cohen, 1994a).
This project, “From Individual Deficits to Complex Instruction” offers a new paradigm of instruction for students with disabilities. This paradigm on the broader level insists that a democratic society needs to remove separate classes for students with disabilities and restructure the curriculum to make this integration possible. It also insists that we need to reconstruct the curriculum from the ground-up to reflect this reality, and we need a pedagogy that allows students to support each other in moving through this new curriculum. I offer complex instruction as the guiding principle for building this new vision of schools.
As a place to start, I offer an hour and a half workshop that I have designed that introduces math teachers (both general education and special education) to this paradigm. It is not a comprehensive training in complex instruction nor in learning disability; rather, it is intended to provide some references and pique interest in these areas and to encourage teachers to work with each other in the process of instructional change. I taught an initial version of this workshop at the California Mathematics Council’s Asilomar conference in December 2013; this new version of the workshop has been submitted to the December 2014 conference as well.
BIOGRAPHICAL NARRATIVE: POSITIONING MYSELF IN THIS PROJECT
I often require students of mine to write mathematical autobiographies, an idea that I took from Tobias (1995); understanding their own history and experience is often key to getting “unstuck” with mathematics. This particular narrative I offer, though, is more than just a mathematical autobiography; it offers my journey both mathematically and in terms of disability, and shows how this relates to the perspectives and ideas that I offer in this project. I am both a person with a disability but also someone that acts as a gatekeeper and a judge of whether others have a disability; I want to problematize the notions of disabled identity through my own autobiography. I also offer some hints of intersectionality here, but for more about my experiences within the labor movement and developing an identity as a queer teacher see my prior Master’s thesis (Sheldon 2010b).
Mathematics had always been one of my easier subjects in elementary school. In reading people’s narratives of their mathematical development, though, I tend to notice that everyone hits a wall. My first was when we encountered Geometry for the first time in 7th grade. In elementary school, Geometry was always the last chapter of the textbook and we never got to it. I remember trying to measure angles in 7th grade and being totally baffled by how to use a protractor, not knowing the difference between the top and bottom row of degree measures, for example. Then in 8th grade, I hit a second wall when I encountered algebra. My teacher used a heavily geometric approach to teaching algebra, and it left me totally baffled – I had no idea what these squares and longs and smalls were, and why we needed to factor things in the first place.
In 9th grade, I entered Geometry class to find something totally unexpected. The textbook we were using was the (at the time) Discovering Geometry by Michael Serra (1989). We were constantly in groups, and rushing to computer labs, and really playing with and exploring Geometry. I remember being totally excited by this approach – only to find myself in a more traditional classroom for the next two years of math where things were all lecture based and we sat in a room and took notes.
My first experience working with students with disabilities was in my sophomore year of high school. I volunteered for an hour a week in a special education classroom. I was actually rather baffled by the experience, which now seems really naïve, but I really had no idea that things like autism existed or that some people didn’t speak until this experience.
Over the next few years, though, I found myself face to face with issues of disability in my own personal life, in ways that shattered my illusions of control. I developed clinical depression in high school, which as a result of some rather illfated medication choices by my doctors and me, eventually developed into a full on bipolar disorder. I found myself having to learn how to ask for disability accommodations and in a constant position of needing to educate people about these kind of disabilities. I found myself on a roller coaster of six months of feeling okay and then six months of depression, and then experiencing side effects from medications that compounded those difficulties. Over the course of the next decade, I ended up in the hospital four times, generally as a result of medications that had been prescribed to me by psychiatrists.
This vantage point is key for this present project – as I have both seen the system from the point of view of someone with a disability attempting to access services but also from the point of view of a special education teacher who’s had to evaluate students for services. I’ve seen, too, how the labeling that might be so useful to get an accommodation on the college level can be very detrimental to a student in their earlier years. And how the paradigm of special education often prevents us from making the very changes we need in K-12 education; much like how the paradigm of accommodations in higher education creates a band-aid that keeps paradigms like universal design from taking hold in higher education classrooms.
My undergraduate years were also a confusing time academically, where I ended up dropping my strictly linear goal of being a computer programmer and ended up exploring a lot of things I never even considered a possibility. I started out in computer science, and then considered Community Studies, American Studies, and Mathematics Education as possible goals, and eventually ended up doing a custom designed degree in Computer Science Teaching. I spent many long hours in the campus bookstore and library, and eventually discovered queer studies, too, and ended up taking almost every course I could think of on queer studies. This queer studies perspective informs my views on disability, which are briefly explored in this project, but are elaborated further in my paper on queer disability (Sheldon, 2013). I spent many long hours researching Master’s programs, and eventually chose to apply to a Master’s degree program in Activism and Social Change at New College of California. I would sleep off the effects of the medications I was on in the morning, go to class in the afternoon, and study mathematics in the evenings. New College eventually closed before I finished my degree, and I took my projects from there into a program in Equity and Social Justice in Education at San Francisco State.
At the time, I was exploring options for part-time work, and it was suggested to me that I should become a teacher’s aide (also known as a paraprofessional). The local district had long since done away with general education aides, so the only real option was to become a special education aide. I started out with working with students with so-called “severe” disabilities. (Though, it’s important when hearing a term like “severe disability” to keep in mind that severity is often measured through the lens of capitalism – as in, can the person support him/herself in a society in which independence and self-support are considered primary moral imperatives).
I was all set to go into a credential program for the so-called “moderate-to-severe disabilities” and already had my application written and ready to submit. But in 2009, I was hired for a month long position as a Resource Specialist Program (RSP) paraprofessional. An RSP program is a program where students are in general education full-time but where specialized instruction is offered in a separate resource room (“pull-out”) or with the resource staff coming into the general classroom (“push-in”). My job was to implement interventions for students with learning disabilities. The school had less than six RSP students, so the teacher was only there on Fridays – and was being required to spend virtually all her time assessing new students to qualify them for services. I, however, was there five days a week and, working only 5 hour days, had plenty of time to sit back and come up with ideas for intervening with students. All of these students struggled in both mathematics and reading. For reading, we had a fluency curriculum with cassettes where the students would listen to the cassette model the reading and then the students would read orally themselves. For mathematics, though, we had no such interventions – students would play games to learn multiplication facts in a way that was rather divorced from actual applications of multiplication – or I would accompany them to the classroom and try to help them with math lessons.
One boy in particular puzzled me, as he could not add, and I soon found that he lacked basic one-to-one correspondence skills – he would see four things and count some of them more than once, and tell me there were six things. The mathematics specialist that came in to work with him and a few of the other struggling students didn’t really have much familiarity with disability.
This experience working at this school inspired me to enter a Master’s program in mild-to-moderate disabilities. I was to find, though, that most of the methods that we were being taught in regards to mathematics failed to really get at the vast, richness of the field of mathematics and instead presented a very technocratic, remedial skills based approach that was but a hollow shell of actual mathematics. On one hand, we took an elementary math methods class that was about getting students into groups and having them solve problems, about teaching a variety of different algorithms, and about getting students engaged and thinking about mathematics. In this class, we built three dimensional figures and worked in groups on problems that sometimes took the entire three hour class period. One time I was so excited about a problem that I spent the entire week working on it and emailed the teacher my solution when I solved it because I couldn’t wait to get back to class to share what I had discovered. That teacher felt that the standard model of “I do – We do – You Do” was backwards and instead that in math classes, the model should be “I introduce – You do – we do – I summarize.”
In the special education methods classes, on the other hand, they taught the standard “I Do – We Do – You Do” model when it came to curriculum design. When it came to interventions, we were taught to use curriculum based measurement, which involves giving a ton of problems and having students work the rote algorithms quickly to see how many they could get done in a minute. (Technically CBM is supposed to be merely about monitoring progress and not a method of instruction, but the format of doing a lot of problems really rapidly tends to set the stage for a more rote-skills based curriculum rather than a conceptual and problem-solving oriented one). We also were taught interventions like rocket math which helped students to memorize their multiplication tables, but didn’t really involve any visual representations or connections to real-world multiplication problems.
Intrigued by this disjunction between the math methods class and what we were being taught in our other classes, I decided to join the 2011-2013 math education Master’s cohort for their full sequence of 4 classes. This was a program for experienced math teachers of three or more years of teaching, so I was a bit of an odd man out still being in my credential program. I did, however, get to see what reform mathematics looked like in practice; we solved mathematical problems in groups from a book called Mathematical Thinking and read articles from the research literature about using problem solving based instruction. We took what we were learning out into the field and tried it out and reported back to the class about how these experiences went. In the first class in this program, I wrote a literature review about problem-solving and special education students. What I found in the literature was a lot of tricks, and a lot of interesting ideas – but also a lot of discouraging articles about how special education students couldn’t do problem-solving. There were a lot of tips, a lot of tricks, but not the comprehensive framework I was looking for. I also found out that a lot of the recommended methods involved direct instruction; for example, one commonly used strategy is schema based instruction (Jitendra and Star, 2011), where students are explicitly taught patterns of problems and then learn to recognize them and use them to solve problems. (For example, there might be 10 types of “problems” they are taught about, and then they practice over and over until they can tell those 10 types of problems apart and know when to use which method.)
In my next-to-final class in the special education program we got to participate in a field-test of a new curriculum for special education teachers about teaching proportional reasoning. In this curriculum, we were assessed on both our proportional reasoning knowledge and our own rating of our confidence in teaching these subjects, and then throughout the semester we worked on computer-based activities designed to increase our knowledge and then practiced applying them to our teaching both in simulated activities and in the classroom. What I found, though, is that most students in this special education credential class were still using a “let me explain to you why you’re wrong” approach rather than letting students reason things out for themselves. So I was still pretty discouraged, having yet to find a method that I was satisfied with.
In my final class in the math education Master’s degree program, one of our textbooks was an interesting little book called “Heterogenius Classrooms” which was written by a San Francisco State professor (Watanabe, 2014). It was about a method called complex instruction, which is based on the premise that all students have the ability to contribute to group problem-solving and that what stops most students from contributing is not their intellectual prowess, but rather the status in the classroom that teachers and other students ascribe to them. This really resonated with me – not to say that our students’ disabilities aren’t real – but that there’s this whole other sociological dimension of what goes on in classrooms that has as much of an impact on our students’ achievement as their own cognitive functioning. At this time, I decided that complex instruction would be the focus of my culminating experience for this program.
RATIONALE AND CRITICAL LITERATURE SYNTHESIS
Common Core and Special Education
As Common Core implementation proceeds across the country, many commentators have worried whether the new standards (and particularly the new tests!) meet the needs of special education students. When I gave a presentation on this to my special education research methods seminar, I asked the teachers in the audience to raise their hands if they felt like their students were prepared for the Smarter Balanced Assessments in Mathematics being rolled out this year in California. Not a single hand went up in the entire room of experienced special education teachers.
A particular area of interest of mine is the Common Core Standards for Mathematical Practice, which lay out a set of eight standards that all students should be able achieve in their mathematical practice. Of particular interest is the first standard, which I reproduce verbatim:
CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. (Common Core State Standards for Mathematics, 2014).
According to the Common Core State Standards, students should be having these rich problem-solving experiences, but the reality in our schools is something different; they tend to do page after page of “exercises” – both in general education and special education, of course, but particularly in special education. In 2006, John Woodward observed that many students have a “mistaken belief that all math problems can (and should) be answered in five minutes or less.” Many teachers hold this mistaken belief as well (Sheldon, 2013), and as a result, “far too often students with LD and those in remedial classrooms spend their time completing worksheets or responding to low-level questions in a direct instruction context” (Woodward, p.47). For example, I observed for a semester in a 7th grade mathematics classroom for students with LD in which students almost exclusively worked rote problems from a workbook called SRA. (The rest of the time, when they were not working the rote problems, they were with a teacher’s aide playing games in a computer lab). Students worked individually at their desks, and the teacher would come around and check the students’ work and tell them whether they were right or not. The curriculum proceeded in a strictly linear fashion, and every day the teacher assigned new pages for them to work in class. Students did not talk to each other about mathematics (although there was quite a lot of “off-task” interaction about almost everything else BUT mathematics).
In classrooms such as the one I observed where students are doing repetitive, low-level mathematics worksheets, many educators blame the low expectations we have for special education students. Advocates for the Common Core, such as Martha Thurlow (2012), suggest that one of the major perils that we fall into in special education is that of low expectations. In my own experience as a high school special education teacher, I often encounter students who have the same goals carried over year after year – such that an 11th grader might have a goal written at the 7th grade level that is never met and somehow is just reintroduced every year without much thought on the part of anyone involved. These low expectations for special education students are perhaps not too surprising, though, given that, according to special educators, students with learning disabilities “typically have deficits in attention, memory, background knowledge, vocabulary, language processes, strategy knowledge, visual-spatial processing, and self-regulation” (Jitendra and Star, 2011, p.13). Generally, what is prescribed to treat these deficits is some form of pull-out or push-in instruction, once the student has been identified as having a learning disability. Alternatively, the student might be removed from the general classroom entirely and put in a special class with a modified curriculum, where the only students in the class have disabilities.
The apparatus of special education requires that we first identify a student with a disability using a scientific criteria. Generally placement is based on a student’s performance on a norm-referenced achievement test administered by a special education teacher or psychologist. These tests are rarely revised; the Woodcock Johnson III, the most commonly used test, was last revised in 2001 with the norms updated in 2007. The basic kit for this test costs $664 (Riverside Publishing), so many smaller districts are using out of date editions and the test is kept under strict copyright controls to keep districts from making photocopies or sharing the questions with parents. The mathematics section of this test covers four areas: calculation, which is paper and pencil arithmetic; math fluency, which is a bunch of short problems done quickly; applied problems, which are oral word problems; and quantitative concepts, which are brief factual questions about mathematics. Each of these questions are designed to be answered quickly, have only one right solution, and correctness is based entirely on a numerical answer rather than the process by which you reach it. Scores on this test are given on a bell curve based on the norms for a student’s grade level; a score of 100 is considered average. The achievement scores are designed on a similar scale as IQ scores and are intended to be compared to them. Depending on which state the student is located in, the cut-off for identification varies; it ranges from one to one and a half standard deviations difference between IQ and achievement. The idea is that a student with this discrepancy has a “learning disability” that prevents their brain from processing information and learning in the same way as other students, whereas a student without this discrepancy does not have such a disability.
These supposedly scientific models of disability identification rarely hold up under close scrutiny. Ellen Brantlinger attacked what she saw as the very premise of special education, that “all children should be at least average” (2004, p. 491). She wrote: “anyone who falls far enough below average gets labeled ‘special needs’ or ‘at’ risk” Additionally, rather than being merely a neutral description of a child’s situation, these labels carry stigma and very real negative consequences to the child. Bratlinger wrote, “Children and parents do not have a choice about whether to be ‘needy’ or ‘risky’, just as they were not consulted about the desirability and personal suitability of handicap, retarded, or disturbed- and before that, feebleminded, idiot, imbecile, moron, or insane” (p.490). She inquired further, “why insist that those below average ‘catch up’? Why penalize those who test below average with stigmatizing names? Why put people unable or unwilling to be the same [as others] in isolating placements?” (p. 492).
Gina Borgioli (2008) carried Bratlinger’s critique of the standard discrepancy model even further, and cited Fletcher et al (2006) in making her point that “research over the past fifteen years has not provided evidence that ‘IQ discrepancy demarcates a specific type of LD that differs from other types of underachievement’ nor has it found that ‘children with expected forms of achievement differ from those with unexpected underachievement beyond the identification criteria’ (p. 135). In other words, the criteria works well to distinguish between the group of students that qualify for special education services and the group of students that do not, but it does not necessarily mean that these two populations of students differ in any significant way aside from what the criteria already presupposes. Unexpected underachivement, therefore, is a form of circular reasoning. Let’s say we have two students, Johnny and Jack. Johnny has an IQ of 120 and achievement scores of 80 in mathematics; Jack has an IQ of 80 and an achievement score of 80 in mathematics. Traditional theories of learning disabilities would say that Johnny has a disability, while Jack does not. In reality, though, Johnny and Jack both struggle in mathematics, and other than Johnny being better at some specific types of skills that are thought to represent innate intelligence, there’s really not much of a difference between them as mathematics learners. Borgioli, therefore, challenged the entire notion of identifying an IQ / ability discrepancy, and thus the way that we diagnose learning disabilities.
Borgioli further hypothesized that these scientific modes of ‘identifying’ learning disabilities locate “the learning obstacle within the brains of the individual student” and offer “the school a convenient explanation for student failure” so that “the fault is placed within the child rather than within the schooling system” (p. 137). Reinforcing Brantlinger’s claim that students are placed in isolating placements, Borgioli suggested that the LD label is used to justify “pulling-out” or “pulling-aside” students for mathematics instruction (p. 139) and that “neither the regular education nor the special education classroom seems to be meeting their [students with LD’s] needs” (p. 139).
Lennard Davis (2010) gave a historical context to these normed models of disability, tracing the very idea of normal back to around the year 1840, where “normal” first came to mean a standard which should be aspired to, as opposed to it’s primary meaning which meant a carpenter’s square. (Hence, the mathematical meaning of normal meaning a 90 degree angle or a perpendicular angle). He finds that this occurred concurrently with the development of statistics; the French statistician Adolphe Queteletet, Davis explained, discovered that the law of error used by astronomers to find stars could instead be used to plot human features such as height or weight. Much like the average on the star map would help you find the star, he felt like the average of a human trait represented “a kind of ideal, a position devoutely to be wished” (p. 5). Stastistics over time were used to quantify all sorts of human traits and attributes, but, as Davis cautioned, “The concept of a norm… implies that the majority of the population must or should somehow be part of the norm” (p.7). The very nature, though, of the bell curve means that it “will always have at its extremities those characteristics that deviate from the norm” (p.7). Eventually, statisticians moved on from height and weight and found themselves looking at attributes like intelligence. What was once a theory about how to find stars became instead the theoretical justification for eugenics; “almost all the early statisticians had one thing in common: they were eugenicists” (p.7). More than merely a coincidence, Davis observed, these are actually closely connected; “there is a real connection between figuring the statistical measure of humans and then hoping to improve humans so that deviations from the norm diminish… of course such an activity is profoundly paradoxical since the inviolable rule of statistics is that all phenomena will always conform to a bell curve.” (p.7). The very nature of a normed test enforces the tyranny of a bell curve and will always create outliers who are labeled as deviants.
But if we need to quantify a student’s academic achievement and place it on a bell curve, this requires that we narrow what can be assessed. The number of simple calculation problems you can do in a fixed period of time is much easier to assess objectively than a skill such as mathematical communication. Our definition of mathematical proficiency, Borgioli suggested, should have “less to do with the ability to follow procedures or conventions and more to do with investigating relationships between ideas and then communicating and justifying one’s thinking to others” (p. 133). In a similar vein, an alternative framework for mathematical proficiency was suggested by the National Research Council in their report Adding It Up: Helping Children Learn Mathematics. Rather than simply being about computation, they imagine mathematical proficiency as a series of five interwoven strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (p. 116). Tests such as the Woodcock Johnson, however, test primarily procedural fluency; even when they claim to measure problem solving, they merely give students short word problems with only one correct answer. Borgioli summed up the restricted lens by which special education views mathematics by stating, “special education policy, research, and practice, are grounded in a traditionalist perspective that embraces behaviorism and a narrow vision of what counts as mathematics and mathematics proficiency.” (p. 142).
Perhaps this pressure towards the norm on the bell curve and the narrowing of what can be assessed could be justified if students were successfully learning mathematics. Returning again to Borgioli, though, she reminded us that “neither the regular education nor the special education classroom seems to be meeting their [students with LD’s] needs” (p. 139) even in the narrow sense of achievement on these narrow standarized tests but also in the larger sense of true mathematical proficiency. Districts are under increasing pressure to include special education students in the general education classroom, but general education teachers tend to be opposed to this inclusion. In traditional teacher-centered instruction, students with disabilities tend to learn very little and are often off-task. The same off-taskness seems to permeate more independent study models as well, as I see quite frequently in the computer-mediated independent study program where I currently work.
One might think that organizing a classroom around whole class discussions would help to remedy this situation. Baxter, Woodward, and Olson (2001) observed five classrooms and found that low-achieving students rarely spoke or gave only one or two-word answers (p. 8) in whole-class discussions of mathematical ideas. They were generally off-task and their cognitive challenges kept them from following the discussion and deciphering what other students were saying. Baxter, Woodward, and Olson were not surprised by this, as even the classroom teachers often had trouble understanding the explanations that other students were giving in class.
Perhaps having students work in small groups would be better for students with disabilities? Baxter, Woodward, and Olson (2001) also studied small group work and found that students with disabilities had challenges in working in small groups as well. Although they participated more fully than in large class discussions, students with disabilities often relied on the more high-achieving students for the complex mathematical thinking and were reduced to performing menial tasks such as handing manipulatives to their partner (p. 11).
The recommendation in the special education literature at this juncture tends to be a model of peer tutoring, such as the model that Sayeski and Paulsen (2010) recommended. In Sayeski and Paulsen’s model, the more skilled student coaches the student with disabilities and then they trade roles and the student with disabilities coaches the more skilled student. A similar strategy is recommended by Scott Baker, et al. (2004), where “the stronger math student coaches first while the lower ability student, called the ‘player,’ solves problems. Roles are then reversed. The peer coach uses a question sheet that has conceptually-oriented prompts, so that they will help the player to develop conceptual understanding. Training students on how to be a tutor is essential for this kind of model to succeed (Cole and Wasburn-Moses, 2010,p. 18).
Peer assisted learning works when helping students to work through problems with definite answers in dyads. But what about solving less routine problems in small groups?
Complex Instruction and Groupwork as an Alternative to the Intervention Paradigm
I turn at this point to research in general education, particularly the research of Elizabeth Cohen, in order to find some clues as to how to productively structure small group work so that all students can participate effectively.
Elizabeth Cohen, a sociologist who held a dual appointment in Sociology and Education at Stanford, looked at this question in her 1994 review of the literature, “Restructuring the Classroom: Conditions for Productive Small Groups.” Cohen was particularly intrigued by something that no one else had really paid attention to: in comparative studies, group work generally came out ahead of individual work when comparing student achievement gains BUT there was huge differences in its efficacy between studies. In her review, she attempted to tease out the crucial elements that make small group work effective. One of these elements is that students must be explicitly instructed in the social skills needed for small groups (1994a, p. 7). Another key element that she found that helped determine the success of group work was the instructor’s choice of task to be given to the students. A group task, as she defined it, “requires resources (information, knowledge, heuristic problem-solving strategies, materials, and skills) that no single individual possesses” (p.8). As a consequence students are forced to turn to each other to complete the task.
In contrast, she argued, many teachers mistakenly have groups working on tasks with definite procedures in which “stronger students [are] helping the weaker students” (p.8) but the weaker students don’t help the stronger students. By choosing group tasks, instructors can create a “reason for the group to interact” (p. 11) and help to create high levels of task-oriented interaction.
Another key element that Cohen identified was the appropriate use of structure in small group work. When solving group tasks, she found, “task instructions can profitably set problems for discussion, specify roles, ask questions, determine procedures” but that specifying the precise steps for solution or designating one student as the teacher and the other as the learner would be counterproductive. Cohen summed up this dilemma as follows:
If teachers do nothing to structure the level of interaction, they may well find that students stick to a most concrete mode of interaction. If they do too much to structure the interaction, they may prevent the students from thinking for themselves and thus gaining the benefits of the interaction (p.22).
In a similar vein, Indigo Esmonde (2009) argued that “group-worthy tasks…need to be accompanied with pedagogical techniques that encourage exploration and explanation rather than procedures and memorization only…”(p. 1028). Most textbooks, though, take a very procedural approach to problem solving; even the most reform-oriented textbooks (e.g. CPM, Discovering Algebra, Discovering Geometry, IMP) tend to take what otherwise would be a great problem and give step by step directions for solving it. Special education students rarely get access to reform curricula; more common in the classes where I’ve worked is curricula such as the aforementioned SRA textbook. And when they do end up with rich problems to work on, teachers tend to break them down step by step and guide the students through the problems – the very trap that Cohen warned about that “prevent[s] students from thinking for themselves” (1994a, p.22).
Cohen, coming from a sociological viewpoint, also identified another key challenge to successful groupwork, the differences in status between group members. She explicitly noted that this is perceived status, not necessarily actual ability. For example, she cited Dembo and McAuliffe (1987) who created a bogus test and publicly announced that certain students had scored high on this test. When given a group problem solving task with other students who had supposedly scored lower on the test, the high scoring students “dominating group interaction on the experimental task, were more influential, and were more likely to be perceived as leaders” (p. 23). She suggests that this perceived academic status is even more of a factor than gender, race, and ethnicity but notes that gender, race, and ethnicity correlate strongly with perceived academic status in most classrooms (p. 23). This mirrors broader societal dynamics, creating a microcosm of social reality in the classroom. These status differences create a self-fulfilling prophecy whereby those with the higher perceived status participate more and are more influential in groups (p. 24). Status, in Cohen’s schema, is not the same thing as popularity, but there is a strong connection between popularity in the local context of the classroom and academic status. This seems counterintuitive at first; many people experienced being academically successful and being unpopular in school. But popularity in the lunchroom is not the same as popularity in the classroom, and the kid that everyone shuns at lunchtime might be the first one people want to sit with when working on an academic task in the context of a mathematics class. Thus, Cohen’s ideas about local status are also true about popularity; status and popularity are not global attributes but rather require a careful analysis by teachers of the actual situation within their classrooms.
Academic Status and the Tyranny of the Norm: If Everyone Should be Average, Shouldn’t That Work Against Status Hierarchies?
How does this theory of academic status fit in with what we have previously discussed about the tyranny of the norm? At first it seems like these theories are incompatible – one being about the pressure to be normal/average and the second being about ranking within the classroom where students are ranked from best to worst. Wouldn’t the classroom tend to favor the average student and have the average student scoring highest on tests and tending to learn the most. Looking more closely at the theory of the tyranny of the norm, however, it turns out that the statisticians discussed in Davis (2010) made some subtle changes to the idea of the bell curve in order to create the possibility of ranking people from smartest to least smart. Davis explained this new way that the eugenicists began to conceptualize human traits:
… if one is looking at human traits, then the extremes, particularly what Galton saw as positive extremes– tallness, high intelligence, ambitiousness, strength, fertility– would have to be seen as errors. The problem for Galton was that, given his desire to perfect the human race, or at least its British segment, tallness was preferable to shortness… Galton divided his curve into quartiles, so he was able to emphasize ranked orders of intelligence…He created what he called an ‘ogive’, which is arranged in quartiles with an ascending curve that features the desired trait as ‘higher’ than the undesirable deviation (p.9).
In a math classroom, students thus face a paradox – they are not supposed to stand out and to simply be average, but at the same time they have to succeed academically, which requires academic status. (As an example of this, I helped with data analysis for an Ed.D. student’s dissertation in Mathematics Education back in 2007, and almost all of his students in both his experimental and control group said that they would “not want to be known as someone who was good in math.”) At the same time, though, there are external rewards for grades and internal rewards within the classroom for having high status.
As a solution to this dilemma of status, Cohen specifically proposes the idea of a “multiple ability treatment” whereby teachers enumerate the many different academic-related abilities that are needed to complete a given task and help students to see that there are many different kinds of smartnesses that one can have in solving a mathematics task. Along with this, tasks are carefully designed to be open-ended (open-ended meaning there should be a variety of methods to reach a solution, and, when appropriate, multiple answers to a problem) and intrinsically interesting; tasks should captivate students and keep them working independently in groups without the teacher needing to constantly redirect students back to task. Students think more like professional mathematicians by solving groupworthy problems, rather than merely repeating what they’ve been taught in endless exercises. (My sole memories of 2nd grade mathematics is of taking timed arithmetic tests, and third grade mathematics was endless individual seatwork where you completed the same problems over and over until you got them right). Instead of motivating students by competition for grades and status (extrinsic rewards), they are motivated by their intrinsic interest in the problems presented.
A final area that Cohen identified that is particularly relevant for special education is that of training students for cooperation. She notes that specificity is important in this training – for example, Lew et al. (1986) trained students in “sharing ideas and information, keeping the group on task, praising and encouraging the contributions of others, and checking to make sure everyone in the group understood what was being taught” (p. 26) and found that this was a necessary element before cooperative groups would produce superior results to individual work. By contrast, Huber and Eppler (1990) had students rate each other on vague / ill-defined attributes such as friendly-hostile and hardworking-careless following a group task and then discuss their ratings of each other; this intervention had no affect on achievement. In other words, simply asking students to give vague, ill-defined feedback to each other was insufficient; students need explicit training on how to work in groups.
Complex Instruction and Detracking
Complex instruction is often suggested as an instructional strategy when schools engage in what is known as “detracking” or “untracking.” Detracking and complex instruction go hand in hand, but are not synonymous; schools may detrack while using other types of strategies or may use aspects of complex instruction without using the entire system. Detracking itself is an umbrella term for a wide range of reforms Ilana Horn (2006) offered some examples of the myriad ways schools have chosen to “detrack”:
In some cases, detracking may only refer to the elimination of ability grouping within a particular class. For example, instead of having college preparatory algebra and regular sections of algebra, a school may detrack by just offering algebra. Elsewhere, it may refer to the elimination of noncollege-preparatory classes, such as general math, consumer math, or pre-algebra. Finally, detracking may refer to a more radical change, a curriculum structure in which all students enter the same college-preparatory math class in their first year of high school (p. 73)
Horn in particular chose to focus on two schools that used the final mode, where all students entered the same college-preparatory math class in their first year of high school. She nicknamed these two schools Phoenix Park and East High. Phoenix Park used a flexible method of grouping where students could work either individually or in groups on problems. By contrast, East High had an open-ended curriculum but teachers were trained specifically in complex instruction (p.78). In both schools, teachers paid careful attention to the selection of problems; “the teachers used problems that required students to make meaning of the mathematics they were using, as they had to clarify assumptions and explore and defend their choices in problem posing and problem solving.” Teachers at East High School “organized their detracked curriculum around what they called group-worthy problems.” These group-worthy problems “a) illustrate important mathematical concepts, (b) include multiple tasks that draw effectively on the collective resources of a student group, (c) allow for multiple representations, and (d) have several possible solution paths” (p.76).
Horn also paid particular attention to the context that allowed these methods to work in the classrooms in the schools in which she worked. She observed, “at both schools, the teachers collaborated on the development and implementation of their respective curricula… both groups controlled the hiring of new mathematics teachers in their department- a common practice in England but highly unusual in the United States.” At Phoenix Park the teachers also arranged the course scheduling so that students had the same teacher for three years, instead of having a new math teacher each year. Teachers still met weekly to “discuss the activities they planned to use and any modifications they needed to make” (p. 77). At East High, students had a new teacher each semester, but they “met weekly in their subject groups and discussed curriculum and its effective implementation… adapting published materials to make them more group-worthy” (p. 77). Both schools “had structures in their workweek that allowed them to consult with each other and learn from their collective experience, breaking through the privacy and isolation that often characterizes teachers’ work” (p. 77).
Detracking and Special Education
Horn’s study showed that detracking can be done without complex instruction, but that it requires the same careful planning of tasks as one would do with complex instruction. Detracking is not necessarily the same as movements for inclusion, but many authors who write about detracking do use special education an illustration of their theories. For example, Jeannie Oakes and Martin Lipton, in their 1999 essay “Access to Knowledge: Challenging the Techniques, Norms, and Politics of Schooling,” use special education as one of their examples of how students are tracked:
Those who promote ability grouping, special education, gifted programs, and the myriad other homogeneous instructional groups in schools claim that these classifications are objective and color blind, rather than, as Goodlad suggests, reflecting myths and prejudices. Advocates of grouping explain the disproportionate classification of white students as gifted or advanced and of students of color as slow or basic as the unfortunate consequence of different backgrounds and abilities. They base their claims of objectivity on century-old (and older) explanations of differences that are neither scientific nor bias-free.
Both students and adults mistake labels such as “gifted,” “honors student,” “average,” “remedial,” “LD” and “MMR” for certification of overall ability or worth. These labels teach students that if the school does not identify them as capable in earlier grades, they should not expect to do well later. Everyone without the “gifted” label has the de facto label of “not gifted.” The resource classroom is a low status place and students who go there are low status students. The result of all this is that most students have needlessly low self-concepts and schools have low expectations. Few students or teachers can defy those identities and expectations. These labeling effects permeate the entire school and social culture. (p. 171, cited in Burris and Garrity, 2008).
Oakes and Lipton, like Cohen, pays special attention to the operation of status, noticing how labeling constructs students’ perceptions of their own and others’ status. Jonathan Mooney has a poignant example of how this labeling worked in his own experience of being a fourth grader with a learning disability:
Even Mr. R couldn’t stop the snide and embarrassing looks my classmates gave every time I left the classroom to go to the resource room. And he couldn’t help that every time I left for the resource room, I walked down the hallway with the kids from the gifted and talented program (GATE). As I walked down the corridor I passed each grade, one by one, in slow motion. Sometimes for cruel fun, the GATE kids would ask me which room I was going to, even though they knew exactly which room was mine. They wouldn’t wait for an answer, but just laughed and called me stupid. (2000, p. 35)
Complex instruction literature rarely mentions special education although it is seems relatively clear to me that this method is designed to work with all students (not just students without disabilities). The special education literature does not address complex instruction. This doesn’t mean, however, that complex instruction won’t work with special education. In fact, the theory behind complex instruction dovetails nicely with paradigms in special education such as Universal Design for Learning.
Complex Instruction and Universal Design for Learning (UDL)
In Rachel Lambert and Despina Stylianou’s 2013 piece, Posing Cognitively Demanding Tasks to All Learners, Lambert and Stylianou discuss how “learners who struggle in mathematics or who have special education placements have access to less demanding mathematics” (p. 501). Although Lambert and Stylianou were not specifically studying complex instruction, they were looking at a classroom that used open tasks which “have more than one entry point” (p. 503) in individual work and then in groupwork. UDL, according to them, involves three components- multiple means of representation, multiple means of engagement, and multiple means of strategic action. They see UDL and cognitively demanding, open tasks as being a natural complement to each other.
Complex instruction relies heavily on cognitive demanding tasks but within the particular context of groupwork. So, in what ways might complex instruction work with UDL? The UDL 2.0 guidelines (2013) suggest that teachers should “provide options for language, mathematical expressions, and symbols”, “illustrate through multiple media”, “use multiple media for communication,” “provide options for executive functions,” and “provide options for recruiting interest.” Cohen’s criteria for a multiple-ability task(1994b, p.68) proposes the following:
Has more than one answer or more than one way to solve the problem-solving
Is intrinsically interesting or rewarding
Allows different students to make different contributions
Involves sight, sound, and touch
Requires a variety of skills and behaviors
Also requires reading and writing
Multiple-ability tasks in mathematics facilitate UDL by presenting information and allowing answers in a variety of formats. Tasks have many different ways in which you can move towards a solution, and multiple modalities can be used to get there. They involve reading and writing, but also options to present information verbally or graphically. They are cognitively demanding, in Lambert and Sylianou’s sense. And in the context of complex instruction, students have direct and immediate support with executive functioning from members of their group and are working on an intrinsically rewarding task that sustains motivation and interest. There may not be as many individual options in a complex instruction classroom as there would be in a classroom strictly following the UDL guidelines. (Students work together on presentations, for example, rather than each student presenting the information in their own chosen modality).
However, in complex instruction, group members are responsible for making sure that all students in their group understand a solution (or multiple solutions) to a problem before moving on; so, if even one student doesn’t understand, the group hasn’t yet solved the problem. What if a student with disabilities uses a different modality than the rest of the group to communicate their ideas? Ordinarily, the students with disabilities have their ideas ignored during groupwork, but with the use of group roles and status treatments, teachers can ensure that these ordinarily ignored ideas get validated and recognized within small group work.
A methodology, Robert Bogdan and Sari Biklen (1998) wrote, differs from methods in that methods are “specific techniques you use” while a methodology is “the general logic and theoretical perspectives for a research project” (p. 31). I spent a lot of time looking at issues of methodology during my first Master’s degree, in which I completed a Master’s thesis that was a combination auto-ethnography, ethnography, and oral history about GLBT teachers in teachers’ unions. This work on methodology was later presented at AERA at a panel on new innovations in qualitative methodology. In that paper I covered four main areas – visibility, researcher subjectivity, reconceptualizing validity, topics of inquiry, and developing new theoretical tools (Sheldon, 2010a). I’m going to go through each of those areas and discuss their application to this particular project.
In their discussion of research methods, Bryson and De Castelle observed that “only heterosexual or faux-heterosexual people are usually welcome to do school-based educational research” (p. 247). Their queer researchers Manifesto included “I will not try to pass as straight in my research work” as one of its primary tenets (1998, p. 249).
Disability, like queerness, has a complex relationship to visibility. For example, in media representations, LGBT folks are often either portrayed as “freaks,” holding up leatherpeople, drag queens, and others as representing the entire community or alternatively portrayed as being “just like everyone else” focusing on those who conform to gender role norms and monogomous relationships. Likewise, media representations of people with disabilities tend to either portray them as objects of pity in need of help or as inspirational tales for able-bodied people – if someone with a disability can achieve success, “why can’t you,” they seem to say.
This question of disability carries over into educational research as well. Very few researchers in special education openly identify themselves as disabled, except for those who have their disability visually marked and can’t avoid such labeling by the operation of ideological state apparatuses. There’s a big difference between a self-identity and being interpellated as disabled; the former being a political act involving agency and the later being something coercively applied. (Interpellation is a critical theory term meaning the way in which society coerces you into a particular identity, often without your consent). As I wrote in Sheldon (2013) about children who are constructed by the special education apparatus as being disabled:
The apparatus of special education (from the legislature to the administrative bureaucracies down to the principals and teachers) interpellates these children as a particular kind of subject, a disabled subject. That is, the apparatus tells the children they are disabled, and the kids look at themselves and see in themselves what they are told, and thus became what they were told they are. Louis Althusser (1971) described this process as “hailing”; he used the example of how a policeman sees someone he thinks is a criminal and shouts out “Hey You!” and in that moment the person realizes he is a criminal and turns around, both actively accepting that identity but also having no choice in doing so. In that moment, the criminal is constructed as a particular kind of a subject. In this case, the children are interpellated as disabled, and come to recognize themselves as such in the process of negotiating that identity.
Researchers in education who are themselves identifying as disabled tend to choose the label of “disability studies in education” rather than “special education” to describe their work – for example, the American Educational Research Association has separate Special Interest Groups for DSE and for Special Education, with only a small overlap in membership.
In my own research, I detail out my own history and identity in my written work, but worry that when I presented the workshop in Appendix A that I tended to gloss over it – I came across more as a geeky special ed teacher than as someone who himself had struggled for years with trying to get appropriate disability accommodations of my own. In this creative project, I present both the original workshop that I offered along with a revised workshop that I will be offering the following year based on my reflections on the initial workshop. This is an area that I particularly want to revisit – how to incorporate my identity into the workshop, but moreover, how to get participants to reflect on their own identities rather than creating a teacher centered experience in the workshop. Esther Newton (1993) told a joke of a postmodern anthropologist who is talking with an informant and finally, the informant says, “Okay, enough about you, now let’s talk about me” (p.3) I want to get the participants talking with each other about identity, not listening to me talk about it.
In Sheldon (2010), I discussed the arguments and issues in putting yourself into your research and critically analyzing your subject position in the process of doing your research. Looking at things from the point of view of deconstruction, there’s certain identities that are marked and unmarked – for example, if a paper doesn’t discuss a researcher’s sexuality, the assumption is they are straight, not gay. If the paper doesn’t discuss a researcher’s disability, it is assumed that they are able bodied and neurotypical. (Neurotypical being a term that the autism/asperger’s community uses to refer to people without autism, but I am using it in a more general sense here to discuss people who don’t suffer societal sanctions or discrimination based on mental disability).
If a paper doesn’t discuss a researcher’s race, it’s assumed that they are white. If it doesn’t discuss their gender, it is assumed that they are male. This implicit identity – white, able-bodied, neurotypical, straight, male – is thought to be the only subject position that can know objective truth. But, as Joshua Gamson (2000) observed, (positivist) science has traditionally been used against those that do not fit the norm (p. 347) – so this supposed objectivity is used in very oppressive ways to those who don’t fit that (mythical) norm.
Honeychurch (1996) suggested that in order to have a queer perspective, researchers must reject paradigms in which the neutral observer comes to know an objective truth. Instead, to research from a queer perspective means to “embrace… a dynamic discursive position from which subjects of homosexualities can both name themselves and impact the conditions under which queer identities are constituted” (p. 342-343). Likewise, a disabled perspective (some might call it a “crip” perspective, in line with the growing field of “crip theory”) means identifying your own perspective and making this explicit in your work, as an antidote to the assumptions that are ordinarily made and as a way of figuring out the implications of your own identity for your work.
There is a caution, though, to be had here about positionality (Sheldon, 2010). Claiming an identity without a critical analysis “slide[s] into claims of essential difference, neglecting to critically examine the social context in which they are formed,” warns Cris Mayo (p. 84). Disability, for example, is a complex identity formed under societal conditions and norms, and comes into existence both in the failure of the built and social environment to accommodate those who are different and in the context of the capitalist imperative to production and the failure of disabled bodies to be properly productive in the expected ways. There’s also the danger of reification discussed earlier, where using a label such as disability can be in danger of making the abstract into concrete, ossified truths. (Even claiming a subaltern subject position does not necessarily free you from the problem of reification.) Rasmussen (2006) observed that researchers need to deconstruct even their own identities: positionality is in some sense just a “chimera” despite having real impacts on our research (p. 38). Constructed, political identities such as “crip” or “queer” are one way of addressing this, but sometimes fall into the trap of claiming a false sense of unity. Lesbians and black gay men, for example, have both challenged “queer” as being a white male identity and in a similar way “crip” sometimes comes under fire for emphasizing physical disability over the more mental forms of disability.
My 2010 paper on research methodology looked particular at the work of Patti Lather (1986) and her attempt to rethink methodology in the context of “openly ideological research.” By contrast, traditional conceptions of validity treat it as being merely about the “appropriateness, meaningfulness, correctness, and usefulness of the inferences a researcher makes” (Fraenkel and Wallen, p. 147). Lather instead is looking at research in a paradigm of criticizing and changing “the status quo” (p. 67).
Her four criteria for validity entailed triangulation, construct validity, face validity, and catalytic validity. Triangulation is looking at multiple sources in order to cross-check one’s research. This is a potential weakness of my current project in that I’m only looking at my own interpretations and perspective on the workshop that I offered – engaging more in auto-ethnography than in ethnography. By contrast, my 2010 Master’s thesis project looked both self-reflectively at my own experiences in the GLBT teachers’ union caucus and used interviews to get in depth at the experiences of others within the caucus. That project melded both auto-ethnography and ethnography in a way that really met the traditional criteria of triangulation.
Lather, though, adds a new twist to triangulation. She believes that it is important to use multiple theoretical paradigms in one’s research (p. 67). For example, in this project, I use the more subaltern interpretive frames of queer theory and disability studies as well as the more traditional lens of special education, mathematics education, and complex instruction to look at the situation facing students with disabilities in general education mathematics classrooms. This theoretical triangulation adds a depth to my project that might have been missed if I merely stuck to, say, disability studies and special education as my paradigms.
Construct validity, in a traditional sense, is about how well an instrument measures the concept that it is supposed to be measuring, as evidenced by other scores or the subject’s behavior in other situations (Fraenkel and Wallen, p. 153-154). By Lather’s definition, however, construct validity is about how theory is affected by the data; instead of sticking with a priori theories, theory must be dynamically shaped by “a ceaseless confrontation with the experiences of people in their daily lives” (p.67). This current project attempts to address construct validity through a constant engagement with other practicing educators; I have attended both research and practicioner conferences to present my work, had long discussions with teachers that I work with in my day teaching job, and presented the draft version of the workshop at a math teachers’ conference. I also have struggled with the day-to-day implementation of my ideas in my own practice as a teacher. So there’s definitely a strong element of engagement with the “rank and file” of teaching.
Face validity involves sharing your preliminary conclusions and theories with the research participants in order to test them against their ideas and conceptions of the setting and situation (p. 67). This is something that’s perhaps harder to design into research than I expected – in my 2010 Master’s thesis project I attempted to run drafts of the work and ideas past my interview subjects and to present them to the caucus before moving onto final drafts, but the complexity of the ideas and the busyness of the teachers’ schedules precluded a more active engagement in this process. In this particular project, there aren’t really subjects per se, but there’s still a process of testing the ideas of this project both against the workshop participants and a continued process of working to interrogate disability in my day to day practice as a teacher and engaging with colleagues, parents, and students in this process.
Perhaps Lather’s most innovative idea is catalytic validity. Catalytic validity is the degree to which participants in research are transformed, gaining “self-understanding” and “self-determination” (p. 67). It is similar to Freire’s “reading the word to read the world;” gaining the tools to collectively transform reality. This project specifically involves the creation and teaching of a workshop on these topics, with the goal of getting participants to critically reflect both on their own ability, experiences, and pedagogy. So it avoids some of the more traditional exploitativeness that most research faces in favor of creating an experience that helps teachers to learn about themselves and learn to transform the world around them. To quote my 2010 paper, “In socially transformative research like queer research, researchers’ goal should be for subjects to be transformed by their encounters with the researcher.” I would add, since we are looking at groupwork and complex instruction, that they should be transformed by their encounters with each other as well, not merely the workshop instructor.
Topics of Inquiry
Capper (1999) mentioned Griffin (1996)’s claim that queer researchers need to define the agenda and focus on the uncomfortable. She suggested the following uncomfortable topics: “cross dressing, transgender people, sex and sexuality in schools, pedophiles who are educators, and anti-gay sexual minority administrators” (p.6). If queer researchers don’t bring up this questions, who will? Probably those who will twist the issues to further stigmatize and pathologize those who are the most vulnerable in our schools.
“Crip” researchers have a lot of similar uncomfortable questions to ask – a brief moment of brainstorming on my part generated the following list: “Why can’t kids with disabilities be in normal classroom? Why do kids with IEPs fail to meet their goals year after year? Why must we have a ‘label’ to get kids the help that they need? Why aren’t colleges and universities required to use universal design principles in developing their curriculum? How can we get LD students through the pipeline to the point where they become education researchers? Must all researchers have doctorates (something that’s often out of reach for most students with LD) in order to have academic legitimacy?” As Reid and Valle ask asked, “What questions might researchers who grew up labeled with LD pursue? How might those questions differ from those that are now being addressed? Do teachers identified as having LD teach differently? How would they reorganize schools if they were given the liberty to do so?” (p. 472). These questions are not necessarily going to be comfortable.
My uncomfortable questions in this particular creative project are: “Why can’t students with disabilities be in the mainstream classroom? Why, given everything we know about universal design, cognitively demanding tasks, and complex instruction, are we not designing our general education classrooms this way? How can teachers be trained in order to teach in a way that all students will understand mathematics? What would happen if we threw out the label of disability as applied to a particular individual and instead took up universal design and complex instruction?”
New Theoretical Tools
Capper (1999) reminded researchers of some of the important principles of queer theory that apply to educational research. She worried that researchers reify sexual categories by attempting to identify LGBT administrators—they must either be or not be LGBT (p.7). Thinking in those terms, Leck (2000) warned, reduces one’s ability to see how “sexuality identity derives its complexities from within diverse social and cultural settings” (p. 324). Similarly with disability studies, we can fall into the trap of reification. In trying to study a disability, we come up with a term for it, and then, as I argued in Sheldon (2010), “ in that the process of sorting their data by categories they are in fact creating the very categories they think they have found!” We must be continually vigilant in the ways that we address disability and how it arises within a social and cultural context.
DESIGNING AND TEACHING THE FIRST ITERATION OF THE WORKSHOP
(Please note that the first iteration of the workshop is printed in its entirety in Appendix A.)
As part of the process of preparing my workshop, I consulted with a mentor of mine, Lisa Heft, who teaches people how to be dialogue facilitators and is known for leading really excellent workshops. It turns out she was about to offer a new workshop she had created on How to Design an Interactive Workshop, and I took time off of work in order to attend. During this workshop, we engaged in interactive activities designed to help us think about what we wanted to teach, and what the big ideas in our proposed workshops were rather than the facts we wanted to communicate. I left with an outline of the first part of my workshop and an idea about how I wanted it to go.
I got to Asilomar having hardly slept the night before because I was so nervous about my workshop. I got there Friday afternoon around noon, only to find out that the first meal that they would be serving was going to be dinner. After dinner, I stayed up late finishing work for my day job and then spent some time on the phone with a mentor talking through my plans for the workshop the next day. In the morning, I wrote and rewrote the workshop until it seemed to really shine.
I got to the session, which happened to be just before dinner, and was surprised by the steady stream of people that were coming in. I eagerly greeted them, introduced myself, and started them on the first set of reflective questions. They seemed to really respond well to these, and then I started the first activity, the discussion of the quotes in small groups. This is where things really started to go awry – in one group a woman wanted methods to work with students with disabilities and she started talking with a man she was sitting next to, rather loudly and animatedly but not on the topic we were currently raising. It became clear that she wanted to focus less on theory about disability and more about methods.
Another group said, “Oh. We know all this already” and left 20 mins into the workshop, when I really had a lot of new ideas I wanted to present. A third group looked at the quote they were assigned and were completely baffled and just got up and left. At this point I started to panic in my head – I jotted down a note saying “fear, confusion, anger” and promised myself I’d get back to looking at my feelings.
So, we got to the second half of the workshop with only about 6 people remaining, and the energy in the room had clearly deflated. I quickly rethought the second part of the workshop, which I had designed as a jigsaw activity, and put people into two groups and each gave them a section of the seminal work on complex instruction to read, and had them discuss it as small groups. One small group kind of went off on a tangent, but had a good conversation. The other group discussed roughly what I said to. Then we got back together and talked things over, and people seemed really intrigued by the ideas and asked me to send them a copy of the articles from the book.
My workshop was designed to occur in three parts – starting with thinking about your own perspective, then looking at disability studies theory, and then looking at complex instruction as a solution to some of the problems the workshop had posed. I do think, though, that pragmatically I should have talked about methods sooner and not spent the first 45 minutes laying the groundwork.
I’m uncomfortable with the idea of centering the workshop solely around methods, though. Susanne Luhmann, in her discussion of queer pedagogy, lays out the problem rather poignantly:
This orientation to pedagogy exceeds education’s traditional fixation on knowledge transmission, and its wish for the teacher as the master of knowledge. The teacher-student relationship at the heart of the transmission model of learning reminds Jane Gallop (1982) of pederasty, where “[a] greater man penetrates a lesser man with his knowledge” (p. 63).
Although few progressive educators today would agree with such a transmission model of learning and teaching, I suggest that it returns like the repressed in the prevalent preoccupation of teachers with methods, or the how-to of teaching. The rationale behind this search for an adequate method is that the teacher’s pedagogical skills—her instructional talents, as well as behaviors—will reflect in the students’ progress of learning. Learning then is relegated to the teacher’s effort and to good teaching, an assumption that gives way to some (fantasmatic) investments in the role of the teacher in the learning process.
Luhmann wants us to think more about the interaction “between teacher/text and student” rather than about the methods by which we choose to teach. I, too, intentionally chose in the design of my workshop to postpone discussion about methods until students had an opportunity to reflect on their own personal experiences and discuss them and until there was a chance to reflect on some theory.
Rather than looking solely at my specific methods in the moments that I relate fro mthe workshop, I want to queer the notion of methods and instead look at broader questions of my teaching philosophy. I ostensibly wanted to draw the ideas from the participants and have a discussion about the concepts being presented in the workshop, but there was also a lot of material I wanted to present, too. Rather than lecture on it, I opted to have them do reading – but the disability studies material I chose was rather dense, particularly for a group that may not have read writing in that genre before. It might have been better to have done a good lecture on what I wanted them to glean and then opened things up for discussion rather than to eschew lecture entirely.
Stephen Brookfield and Stephen Preskill (2005) caution us that one of the traps that advocates of discussion methods often fall into is setting up a false dichotomy between lecturing and discussion” (p. 44). They suggest that you can use lecture to model democratic dispositions by incorporating into lectures the questions that you are posing, ending with the questions that your lecture has raised, to deliberately introduce alternative perspectives, and by introducing buzz groups into lectures. So in redrafting the workshop, I looked carefully at my own reluctance to use lecture and about how sometimes it makes sense to tell the participants what you want them to know rather than to always be trying to draw it out of them or set up activities in order to get them to that point – recognizing, of course, that no one ever really hears exactly what you tell them and there’s always going to be something different in their head than in yours even when you lecture.
ATTENDING THE SMARTER TOGETHER! COMPLEX INSTRUCTION WORKING CONFERENCE
Hoping to gain more of an insight into the dynamics of complex instruction, I attended the Smarter Together! Working conference, which was a conference of elementary teacher educators who are looking to use complex instruction in their math methods courses that they teach to prospective elementary teachers. About a third of the conference was spent working in groups on problems that could be directly taken and used in an elementary school. Another third was spent on discussions about how to use the methods both with students and with prospective teachers. The final third of the conference was spent creating tasks that we could use with both prospective teachers and students.
My role in this conference was a hybrid participant/observer – sometimes I observed and took notes, sometimes I contributed ideas, and sometimes I was entirely engaged with a group on solving a task. One task from this conference particularly exemplified how what seems like a rather mundane, one answer task can turn into something for students to solve together. The task is called, rather appropriately, “Ordering Numbers.” The task is introduced by explaining to the students the multiple abilities they might encounter with the task – for example, thinking creatively about numbers, logical reasoning, visual reasoning, communicating ideas. The emphasis was that everyone in the group was needed to solve the task and could bring abilities to the table that others didn’t have. Academic abilities, the instructors clarified, not just “can write neatly to write up the notes” but rather different ways of approaching the mathematics of the task.
We were also primed with a set of group norms, for example, that anytime you want to say “I don’t understand” you have to change it to “I don’t understand… yet!” Students are also given role such as facilitator and recorder and taught what those roles entail and have an opportunity to practice those roles. So, we were given 8 cards, each with a different representation of a number on it. For example, .666 or 5/8 or a rectangle with 12 squares in it and 9 of them shaded in. (See Appendix C for a copy of the task.) We were told to each write our name on 2 cards, and that only we were allowed to move that card in the number line. We were also told to use as many different strategies as we could in rearranging the cards.
Our group started off by converting things to decimals – but we soon found that difficult when we hit 5/8ths – I mean, who knows what 1/8th is in a decimal, and long division is tedious and fraught with error, and perhaps not something you’d expect a third grader to know how to do. Then we started looking at relative sizes – for example, was 4/7ths less than 5/8ths. We compared some simpler examples, say ½ and 2/3s and conjectured that 4/7ths would be less than 5/8ths. At one point, I had an epiphany and grabbed a post-it noted and labelled it ½- creating a new card. We then labelled it 4/8, 3.5/7ths, 1.5/3rdsand 4.5/9s, thus being able to compare all of the fractions to this ½ that was created. This turned out to be a really valuable insight, along with the realization that we could put a decimal amount in the numerator of a fraction – something that would probably leave a lot of elementary teachers aghast! And then with the visual representations – we came up with ways of working with the visual representations so that we could keep them visual.
At one point the teacher called for a “huddle” where she called over all of the recorders, and then gave them some coaching on a particular strategy they could take back to their groups. We talked later about how you might be able to selectively use this huddle to coach a student that was particularly low-status in the class by making sure they were assigned the role that you call over – a way of intervening in status in a more subtle way than coming over to a group and jumping in on their process. (One of the key challenges in groupwork is that whenever the teacher comes around to a group, instruction becomes teacher-centered for that moment rather than student-centered. So you need a “bag of tricks” for intervening in status rather than always walking up to a group and creating a teacher-centered moment.)
For the final part of this task, we were asked to pick any two numbers and come up with a list of all the numbers between them. We chose to use a visual representation to do this, and then were able to show that one of the boxes could be filled in any amount except totally empty or totally full in order to represent the numbers between the two fractions in question.
Afterwards, we are asked to debrief the task and to talk not just about the task, but how the roles worked in our groups and how status played out within our group. We also talked about how the design of the task led to everyone participating and to many different ways of arriving at the answer, even though there was a correct answer to the problem.
DRAFTING THE FINAL WORKSHOP
The final aspect of this project was to create a new version of the workshop to be offered at CMC/Asilomar in December 2014. There are several goals that I had for the new iteration of this workshop; I wanted to be sure to emphasize the broader issues of collaborative planning time and working conditions that the research suggests, to consider how to get more participation from general education teachers in the workshop, to find ways to bring complex instruction into the workshop sooner, and to have participants experience working in groups on a mathematical task early on in the workshop.
In drafting the version in Appendix B, my goal was to incorporate the insights that I’ve gained from attending the smarter together working conference, and to incorporate the additional six months worth of study and research that I had done in this area since the version in Appendix A was taught. I also wanted to strike a balance between presenting ideas and drawing ideas out of participants in discussion. Finally, I wanted to redraft the marketing for the workshop in order to attract more than just special education teachers into the audience at the conference.
This creative work is the third major project of its time that I’ve embarked upon during my past fifteen years of formal education; the first being about queering computer science education, the second being about gay, lesbian, bisexual, and trans activism in teachers’ unions, and this present project being about deconstructing the apparatus of special education and offering an alternate vision of what mathematics education could look like in our schools. All three of these projects were at their core political projects, that attempted to move beyond positivist models of the neutral observer and to take an approach to studying problems that I had a significant investment in and in which I passionately cared about the outcome because the project had real, concrete, practical implications for the communities and people that I care about in my life.
There’s an old saying that you should “shoot for the moon, because if you fail, at least you will end up among the stars.” I started with the goal of taking apart the apparatus, but as I refined the project it became more and more focused on this one particular pedagogical method; narrowing the scope but also offering something that would be of immediate use to practitioners in the field of mathematics education rather than something that would be of interest only to the most committed scholars of disability studies in education. I was pleased therefore with the results of this project; it offers both a significant contribution to both the literature on disability studies in education and to the literature on complex instruction in mathematics.
My goal for both this workshop and this creative work is to pique interest in complex instruction; to make teachers and scholars start to wonder some of the same things I’ve been wondering: What if I give groupwork another look and take advantage of the research that shows the conditions under which it works? What if I stop focusing on pinpointing interventions for struggling students and instead change the nature of the curriculum to accommodate all learners? What might a school look like in which this sort of collaboration was designed into the very fabric of the school, much like Phoenix Park and East High? This project offers some starting points and ideas along the path of constructing a classroom, a school, a school system, and an educational system in which we can truly move from individual deficits to complex instruction.
References / Works Cited
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Selected Additional Bibliography
Lockhart, P. (2009). A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. New York, NY: Bellevue Literary Press
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Sullivan, P. and P. Lilburn (2002). Good questions for math teaching: why ask them and what to ask. Sausalito, CA: Math Solutions Publications.
APPENDIX A: FIRST ITERATION OF WORKSHOP
Workshop: From Individual Deficits to Complex Instruction
While people are coming in, have them fill out an index card with name, email, city, grades/subjects taught, and favorite math content topic.
Opening: Explain the purpose and background of the workshop – that I’m intrigued by the plight of those who “can’t do math,” particularly kids that are identified as having disabilities. I’m in the process of finishing up a Master’s degree in Special Education focused around this issue, and have put together this workshop offering my thoughts and perspectives around the problems with special education and some proposed solutions I have for addressing those problems.
Tell Story: When I was in 9th grade (around 1996), I was required to take a Physical Education class. There were (not surprisingly) very strong norms of masculinity, and I was routinely called a “faggot” by classmates for not conforming to their gender stereotypes. One day, we had a swim class and were learning the side stroke. It turns out I had a nearly perfect side stroke, while no one else could do it. So, the teacher made me demonstrate it for the entire class. The whole time the class was laughing and jeering, making fun of me – I remember every moment as though it was happening now. It was definitely not okay to be different from the norm.
Think about a time when you were significantly below or above average at something in school and you suffered social or academic repercussions because of it. Think about the details of the experience. What was the experience like for you? Remember your sensory impressions, your emotions, and the details of the experience. Take a moment to fix these details of the experience.
Pair – Share.
Now, close your eyes and imagine a student you had who wasn’t able to succeed in your math class, no matter what you tried. Think about what areas they struggled in – what their deficits were. Think about the ways in which they failed to conform to the expectations you had – of the average mathematics student. Think about the responses of the other staff – other teachers, administrators, specialists such as psychologists or special ed teachers. Think about the responses of other students. If they had an IEP, imagine what the IEP might have said.
What was this experience with this student like for you? What were your feelings? Emotions? Is there a particular incident that comes to mind?
Take a moment to fix these details in your mind and open your eyes. When you are ready, turn to a partner and share.
What’s important to note here is that this pressure to be normal, to conform to a set of norms, is not merely a sociological phenomenon. It’s tied closely to the history of statistics – to the bell curve – and to the idea of a normal population.
I’m going to put you into groups, and each group is going to read a quote about the concept of norms and averages. I want each group to relate their quote back to their two experiences we just discussed. Then we will get back together and I will have you summarize your quote and how you connected it. back to your experiences.
Hand out quotes, participants discuss them, bring everyone back together. Have each group read their quote and present it back to the larger group.
Ask participants to brainstorm a list of different strategies they’ve used with students with learning disabilities. Which of these use a deficit model? Which are premised on getting all students up to average? Do any of them “water down” the curriculum?
The remainder of this workshop is going to focus on a method developed by a sociologist and educator at Stanford. This method is called “complex instruction” and involves putting students into groups in order to work on mathematical tasks, but with some new strategies that we’ll go over – multiple ability tasks, a multiple ability orientation, and training in group roles.
We’re going to do a jigsaw activity now. First, we’ll start in groups of three or four. Then, you’ll decide who’s going to become an expert on each of the topics. These topics are: creating the task, the multiple ability strategy, and group roles. Pick one person for each of these – if you have someone left over, assign them to creating the task or to the multiple ability strategy. Remember who’s in your current group, as I’m going to ask you to come back to them after.
[After jigsaw activity]
Take a few moments to share any final thoughts you have about the experience with the person next to you. Then if you like, you can share it out with the larger group.
APPENDIX B: FINAL VERSION OF WORKSHOP
This title and description are what were submitted to the California Mathematics Council as a proposal for their December 2014 conference.
Title: From Individual Deficits to Complex Instruction
Description: Everyone has had a student that didn’t succeed in mathematics no matter what they tried. Ordinarily, we would focus on the student’s deficits and refer them for specialized intervention. This interactive workshop offers an alternative by inviting teachers to reorient curriculum around multiple-ability, groupworthy tasks. Teachers will experience complex instruction and reflect on how to use this approach so that all students can meaningfully participate in their classroom.
Introduce myself, opening activities (5 mins): Have participants fill out 3×5 cards with contact information and some information about what they teach. Talk very briefly about my own credentials and the purpose of the workshop.
Opening Activity (15 mins): Imagine a student you had who wasn’t able to succeed in your math class, no matter what you tried. Think about what areas they struggled in – what their deficits were. Think about the ways in which they failed to conform to the expectations you had – of the average mathematics student. Think about the responses of the other staff – other teachers, administrators, specialists such as psychologists or special ed teachers. Think about the responses of other students. If they had an IEP, imagine what the IEP might have said. How did this student do in large class discussions? In groupwork? When asked to work individually on assignments?
What was this experience with this student like for you? What were your feelings? Emotions? Is there a particular incident that comes to mind?
Take a moment to fix these details in your mind, and write down a few notes if you like. When you are ready, turn to a partner and share. You will each have a couple of minutes to share, and then we will have an opportunity for anyone who wants to to share key insights from their discussion with the larger group.
My own background (5 minutes): [If someone other than me is facilitating this workshop, they might choose to tell a bit of their own story here.] I’m particularly intrigued by the puzzle of low-achieving students in mathematics classrooms. As a special education teacher, I was primarily taught intervention models – pull the kid out of class to work with them on a skill, push into their class and help them individually when they’re working on a skill. What I found, though, was that the students really needed a curriculum and pedagogy that supported them learning mathematics through interaction with other students rather than by having the teacher try to remediate the things a student struggled with. Hence, I turned to the study of how to make groupwork effective with low-achieving students.
Groupwork Reflection Activity (10 mins): Take a moment to think about your experiences using groupwork as a teacher. If you haven’t used this method in your classes, think about your experiences as a student. Think in particular about a time in which groupwork did not go well – think about the details of what happened in that group, and think about your reactions to the situation. Were there certain students for which groupwork did not work well? Did certain students dominate the groups? What kind of assignments were students asked to do in groups? Were they doing repetitive homework exercises, or were they solving more complex and intrinsically engaging problems? How did groupwork work for students that were struggling – students with disabilities, english language learners, other low achieving students? Now, take a moment to fix those details in your mind, and then talk to a partner about the experience.
Brainstorming activity (5 minutes): To summarize what you just talked about with groupwork, let’s brainstorm as a whole group a list of things that are challenging about using groupwork in a classroom, particularly when using it with low-achieving students. [Write them up on chart paper.]
Cohen introduction (5 minutes): A lot of people have had similar experiences with groupwork. Elizabeth Cohen was a professor of sociology and education at Stanford who was particularly interested in why groupwork seems to really work in some situations and not so well in others. She wrote a literature review in 1994 called “Restructuring the Classroom: Conditions for Productive Small Groups,” and summarized her ideas in a practical book for teachers called “Designing Groupwork.” A key element that came up over and over again was the issue of status – certain students were ascribed higher academic status by their classmates, and high status students participated more in classes, had their ideas listened to, and learned more in their classes. This tended to also correlate with their popularity in the classroom setting, although it’s not identical. She developed a series of interventions; one of them is called the multiple ability orientation. In the multiple ability orientation, the teacher highlights specific ways that students can be smart at mathematics. I’m going to have you work in groups on a task that’s designed for elementary level students and will introduce it to you as I would introduce it to those students.
Ordering Numbers Task (20 mins): [See Appendix C for Ordering Numbers Task, and introduce task using multiple abilities listed on the coversheet for the task.]
Debrief task (5 mins): Use debrief questions from Ordering Numbers Task Coversheet: “What did people in your group do that helped the group work on the mathematics?” “What are the different ways of being smart at mathematics that your group used in carrying out the task?”
Something to consider on your own: How did status affect the operation of your group? Did certain people have more or less status in your group?
Groupworthy Tasks Activity (10 minutes): For this multiple smartnesses approach to work, however, tasks need to be designed to be groupworthy – that is, something that actually uses multiple smartnesses and is, as Cohen suggests, challenging enough that even the highest achieving student in the class can’t solve it on their own and is forced to turn to members of their group to figure it out. These kind of groupworthy tasks are essential if we are to take seriously the Common Core and have students solving complex real-world problems and “perservering in problem solving”; if we want students to do REAL mathematics rather than mere exercises. (Pass out page 68 from Cohen, 1994a).
Is there anything you would add to this criteria? What else might be important in developing a groupworthy task in mathematics?
Let’s pair up and take a look at some tasks, and evaluate them in terms of this criteria. (Give each pair a CI task to look at, drawing on various online resources such as cimath.org)
Institutional Support Activity (10 mins): Think about how your current school, or other schools in the past that you’ve worked in were structured. Were you given the time you needed to develop curriculum together? Would you have had time in your day to sit down with other teacher and develop groupworthy tasks? Would the way instructional time and bell schedules were structured support doing complex instruction in your classroom? Think about your past experiences and what an ideal school environment would look like, and what kind of changes would have to be made to your working conditions and then turn to a partner and share your ideas.
Conclusion (2 mins): Take a moment to think about one thing that you’re going to take away from this workshop that you can use in your classroom, and then turn to a partner and share your one thing.
Thank you all for joining me; my contact information is on the handout, and I look forward to hearing from you about your ideas or about how you end up using the material in this workshop. I also invite you to check out resources on the handout, including cimath.org – the website that the research group I work with has put together.
APPENDIX C: THE ORDERING NUMBERS TASK
CI Task Information
Please include any information on this form that might help others implement the task you’ve designed. Feel free to add more information or leave areas blank.
Larisa Velasco & Marcy Wood
Objectives (mathematical and/or pedagogical):
Use multiple strategies to compare fractions and decimals.
Common Core Content Standards Addressed:
3.NF.A.3a Understand two fractions as equivalent if they are the same size.
3.NF.A.3b Recognize and generate simple equivalent fractions.
*3.NF.a.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size.
4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models.
*4.NF.A.2 Compare two fractions with different numerators and denominators.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size.
Common Core Standards for Mathematical Practice Addressed:
MP6 Attend to precision
MP3 Construct viable arguments and critique the reasoning of others MP1 Make sense of problems and persevere in solving them
Set up Information
Everyone records (make sure everyone is writing and understands all of the strategies)
Everyone contributes (only the person who “owns” the card can move it)
I’ve done this with and without roles
Making sense of pictures
Making sense of fractions, decimals, and percents
Ordering based on quantity
Relying on others
Materials to prepare
Copy and cut up number cards so that each group has one set
Handouts that should accompany the task
None – just number cards
I use the multiple abilities orientation as my launch. I found that it helps to emphasize that everyone should thoroughly read the task card before they start.
Comparing to benchmark fractions (1/2 and whole)
Various interpretations of the visual representations (9/3, 9/12, 3/9, 3/12, 12/9, and 12/3)
Any other interesting moves
Add to sentence strips
“What did people in your group do that helped the group work on the mathematics?”
Refer to the participation quiz to highlight moves that were especially productive
Any specific directions?
As participants engage in the task, be sure they are only moving cards with their names on them. Also watch to see whether there are groups in which one person has all of the cards in front of him/herself. Has this person taken over the task?
Possible variations – how might this task be adjusted for different content or grade level?
This task can be easily adapted for different content and grade levels. For example, the number cards can be changed so they are all fractions or unit fractions. Also, I have made a variation with multiplication expressions, but this can also be easily changed to work with small quantities for kinder or for addition expressions. There is also a variation that has fractions represented using flags.
By Larisa Velasco and Marcy Wood
TASK: As a group, arrange the cards so the quantities they represent are ordered from least to greatest. Your group must use a different strategy each time you place or rearrange any cards. Find as many unique strategies as you can.
1. Hand out all of the cards. Each person must have at least one.
2. Write your name ON THE FRONT of your card(s).
3. You may ONLY touch or move your card(s). No one else may touch or move your cards.
AFTER the cards are arranged:
As a group, choose any two cards. Using the strategies you developed as you ordered the number cards, make a list of all possible numbers between those two cards. Be sure everyone in your group can explain all of the variations.
Individual Final Product:
Each person must describe in writing each different strategy for ordering the quantities on the number cards.
Explore until time is up.
Everyone takes turns.
(This is a placeholder page to insert the complex instruction task cards).