Situated Learning and Authenticity

As I continue to think about pedagogy as part of my participation in the Cluster Pedagogy Learning Community, I want to revisit the one of the math ed articles that I think most about. I share it partly because it terrifies me. Reflecting on this article made me realize that an individual can be a dedicated, well-intentioned teacher who implements high-quality practices and who still has little influence on how/if a student uses the content from the class. The article in question is a 1985 investigation into the mathematics used by Brazilian children as they worked as street vendors as compared to the mathematics they used in the classroom, entitled Mathematics in the Streets and in Schools. Check out the article here.

Summary of the Article

In this study, Carraher, Carraher, and Schliemann sought to understand the mathematical thinking of 5 Brazilian children. They first performed informal tests, visiting the children in a street market and posing a series of questions that required the children to perform computations related to their work. We’re talking, “I want 3 coconuts and pay with a $20, how much change should I get?” types of questions. Each child answered at least 7 questions (the child who answered the most did 19). Among all 5, only 1 computational error was made on a single question. Their performance was almost perfect. 

The researchers then changed the setting for a more formal test. The exact same questions that the children had already answered correctly were used in this formal test. They were either posed as traditional word problems (“Maria bought 3 coconuts. If each coconut cost…”), or as basic problems without context (maybe something like 20-3×4, if 4 was the cost of a coconut). Performance on the word problems was still high, though not nearly as accurate as the informal test in the streets. The context-free performance tanked. It wasn’t good at all. The researchers also found that children tried to use completely different strategies in those different settings. Reminder: these were the exact same set of children.

The authors used the results to specifically question the practice of teaching children algorithms and rules in school and then expecting them to use those tools in their day-to-day lives. This won’t surprise anyone who has really thought about their mental computation strategies. I know that I often use different strategies depending on the situation. I rarely try to perform a standard algorithm in my head.

For me, this implication is powerful, but feels very specific to mathematics education. An overarching implication that this article highlights is how interwoven learning is with the context in which things are learned. Do children fail to use school-learned algorithms in their day-to-day lives because those algorithms were taught without context (I’m sure that’s part of it), or do children do school math at school and use other strategies that have been developed in other contexts outside of school? This question haunts me. The fact that children used completely different strategies in the different settings suggests that something as seemingly objective as mathematics may still be context driven. The reason that this scares me is that even when school mathematics improves (and it has, though rote memorization activities still abound), school math might continue be the knowledge and skills that one uses in school, with little impact on how an individual uses the mathematics in outside of school. 

Now, I think we have plenty of evidence that transfer does, in fact, exist and that things are not so bleak. This study considered a very small sample. This article, however, serves as a regular reminder to me that I need to be thinking about how closely the context of the learning matches with the contexts in which I hope the learning can be utilized. It’s this article that keeps me thinking about the authenticity of the learning experience.

Implications for CPLC

I teach people who want to be elementary school teachers. They spend a lot of time studying pedagogy and specific teaching practices. I often wonder, “What if my students develop a nice knowledge base, with associated practices, that their brain decides to label University Pedagogy? And then they go into the field and work with practicing teachers and develop a whole new knowledge base and practices that their brains label as School Pedagogy?” Which pedagogy will they use (spoiler: it’s the one they used in the setting that most closely matches where they will use it)? It could (and should) be argued that as long as the University Pedagogy matches the School Pedagogy then transfer will surely happen. That’s great, but perfect alignment between those settings serves more to reinforce the status quo than it does to push thinking in different directions. I want to equip my students with the knowledge they need not only to work within the system, but to challenge it when necessary.

What is the utility of a teacher preparation program if practicing teachers learn most of their best stuff outside of their program? I don’t think I’m off the rails here. Do you know how many practicing teachers, myself included, who rave about how much they learn and grow in their first couple of years teaching? That’s not a bad thing, of course, as learning should be continual. I worry, however, about how much of their university learning experience never actually makes it into their day-to-day practice. Just like the Brazilian street merchants, they’ve developed their own strategies that work just fine for them in the context in which they need them.

So what advice can I give to the CPLC cadre as I reflect on this piece (and bemoan all of the things the piece reminds me that I could be doing better…)? I think as we consider the three pedagogical approaches we’re investigating, we can find not only an opportunity to improve our practices generally, but also to implement practices that encourage greater transfer. Again, we could improve our courses tremendously, but if we do so in a way that is heavily context dependent then students will develop a wonderful knowledge base that becomes almost inaccessible outside of the university setting.

Interdisciplinarity

The value that I see in interdisciplinarity is the opportunity it provides educators to reflect on their discipline. I find that the deep specialization that one experiences while pursuing a doctoral degree can have the potential to shut a person off from broader contexts in which a discipline exists. I, for example, identify heavily as an elementary mathematics educator. I could happily lose myself in issues directly related to the teaching and learning of mathematics. In seeking to improve my courses I might use the best resources from national organizations and plan a series of related experiences that would result in students who were were-versed in all things math education. 

But then one of those students might come back and say something like, “I feel really ready to teach an amazing unit that helps children to develop meaning around fractions, but do you have any ideas on how to support a child who has experienced trauma?” My discipline exists in a complex context and those who research in my discipline are not the sole arbiters on all things that impact our discipline. An elementary school teacher would be well-served by becoming trauma informed, and a mathematics educator may not be the right person to offer that support. Embracing interdisciplinarity is not about mastering everything that could be important to my students, but rather, seeking to understand the complexities of the contexts in which the learning is supposed to serve the students. Once I contemplate those contexts, I feel the need to reach out to others, who have different areas of expertise, to ensure that my students are learning everything that they should. 

I was in a recent unconference session at a CPLC meeting in which we were discussing ungrading (part of our goal, of course, was to see how many words we could put the prefix unin front of). One of the participants discussed the difficulty of implementing ungrading approaches due to the amount of content that his students just needed to know. We all nodded our heads because we have all felt the pressure of making sure our students leave our programs with specific content that’s just essential in our field. Interdisciplinarity helps us to think about (rethink?) the primacy of content within our disciplines. 

I won’t argue that content knowledge is unimportant. I believe, however, that too often we ignore how we expect students to use that content knowledge. As an example, I think it’s pretty important that teachers can discuss, in depth, the ins and outs of rubrics. That is content knowledge that matters to a teacher. This is content that I have studied and thought a lot about. I do not, however, walk over to a colleague’s office, peek my head in the door and say, “So, a single point mastery rubric lacks some of the development that supports objective evaluation of student work, but they’re far superior than traditional rubrics for providing formative feedback,” and then give my colleague a high-five and leave. That’s not how that content knowledge is used. Being able to simply rattle off a statement like that isn’t evidence that I implement high quality assessment practices. Possessing knowledge is great, but knowing how and when to use it is even more important. Interdisciplinarity can be an invaluable approach to reflect honestly about the contexts in which our disciplines exist. Authentic learning experiences that lack interdisciplinarity are probably not as authentic as we think, and thus, will be unlikely to support transfer. 

Open Education

Open education has interesting potential, but it’s impact on transfer seems entirely dependent on how it is used. Approaching open education with a desire to increase the authenticity of the learning experience can do far more than simply dedicating oneself to utilizing open education (see my last post about metaphors for learning and epistemologies). In your discipline, how do people find information? What do they do if they face a situation that doesn’t conform to the norm? How do they use the information they find? Are there common mechanisms for sharing information that they can contribute to? 

As I’ve written previously, my use of OER and open pedagogy comes from a desire to engage preservice teachers with dynamic sources that most closely align with how teachers, especially early career teachers, seek out knowledge. I want my students to feel empowered to contribute their experiences to the broader knowledge base. Open education has helped me think through some of these issues. Again, careful reflection on the contexts in which I hope my students will utilize their learning has helped me to see benefit to open education. While there are many great reasons to dedicate oneself to open ed, the potential for improving transfer should be part of the consideration.

Project-Based Learning

One of the primary benefits of exploring project-based learning (PBL) is its potential to connect the learners to meaningful issues in the community. Even better, PBL can connect students directly to the community. As such, it is in a wonderful position to improve instruction in a way that encourages authentic experiences. Not surprisingly, authenticity is considered one of the essential design elements of a gold standard PBL. As such, this approach seems to be a natural fit for those concerned with the potential of the learning experiences to support knowledge that transfers. I won’t ramble about this approach. Good PBL focuses on authenticity in ways that other approaches might inadvertently sidestep. It’s hard to do PBL without dedicating oneself to issues of transfer.

We’re Doing That

It’s easy to assume that transfer happens whenever good teaching happens. Any shift away from traditional lecturing is assumed to lead not only to improved learning, but transfer. I mentioned the wrestles I’ve had with transfer and authenticity at one of our first CPLC meetings with a few colleagues (it was a short conversation) and their reply was, “Oh yeah, we’re doing that. We do case studies and simulations and stuff like that.” The reply caught me off guard. This topic, which feels so daunting to me, was so easy dismissed. Granted, I probably failed to communicate my thinking clearly, but still. I was coming at this from the position of trying to increase and improve field hours in my program and moving my courses off campus and into local schools. I think case studies and simulations can be amazing, but they also don’t seem to capture the complexity of transfer. Transfer is tricky. It’s not automatic. Even when we improve learning, we may not improve transfer. One of the things that this article reminds me of is that if I think I’m doing authenticity well, it’s probably still worth taking more time and seeking to understand the contexts in which the learning is to be used, especially as those contexts change and our understanding about those contexts change.

The Multiplication Facts of Teacher Education

This is the first entry in a series in which I explore some common problems in mathematics education that we’ve hopefully learned a lot about (though many still persist), and how lessons we’ve learned from these mistakes could help mathematics teacher educators from making similar mistakes in our work preparing teachers. In this first installment I want to look at a possible connection between instruction that emphasizes memorizing multiplication facts (at the teacher level) and instruction that emphasizes learning classroom management (at the teacher educator level). 

Multiplication Facts

A few months ago I was having a conversation with my wife that we had had a number of times before. I was recalling how parents of my 5thgrade students would approach me, concerned about their children’s mathematics. I knew that if I asked probing questions, I was likely to get the same response, “No matter how much we practice, [insert child’s name] just doesn’t have their multiplication facts memorized.” The reason this particular conversation stands out is that shortly after sharing my frustration with the predictability of that response, we met some friends for dinner. No more than 5 minutes into the dinner a friend asked, “You teach math, right?” 

“Sure,” I responded, not really wanting to get in to the intricacies of what my current job involves. 

“My son is really struggling in math and I wanted to know if you have any tricks,” she continued. I knew what was coming, but I smiled and waited for her to say it. “Even though he’s been practicing them for years, he still doesn’t have his multiplication facts memorized.” My wife burst out laughing, which is not really the appropriate response when someone mentions their child’s struggles with mathematics, but actually seeing this phenomenon that I had just described surprised her.

I’ve often wondered why multiplication facts get so much attention. When one considers the breadth and depth of mathematics, why have multiplication facts emerged as the metric by which so many parents measure their children’s mathematical abilities? Don’t get me wrong, I understand that they’re useful, but just once I’d love for a parent to come up to me and say, “You know, my child is really good at computation but struggles to pull relevant mathematics out of everyday experiences. Do you have any suggestions?”

Here’s why I think it’s problematic for anyone (teachers, parents, students, curriculum developers) to place too much of a premium on memorizing multiplication tables (or any math facts, for that matter):

  1. While rapid recall may make computation easier, it does not guarantee that a child is skilled in problem solving. In other words, a child could use memorized facts to compute 12 x 15, but have no idea that the monthly subscription they’re paying $15 for is costing them $180 a year.
  2. It assumes that a child needs to have mastery of basic facts before he/she can solve more complex problems. In my experience, that just isn’t the case. When I had students who struggled with their multiplication facts, I just handed them a multiplication chart or a calculator and we jumped right in and solved more interesting problems. I found that by the end of the year, students who had needed the chart or the calculator relied on them far less. Children can do complex and exciting problems even if they don’t have all the tools you think they might need. In other words, mastery of facts does not have to be a precursor to authentic mathematics investigations, but rather, facts can be better memorized through authentic mathematics investigations.
  3. Rote memorization of facts is boring. Overemphasizing math facts teaches children that mathematics is tiresome and only concerned with getting the right answer. Only those children who excel at memorization develop a fondness for mathematics. Some of those children will be let down when they find that mathematics is so much more and their ability to memorize facts and procedures will only take them so far.

Research on Multiplication Facts

Okay, so enough of my opinion, what does the research say about the role of memorizing multiplication facts?

In general, research aligns with people’s view that having the multiplication facts memorized is a positive thing. Research also suggests, however, that how one learns those facts matters. Those calling for increasing memorization of math facts often point to research related to cognitive load. Other researchers argue for a broader conceptualization of fluency.

The notion of cognitive load comes from cognitive processing theory and argues that a person’s cognitive resources available for problem solving are finite. If a person’s resources are dedicated to performing simple calculations, then they have less overall capacity to dedicate to more complex tasks. Thus, having the multiplication tables memorized would free up an individual’s capacity to engage in more complex mathematics (see Chinnappan & Chandler, 2010 for a nice overview). While the theory is reasonable, it has also been used to justify drill and practice approaches, timed tests, or a litany of other “just memorize these” approaches to fact fluency (so much so that Codding, Burns, and Lukito (2011) performed a full meta-analysis on the interventions).

NCTM’s Principles to Actions takes aim at these approaches as it reminds educators that the belief that “Students can learn to apply mathematics only after they have mastered the basic skills” is unproductive (p. 11). It goes on to share the importance of building procedural fluency from conceptual understanding. It argues that fluency is far more than speed and accuracy with computations. “Fluency is not a simple idea. Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they under- stand and are able to explain their approaches, and they are able to produce accurate answers efficiently” (p. 42). When viewed in this way, fluency would include memorized math facts, but would extend far beyond that.

One of the best explanations of fluency comes from Jo Boaler. Rather than try to summarize her work, I’ll just connect one of her papers here and draw attention to one section in which she states, 

“What research tells us is that students understand more complex functions when they have number sense and deep understanding of numerical principles, not blind memorization or fast recall (Boaler, 2009). I am currently working with PISA analysts at the OECD. The PISA team not only issues international mathematics tests every 4 years they collect data on students’ mathematical strategies. Their data from 13 million 15-year olds across the world show that the lowest achieving students are those who focus on memorization and who believe that memorizing is important when studying for mathematics (Boaler & Zoido, in press).”

So my 5th graders’ parents were right in desiring their children to fluently use multiplication facts, but the simple memorization benchmark they were using was too narrow and ultimately problematic.

So, in mathematics education we’ve learned that being hyper-focused on useful knowledge (multiplication facts) can lead to poor pedagogical choices, such as blind memorization and timed tests. We should focus more on developing that knowledge through engaging explorations of meaningful content. 

Classroom Management

As I reflected on the pervasiveness of overvaluing memorized multiplication facts, I wondered if we make a similar mistake in teacher education. I wondered if we, even implicitly, put a premium on an action simply because it was visible. I wondered if we made something a routine to simply carry out, rather than part of a rich and interconnected undertaking. As I reflected, I recalled a number of experiences in which classroom management felt that way for me. Just as we communicate to students that being good at something as complex as mathematics can be demonstrated through a basic skill, sometimes we communicate to teachers that being good at teaching can be demonstrated through having a well-behaved class.

The first experience worth sharing was a more general experience from my early years as a public school teacher. I had yearly evaluations with my principal and those often included some discussion of a “management system”. I never felt I could be completely honest during that discussion. You should know that my principal was amazing and never made me feel like I had to give a very specific type of answer, but I still lied to her during those conversations. What I wanted to say was that I tried to develop good relationships with my students, that I told jokes, and that I sought out engaging curriculum. I felt that those things could be credited with the vast majority of the “good” behavior in my class. That’s not what I told my principal, however. I told her about points systems and rewards. About reflection time for misbehavior. I told her about individual and group accountability systems. I thought that was what she wanted to hear. That system felt more tangible and could help me maintain control. What does it say about unspoken norms in education that I felt I couldn’t be honest about my classroom management? What does it communicate when we privilege systems that control behavior over simple relationship building?

It could be argued that I was simply being insecure and misreading the expectations of my principal. That’s a fair point, but principals consistently perceive teachers as being better at classroom management when they can control disruptive behavior (Brophy & McCaslin, 1992). I later had experiences that reinforced the notion that control of student behavior was essentially the same as good teaching. A few years into being a 5thgrade teacher I had the wonderful experience of facilitating a multi-year professional development program in a local school district. So much of that experience was positive, but I had one interaction that I’ve thought a lot about. A fourth grade teacher asked if I would come teach a lesson in her classroom. She noted that her students struggled with the long-division algorithm (which is a post for another day…) and she wanted me to… you know… fix them. I told her that I would be willing to teach a lesson that helped them connect conceptual models to the procedure that they had been studying, but that it would take more than a lesson for students to overcome that particular obstacle. She seemed okay with that. 

She informed me that the math instruction in her class was leveled and that she had the “high” students (again, a lot to unpack there). I asked to teach the whole class as I thought the experience would better serve the students if they could see the variety of strategies and ideas that come out of a heterogenous group. She later told me that she could only get the “high” and the “low” students. All this is to say that I was not walking into prime teaching conditions. Anyway, I was pleased with a lot that came out of the lesson. I thought the students made some powerful connections and engaged in some strong math talk. When I debriefed with the teacher and asked her about what she saw students do and heard them say she immediately noted pockets of “misbehaving” students. I conceded that a few got off task, but then started sharing what I had heard some of those exact students say during the exploration. I wanted the focus to be on student learning. She hadn’t heard the students say/do the things I pointed out. Though she didn’t say it directly, it was clear that the lesson had been a failure because I hadn’t sufficiently controlled the class. 

This was a seasoned teacher who was very good at so many things, yet she still equated the quality of a learning experience with the directly observable student behavior rather than the work students produced. Student compliance was more important than competence. This mindset is still prevalent and impacts teacher education. Take a look at this article posted on NEA’s site. Wouldn’t it be nice if high quality management could be reduced to “6 tips”? Those would be the “math facts” of teaching. Just memorize these things and you’re good to go!  Now look at those tips. It’s all about control, reward and punishment, and focusing on misbehavior. My preservice teachers (PSTs) come to me having developed these types of beliefs about management simply because they attended 12+ years of school. In fact, controlling student behavior “is one of the most persistent perceived needs of preservice teachers; for many it is practically the sina qua non of teaching itself’’ (Ayers, 2004, p. 89). 

Research on Classroom Management

Now, as with multiplication facts, I am not arguing that classroom management is unimportant. Clearly, disruptive behavior harms learning. Again, as with multiplication facts, how we approach management matters more than that we simply have it.

I appreciate the review of the literature in Marzano and Pickering (2010) The Highly Engaged Classroom. They note that students are likely to be engaged (a much better goal than “well-behaved”) when teachers and classrooms attend to the following (see chapter 1):

  • How students feel
    • Students’ energy levels and how to maintain that energy (pacing, curiosity, physical exercise, etc.)
    • A teacher’s positive demeanor (humor was specifically identified)
    • Student’s perceptions of acceptance (by teachers and peers)
  • Students’ interest level
    • Employing game-like activities
    • Initiating friendly controversy
    • Using unusual information
    • Developing effective questioning strategies
  • Students’ perception of content’s importance
    • Implicit and explicit goal setting
    • Cognitively complex tasks with real world application
  • Student’s perception of efficacy
    • Students develop a vision of what they can become
    • Students maintain a growth mindset

Each of these bullets has a rich research background that is worth checking out. I am encouraged that none of these focus on controlling behavior or developing systems of reward and punishment. These are well researched approaches that support student learning. Management matters, but PSTs receive messages that management and student compliance are essentially the same. They learn that a well-behaved classroom is a key indicator of high-quality teaching. A teacher education program that fails to confront such notions cannot adequately prepare PSTs for the reality of teaching. Worse still, if PSTs fail to confront those issues they will likely engage in practices, though well intentioned, that discriminate and harm children from racial and ethnic minority backgrounds (Weinstein, Curran, & Tomlinson-Clarke, 2003).

There will always be the temptation to oversimply complex things. We will always feel the temptation to use easily accessible data points to determine success. Giving in to such impulses harms learning. Children need to know that their growth and development in mathematics goes far beyond their ability to memorize basic facts. PSTs need to know that their growth and development as a teacher goes way beyond their ability to keep a class under control. 

References:

Ayers, W. (2004). Teaching the personal and the political: Essays on hope and justice.New York: Teachers College Press. 

Boaler, J. (2015). Fluency without fear: Research evidence on the best ways to learn math facts. Reflections40(2), 7-12.

Brophy, J., & McCaslin, M. (1992). Teachers’ reports of how they perceive and cope with problem students. The Elementary School Journal, 93(1), 3–66. 

Chinnappan, M. & Chandler, P. A. (2010). Managing cognitive load in the mathematics classroom. Australian Mathematics Teacher, 66 (1), 5-11. 

Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta‐analysis of mathematic basic‐fact fluency interventions: A component analysis. Learning Disabilities Research & Practice26(1), 36-47.

Marzano, R. J., & Pickering, D. J. (2010). The highly engaged classroom. Solution Tree Press.

Weinstein, C., Curran, M., & Tomlinson-Clarke, S. (2003). Culturally responsive classroom management: Awareness into action. Theory into practice42(4), 269-276.

Photo Citation: Mayor Kevin White photographs, Collection # 0245.002, Subject file, Box 214, Folder 55, Boston City Archives, Boston