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In the Moment

I have always been a little bit of an anomaly in my career field. Although I’ve worked on various education projects and materials for over ten years now, but I’ve never been a classroom teacher. I was a math major as an undergraduate student, with little interest in education at the time. It was only through a random search for “mathematics” in a job search engine upon graduation that I ended up with a temporary contract position in educational publishing. After that, I went to graduate school to get a masters in learning sciences, then to the university for ten years of work in curriculum development, and finally, this year, I started a PhD in educational psychology and educational technology. I taught a couple of recitation sections of a math course as an undergraduate senior, and I’ve taught brief lessons in various classrooms on occasion, but other than that, I have no teaching experience.

People ask me sometimes why I didn’t choose to go through a teacher ed program and teach for a few years once my interest in education started to develop. There are a couple of related answers to that question. One is that it felt somehow insincere to enter teaching with the explicit intention of leaving it after a couple of years. Another is that I didn’t (and still don’t) think I’d be particularly good at it. I find it difficult to develop good rapport with kids, and I am not a quick thinker. When I’m in a conversation that doesn’t go the way I expect, I am almost never able to respond intelligently in the moment. It takes me a while to work through things. That’s not possible when 30 students are sitting in front of you, waiting for reasonable answers to their questions.

The constant need for in-the-moment decision-making, in my opinion, is among the most difficult aspects of teaching. The very thought of it is one of the major reasons I never pursued teaching. The willingness and ability to handle the constantly changing course of conversation in classrooms is the thing I admire and respect most about teachers. I have always been too scared of the idea to even try.

As a researcher interested in processes of learning, I’ve always been curious about the ways in which teachers learn to be agile and responsive in their teaching. I know that it’s something that develops over time with practice and experience, but are there ways of helping that development along? How does that play out in teacher education?

This week I read a couple of pieces that answer that question, at least in part. Crespo, Oslund, and Parks (2011) asked pre-service teachers to imagine and write out narrative scripts of how they thought their lesson plans would play out in their classrooms. What would students give as answers to particular problems? What would the teacher say in response? How would the conversation progress toward the mathematical goals of the lesson?

The researchers explored the idea of having preservice teachers write out enactments, rather than focus exclusively on lesson plans, because they felt that the former would allow a different window into teacher thinking. “Consider the difference between writing out a teacher action plan and acting out that plan (even if on paper). Although both of these serve to map the intellectual journey and destination of a math lesson, they do so from different perspectives. The first takes a bird’s eye view perspective, while the second takes a ground level viewpoint” (Crespo, Oslund, & Parks, 2011, p. 121).

I found this contrast fascinating, because it clearly articulated the difference between planning lessons and teaching lessons in a way that starts to get at the challenges of in-the-moment thinking that happens while teaching. Writing out scripts of possible enactments is an interesting way to prompt teachers to think about the possibilities of that in-the-moment work, in a context that doesn’t have the same time pressure for thinking through responses.

The Crespo et al. (2011) article discusses script-writing as a method for gaining a window into teacher thinking. That is, they discuss it as more of a research tool than a teacher education tool. When I brought up the prospect of using scripting as a technique for helping teachers develop skills in responding to students in the moment, my classmates and instructors rightfully brought up a concern that writing out a script might make teachers less apt to deviate from that script if something plays out differently than they expect in the classroom. I understand this, and so don’t really believe that hypothetical scripting of lesson should be part of daily lesson planning. I do, however, still wonder if making the creation of multiple possible scripts for the same discussion, done in the right context at the right time, could be beneficial to developing flexible thinking, that in turn helps teachers to be flexible in moment-to-moment thinking.

The second article I read this week that spoke to this particular issue was Lampert et al.’s (2013) study of rehearsal as a teacher education technique. Lampert et al. explain that rehearsal “can involve novices in publicly and deliberately practicing how to teach rigorous content to particular students using particular instructional activities” (p. 227). During rehearsals, preservice teachers teach a lesson to their peers and teacher-educators, with the teacher-educators calling out feedback and sometimes interrupting and pausing the lesson to discuss what just happened and ways a situation might have been handled differently. Rehearsal, like scripting, was appealing to me because it represents a way to examine and develop in-the-moment thinking with a built-in capacity to give teachers a bit of extra time to think things through and consider possibilities.

What do you think? Is one of the keys to developing skills in responsive, in-the-moment decision-making providing ways to slow down time a bit, as in scripting and rehearsal? What other ways of developing skills have you heard about or experienced?

And if you’re a teacher: Does it get easier over time? What has helped you feel prepared for the unknown trajectories of conversation in a classroom?

References

Crespo, S., Oslund, J. A., & Parks, A. N. (2011). Imagining mathematics teaching practice: Prospective teachers generate representations of a class discussion. ZDM – International Journal on Mathematics Education, 43(1), 119–131. https://doi.org/10.1007/s11858-010-0296-z

Lampert, M., Franke, M. L., Kazemi, E., Ghousseini, H., Turrou, A. C., Beasley, H., … Crowe, K. (2013). Keeping It Complex: Using Rehearsals to Support Novice Teacher Learning of Ambitious Teaching. Journal of Teacher Education, 64(3), 226–243. https://doi.org/10.1177/0022487112473837

 

Are we removing a barrier, or just shifting it?

About six months after completing my master’s degree, I went to a holiday party thrown by one of my former classmates. In my particular program, master’s students take many classes with first-year Ph.D. students. The attendees at the party were a mix of the Ph.D. students I knew — now in their second year of study — and new master’s students that I had not met before that night. At some point, one of the Ph.D. students, who was my officemate the year before, made a comment to one of the master’s students that I still remember.

“We have ten weeks to finish the work of a quarter, right? Katie was always done in seven.”

While finishing three weeks ahead was an exaggeration of my general pattern, I couldn’t deny that I was usually ahead of the pack in terms of finishing course requirements. It wasn’t that I was faster than everyone else when it came to reading or writing. I may be a bit above average on writing speed, but I would bet that I’m below average on reading speed, and usually slow to organize my thoughts, as well. Speed wasn’t the issue. And I really struggled with a lot of the coursework in grad school. Ease versus difficulty wasn’t the issue, either.

The reason that I tended to finish early, and still tend to do this, is this: I live in perpetual fear of running out of time.

While I do have isolated bouts of procrastination, it definitely isn’t my biggest vice. I start projects the day they are assigned just in case I end up making a false start and need more time than I anticipate. If I happen to have extra time during one week, I tend to read ahead in my courses just in case something big comes up the next week. I love to be done early. That’s the only time I feel like I can be justified in not working on something.

I have long known this about myself, but I started to think about it in a bit of a different way after revisiting the work of Carol Dweck (1999) and Jo Boaler (2012).

Dweck’s (1999) work focuses on the idea of fixed versus malleable conceptions of intelligence. Student who believe that intelligence is malleable and they can become more intelligent over time are more likely to choose and persist at challenging tasks. Student who believe that intelligence is a fixed tend to avoid challenging tasks. Among Dweck’s more prominent findings is the fact that the kind of praise given to students when they succeed can have a strong effect on their conceptions of their own ability and their reactions to failure. Students who are praised for being smart develop a fixed theory of intelligence, and tend to react to failure by starting to believe they are not smart after all. Students who are praised for their effort develop a malleable theory of intelligence, and tend to persist in the face of challenge and try again upon failure. Building on Dweck’s theory and findings, Boaler (2012) has conducted studies of interventions aimed at helping students develop a malleable view of intelligence, or a growth mindset.

Dweck and Boaler are among my academic heroes. A big part of my reason for starting a career in education research is my desire to help put an end to the prevalent I’m just not a math person narrative, and hopefully help to prevent a similar I’m just not a computer science person narrative from becoming as common. Growth mindset interventions have a lot of promise in this regard.

At the prompting of one of my instructors earlier this year, I took some time to think about whether I have a growth mindset. I was pleased to come to the conclusion that I think I do. There isn’t much that I don’t think I could learn, if I took the initiative to do so. What I found interesting about this reflection, though, was the caveat I felt I had to add:

I could learn just about anything…. as long as I had enough time. Some things might take me a very, very long time.

This was interesting to me from both personal and professional perspectives. On a personal level, it helped me to explain my perpetual anxiety about running out of time and how that coexists with a solid sense of my own academic ability. I think I’m perfectly able. I just think that there’s always a risk of things being slow and difficult. Not impossible, but slow and difficult. I’ve never articulated that belief before. It’s helped me to reconcile my belief in malleable intelligence with the existence of sports stars, musical prodigies, and the like. Everyone can learn anything they like, with sufficient time, resources, and motivation — but this does not mean that each person will have to do the same amount of work to get there. There is such a thing as a natural proclivity. The power of growth mindset is that it shifts this idea of natural proclivity from separating the world into cans and cannots — something that is false — to identifying the those who have a quicker path to being highly skilled among a huge population of capable people.

From a professional perspective, this realization helped me to think more carefully about the problems that growth mindset work does and does not help to solve. I think many students carry around a belief that they are not good — and cannot be good — at mathematics, and that particular issue can be addressed with growth mindset work. Even so, the interventions won’t remove all potential anxieties about mathematics. Even if a student comes to believe he or she can indeed learn mathematics, that won’t necessarily lead to the belief that it will be easy, enjoyable, or worthwhile.

On one hand, perhaps this is ok. The goal has never been for all students to become mathematicians — just to help them attain a certain level of mathematical fluency and efficacy. One can hope that any overpowering anxieties about slowness and difficulty could be kept at bay through K-12 education. At that point, perhaps a greater number of students will pursue mathematics-related careers, and the rest will have reached a point of mathematical literacy that makes them effectively able to interpret and use mathematics to understand day-to-day issues.

On the other hand, there’s something about this line of thinking that bothers me. Is it a slippery slope? Is “Everyone can do math, but some people can do it more easily and faster than others” really a better narrative than “Some people just aren’t good at math”? I’m not so sure. My perpetual assumption that I need to plan for things to be difficult and time consuming isn’t something I want all kids to be carrying around.

The simplistic solution is to just change the narrative to “Everyone can do math,” and leave it at that. The decision of whether or not to pursue mathematics is then up to the student, and hopefully any decisions made won’t be based on worries about fixed intelligence.  I just worry that the second part, “but some are better than others,” will be carried around after it, implicitly, if we don’t address it head on. Then we won’t really have fixed the problem — just shifted the barrier a bit.

I am sure you can tell by my rough writing here that I have not worked out all of my thoughts on this. But it’s something I’ve been thinking about, particularly when considering how I can use lessons learned from mathematics education to help to prevent a not a CS person narrative from taking hold as CS is introduced into K-12 schools.

References

Boaler, J. (2013). Ability and mathematics: the mindset revolution that is reshaping education. FORUM, 55(3), 143-152.

Dweck, C. S. (1999). Caution—Praise Can Be Dangerous. American Educator, 23(1), 4–9.

Digital, Dimensional, or Dynamic?

Yesterday I had a conversation with some fellow curriculum developers about the potential impact of translating hands-on activities into a digital environment. It was quite interesting and engaging — and like most interesting conversations I have these days, it left me much more confused than I had been when the conversation started.

We started by talking about 3-dimensional geometry, and the potential perils of studying it 2-dimensional representations on paper or on a screen. Though none of us had research at our fingertips that spoke to this (although I’m reasonably certain such research exists), at first we generally agreed that there are likely important benefits related to manipulating physical, 3-dimensional objects when studying 3D geometry and volume.

Then someone mentioned the idea of a digital environment with a 3D rendering of a physical object. What about a cube, for example, that can be rotated and viewed from any angle? Does this have the same benefits as handling a physical cube?

In other words, is 3D versus 2D the difference that matters? Or is it dynamic (and thus manipulable) versus static? Or is the really important difference the one that started this conversation in the first place: physical versus digital?

If the critical factor is 3D versus 2D, and true 3D representations are what work best for learning, then certainly the prospect of designing a curriculum that is delivered entirely digitally is a problem. This sort of argument is often used against the trend toward digital curriculum products in classrooms, especially for young kids. But this conversation with my colleagues helped me articulate something that has always bothered me about that argument. It’s this: if 3D versus 2D is the critical difference, than digital curricula aren’t the only curricula with problems. Traditional print curricula so often rely on 2D representations of 3D objects — static ones, on paper. So if there are going to be 2D representations anyway, aren’t the digital, dynamic, 2D representations just as good as the paper, static, 2D representations — and perhaps even better?

This chain of reasoning led me into thinking about digital alternatives to other physical objects used in classrooms. If the object of study isn’t something 3-dimensional, do the same kinds of concerns apply? Take base-10 blocks, for example. The physical blocks are 3-dimensional and the digital blocks are 2-dimensional, but that difference doesn’t seem to matter in this case, since neither version is a representation of a 3D object, but rather of an abstract number. Both versions are manipulable, but the particular manipulations are different. With the physical blocks, if children want to think of 1 ten as 10 ones or vice-versa, they must exchange a long (or ten block) for a cube (or one block). In many of digital versions of the blocks, if children want to think of 1 ten as 10 ones or vice-versa, they are able to break apart 1 long or put together 10 cubes. In that sense, the digital blocks are more dynamic than the physical blocks.

Sarama and Clements (2009) further point out that the action of breaking a long is closer to the mental actions children must make on numbers to add and subtract than the action of exchanging a long is. Since the first step toward abstract understanding of a concept is typically interiorization, or the ability to carry out a process through manipulations of mental representations (Sfard, 1991), this “closeness” to the mental manipulations of numbers could be important.

A study by Bouck et al. (2017) gives another example of how a digital tool accelerated the process of interiorization, in this case for students with mild intellectual disabilities. Previous research had shown that leading these students through a concrete → representational → abstract sequence was effective in supporting learning of addition of fractions with unlike denominators. The concrete phase involved physical manipulatives, the representational phase involved creating drawings, and the abstract phase involved symbol manipulation. This study found, however, that when digital manipulatives were used, the representational phase could be skipped entirely. For three out of four students in the study, a virtual → abstract sequence was effective. This suggests that the digital manipulative helped students to interiorize the process of fraction addition.

My point in giving these examples, I suppose, is to illustrate why I’ve come to think that the debates about digital versus physical mathematics manipulatives are placing too much emphasis on physical versus digital as the difference that matters. In some cases, digital versions have affordances that physical versions do not. I’m sure it’s also that case in reverse.

I’ve been struck, in the past, as I’ve noted that there really aren’t many studies that pit physical and digital versions of manipulatives against each each other in terms of effectiveness. It seemed like such an easy and straightforward area of inquiry. Now I think I’m coming to understand why so little research is formulated that way. Studies that show a difference between physical and digital tools for learning would only open more questions about which affordances of each version led to the differences. Digital versus physical isn’t really what matters.

References

Bouck, E. C., Park, J., Sprick, J., Shurr, J., Bassette, L., & Whorley, A. (2017). Using the virtual-abstract instructional sequence to teach addition of fractions. Research in Developmental Disabilities, 70(June), 163–174. https://doi.org/10.1016/j.ridd.2017.09.002

Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145–150. https://doi.org/10.1111/j.1750-8606.2009.00095.x

Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22, 1–36. https://doi.org/10.1007/BF00302715

 

(in)visible

Recently I’ve been thinking about the implications of feeling visible or invisible, and how the implications vary by context. On one hand, invisibility can be a sign that a community finds a person or population uninteresting, inconsequential, or even necessary to hide. (Think Mr. Cellophane.) On the other hand, invisibility can be a form of protection. (Think of invisibility as a superpower.)

I started thinking about this after reading two papers on the role of race in mathematics education. First came Maisie Ghoulson’s (2016) striking piece on intersectionality, where she demonstrates the ways in which Black girls and women are invisible in mathematics. The invisibility isn’t inevitable. Rather, it is constructed. Ghoulson points out that Black women are used as props to justify research that excludes them. When researchers discuss the difficulties facing the Black community as a whole and then pivot to discussing the particular issues facing Black men, for example, Black women drop out of the discussion. Because of this, we don’t know how Black women are doing in mathematics. We presume they’re doing fine, when really, we have simply made them invisible.

This piece, naturally, got me thinking that invisibility is perilous.

A few days later, though, I read a study by Niral Shah (2017) about the ways that racial narratives relate to mathematics education. There’s a prevalent narrative, for example, that Asian students are particularly good in mathematics, and another that says Black students are not. Shah’s study found that these narratives about the role of race in mathematics education are quite often evoked in relation to each other in ways that imply a hierarchy. One of the excerpts from his study, for example, was this, from a Black student: “Like I’m Black but I’m good at math. So are you not going to ask me for help because I’m not Asian?” (Shah, 2017, p. 23). This statement places Asian students high on the hierarchy and Black students low, giving each group not just an absolute position, but a position relative to each other. Meanwhile, narratives about White students did come up occasionally, but not nearly as often. The White students, in this case, were invisible — which, in a sense, gave them a pass from being ranked according to their race.

So, perhaps invisibility is not always perilous? What makes the consequences so different for Black girls than for White students? I know, of course, that the contexts of the invisibility are quite different. Invisibility from attention in research is different from invisibility in racial narratives. That alone, I suppose, could be enough to make this comparison uninteresting. But something about this explanation just wasn’t satisfying to me. It was as if the comparison was nagging at some dusty memory in the back of my mind that I couldn’t quite identify.

I was lucky enough to have the chance to speak to Dr. Shah about his article a few days ago. I posed this comparison to him and asked what he thought. He said lots of interesting things in his response, but the thing that really gave me a light bulb moment was this: Dr. Shah suggested that perhaps thinking of White students as invisible isn’t the most illuminating way of thinking about visibility in his study. Rather, maybe the effects of the racial narratives are to make Asian students hyper-visible. Further, Dr. Shah said he wasn’t convinced that being hyper-visible was beneficial to anyone.

This made perfect sense to me, and helped me think differently about the narratives about Asian students and those about Black students. For the Asian students, the narratives could lead to high expectations based on race that, for some, could be hard to attain and lead to undue pressure. For Black students, the narratives could lead teachers to devalue their contributions to mathematics discussions, which in turn could lead to a kind of invisibility.

Thinking about this also surfaced that nagging memory I suspected was in my head somewhere. When Dr. Shah used the term hyper-visible, I suddenly remembered writing this Facebook post on the day that Hillary Clinton got the Democratic party nomination for President:

invisible screenshot

I know what it means to be hyper-visible, and while the consequences are different than being invisible, they can still have detrimental effects.

Once again, I find myself with a new frame for thinking about a complex issue related to teaching and learning. Grad school is good for that.

What do you think? What are the potential effects of invisibility and hyper-visibility? What does this mean for teaching and learning in classrooms?

References

Gholson, M. L. (2016). Clean Corners and Algebra: A Critical Examination of the Constructed Invisibility of Black Girls and Women in Mathematics. The Journal of Negro Education, 85(3), 290–301.

Shah, N. (2017). Race, Ideology, and Academic Ability: A Relational Analysis of Racial Narratives in Mathematics. Teachers College Record, 119(7), 1–42.

Down Memory Lane

The past two weeks, I have had a hard time concentrating on any one task. It’s the point during the semester when several papers and projects are coming due at once. I’ve started drafts of many of them. Try as I may to concentrate on one at a time, my subconscious never seems to be working on the same one as my conscious brain. In the middle of a sentence about computational thinking, I suddenly realize one more thing I should have said about digital curriculum resources. Oh, and that thing I just read about qualitative research methods? I really need to write down the way in which it’s connected to that study of abstraction I read last week.

I was reflecting on this phenomenon tonight, and thought of a blog post I wrote many years ago — on November 22, 2010, to be exact. The following text comes from the post, where I’m trying to decide why so many university professors wear similar clothes day after day:

“Over the past few years, I have had the privilege of working with some of the most brilliant minds in my field. My bosses are the rock stars of mathematics education. And I can tell you the one thing they have in common, the one thing that has set them apart: they are always thinking. Always. Their minds work like conveyor belts; when they solve one problem, all the rest are waiting on the belt. They never stop thinking about their work.

This means that the rest of the things that us average joes think about — like the clothes that we wear — don’t get any time in the conscious brain. Academians just operate on autopilot. You know how, when you wake up late and have to rush out, you end up having no memory of choosing your clothes? I imagine that it’s like that for academians all the time.

People in all walks and phases of my life have branded me as a smart person. I was on top of my class in high school, and no one was surprised when I graduated from college cum laude or got accepted to a prestigious graduate school. Many of these people are probably expecting me to get a doctorate someday. It’s just what smart people do. But this line of thought has only made me more sure that academia is not where I belong.

The people that succeed in academia are the ones that choose their clothes on autopilot. I, on the other hand, go to autopilot while I am running so that I can spend that time deciding what I will wear that day.

I could never become a research scientist or professor. I love cognitive science and I love math — but I also love the moments when I can stop thinking about them.”

 

Are you laughing? I laughed. I wonder when I crossed the line. I never stop thinking about math and cognition and learning now. But hopefully that means that I have ended up where I belong.

Abandoning the Perfect Match

As you have probably noticed by now, the role of context and applications in mathematics education is a key interest of mine. Over the years I have come to believe more and more firmly that not all mathematical learning has to be in context, and that it is counterproductive to attempt to teach everything in context. Forcing context on mathematics often leads to meaningless and contrived problems that almost certainly do nothing to enrich learning.

On the other hand, I also believe that when clear and relevant connections can be made between mathematics and other content domains or broader everyday-life issues, they should be. Mathematics, after all, was developed to solve real problems, and some connections can be productive. It is helpful, for example, for children to think about putting things together when they do addition and creating equal groups when they divide.

This seems like a very pragmatic, and maybe even obvious, position to take: Use context when it makes sense. Yet many times in the course of my career I’ve been struck by how difficult it is to strike the right balance. Take, for example, the relatively simple mathematical topic of graphing a linear inequality. (Full disclosure: What follows is a retelling of the process I went through to write a lesson on this topic during my curriculum development years.*)

To graph something like x > 3, you draw a number line, draw a circle on the 3 mark, and shade everything that’s greater than 3, like so:
NumberLine1

And it’s pretty simple to come up with a context for an inequality. For example, suppose a library has a rule that a patron can check out no more than 4 books at a time. If b is the number of books, we can represent the situation with the inequality b ≤ 4. A graph of that inequality looks like this:
NumberLine2

But wait… if this is a representation of the context, then we really shouldn’t have any values of b less than 0. You can’t check out -2 books. So maybe this:
NumberLine3

Well, and you can’t really check out a fractional part of a book, so what we really want is this:
NumberLine4

Okay. Now we’ve got a reasonable representation. But creating one brought out some other mathematical ideas, like defining domains and discreteness versus continuity, that I didn’t expect when I set out.

I don’t mean to argue that it was too hard to make a context work in this instance. My colleagues and I were able to finesse these ideas into a lesson that we thought did justice to both what we intended to be the main mathematical idea (graphing inequalities) and these other mathematical ideas. Rather, my point is that the connection between the mathematics and the context wasn’t simple. There wasn’t a one-to-one correspondence.

My point in telling this story is this: Lately I’ve been thinking that perhaps a lot of my troubles — and to the extent that I can generalize, the troubles of the mathematics education community — in connecting mathematics and context is that we’re focused on finding a perfect match. We dismiss potential ways of connecting mathematics to life outside of school because we’re afraid that either the mathematics or the context becomes skewed, oversimplified, or contrived. But maybe it doesn’t need to be. Maybe instead of adjusting the math or the context to achieve a fit, we can acknowledge that the match is imperfect, and take it for what it is.

I started thinking about this last week after reading a book chapter describing a researcher’s experience teaching a class on mathematics for social justice to high school seniors in a Chicago community. Gutstein (2012) describes how he structured the course around “generative themes” that were suggested by the students in the class. I was particularly intrigued by his discussion of a unit on the spread of HIV/AIDS, where students used complex mathematical models to simulate the spreading of the disease and had in-depth discussions about some of the more striking statistics showing the disproportionate effects of the disease on certain populations.

At the time, he was concerned that the mathematics and the context were not well aligned: “[S]tudents did not need to model HIV/AIDS to understand the disease, grasp the disproportionate impact on certain populations, or not blame Black women” (Gutstein, 20102, p. 35). Later, he considered the idea that this may not be a problem, but rather just an example of a different kind of connection: “At times, we use mathematics to explain social things (like the election being stolen), at other times, we use social analysis to explain mathematics (like high AIDS rates). The point here is, I think, that you cannot easily explain one without the other” (p.35).

Sometimes, mathematics directly helps solve a social problem (like explaining the results of the 2008 election). Other times, a context helps establish why mathematics might be useful (like AIDS statistics motivating a study of mathematical models). In either case, though, the match isn’t perfect. Still, the lack of a perfect match does not diminish the connection’s utility.

As for most of the musings I write about in this blog, I am not sure where this realization leaves me or how I would do anything in my curriculum development work differently if I could go back and do it again. I don’t have any easy strategies for opening up my own thinking about connections to context. Instead, I’ll just end with one more example that struck me as I read today.

In an article about ways in which the mathematics education research community can be more attentive to issues of intersectionality — that is, the particular issues and challenges faced by those in two or more overlapping marginalized groups, such as Black women — Bullock (2017) says the following: “The idea of multiple burden speaks to intersectionality’s key concern that racism, sexism, and other forms of oppression, when considered in parallel, appear additive, but those who experience these oppressions in combination endure multiplicative effects” (Bullock, 2017, p. 31).

This article has nothing whatsoever to do with any of the issues I discussed in this post. She wasn’t aiming to make a connection between mathematics and social justice issues (at least not in the same way Gutstein (2012) was). Still, her framing of the issue in terms of a mathematical idea — specifically, the difference between additive and multiplicative growth (5 + 5 versus 5 * 5, for example) — really brought her point home for me. I understand intersectionality better than I did before.

The connection between intersectionality and multiplicative growth isn’t perfect. But it is powerful.

References

Bullock, E. C. (2017). Beyond “ism” Groups and Figure Hiding: Intersectional Analysis and Critical Mathematics Education. In A. Chronacki (Ed.), Proceedings of the 9th International Mathematics Education and Society Conference (MES9, Vol 1) (pp. 29–44). Volos, Greece: MES9.

Gutstein, E. R. (2012). Mathematics as a Weapon in the Struggle. In O. Skovsmose & B. Greer (Eds.), Opening the Cage: Critique and Politics of Mathematics Education (pp. 23–48). Rotterdam, Netherlands: Sense Publishers.

 

*Shout out to my colleagues Andy Isaacs (my most loyal reader and commenter!) and Sarah Burns, who were involved in this discussion.

Noting the Novel

When the Common Core State Standards for Mathematics (CCSS-M) were first released, one of the aspects of the reactions that I found most interesting was the diversity of reasons for opposing them. Education researchers and developers, like myself, were dismayed at the nearly exclusive focus on arithmetic in elementary school, or the undue burden placed on the middle school grade band when content was both moved up from elementary school and down from high school, or likely some other issues related to content distribution. Teachers shared some of these content concerns and also worried the standards would only place greater emphasis on standardized tests. Conservative politicians disliked the standards because they were a strong step in the direction of federal control of education. Most of the criticism I heard from the general public, though, related to the idea that the standards made mathematics unduly complicated or even incomprehensible.

This video is one example of the kinds of critiques that floated around the Internet. It’s not very long, so if you can, take a minute or two to watch it. In a nutshell, it compares the steps for the subtraction algorithm most adults learned in school to the steps of a different approach, when applied to a particular problem. The new approach — a method that I and many of my math education colleagues call counting up — comes off looking complicated and nonsensical in comparison to the traditional method.

A friend of mine posted a link to the video on my Facebook wall and asked my opinion. Here was my response. (Spolier alert: I found videos like this super annoying at the time.)

response1

I was annoyed, first of all, that the video made it sound like this method was the new standard, which isn’t true. The standards still require students to add and subtract with the same traditional algorithm as always by the end of fourth grade. It’s just that before that, there are standards that specify that students should add and subtract (and later, multiply and divide) using more generalized strategies that help illuminate conceptual understanding of the operations. For example, consider this first-grade standard:

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. (CCSSI, 2010, p. 16)

That’s as specific as the standards get when it comes to prescribing computation strategies (until they require the standard, familiar algorithms later). So the video is based on a false premise. That’s the first problem.

Later, a comment from a friend on the same Facebook thread led me to respond to the second, and probably more important problem I noted at the time: the fact that the method presented in the video is based on kinds of everyday reasoning that make perfect sense in some situations.
Response2

So, the second thing that really bothered me was the way that the counting-up method was (likely deliberately) set up to look confusing, when the same kind of argument could easily be made in the other direction. I felt that videos like this were masking the value of alternative strategies when I have, in my own personal and professional experiences, seen such strategies really help students see mathematics as something other than a set of procedures to be memorized and applied.

Those were my arguments pushing back on criticism of alternative computation methods at the time. I still agree with them, but also admit that they don’t always change people’s minds. My kid doesn’t like this stuff, they tell me. You’re saying it’s better and equally clear, but it’s not working for my kid, and what’s worse is that I can’t help with homework. It makes both of us feel helpless.

I’m sympathetic to these concerns, and haven’t had a better response in the past than, Hang in there. I knew it was true that these alternative methods and ideas were not magically going to make every kid suddenly understand computation in a new way. Education is not so simple. I was claiming that methods like counting-up subtraction often made sense — but what about the cases when it doesn’t?

I read a few things in the past few months that got me thinking about this. First, I read the first three chapters of Jerome Bruner’s Acts of Meaning. In his third chapter, as he constructs an explanation for patterns of language acquisition in young children, Bruner (1990) points out that from infancy, children have a strong tendency to devote more attention to things that are novel:

[W]hen [children] begin acquiring language they are much more likely to devote their linguistic efforts to what is unusual in their world. They not only perk up in the presence of, but also gesture toward, vocalize, and finally talk about what is unusual. As Roman Jakobson told us many years ago, the very act of speaking is an act of marking the unusual from the usual. (p. 78-79)

Bruner’s text is dense and sometimes difficult to interpret, but this passage made perfect sense to me. We do attend to what is novel. It’s one of the ways we sort through the enormous amount of information we perceive and decide what needs attention.

I didn’t connect this to the Common Core criticisms, though, until just a day or two ago. I read a study about treating bilingualism as a resource in mathematics classrooms that mentioned the very passage from Bruner that I quoted above. In response to the Bruner quote, Dominguez (2011) says, “This may explain why in most mathematics classrooms, students barely talk—they rarely do things with words—as they may be dealing primarily with the usual, that is, repeated exercises of the same kind” (p. 310). This response, like Bruner’s original point, made a lot of sense to me. Familiarity is not always a good thing. It doesn’t always lead to good experiences.

Perhaps this is another way of thinking about the potential benefits of conceptually-based computation methods like counting up. I’ve been focused on explaining why it is not as confusing as some might think. But the other side of that coin is that for some students and in some circumstances, it will be new and confusing. But in the right dosage, that’s ok — even a good thing.

New is what gets people talking. Talking is what gets people learning.

I know it will probably seem like a hollow response to the parents wanting to help kids with their mathematics homework and feeling incapable of doing so. But it at least feels a bit more well-rounded than my previous response insisting that these conceptual methods are not so confusing.

References

Bruner, J. (1990). Entry into Meaning. In Acts of Meaning (pp. 67–97). Cambridge, MA: Harvard University Press.

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/Math/ on 1 March 2017.

Domínguez, H. (2011). Using what matters to students in bilingual mathematics problems. Educational Studies in Mathematics, 76(3), 305–328.

 

Intersecting Interests

I find myself without a particular topic in mind for a post this week, so I thought I’d share a little (read: enormous) project I’ve been working on since the start of the semester.

Brief preface: the main project of one of my courses is aimed at helping us to develop our research interests. Over the course of the 15 weeks of the semester, we’re gathering and annotating scholarly articles, conducting a historical overview of how our topics have been studied over the last 25 years, and documenting important scholars, journals, and associations in our fields. From what I can tell, most of my classmates are keeping track of this stuff through lists, folders, and spreadsheets — all totally normal and sensible organizational tools. But I’m weird. And I like math, and colors, and things I can drag around on screens. So, I took a little bit of a different approach to organizing my project information.

I present to you: my Venn diagram of research interests! (OK, so you already saw it above. That was the easiest way for me to make it big enough to see. But now you know what it is.)

The circle with the blue box and bubbles represents my first big area of interest: dynamic visualizations in math. In that circle live a lot of questions about the use of draggable interactives in mathematics, like digital versions of manipulatives and smart graphs that, for example, retain certain properties as you drag them. How does the use of stuff like this change the way kids think about math? How do we best leverage its potential? When are the times where it’s best to just put the tech away?

The circle with the red box and bubbles represents my second big area of interest: digital curriculum materials and how teachers interact with them. In that circle live questions about how digital materials can best support teachers in designing instruction for their specific contexts. How might feedback on instructional design decisions help teachers develop pedagogical design capacity?

 

The circle with the yellow box and bubbles represents my third big area of interest: the relationship between computational thinking (CT) — in brief, the thinking practices used by computer scientists, like abstraction, conditional relationships, and pattern recognition — and elementary mathematics. There’s a current push to bring computer science education to all kids in K-12, and when it comes to elementary school, I think integration of some CT ideas into mathematics is a great place to start. But what does math+CT instruction look like? How do teachers implement it? What is similar in mathematical thinking and computational thinking, and what’s different? When are those similarities and differences important?

The most interest part about exploring literature related to these topics has been discovering the ways in which they interrelate. One of the benefits of using digital manipulatives, for example, is that they seem to help kids abstract mathematical concepts (Moyer-Packenham & Westenskow, 2013) — and abstraction is a CT practice! That is what is living in the green section. Digital manipulatives (blue) + math and CT (yellow) = abstraction (green). The same research team who noted the connection between digital manipulatives and abstraction also did a study of how and when teachers choose to use digital manipulatives (Moyer-Packenham, Salkind, & Bolyard, 2008), and this study got me thinking about how the choice to use a digital manipulative is a critical part of the instructional choices teachers make when enacting curriculum. That is what’s living in the purple section. Teacher design decisions (red) + digital manipulatives (blue) = digital manipulatives as part of instructional design (purple). Last but not least, I recently read a study looking at how teachers use and adapt simple debugging tasks as part of their mathematics teaching (Kalogeria, Kynigos, & Psycharis, 2012). The tasks were not framed as debugging tasks by the researchers, but I definitely think they could be framed that way. So, that’s the orange section. Debugging tasks (yellow) + teacher design decisions (red) = how teachers use debugging tasks (orange).

As my advisors have told me many times already, I can’t expect to explore all of these questions in the four years I am in graduate school. But if I do end up in a research position later, these are the interconnected ideas that I hope will drive my program of research. If any of my faithful readers have recommendations for readings related to such things, I’d be glad to hear them.

To end, a fun exercise: What do you think goes in the center of the diagram?

References

Kalogeria, E., Kynigos, C., & Psycharis, G. (2012). Teachers’ designs with the use of digital tools as a means of redefining their relationship with the mathematics curriculum. Teaching Mathematics and its Applications, 31(1), 31-40.

Moyer-Packenham, P., Salkind, G., & Bolyard, J. (2008). Virtual Manipulatives Used by Teachers for Mathematics Instruction: Considering Mathematical, Cognitive, and Pedagogical Fidelity. Contemporary Issues in Technology and Teacher Education, 8(3), 202-218.

Moyer-Packenham, P., & Westenskow, A. (2013). Effects of Virtual Manipulatives on Student Achievement and Mathematics Learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35-50.

 

A Painful Truth

I was in middle school when the musical RENT was big. My friends and I were obsessed. We wanted to be just like Mark and Roger, living the Bohemian life of sacrifice for art in New York City. We stood in hours-long lines to get rush tickets, sang the soundtrack at the back of the bus, adopted No Day But Today as a motto, and swore we’d never sell out by working for the man. RENT was everything. RENT was life.

Gradually RENT fell out of my day-to-day existence, but even into adulthood it still held a special place in my heart. That’s why it was so painful when I read this article a few years ago. In short, the article points out that RENT looks quite different from the point of view of a working adult. The sell-out character, Benny, isn’t the villain our middle-school selves believed him to be. He’s a hero who tried desperately to save his friends from themselves.

Oh man, that hurts, I thought. It hurts because it’s so, so true.

The hard thing about reading the article wasn’t its scathing tone. It was the way it completely destroyed something I believed in for a long time. And did it in a way I couldn’t refute. Oh, it hurt. It really hurt.

I had a similar experience this week when reading a book chapter about equity in mathematics education. In a critique of the common rhetoric used to discuss equity, Alexandre Pais (2012) calmly and completely deconstructed the claim that students need to learn mathematics to be functioning citizens: “Mathematics is posited as indispensable knowledge and competence to participate in the world – the idea that through mathematics we become empowered citizens” (p. 63). But the idea that students must use mathematics in their daily lives, he said, is a lie we construct to conceal an ugly truth: “It is not that school mathematics is powerful because people use it in their daily lives; mathematics is powerful because it gives people school and professional credit” (p. 65).

Oh man, I thought. That hurts. That hurts because it’s so, so true.

When I read that sentence I had to close my eyes and pause for a minute before I went on reading. It really felt like a gut punch. Since that moment, I’ve been trying to sort out why it made me feel that way. Here is what I’ve been able to figure out.

First, I have been struggling with the idea of mathematical utility in everyday life for a long time. After years and years of writing word problems designed to help kids practice particular mathematical skills, I became thoroughly disenchanted with the process of working from mathematics to context. Most of the time that just produces contrived problems. For a while I was focused on working in the opposite direction — starting with contexts and writing problems that used mathematics to solve related problems. This was a fun project, and interesting, but also difficult. The truth is, there are limited professional and everyday contexts that use mathematics in a meaningful way — at least when the mathematics under consideration is the mathematics typically taught in schools.

So it wasn’t the first half of Pais’s (2012) statement that hurt. I had already come to the same conclusion on my own; I knew that school mathematics wasn’t all that useful in everyday life. But I made peace with that part by insisting that there were other benefits to learning mathematics. I traced ways that my math training has helped me to think more critically about problems and to cut through the noise to get to the heart of matters. Surely other students reaped the same benefits. Plus, I thought, math is just interesting and beautiful. Not every student is going to feel that way, but every kid should get the chance to discover whether they do. Right? There are reasons for learning school mathematics.

This is why I reacted as I did to Pais (2012). That strongly held belief was shot down by the second half of Pais’s statement. Sure, there is a reason to learn mathematics. But it isn’t any of the beautifully crafted reasons I held dear. It’s because facility and achievement with mathematics earns students “school and professional credit” (Pais, p. 65).

The reason kids need to learn mathematics is because it earns them credit in the system. Not because of all the lovely reasons I had constructed in my head.

Ouch, I thought. It hurts because it’s true.

My first defense mechanism was to think that just because this was the reason kids had to learn school mathematics didn’t mean it was the only reason they should. But I kept going back to the main point of Pais’s (2012) essay, which was the falseness of the narrative of about “mathematics for all.” Schools cannot claim to treat “mathematics for all” as their goal when the whole system depends on some students passing and some failing. As long as we keep pushing “mathematics for all,” school mathematics will remain a gatekeeper in education. In that sense, should is not any better than must as a preface for a reason to learn mathematics.

Ouch again.

Pais (2012) acknowledges that his points in some sense lead to a deadlock. What should we do? Just stop teaching math instead of trying to improve mathematics education? Put an end to schooling altogether? The sensible first act, Pais says, might be to stop acting. The best thing might just be to stop and think. I did some of that, and here are a few things I have to say.

One of the goals of my professional life to this point has been to help make mathematics education more meaningful for kids. Recently I’ve talked about how I think meaning can be achieved not through connectedness to everyday life but through connectedness internal to mathematics — by highlighting big ideas in mathematics. This line of thinking is not really in conflict with Pais’s (2012) point. This line of my work doesn’t claim that mathematics is good for anything but learning more mathematics. So as long as schools still exist and mathematics is taught, I don’t think I’m doing harm by trying to make it more meaningful.

A big part of my motivation for this work, though, is that I am deeply bothered by how common it is for people, children and adults alike, to proclaim that they are “not math people.” I want everyone to feel like a math person. Until today I held strongly to the belief that the reason most people say “I’m not a math person” is because they were never taught math in a meaningful way. Pais’s (2012) essay has forced me to admit that there is much more to it than that. Yes, in many cases, the idea that one is “not a math person” likely comes from poor experiences in mathematics classes. And yes, I still think the “not a math person” is a societal narrative worthy of change. But I think it’s important for me to recognize that “not a math person” might mean “not a school math person,” and to insist that all students should love and appreciate school mathematics is something that will only perpetuate inequity.

It hurts. But it’s true.

Reference

Pais, A. (2012). A Critical Approach to Equity. In O. Skovsmose & B. Greer (Eds.). Opening the Cage: Critique and Politics of Mathematics Education (pp. 49-86). Rotterdam, Netherlands: Sense Publishers.

Realizing the Register

For several years, starting shortly after the release of the Common Core State Standards for Mathematics (CCSS-M; Common Core State Standards Initiative [CCSSI], 2010), I spent every day of my working life designing, writing, testing, and editing activities meant to address the K-5 content standards. Often, this work was fun. There were a handful of standards, however, that tended to make the work miserable.

Take this fifth-grade standard from the Operations and Algebraic Thinking strand (5.OA.2):

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. (CCSSI, 2010, p. 35)

True confessions: I came to hate this standard. At first read, it doesn’t seem so bad. However, attempting to develop extended activities on it — a full lesson, and plenty of follow up practice — was a slog. I (and all the members of the team working on this grade level’s materials, from what I remember) quickly ran out of ways to make this work interesting. Grouping symbols were new to the grade level, but the translation and interpretation of expressions from words to symbols was not. The standards in first through fourth grades required students to write equations to represent word problems, a task related to this standard but more interesting because the translation in those cases required more abstraction. Story-to-symbols can be interesting; words-without-context-to-symbols seemed to be a step backward. I just couldn’t understand the point of this standard. Any time I wrote another prompt for students to write about how, for example, (5 + 7) × 2 was twice (5 + 7), I said a silent apology to the future fifth graders of America.

A few weeks ago, though, I read some articles that made me think a bit differently about this standard. The articles focused on the mathematics register, or “meanings that belong to the language of mathematics (the mathematical use of natural language, that is: not mathematics itself), and that a language must express if it is being used for mathematical purposes” (Halliday, 1978, p. 195). The math register does not consist only of vocabulary, but also of language structures and modes of argument that convey meaning (Halliday). The notion of the math register has been discussed and expanded in various ways since Halliday introduced it. Of particular interest to me is the idea that the mathematics register includes symbolic notations and grammatical structures that map onto each other in opaque ways; for example, one way to translate a2 + (a + 2)2 = 340 into spoken or written language is, “The sum of the squares of two consecutive positive even integers is 340” (Schleppegrell, 2007, p. 145).

As I read the Schleppegrell piece, I was struck by the illustration of how complex the translation from mathematical symbols to words and back can be. I thought again about 5.OA.2. The examples given in the standard are trivial (after the first few practice rounds, at least), but in general, there are plenty of examples that are not, particularly when variables start to be used to express general relationships. The text of 5.OA.2 does not allow for the use of algebraic, rather than numeric, expressions, but suddenly I wondered if this standard was intended to support readiness for more complex sorts of translation and interpretation of expressions later.

Based on the information I have at hand, my best answer to that question is “maybe.” According to the draft progression document for the Operations and Algebraic Thinking (OA) strand of the CCSS-M, the intention of 5.OA.2 is to prepare students to work with algebraic expressions in later grades: “[S]tudents in Grade 5 begin to think about numerical expressions in ways that prefigure their later work with variable expressions (e.g., three times an unknown length is 3 × L)” (Common Core Standards Writing Team, 2011, p. 32). They go on to further describe the work in Grade 6:

In Grade 6, students will begin to view expressions not just as calculation recipes but as entities in their own right, which can be described in terms of their parts. For example, students see 8 × (5 + 2) as the product of 8 with the sum 5 + 2. In particular, students must use the conventions for order of operations to interpret expressions, not just to evaluate them. Viewing expressions as entities created from component parts is essential for seeing the structure of expressions in later grades and using structure to reason about expressions and functions. (Common Core Standards Writing Team, 2011, p. 34)

It seems clear that the standards writers were attempting to anticipate an increase in complexity in interpretation of expressions. I’m not entirely convinced they were focused on the communication of meanings as Schleppegrell (2007) was; rather, they seem to be focused specifically on recognizing mathematical structures and reasoning about them, rather than ascribing them with meaning in natural language. Still, there’s some evidence of an intention behind standard 5.OA.2 that I didn’t understand until I read about the mathematics register.

I spent countless hours thinking about 5.OA.2 and many of the other standards. I have thought about them together and separately; as activities in their own right and as learning goals for those activities; in their most literal and most figurative senses. I did not think there were many more ways to think about or interpret them than those I’d already turned over in my head at least once. But this experience proved me wrong. Now I have new way to think about why this standard might matter to students’ understanding of mathematics.

I’m not sure how this new information would affect my curriculum development work if I had a time machine and could go back and try again. Maybe it wouldn’t change anything about the practical implementation or the student exercises. But I suspect it would change the way I wrote about it in the teacher materials.

What do you think? Is building readiness for interpretation of algebraic expressions a worthwhile goal? Is it what 5.OA.2 was meant to do?

References

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved 06 October 2017 from http://www.corestandards.org/Math/.

Common Core Standards Writing Team. (2011). Progressions for the Common Core State Standards in Mathematics (draft): K, Counting and Cardinality; K-5, Operations and Algebraic Thinking. Retrieved 06 October 2017 from https://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf

Halliday, M. A. K. (1978). Sociolinguistic Aspects of Mathematical Education. In Language as Social Semiotic: The Social Interpretation of Language and Meaning (pp. 194-204). Baltimore, MD: University Park Press.

Schleppegrell, M. (2007). The linguistic challenges of mathematics teaching and learning: A research review. Reading and Writing Quarterly, 23(2), 139-159.