A New View of “Most Common Mistakes”

Lately, I’ve been thinking about the way we as teacher talk about students and their understandings and skills and their struggles. We often describe students as “low” or “high” depending on their performance on formal assessments (but we often fail to acknowledge that those are snapshots in time, informed by all of the students’ experiences). I’m sick of hearing teachers talk about “filling in the gaps” or how their students are always making the same mistakes. Maybe we need to stop viewing student mistakes and misconceptions as barriers to learning and start seeing them as the necessary rungs of the ladder to understanding instead. Some students can scale the ladder quickly, getting boosted up through certain misconceptions by prior/alternate experiences with math, whereas some students need to build up more muscles before they can move past a certain misconception. And sometimes, students appear to have passed a particular rung because they’re able to do other math, but then it later becomes apparent that the muscles needed to avoid falling on that rung (making that mistake) were underdeveloped and need more strengthening. So when we see students making “the same mistake” over and over again, we need to ask ourselves what experiences do they need that will create the kinds of cognitive dissonance that will unearth the limitations of their current understanding and grow their brains?

Right now, I’m focused on the “conceptual errors” students make in the predictable most common mistakes. I’m not concerned with the copying errors or rounding errors or even the computational errors (unless they’re related to the concept at hand).

I have spent a lot of time analyzing student thinking closely by detailing what do students know and not know based on exit tickets (low-stakes problems solved at the end of class and collected. I give feedback, but no grades on these). I’m currently in my 8th year of teaching middle school math, and I always notice patterns in the kinds of mistakes students make. There are certain trends that I can tell students in class that I can predict the ways they might go wrong even before they start! I even draw their attention to the connection between these “most common mistakes” (MCMs) and the wrong multiple choice answers (the distractors) – those other options aren’t random! They’re usually carefully chosen to highlight a student that has chosen the incorrect strategy to solve a problem.

Lately, I’ve been rethinking the fact that we call these mistakes at all. If they’re so predictable, maybe they’re not the same as other kinds of mistakes. Maybe they’re actually a necessary phase of the learning process.

I first heard about the Landscape of Learning from Kara Imm, but my understanding is that Cathy Fosnot developed it.

When you look at it, it’s made up of big ideas, strategies, and models that build on each other. There’s no one linear path through the landscape, and sometimes students will have mastered some of the big ideas, strategies or models, but not others. While some of these build on others (i.e. you probably can’t use double and halve if you don’t know how to double), other strategies, big ideas, and models are independent of each other, and depending on a student’s prior experiences, they may each have different strengths than other students.

I’m now wondering if these predictable misconceptions also belong on the landscape for learning as stepping stones because they inform our recognition of student understanding (and misunderstanding). These predictable mistakes are often ways that students overgeneralize “rules” or mis-apply new ideas in the wrong contexts. For example, every teacher who’s taught fraction addition to students has probably seen someone take 1/4 + 1/3 and get 2/7. And we can see WHY students are getting that number! But by calling it a mistake, we’re not recognizing the way in which that misconception builds on addition of whole numbers; we’re just acknowledging how it falls short on recognition of fractions as numbers. I am proposing that we shift the conversation here from saying “What’s lacking in this student?” to “What can this student do?” and then build on where the student currently is. If we recognize these predictable mistakes as stepping stones that some students need to have access to the main roads through the landscape of learning, then we are better able to see the pathways that students need strengthened in order to master big ideas, strategies, and models.

We can better understand why students hold certain misconceptions (and thus what experiences they need to complete their understanding) by seeing which big ideas, strategies and model were only partially connected to this misconception.

Going back to my previous example of 1/4 + 1/3 = 2/7, we might recognize that students don’t seem to realize the big idea of needing a common whole to add or subtract fractions. We might see that the clock model might help students move through this misconception, because they know 1/4 of an hour is 15 minutes and 1/3 of an hour is 20 minutes, and they can see on the clock that the combination of that is 35 minutes (or 35/60) of 7/12 of an hour. (In this case, because of the numbers I chose, the clock model works better than the money model, but for other numbers such as 1/4 + 1/5, that model would work better). The connections between these ideas and the fraction bars model may be less obvious – but that’s another model that students might rely on when adding fractions with unlike denominators. Additionally, a student could use landmark fractions to help them estimate that 2/7 does not make sense, since the sum of 1/3 + 1/4 should be greater than 1/2. All of these different connections can provide possible pathways from the misconception (add numerators and add denominators separately) that might help guide students back onto the landscape of learning and away from this common mistake.

I also think that if we understand that students are having a difficult time with addition of fractions with unlike denominators, this might help us step back and question whether the student has mastered adding fractions with common denominators – whether they understand that they’re just counting up pieces or if they are still adding the denominators. For example, when they see 1/3 + 1/3, do they say 2/3 or do they say 2/6? That will inform where the disconnect in their landscape is and help a teacher figure out the necessary experience for the student to gain this understanding.

In thinking about the class discussions I facilitate, I often think posing the student mistakes, asking them to make sense of what someone might have been thinking if they made that, and then discussing where it falls apart in the reasoning or why we need to do something else – and then being able to convince our skeptics – results in a much richer classroom discussion where students are actually growing.

I think we need to reframe the way we discuss mistakes as wrong answers and see them through the lens of stepping stones on the learning journey. We need to honor their importance in growing our minds, and that without those mistakes, we never realize that there is more need to grow because we can’t see the limits of our own understanding. According to the definition of mistake, it is “an action or judgment that is misguided or wrong.” I think calling these ideas mistakes indicates that they’re wrong and misguided when in reality, they’re often necessary stepping stones that are pushing us on the trajectory of learning. I think it’s time we stop calling them mistakes, and start naming them as partial understandings and consider how they connect to the understandings we want to cultivate.


Teacher Education in response to @teachbarefoot #MfAProud

This week, I saw a tweet by Taylor Grant:

and I have a lot to say about teacher education. My own experiences in grad school at an Ivy League institution left me woefully underprepared for my first year teaching in the South Bronx, to the point that my first principal gave me a “D” rating – D for doubtful! As in, it was “doubtful that I would be a good educator.” He told me he didn’t think I had what it took to be an effective educator, and that perhaps, I should reconsider my career choice.

Seven years later, I received a “highly effective” from my principal on last year’s rating. The five years in between, I received either satisfactory or effective from the second principal (I’ve worked for three now). Some of it was the fact that I was woefully unprepared for my first year of teaching, some of it was the extra level of support that I received in the second and third year at a new school and some of it was me deciding to use my research skills from SLC to find out everything I needed to know since no one was going to teach it to me! Plus, some of that was having the support of MfA and the MfA classes and cohort meetings and my mentor who took several of her observation sessions to teach me how to lesson plan instead of watching me deliver a poorly designed lesson.

However, I don’t think that just being in the classroom and teaching is what taught me the skills I needed to become a highly effective teacher. Rather, I think it was the mentoring I sought out, the articles I read independently, the courses I took through Math for America that helped me improve as a teacher. If I were to redesign the way teachers were educated in this country, I have some very specific ideas about how it should be done based on my own experiences.

Honestly, I think becoming a (secondary math) teacher should be a five-year process.

Summer 0: I think the How to Learn Mathematics course by Jo Boaler (the MOOC I did) should almost be a pre-requisite for entering teacher training programs for math educators. The combination of dispelling the myths about speed and instilling a growth mindset in ourselves and our students is so vital that I think it lays the foundation for everything to come afterwards. It doesn’t have to be her specific course (I think there’s room for it to be taught “live” by the professors of the education college itself), but I think it would draw heavily on her work. While it’s possible to make it a full eight week course, I also think it’s possible to shorten if the person entering the course could devote time each day to working on it rather than working on it in small chunks. I think this course should also include reading a Mathematician’s Lament, and developing an appreciation and understanding for the richness of math going far beyond just algorithms and calculations and manipulating symbols on the page.

Year 1:

I think this year should be primarily preparation to enter the classroom, giving some background knowledge on education pedagogies, philosophies, and psychologies. I also think it should be accompanied by observations where pre-service teachers get an opportunity to do observations in many different schools and then prolonged study in one classroom with the same group of students and teacher(s).

A)  Teacher Observations Seminar – I think teachers should be given training in how to observe students and teachers. They should learn how to take low inference notes and they should also be taught how to use rubrics for various types of observations, including paying attention to questioning, classroom discourse, classroom climate, teacher moves, etc. They should start with watching videos and norming some of their notes and observations based on those and then go into real classrooms. I think they should start with observing many DIFFERENT classrooms (both strong and struggling teachers) and discuss the positives and the areas for improvements they notice. For the longer placement in one classroom, though, I think they should be placed only in classrooms with identified Master teachers (more on that later) who are also allowed to be cooperating teachers/mentors later on. I would say they should be in classrooms starting on day 1 of the local school’s first day of school, and then they should observe teachers in different rooms & schools until approximately November. Around November, they should select one or two classrooms to be placed in for the rest of the year: they’ll switch in February, after mid-winter recess to their second placement for observations. In this case, they’ll shadow a master teacher, but they won’t teach any classes. They will, however, be available to tutor students one-on-one during the master teacher’s classes or under direct supervision.

B) Child Development – I think to the course I took with Barbara Schecter in grad school at Sarah Lawrence College, and I think about the first-hand education I got about child development from reading Vygotsky and Piaget and how helpful that is in understanding appropriate levels of child development (i.e. the level of abstraction that children are ready for at various ages). So I think there should be a course focused on child development, adolescent development, and child psychology. What I think is so often lacking from these classes though is a connection to real children and to students and learning, so I think it would be interesting if students in this class were required to write a case study about a specific child or a few children that they focused on in their observations. I would start this class with an overview of some of the literature from Dewey to Piaget to Vygotsky to Eleanor Duckworth (and her article, The Having of Wonderful Ideas). They’ll be asked to write about how their observations in the classrooms align with some aspect of their readings, and to conduct interviews with students to ascertain a better understanding of how students develop their thinking. I think hearing from students directly will also be very informative for these pre-service teachers.

C) Landscapes of Learning Mathematics or From Cardinality to Calculus – I think every math teacher needs an overview of how students learn math, what students at various grade levels are expected to know and do and how the learning builds on to itself. I think of a year-long course I took with Kara Imm at MfA about how mathematics develops from subitizing (which I’d never even heard of before!) through counting, then adding, subtracting, multiplying, dividing, proportional reasoning, algebraic thinking, and so on. I think it’s important for teachers to experience learning the “foundational” mathematics deeply, and I remember using Reknreks and number strings and problems from Cathy Fosnot’s Young Mathematicians at Work series for this course. I think it’s valuable to start with young children and work our way up through K-12 math learning and ending with what “higher levels of mathematics” looks like at the college level for math majors and science/engineering majors as well. I think it’s important for math teachers to have a good picture of where their students are situated among the large landscape and without seeing the big picture, how can they understand their particular role? I also think this course can distinguish between the standards of what math topics/skills/understandings are supposed to be developed at specific times, and how children actually learn math. I think DOING math and experiencing it as a learner should be a major focus of this course – but it should be about doing ALL of the math from elementary school through high school/early college. It’s more of a “survey” of the math, with roughly two or three class sessions dedicated to each grade level/course’s major work. Embedded within this learning of the math is also pedagogy of how to teach these topics. Yes, in my vision the specific types of instructional techniques are specific – I think experiencing manipulatives, number strings and number talks, math congress, and instructional routines from fostering math practices (such as contemplate then calculate, 3 reads, connecting reps, etc.). I think the avenues for thinking should also be included in this course, as it helps us make sense of problem solving. I also think reading Liping Ma’s book Knowing and Teaching Elementary Mathematics should be the launching point of this class so that the teachers understand how a profound understanding of mathematics is required to teach even seemingly “simple” math. I also think this course should include analyzing student work to ascertain their understandings and their misconceptions and look at student mistakes. I think this would be a great class for the pre-service teachers to get practice giving students written feedback.

D) Understanding student behavior as communication – I think about the courses I did with Ramapo for Children that was based upon Ross Green’s work with collaborative problem solving, and I think teachers immediately need training in these skills of identifying unmet needs and lagging skills and how the “teacher toolbox” is organized. By having this course in the first semester, when teachers are doing their observations, they can be asked to identify teacher responses to student behavior through these lenses. I think there can be a focus on each of the four aspects of the toolbox (relationships & role-modeling; clear expectations, structures and routines; adapting for individual needs; and responding, reflecting and repairing) and how the pre-service teachers may want to structure their own classrooms accordingly. I think this course should begin by discussing potentially challenging student behavior and the underlying assumption that when students CAN do better, they DO better, and if they’re acting out, it’s because of an unmet need or a lagging skill – not a malicious attempt to make a teacher suffer. I recall the classes Rachel Lissy taught at MfA on this topic, and how she would always start the first session by saying that if you are a teacher, you are going to have to deal with challenging behavior – because that’s how children communicate – and I think that’s an important message to share with teachers right from the get-go.

E) Acting and Improv skills – I think the final course for year one should focus on the  ability to modulate your voice, the way you hold yourself, the non-verbal communication we’re unaware of, and the development of the “teacher voice.” I think we often act as though the teacher voice is something that needs to be developed with real students, but I actually found that the way Teach Like a Champion breaks it down made it learnable and then the courses I took at MfA on improv skills for the teacher also made me learn how to find my teacher voice to the point where I rarely have to raise my voice anymore and yet students in the hallway who don’t know me will stop running and walk, etc. I also think this would be a good place to practice “faking it until you make it” with the right types of improv assignments to strengthen your skills of responding to anything students might say.

And then, maybe it’s because I went to Sarah Lawrence College, where we believed in an in-depth and long-term study of the same topic was valuable, but I think these five courses should be a year-long study accompanied by observations in classrooms. My vision for year one of this program would be that each of these classes would meet one evening per week, and during the school day, each of the pre-service teachers would be in classrooms. So it would be very intense study (school all day and into the evening), but I think it would be worth it.

Summer 1:

Over the summer between year 1 and year 2, I think the pre-service teachers should become teaching assistants in summer schools during the day.

They should also begin taking courses on students with disabilities, students who are English Language Learners, gifted students and other types of student diversity (students of color, LGBTQIA students, etc.). I think these should be a few parallel courses that discuss ways to ensure these students are not seen by their deficiencies or differences, but rather are seen as whole people and what teachers’ roles are in supporting them. I think the summer courses should be providing pre-service teachers with background about racist, sexist, and homophobic/transphobic structures in the school system and give these pre-service teachers tools to challenge them. That summer, they should also get an introduction to restorative justice and how to deploy circles or peer intervention strategies to support students who struggle to follow classroom expectations. I think much of what’s currently covered in the DASA course in NYS, the child abuse and suicide prevention courses should also be rolled into these courses.

The second year would begin their student teaching placements. I believe they should do the fall semester in one class and the spring semester in a different class, perhaps even switching between high school and middle school. I do think it’s important for them to do two weeks of observations in the teachers’ rooms focusing on the routines and classroom expectations that the teacher has, as well as learning student names and getting to know the kids. I think each of their courses this year should include things they should be expected to try out in their teaching experiences. I also think they should take it slowly, starting off with teaching one aspect of a class (such as going over the warm-up or giving instructions for a gallery walk), and build up to teaching a whole lesson.

In the second year, I think the courses should be as follows:

A) ALL-ED (all learners learning every day): this would be a course specific to teaching how to make differentiated learning practical. I think this course  would be vital for helping the pre-service teachers learn how to meet the needs of all of their learners, and also to identify the “edges” or “extremes” in the classroom that need support or challenges. The pre-service teachers would be required to identify two or three extremes in their host classroom and design structures/routines to support them and try them out.

B) Problem Posing and Creating Mathematical Headaches – I think about the work that Dan Meyer has done and how important it is to hook students in the need for a solution before teaching students about a new topic. I think the idea of focusing on how to structure a lesson (or a unit) as the creation of a mathematical headache and then discovery of something to cure that headache. I think learning how to do that is important. I also think this course would involve learning how to teach students by sharing patterns with them to evoke their noticings. I also think this class would include instruction on notice/wonder and how to use that to use students to drive instruction.  Pre-service teachers would be required to design a lesson through this course, they would get feedback on it from the other members of their class before trying it out.

C) Rich Tasks, 5 Practices & Talk moves for the teacher – Nothing has had such a profound impact on my teaching as discovering the 5 practices, learning about rich tasks, and learning how to facilitate student talk in a way that was NOT focused on “IRE” (or initiate, response, evaluate) talk models. I think this class would include reading the Smith and Stein book on the 5 practices for facilitating productive discussions in the math classroom, as well as discussing the different kinds of math talk. I think there would be a task sort and identifying the different levels of richness. Pre-service teachers would get the opportunity to discuss the importance of anticipation, go beyond just “show and tell,” and understand the importance of teaching math through problem solving rather than teaching math to problem solve. I think the 5 practices are so profound in being an effective teacher that it’s vital they get a chance to use them before entering their own classrooms. I’m also thinking about the article “Never Say Anything a Kid Can Say” as one of the articles included in this course. I also think conferencing/conferring with students and giving students feedback should be part of this course.

D) Instructional Routines – I think this course would focus on the different instructional routines that Amy & Grace have developed, and give the teachers ample experiences designing and teaching them to their peers. They would have the opportunity to be coached through the experiences, and while they wouldn’t get experience doing it with real students in this class, they would get experiences using the adults as students to uncover the mathematical thinking. I also think that this second year would be when they would do their first student teaching placements, so they would be required to try out some of these routines in their host classrooms.

E) Problem Solving and Deepening Math Understanding – I think at this stage, either people could focus in on middle school or high school math OR they might do both, but switch each semester to match which grade level they were teaching. I think here would be an opportunity for the pre-service teachers to deepen their own mathematical understanding by taking seemingly “simple” topics and learning the full trajectory of those topics. For example, doing PCMI-style explorations of terminating vs repeating decimals and understanding that the prime factorization of the denominator will determine both whether a decimal will terminate or repeat and how long the period is might be an example of a problem that would relate to a seemingly simple aspect of math that middle school students are instructed in (i.e. decimal/fraction conversions), and yet take it to a deeper level of generalized understanding. I also think this course should highlight the work done by Nix the Tricks and give teachers experience with when and how certain ideas expire (like PEMDAS). The pre-service teachers should get exposure to the limitations of mnemonic devices and exposure to the richness of big ideas in math so they can better understand how it all fits together. I’m thinking about that article on big ideas in math as also being the launching off point for this class.

I don’t know whether this idea belongs in this course or if it needs its whole own course, but I also think the idea of “a survey of math topics” would also be good – I never learned topology, and yet I think there are so many ways in which it could be incorporated into math earlier than college. So I think there should be a brief overview of a variety of different branches of mathematics so that math teachers can support student interest and joy in mathematics.

F) Student teaching seminar – In addition to the other courses, I think the pre-service teachers need a place to reflect on their work in their host classrooms. I think the cooperating teachers should be required to attend this course as well (I haven’t decided if they should be there every week or just certain weeks), so as to provide some background on the students or the class/school itself that the pre-service teacher might not be aware of. I also think having Master Teachers (experienced educators!) in the room will support the new teachers in deepening their understanding rather than perpetuating myths or misunderstandings. I think this course would also include viewing videos of the student teachers and giving each other feedback on the work they’re doing in their host classrooms.

I think at the end of this year, teachers should graduate with their degrees and be certified as “apprentice teachers.” I think as an apprentice, you should be allowed to teach a reduced schedule with a master teacher mentor.

And here’s where my design ideas go into overdrive. I think it would be very important for the fledgling teachers to get meaningful mentorship. I think they should be paired with a master teacher whose pedagogy matches their own educational philosophy. They should be required to do a three year placement (I think with the same person the whole time, but I could see how flexibility might be required for any number of reasons). Let’s say that 4 classes is the standard teaching load. I think in the first year, the apprentice teacher should be lead-teacher on one, observe three of the Master Teacher’s. Ideally, programming would work out such that the one they lead is an “identical prep” to one their Master Teacher is teaching and they would be doing it after the MT’s class. They would have identical schedules and the apprentice teacher would shadow the master teacher.

In year two of the apprenticeship, the apprentice teacher would lead two classes and the MT would lead two, with again co-planning time required and available.

In year three of the apprenticeship, the AT would have a full course load and the MT would have a reduced course load of only two classes. The MT would be in the AT’s classroom for two classes, observing and giving feedback and planning. But they would take a much more background role by this year than previously.

I think after these three years, an apprentice teacher could “graduate” to teacher. A teacher would be eligible to be a lead teacher for their own classroom, but NOT host a student teacher or be a master teacher. I do think a teacher could be someone who the pre-service teachers observe.

I think by year 5 of teaching (two years of being completely independent) would be the first opportunity a teacher would have to apply to become a master teacher. I think there should be an application similar to MfA’s application, focusing on knowledge of content, of students, and of pedagogy in order to be selected. I think current Master Teachers should approve the application of Master Teachers, and it should involve a classroom visit as well as a video lesson. I also think a student recommendation as well as a supervisor recommendation AND a peer recommendation should be required. I think becoming a Master Teacher should be a prestigious accomplishment.

In my ideal world of teacher education here, I think Master Teachers should have a reduced teaching load, should often mentor and host either student teachers or apprentice teachers, and I think THEY should be the ones who are recruited to teach the pre-service teachers their evening classes! I don’t think education professors who are often researchers should be doing that instruction, at least, not in any of the methods classes.

I’m sure that there are things I’ve missed in this ideal teacher education program. And I acknowledge that making it require two summers as well as two full years of full-time studenting makes it potentially difficult for people to afford to do that. So I’m by no means suggesting my ideas are perfect. In fact, I would say this is the first time I’ve actually written out my ideas, though I’ve had these ideas swirling around in my head for some number of years. I also acknowledge that some of the ideas I’ve lumped into one class might be too ambitious to include in just one class.

I look forward to hearing your thoughts on this proposed design and refining it through your feedback!