Elementary Mathematics Education

Mathematics Teacher Education OER

In-depth blogs that tackle a lot of topics in meaningful ways.

National Council for Teachers of Mathematics– this blog is hosted by a premier mathematics education organization

Dan Meyer– one of my favorite blogs. You’ll find very practical resources as well as posts that handle bigger issues.

Marilyn Burns– amazing researcher and children’s book author. She transitioned to blogging extremely well.

Youcubed– though the entire site is amazing, this link focuses on writings that support teaching and learning

Illustrative Mathematics– great organization that has posts from a variety of writers.

Investigations– this the site from a solid textbook company that supports its work through blogging.

What do we know about how children think and make meaning in mathematics?

Embrace the Challenge– here’s a blog that does an amazing job at closely observing children’s thinking

Math is Figureoutable– solid site overall. The author does great work challenging the notion that we need to teach children algorithms.

Thinking Mathematically– Mark Chubb is an instructional coach who posts up-to-date research ideas in a very practical and student-focused way.

Graham Fletcher– in addition to providing some nice curriculum, Graham does great work in breaking down content and how children make sense of it.

What types of curriculum support in-depth student learning?

Math Coach’s Corner– This site focuses on providing practical resources for teachers

Talking Maths with Kids– Nice mix of practical resources with emphasis on student thinking.

Games– the whole site is great, but I’ve linked to the blog section to learn more about the games and their intent.

Joe Schwartz– practical ideas, often accompanied with student work samples.

Brian Bushart– nice vision of contemporary research with skill in creating new curriculum.

How do we teach mathematics in a way that supports equity?

Being Black at School– though not math specific, this resource is geared toward identifying and supporting the unique strengths of black children in schools.

Marian Dingle– she is wonderful at providing some very human reflection on teaching. She provides great insights how existing systems can be harmful to many students

Neplanta Teachers Community– This group approaches education with a social justice lens.

Hema Khodai– Hema is a practicing teacher who provides rich reflections related to issues of mathematics and race.

Lauren Baucom– practicing teacher and DESMOS fellow with great general reflections as well as reflections specific to mathematics and race.

Other

Tracy Zager– Tracy’s writings vary, though all her posts demonstrate a wonderful understanding of the research in mathematics education.

Visuals and Representations– Berkeley Everett is a K-5 math coach who has created an amazing bank of visuals to support learning.

Robert Kaplinsky– prolific writer with some great posts about mathematics education.

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.

Revisiting 2 Metaphors for Learning

Setting the Stage

I have been privileged to join a large group of faculty and staff at Plymouth State University in the Cluster Pedagogy Learning Community (CPLC). The aim of this learning community is to support instructors as they seek to improve student learning. We are considering learning through specific learning theories (constructivism and connectivism), through three broad pedagogical approaches (interdisciplinarity, project-based learning, and open education), as well as our general education program’s outcomes, or Habits of Mind (Purposeful Communication, Problem Solving, Integrated Perspective, and Self-Regulated Learning).

As part of my participation in this learning community, I read an interview with Cathy Davidson, author of The New Education. After outlining the failure of both edtech and a skills-centric approach to saving higher education, she noted the need for higher education to emphasize “learning how to learn”. This focus, she argues, is necessary for preparing students for a world in flux. Adaptation and flexibility, persistence and problem solving become critical traits in a world that changes so rapidly.

In considering my participation in the CPLC and some of my initial reading, I have found myself revisiting papers and studies that have been critical to my learning and development as a mathematics educator. I keep finding myself thinking about how those same articles, which helped me so much in mathematics education, might inform my work in higher education. One of the first papers I want to revisit is Anna Sfard’s amazing discussion of different ways in which we conceptualize learning. It’s a pretty heavy read, but definitely worth it if you have some time to dig deep. My intent is not to summarize the entire paper, but simply to introduce main ideas that could help us as we continue to rethink how we approach curriculum in higher education.

The two metaphors that Sfard discusses are the acquisition metaphor and the participation metaphor. She makes the argument that, at least in mathematics education, both perspectives on learning are necessary. 

The Acquisition Metaphor

This metaphor captures common Western views on learning and knowledge. In this metaphor, knowledge is a cognitive construct that one can obtain and possess. When we “learn something” we have somehow captured it and sufficiently trapped it in our brains. Now, a whole variety of learning theories discuss different mechanisms for how we come to possess that knowledge. Some view the learner as a passive recipient while others view the learner as an active co-constructor of the knowledge. Behaviorists and constructivists differ greatly in how they define learning, but both view knowledge as a cognitive construct that one must acquire. 

We see this tradition all over higher education. Colleges, departments, and programs are often created around the shared “stuff” that students are expected to learn (read: acquire). Course numbers often have a specific discipline code, categorizing the type of stuff one is to acquire in that course. Many professional preparation programs have culminating exams to ensure that graduates have sufficiently procured the necessary knowledge to be trusted in that field. Higher education (and education more generally) is not struggling to leverage this particular metaphor. One of the primary reasons I share this article and discuss its implications is because it’s very easy to get excited about a particular learning theory without understanding its inherent connection with a learning theory one is trying to shed. I have seen this in mathematics education. In moving away from behaviorism and toward constructivism and social constructivism we have made amazing strides and breathed new life into the discipline. I have seen engaging curriculum and improved assessments. If we make similar changes in higher ed, I have no doubt that student experiences and learning will improve. What will not change, however, is the baggage that comes by overemphasizing any learning theory that rests wholly within the acquisition metaphor.

One of the clearest problems with an acquisition perspective on learning is that when we treat knowledge as a commodity that one can obtain (again, Sfard outlines this well), it becomes easy to separate the “haves” from the “have-nots”. In education, those who have successfully obtained more knowledge than another is worthy of accolades, honors courses, and/or scholarships. Those who have obtained less are identified as in need of remediation, intervention, or is at risk of severance. Unspoken hierarchies develop. Traditional power structures are reinforced. Deficit perspectives abound. The long-term impacts of sole focus on knowledge acquisition, even when fueled by student-centered learning theories, are ultimately harmful for so many of our students.

Participation Metaphor

As a worthwhile counterpoint to the acquisition metaphor, the participation metaphor provides a vastly different conceptualization of knowledge. In this metaphor, knowledge is not the possession of facts, but rather, the ability to engage within a specific community and to contribute to an activity in a relevant way. Knowing is replaced by doing, and the role of context and community are highlighted. Now, this is far more than listing skills one should possess (there’s that acquisition language again). Knowledge is manifest through authentically belonging. For more detailed information, it’s worth checking out socio-cultural learning theories, especially as they highlight notions such as situated cognition and cognitive apprenticeship (thanks Brown, Collins, and Duguid), and communities of practice (Lave and Wenger). 

There is something liberating about the participation metaphor. Regardless of the learning theory that guides me, I could never possibly facilitate the learning of everything my students would need to know in order to be successful in their field. I cannot adequately distill a list of topics and skills students need to acquire and then cover it in a series of courses. When I stop putting pressure on myself to ensure that my students’ heads are crammed full with all the important stuff from my field, then my focus changes. It’s not that my job gets easier, but I can at least avoid all the pedagogical pitfalls of trying to cover everything.

So, before I launch into how my focus changes, let me pose a few questions. How might your practice change if your end goal was to prepare students to find a place in which they can work with others and make meaningful contributions? What types of learning experiences would you value? What would you, as an educator, need to understand about the communities your students intend to join? What communities might you seek to join? 

As I have reflected on the participation metaphor, there are a few changes I’ve tried to make. I’ve focused more on the authenticity of student experiences, access to generative resources, and identity development. I won’t go into detail about each as each deserves, and will probably receive its own post, but I will give a brief overview as that will help illustrate the benefit of thinking through both metaphors. As I imagined my students becoming meaningful participants in their various communities, I worried about the alignment between the university classroom experience and the elementary school experience (the setting for which I prepare my students). What will happen if students get out in the field and come across messages from practicing teachers that directly contradict what they learned in my courses? Would that be an indication that my courses were out of touch? I’m certainly willing to entertain that possibility. Would they view that as a flaw in the practices of their colleagues? Either approach could be detrimental as it acts as an obstacle to joining their community. This raised a whole host of other questions. Is my job simply to maintain the status quo in education so that the preservice teachers I teach can move seamlessly into the workforce? If I do challenge the status quo, how can I do it in such a way that my students do not find themselves ostracized for not fitting a desired mold? The answer, for me, was to seek out ways to join my students in the field by moving my courses off site. Not only would this give my students a more authentic experience, but it would also help me to better understand the issues that different communities are wrestling with. I started the process of trying to join the very communities my students might later join. By serving within those communities together, we gained a stronger collective sense of how to navigate, challenge, and support the broader system. 

Moving courses off site was a great start, but I still worried about how well-connected their university experiences were with the realities of communities they sought to join. I thought back to my time as an elementary school teacher and remembered that though I kept all of my textbooks from my own program, they all sat on a shelf behind my desk, collecting dust. I never looked back to my university resources. I looked forward to new resources or used resources that other teachers were using. My textbooks felt like a remnant of my past. Growth came not from rereading a textbook, but from finding new resources to bring to bear. The more I reflected on this, the more I knew that utilizing open educational resources (OER) would be critical for my students’ ability to engage in their respective communities in meaningful ways. Using OER wasn’t simply about ditching a textbook or saving money (though both of those things were fun). It was about connecting students to resources that grow and get updated. It felt more authentic for teachers in the field (target community) to read blogs and access Twitter. I wanted my students to view learning as dynamic and ongoing, and a textbook seemed too static. Funny thing though, if I’m completely honest I would tell you that the textbooks I ditched were better than the OER with which I’ve replaced them. At least from an acquisition perspective. They contained all the information I’d hope the students would come to possess, packaged nicely with illustrations. The fact that students barely read/understood/applied the chapters in the text speaks volumes for not relying solely on an acquisition perspective. 

Why Both Metaphors

I have some more examples, but I’ll save those for a later post. At this point, some of you still reading (I applaud your persistence), may wonder why we need that acquisition perspective at all. I mean, if the participation metaphor led to strong pedagogical changes, why not kick that acquisition metaphor to the curb? Permanently. I understand the desire. I think I’m a better educator when I focus on the participation metaphor. 

At the crux of the matter, however, is transfer. Sfard explains how learning theories that emphasize the situated nature of learning and the importance of interacting within a specific community fail to explain how individuals are able to carry knowledge or tools from one situation/community to another. The best explanation for that transfer is to view knowledge as a cognitive construct that one can possess and take into different scenarios. The fact that transfer exists tends to weaken the supremacy of the participation metaphor. I should note that I think we assume transfer happens far more that it actually does, but again, that’s a post for another day. I see this transfer happening at 2 levels.

At a more superficial level, when university students learn the basic ideas and buzzwords of a particular community, it gives them a sense of legitimacy that invites some degree of participation within that community. Possessing the right knowledge can give someone a foot in the door. At a deeper level, if there are core concepts that one needs to have acquired in order to thrive within a particular community, it would behoove us to identify what those are and to ensure that they are, in fact, acquired. We have some great research, for example, that suggests that mathematics teachers who have a deep and interconnected knowledge of mathematics turn out to be better mathematics teachers. The acquisition metaphor is helpful in that regard. It’s important to note, however, that a deep and interconnected knowledge of mathematics alone will not make someone a good math teacher. That knowledge will not be enough to ensure that colleagues want to interact with that individual and support and/or learn from that individual. Here’s where looking at things from a participation metaphor could be powerful. The overall experience of university students would be enhanced if faculty and staff considered goals using both the acquisition and the participation metaphors.

How Does this Connect to the CPLC?

One of the main reasons I wanted to write this piece and to reflect on this particular article is that I see so much of the participation metaphor already shining through in the goals of the CPLC. PSU’s Gen Ed outcomes, the Habits of Mind, are written in a way that really supports both metaphors. I am a big believer, however, that when educators are explicitly thoughtful regarding their ontology and epistemology then they can be more purposeful in the pedagogical decisions they make. In our work we can easily bat around a number of good ideas or… *sigh*… best practices (you know, because we’ve definitely identified all the best ones). People may make changes to a course because someone keyed them in on this really cool thing they’re doing. Educators are amazing at recognizing good pedagogy and trying it out. The problem with that approach is that when those choices lack a firm anchor (as a clear epistemology provides), then educators find themselves vulnerable to getting swept up in the next fad. Strong pedagogical practices bear fruit only as they are thought about, implemented, revised, shared, tailored to a new group, and revised some more. The fad approach leads to the premature abandoning of amazing ideas that just hadn’t developed yet. Talk of theory, though not sexy, helps to keep us grounded. I am so excited to work with colleagues across disciplines to improve our work. As we do so, I’m happy to be the person sitting off to the side who periodically shouts out, “Let’s not forget the theory that drives us!”

I have found great traction as I have reflected on the differences between the acquisition and participation metaphors. I am confident that those delineations can be powerful for others engaged in this same work.

If you have the time, go back and read that source article.

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

So, what really is a fraction?

This should be an easy question to answer. I mean, fractions aren’t exactly new. As I’ve dug into this question, I learned that the answer was not as straightforward as I had assumed.

I’ve gotten pretty used to disagreeing with people. It seems there are no shortage of issues about which to disagree. One of my recent disagreements was with my students about the value of stuffing as a Thanksgiving side dish (many of my students were completely wrong, believing that it has any merit at all). Normally I don’t think twice about the disagreement. When I disagree with people I genuinely admire, however, it causes me to pause and think deeply about my position. I recently had just that experience.

While participating in the Thursday evening #ElemMathChat on 1/24 I was caught off guard by some approaches to defining fractions. While exploring some different fractional representations, some presented ratio relationships as fractions (it was something like “I see the fraction 2/3 because there are 2 pears for every 3 oranges”). My first instinct was to say, “I don’t think that’s a fraction. I think that’s a ratio.” Mark Chubb (@MarkChubb3), who was moderating the chat, shared this documentas a reference, supporting the claim that what was shared was, in fact, a fraction. This was the first time that I’ve seen part-part relationships lumped together so clearly with other definitions.

Now, the fact that ratios use a fraction bar is obvious. So why was this an issue for me? I suppose the question I am wrestling with is whether or not every rational number that uses fraction notation should be called a fraction.

Common Ground

I should start by acknowledging all the ways I agree with Mark and the Paying Attention to Fractions document. Focusing on fractions exclusively as parts of a whole is limiting. If students only conceptualize fractions by chopping up pizzas and pans of brownies, they will be woefully unprepared. I applaud the document’s aim of helping teachers understand the complexity of fractions. Let’s take a look at some of that complexity. 

Consider for a moment the different meanings of fractions in the following statement:

A 1-gallon milk jug that 1/4 full, contains 1/4 of a gallon of milk.

Even though the exact same fraction occurs twice in that statement, they have vastly different meanings. 1/4 of a gallon is a specific quantity and I could place that quantity on a number line between 1/5 and 1/3. 1/4 full, however, is not a quantity. It establishes a relationship. If a 2-gallon milk jug were 1/4 full, it would have 1/2 of a gallon in it. The exact quantity would depend on container (what we would call the “referent whole”). No wonder fractions are so challenging for students! A simple statement using the exact same fraction could employ completely different meanings.

This is just a fraction of the complexity I could mention (yes, that was intentional). The Paying Attention document does a wonderful job reminding teachers that fractions are more complex than parts to whole relationships and it does well identifying valuable models worth exploring. 

Let’s Settle this! What’s the Definition?

When I first started asking this question I went back to all the books I’ve used to learn about fractions. Here are some of the ways fractions are discussed:

NCTM’s Principles and Standards for School Mathematics notes, “During grades 3-5, students should build their understanding of fractions as parts of a whole and as division” (p. 150). And then later it states, “In the middle grades, students should… recognize and use fractions not only in the ways they have in lower grades—as measures, quantities, parts of a whole, locations of a number line, and indicated divisions—but also in new ways. For example, they should encounter problems involving ratios, rates, and operators” (p. 216).

NCTM’s Developing Essential Understanding of Rational Numbers: Grades 3-5 provides a clear definition as it states, “a fraction is a symbolic expression of the form a/b representing the quotient of two quantities (provided, as always, that the divisor b does not represent zero)” (p. 15). In other words, if it looks like a fraction, it’s a fraction.

From the CGI world, in Extending Children’s Mathematics: Fractions and Decimals it notes that “fractions do not have a single meaning, such as n parts out of m parts. They can stand for amounts of stuff, such as 3/5 of a candy bar. They can stand for relationships between amounts, such as 3 candy bars for every 5 children, or the processes involving amounts, such as 3 candy bars shared among 5 children. They can also of course be interpreted in terms of mathematical models, such as points on a number line or ordered pairs of numbers satisfying certain properties, as in the formal mathematical definition for rational numbers” (p. xxi-xxii). The authors go on to emphasize the importance of the multiplicative relationship that exists in any fraction.

The Paying Attention text seems to lean most heavily on Van de Walle’s work (which I think is pretty great). Elementary and Middle School Mathematics: Teaching Developmentally has the same definitions for fractions as the Paying Attention document, explicitly including part-part relationships.

Sowder, Sowder, and Nickerson’s Reconceptualizing Mathematics for Elementary School Teachers separates the “part-whole meaning” from the “division meaning”, from the “ratio meaning”.

Finally, Beckmann’s Mathematics for Elementary Teachers is one of the only resources I scoured that eschewed the ratio definition. “Then the fraction A/B of our whole is the amount formed by A parts (or copies of parts), each of which is 1/B of the whole”. 

It seems that nearly all of my favorite resources support calling ratios “fractions”. I should have stopped there and been happy that I had learned something new, but every time I thought of a 4thgrade teacher showing a 2 apples for every 3 oranges relationship, writing it as 2/3 and saying, “And here’s another fraction!” I got really worried. So, I went back to those very books to see how part-part relationships were handled. What I found was that once they defined fractions these resources almost immediately separated the term fraction from ratio and treated the terms differently.

Consider the following outcome from the Van de Walle chapter on Ratio, Proportions, and Proportional Reasoning: “Describe the essential features of a ratio, including how it relates to fractions, and articulate ways to help students understand and be able to use ratios.” What does the term fraction mean here? Does that statement even make sense if I neatly fold ratios (part-part and part-whole relationships) in with fractions? Clearly the term fraction is offset from the term ratio, recognizing similarities and differences in which those quantities exist. In fact, the rest of the Van de Walle chapter that I referenced earlier (that explicitly includes part-part relationships) goes on to exclusively use the word fraction to reference part-whole, measurement, division, and operator examples.

Let’s make matters a bit muddier. In the same NCTM Developing Essential Understanding series that I quoted, comes this gem from the Ratios, Proportions, & Proportional Reasoning: Grades 6-8 text.

            “Essential Understanding 4. A number of mathematical connections link ratios and fractions:

  • Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning.
  • Ratios are often used to make “part-part” comparisons, but fractions are not [emphasis mine].
  • Ratios and fractions can be thought of as overlapping sets.
  • Ratios can often be meaningfully reinterpreted as fractions” (p. 12).

To summarize this whole section, fractions include part-part relationships… just kidding, they don’t. It’s always fun to find that something that should be clear and easy to define is not so clear and easy. This excerpt helped me to understand why it irked me to see part-part relationships so neatly couched under the umbrella term of fractions. The more examples I read the more I realized that despite defining nearly every expression that follows the a/b format as a fraction, the term fraction simply isn’t used that way. Ratios and fractions are words that are clearly separated in secondary mathematics.

Operations with and Manipulations of Fractions

All of this, of course, begs the question, does this really matter? Isn’t this a minor semantic difference?

Despite my fundamental agreement with the Paying Attention document and the purpose behind it, the more I thought about it the more I realized that when it came to formally calling something a fraction I drew the line at part-part relationships (I’m sorry, I couldn’t help myself). Though a part-part relationship expressed in fraction notation certainly looks like a duck, it simply doesn’t quack the same, and that’s the problem I just can’t get past. When fractions are already so complex and nuanced, why add to the confusion by calling things fractions that follow fundamentally different rules as soon as operations are involved?

Let’s use fraction addition to clarify. When adding fractions we need a common denominator, at least to be able to express the answer. If I run 1/2 a mile and rest (yes, I’m a bit out of shape), and then run 1/3 of a mile before collapsing, I’ve run 5/6 of a mile total. I can express that as 1/2 + 1/3 = 5/6. Most of the other definitions of fractions fit with this part-whole approach. 1 divided by 2 plus 1 divided by 3 is 5/6. Whether fractions are “measures, quantities, parts of a whole, locations of a number line, and indicated divisions” they follow the same rules when adding fractions. Part-part relationships do not.

 Consider what happens when I accidentally make my chocolate milk too chocolatey (this is contrived scenario, of course, as “too chocolatey” is not a real thing). I make some chocolate milk that is 1 part chocolate syrup and 2 parts milk and decide it’s too chocolatey. I make a second batch that’s 1 part chocolate syrup and 3 parts milk. This one is too bland. So I mix them all together for a mixture that is 2 parts chocolate and 5 parts milk. I might even write that as 1/2 + 1/3 = 2/5. That expression conflicts with everything we know about fraction addition. It doesn’t work unless we clearly put some labels in there to separate it from the other fraction addition. It’s a really cool issue to explore with middle grade students simply as a means of differentiating between fractions and ratios.

What we call fraction addition doesn’t include ratios because adding ratios involves completely different rules. That’s important to note because as students see operations with fractions in the later grades, they rely on the assumption that the fraction does not represent a part-part relationship.

Summary

After wrestling with this for a while and reading everything I had on my shelves, here are some key takeaways:

  • It could be argued that any number that uses a fraction bar is technically a fraction and I should whine less. 
  • When the term fraction is used it almost always refers to a number that shares the same overall properties as a part-whole relationship.
  • Part-part relationships are critical for the development of students’ mathematical understanding, and the term “ratio” does a wonderful job capturing that particular relationship.

So here’s my plea. As we work with elementary age children and explore the various meanings and representations of fractions, let’s be careful and purposeful in the language we use. Yes, part-part relationships use fraction notation, but a determination to call them fractionscan sow confusion in an already complex topic. When we have terms like part-part relationship, rational number, and ratio, we can be more precise with our meanings and allow our students to develop meaning around the similarities and differences between fractions and ratios.

Shoot for Boring- What do we learn?

I opened this semester using a task for my elementary math methods course that I’ve used many times before. Each time I present it I’m struck by how engaged the PSTs are. They persevere for an extended period of time and I frequently hear that they continued to think about the task and work on it after the class has been dismissed. From an engagement standpoint, the task isn’t all that impressive. In fact, the side of me that really loves socio-cultural perspectives on learning gets mad every time I use this task. It’s not “real-life”, I mean, not really. It’s not a situation they’re going to run into again, thus giving students prior knowledge that would transfer. So why was it engaging? Why did they persevere? As I share the task I encourage the reader to avoid making a “Here’s why it’s engaging” list or “Here’s why they persevered” list. I feel like we all know what those lists would look like and that they may ultimately be hollow. Instead, I’m interested in how you might alter the task in a way that would kill engagement or cause persistence to falter. It’s a weird goal, I know. Attending to what hurts persistence can tell us just as much as attending to what helps it. Shoot for tiny alterations. The most minute alterations would tell us a lot about lesson design. In my experience, it’s the small differences that make it so one lesson thrives while another bombs. I mean, from one day to the next I don’t say, “You know, I think this technology thing is a fad and I don’t really want to build it into my lessons.” Telling me that technology use increases engagement and can support perseverance is the sort of broad-strokes advice that isn’t super helpful. I get it. That’s not a tool I’m likely to mistakenly toss aside. I’m interested in those small tweaks that I may not even be aware of. Bringing those to the surface is what improving teaching is all about.

So, here’s the task.

Launch: I start by telling a (mostly) fabricated story about watching my little niece play with blocks. I notice that she was building a pattern, as kids are wont to do, so I watched more closely. I present the pattern as my adult brain made sense of it. As a pattern that started with a single blue block and then grew outward in concentric rings. I label each new ring a “step”. I use a simple presentation to show the first 3 steps as they develop.

I ask students to list any questions they might have if they were watching this pattern unfold, giving them a voice in the problem formulation. If they didn’t ask it, I’ll usually ask a question about whether or not my niece would run out of blocks. I then ask PSTs to find how many total red blocks would there be in the entire figure once step 5 has been completed. I generally have multi-link cubes, tiles, and graph paper available. I let students work alone or in groups.

 

Side note: My elementary math methods students generally don’t have go-to strategies regarding linear or quadratic functions, so this task tends to put them at the edge of their comfort level without pushing them overboard.

 

Explore: As students problem solve, I walk around asking questions about how they found their answer, making note of the different strategies they tried. I only redirect their thinking if I think it will lead to unproductive failure. I’m more thank okay with incorrect answers, but some incorrect approaches do not support productive struggle.

Discuss: I choose students to share with the attempt to build toward greater efficiency. So, if I had a student who built the whole figure and then counted each red block, I have them go first. I ask if that strategy will always work. We talk about strengths of the approach as well as any potential obstacles. A student who built it (or drew it) and only counted up a quarter of the figure and then mulitplied by 4 might go next. We identify similarities and differences, strengths and challenges. Some students start noticing patterns, we get those out there. Some start tracking information through tables, we talk about those. Pretty foundational stuff.

 

We then repeat the launch, explore, discuss cycle 2 more times. During the second cycle they answer the same question for step 10 (does it work just to double step 5?) and on the third, they tackle step 100. Each Discuss section involves making connections, justifying work, asking questions to presenters (student to student interactions), and challenging one another’s thinking.

 

There are quite a few fits and starts. Strategies are tried and then abandoned. No one builds step 10 in its entirety. Many find that each new row of red grows by 8 blocks. That makes it easy to track the new red and then add each layer together. It’s fairly efficient for 10 steps, but not 100. By step 100 they either need an algebraic expression (formal or informal), or enough patience to number crunch.

 

So… Nothing special. Students really persist, though. Why? Is it just because they’re wonderful students? I mean, probably, but that can’t be the only reason.

 

So, let’s hear those alterations. Shoot for boring. What would shift this from a dynamic experience to a mundane or tedious one? I’ll start the ball rolling with some obvious, broad-strokes alterations.

  1. Jump right to step 100. Skip the steps 5 and 10 cycles. By requiring abstraction right off the bat I would lose all of my less confident students. Frustration would replace persistence.
  2. Remove the tools and manipulatives. This would also implicitly communicate that the only type of thinking that is valued is abstract representation.
  3. Expect all students to work independently, because, you know, who cares about Vygotsky? No need for all this ZPD stuff.

 

These would surely be terrible alterations, but they’re also not ones I’m likely to make. Can you think of subtle changes that could really impact engagement and persistence?

 

Other thoughts about the task generally:

Things I don’t like about the task:

  • The problem is not compelling. If she runs out of blocks, who cares? Is this something that really needs to be solved?
  • As far as PSTs’ mathematics is concerned, it doesn’t function well as a stand-alone experience. Students do not gain deep insights into different functions. They would need follow up experiences to deepen and connect their knowledge.

 

Reasons I’ll likely use it again next semester:

  • Did I mention students persisted?
  • It gives me great insights into their problem-solving approaches. Some students needed very concrete approaches. Some loved to delve into the abstract but got lost in their own thinking. Some were brilliant, but lacked confidence. As a formative assessment piece, it was very rich.
  • It engages students in a number of important mathematical practices.