Over the last year, Dr. Lisa Lunney Borden and I have been working on a model for integer operations that she introduced me to a while back. Our goal is to amplify her research for classroom mathematics teachers. Right now, the idea consists of three pieces, each at varying stages of development.
- A paper
- A platform
- A set of plans
This post is a draft of the paper that will be sent to a journal at some point in the future. We hope that you might provide us any feedback you have by commenting directly on this post or pinging us on Twitter (@NatBanting & @LLB_315).
Right now, Mathigon is working to build The Bucket of Zero directly into their Polypad. Once we get the foundation laid for this, we will send it into the world for testing and feedback. For now, Lisa created these multiplication of integers and division of integers Polypad investigations with her pre-service teachers.
A set of plans
In the near future, the goal is also to develop a series of guided investigations using the new-and-improved Polypad bucket. Mathigon has an in-house collection of lessons/tasks/investigations that are free to access for teachers, and this feels like the natural place to start building.
We are excited to slowly get this idea in front of mathematics learners. Please do let us know if you have any feedback!
Nat & Lisa
The Bucket of Zero: Verbing Integer Operations
Our role as teachers and teacher educators often has us thinking about ways to establish mathematical environments that foreground the process of making mathematical meaning and building strong conceptual understanding. This focus on core concept is critical—for both students and future teachers—because in its absence, the topics of school mathematics can quickly dissolve into a series of disconnected rules. These rules are often accessible to sensemaking, but unless mathematical activity starts in a meaning-filled manner, sensemaking is often backgrounded. The rules can then act to obscure a concept by asking us to blindly execute, instead of wondering why a specific instruction makes sense.
This article is unique in the sense that it is a discussion of how we made sense of one particular model to support the conceptual understanding of integer operations. Although we were inspired through different sources and motivations, and operating in different circumstances, an online collaboration nurtured this bucket of zero model into one that holds considerable power. What was very interesting about this particular collaboration was how each of us were inspired by different ways in which the bucket of zero enlightened the concept of integer operations. However, as we discussed the model together, the common thread that granted this model a particular potency was its explanatory, conceptual power when it came to integer operations. That is, the bucket of zero supported our understanding of the concepts of integer operations better than any model we had interacted with previously. Here, we attempt to weave our stories together in a way that does justice to the process of our knowing, highlights the generative nature of teacher collaboration, and also provides the model as a tool for teachers and teacher educators attempting, alongside us, to foreground conceptual understanding when teaching integer operations.
Nat’s Unexpected Search for a Model
I am willing to propose that, as a teacher of mathematics and now as a mathematics teacher educator, there is no more satisfying feeling than observing a learner stumble into sense, and this story is a chronicling of such an occasion. In fact, it is a particularly powerful one, at least personally, because, in this instance, I was the one stumbling. As I reflected on how my knowing unfolded, and continues to unfold through this writing, I realized that a lot of this experience reminded me of why I fell in love with teaching in the first place. If you take the time to pursue sense, you are likely to be surprised, and teaching mathematics is loaded with lurking epiphanies (e.g., Banting, 2018; 2020).
My search for a model of integer operations began as I prepared materials for an undergraduate content course designed for future elementary and middle school teachers. The mandate of the course was to introduce preservice teachers to the topics of their future classrooms by emphasizing conceptual understanding. It was designed under the assumption that the students had all interacted with the topics of the course before (i.e., number operations, fractions, proportional reasoning, 2D geometry, introductory algebra, etc.), but, perhaps, not in a way that provided opportunities to develop a deep understanding. Therefore, my focus was on designing novel and meaningful encounters with these familiar topics.
I had taught integer operations for many years as a classroom mathematics teacher, but, as a secondary teacher, I was never expected to initiate this understanding. By the time students came to me in the ninth grade, integers were no longer included in the curriculum as a stand-alone topic, and there was really only a cursory review of the skills as opposed to an in-depth dive into the conceptual underpinning. I had used models and metaphors with students while reviewing the skills, but the focus was on executing rules assumed, out of necessity, to already have a conceptual base. This focus, in and of itself, does not bother me. The time burden on teachers is immense, and I believe it was more than fair to devote precious instructional resources to the specified topics of a curriculum. However, what struck me particularly hard as I prepared a class for my undergraduates on integers and their operations, is that I didn’t have a robust model for the addition, subtraction, multiplication, and division of integers. It wasn’t that I just never had time to think deeply on the topic with my high school students; I didn’t know a model that demonstrated all possible integer additions, subtractions, multiplications and divisions. That is, I didn’t have a way to model addition as ‘combining,’ subtraction as ‘taking away,’ multiplication as ‘gathering equal groups,’ and division as ‘creating equal groups.’ I was forced to the realization that I had never stopped to make real sense, only partial sense that was eventually abducted by an adherence to the rules. Davis (2008) refers to such moments as huh moments, where teachers revert to the rule; this was my huh moment. Unsatisfied with a reversion to rules, my search for a comprehensive model for integer operations began. I started by digging through an old notebook where I had scribbled notes from one of Lisa’s conference sessions where I re-discovered the notion of a bucket of zero.
Lisa’s Passion for Verbs and Concrete Models
Making sense of mathematical concepts in ways that are visual and tactile has been a focus of mine since my undergraduate degree when my Honours thesis supervisor guided me to explore visual models that show mathematical concepts. I became obsessed with searching for visuals that showed proofs without words and finding concrete models for concepts that children might need to learn in school. My advisor knew I wanted to be a high school teacher and supported me to explore these varied representations of mathematical concepts, knowing it would be good grounding for me in my future career. This passion continued throughout my teacher education degree. What I did not know then is that in a short time, I would find myself teaching in a Mi’kmaw community school where my interest in concrete and visual approaches to teaching mathematics, I would come to learn, aligned well with Mi’kmaw ways of knowing or L’nui’ta’simk. In the ten years I spent working in that school, I continued to think about ways to make mathematical concepts meaningful for my students. I constantly asked myself how I could ensure my students could hold math in their hands and make meaning from this engagement. I also began to recognize that when my students could use more verbs and talk about mathematics as actions and processes, the learning was richer and concepts were better understood. After 10 years, I made the difficult decision to leave the school and pursue my PhD in Mathematics Education. I did so with the blessing of the community and a commitment to use my new role to continue thinking about mathematics education that would support Mi’kmaw students, and other Indigenous learners, who are often underserved by our system.
As I engaged with the community in my doctoral research, we collaboratively developed a model for transforming mathematics education for Mi’kmaw students that placed meaningful personal connections to mathematics at the centre with four key areas to attend to: 1) learning from language; 2) rooting in community values; 3) connecting to community knowledge; and 4) making use of community-based ways of learning (Lunney Borden, 2010; 2016). For the purpose of the bucket of zero, I particularly draw upon learning from language and the ways of learning which are connected to community values. Mi’kmaw is a verb-based language, this means that actions and processes are centered more than nouns or objects. This understanding led me as a teacher to use more verbs in my teaching and to focus more on actions in mathematics. For example, rather than asking about the slope of a line, I would ask students to describe how the graph is changing. Students descriptions of “go over” and “going up” a certain number of units could easily be connected to changes in the table of values and eventually students would come to the notion of slope without having to engage with the dense noun clause, “Slope is the ratio of the change in the y-value to the change in the x-value.” I coined this process of using more verbs as the verbification of mathematics (Lunney Borden, 2011) as a way to contrast mathematics tendency towards nominalization – turning actions and processes into nouns – however I more commonly refer to it as verbing math.
When I ask myself in planning, “How can I verb this concept?” what I really want to know is how can I strip away some of the rules we, as math educators, know about these concepts and consider a task that is action oriented and generative, in that the concept can emerge through playful engagement with an idea. This requires a deep level of understanding of the meaning of concepts so that one can consider how this will emerge. For example, as Nat has discussed above, we can think of addition and subtraction as joining, separating or comparing sets or lengths. This allows us to engage students in these actions to lead them to conceptual understanding of these operations. Similarly, in thinking about multiplication we can think of it a building equal sets, jumping equal distances on a number line, or building equal rows in an area model. Division can be thought of as forming equal sets from a whole, equal partitions of a length, fair sharing, or finding the dimensions of an area model. These processes allow me to invite students to engage with these concepts through actions that generate concepts. It was through thinking about these actions that I developed the bucket of zero idea.
Nat reached out to me and expressed his desire to learn more about the bucket of zero as a way to model integer multiplication and division in a concrete way that would allow students to understand the concept rather than just learn the rules. Through a series of social media interactions, we worked through the bucket of zero, as it had worked for me.
Nat’s Experimentation with the Bucket of Zero
Included in my notes from Lisa’s presentation was a phrase that I circled several times, obviously in an attempt to remember it down the road: “Reclaim the action behind the mathematics”. Recognizing this as an opportunity to heed this advice, I began to re-examine my current model for integer operations and draw parallels to the bucket of zero model that Lisa had presented.
Up to that moment, I had used a pile of two-colour counters to model my concept of ‘addition as combining’ and ‘subtracting as taking away’. Students were given a collection of these two-colour counters, small discs coloured red on one side and yellow on the other. In this model, each yellow counter is designated as a positive and each red counter is designated as a negative. Taken together, a single red counter and a single yellow are neutral; they are known as a zero pair. The nice thing about this model is that when students are asked to add and subtract, that action translates into the model as combining and taking away. In other words, the action of the model matches the concept. For example, if asked to compute 5 + (-2) with a pile of counters (see Figure 1), the collection starts with five yellow counters, and the solution is obtained by combining that group with a group of two red counters, neutralizing any zero pairs, and counting the value of the new collection. Similarly, if asked to compute 5 – (+2) with a pile of counters (see Figure 2), the model starts with five yellow counters, and the solution is obtained by taking away a group of two yellow counters, neutralizing any zero pairs, and counting the value of the new collection. The model aligns with my conceptualization of addition and subtraction, and, through the action, rules can be thoughtfully composed. In a similar manner, the bucket of zero also uses red and yellow counters, and the expression determines both the type of counter (red or yellow) and the action to undertake (fill or remove). After the action is complete, the solution to the expression is the ‘value’ of the bucket—the numbers of red or yellow counters after all zero pairs are neutralized.
At first glance, I figured that the bucket of zero would be identical to what the collection model in terms of reclaiming the meaningful action behind the operation. Initially, it felt like the transition to the bucket of zero could simply be accomplished by drawing the image of the bucket around the counters (see Figure 3 & Figure 4). That is to say that the bucket of zero contained identical actions; adding still manifested itself as a ‘combining’ and subtracting was still executed as a ‘taking away.’ However, after some playing, I soon realized that the bucket of zero accentuated a critical piece of the notion of integers. Namely, that there are many ways to make zero.
This became obvious when modeling operations like 5 – (-2). To model this expression with a collection, I started with five yellow counters, in exactly the same way I began modelling the operations in Figure 1 and Figure 2, but I ran into a snag when attempting to remove two red counters because there were none to remove. In comparison, the bucket of zero model began with ‘filling the bucket with zero,’ which encouraged me to think about the many ways in which any starting balance could be created. The ease at which the bucket of zero welcomes the addition of zero pairs accentuates the fact that we do not need to begin with a collection of exactly five yellow counters, but, rather, the bucket must be ‘filled’ with a value of five. That is, there must be five more yellow counters than red ones (Figure 5). It follows that there are an infinite number of ways to fill the bucket. Of course, zero pairs can also be added to the collection of counters, but, for me, the collection model directs my attention (and places emphasis) on the absolute size of the collection, thus triggering me to pay attention to a zero pair as ‘two counters.’ In contrast, the bucket of zero directs my attention to the balance in the bucket, triggering me to pay attention to the zero pair as ‘worth nothing.’
The prompt 5 – (-2) is then resolved by filling the bucket with zero and then establishing a bucket value of five by adding an additional five yellow counters. Subtracting negative two is accomplished by removing two red counters from the bucket, and the solution is then obtained by neutralizing the remaining zero pairs and determining the new value of the bucket (Figure 6). It is important to reiterate that, while the bucket of zero accentuates the notion of multiple zeros, the same notion of multiple ways to create zero can be modeled with the collection model as well. That is, both models support the concept of adding and subtracting integers; both the collection model and the bucket of zero are robust in the sense that the available actions with the model aligned conceptually with my underlying actions of the operations I was modelling. In short, they allowed me to make meaning. It could therefore be argued that the choice of model is a personal one, and should be made based upon which is best suited “for the immediate purpose of developing their understanding” (Greeno & Hall, 1997, p. 365).
At this point, my actions with the addition and subtraction of integers had convinced me that the bucket of zero model was essentially identical—save for the accentuation of multiple zeros—to the collection model I had used with students for years. What is further, the collection model also seemed, at least initially, to support my concept of multiplication as ‘gathering same-sized groups’ and division as ‘creating same-sized groups.’ For example, if asked to compute 3•(-4), the solution is obtained by gathering three groups, each of which is composed of four red counters. Similarly, if asked to compute 10 ÷ 2, the solution is obtained by creating two same-sized groups from a collection of ten yellow counters, and if asked to compute -10 ÷ 2, the solution is obtained by creating two same-sized groups from a collection of ten red counters. It did not matter whether these three groups are gathered in a bucket or in a pile; the concept of gathering was actualized through the use of the model.
However, the collection model did not provide a way for me to make sense of exercises like (-4) • 3 or 10 ÷ (-2). That is, both of these exercises involve negative groups. Not groups composed of negative objects (red counters), but negative groups. In the past, I told students that we handled an exercise that asked us to gather negative four groups each composed of three yellow counters, by flipping the counters and gathering four positive groups each composed of three red counters. The same flipping action was also used when asked to create negative two groups from a collection of 10 yellow counters. However, as I thought about how to share the model with my undergraduates, I realized that the move of ‘flipping’ had no part in my conceptual understanding of what multiplication and division was. It felt like I was flipping signs because the rules dictated that I needed to, not because it made sense to do so. For example, if asked to compute (-4)•3, the model treated this as identical to 4•(-3). Similarly, if asked to compute 10 ÷ (-2), the model treated this as identical to (-10) ÷ 2. Apparently, this used to satisfy me, but it longer did. Now I got the distinct impression that the rules were explaining the model and the model was not providing explanatory power to the rules. This was further illustrated by cases like (-3)•(-4) and (-10) ÷ (-2), where there seemed to be no place for the negative sign to migrate to.
In each of these cases, I found myself justifying the model with a standard set of rules for using the model, where it should have been the exact opposite phenomenon: The rules for multiplying and dividing integers should emerge through activity with the model. Of course, there are other conceptualizations of multiplication and division that accompany different models (see Davis & Renert, 2013), but this had now become more than a search for a model for my undergrads; this search was personal.
Lisa’s Verbification with the Bucket of Zero
As a teacher and teacher educator, I have always paid attention to classroom models that expose the movement of mathematics. The bucket of zero also began with search for a generative action that would allow me to help my pre-service teachers understand the rules for multiplying integers. Like Nat, I had long used collections of two-colour counters for modelling the zero property for addition and subtraction, but my models for multiplication and division were similar to Nat’s in that they did not really allow meaning to emerge. My search for something more substantive was inspired by a few ideas. I had been considering the “huh” moments for teachers. I had also been exploring the website of Zolten Dienes (2000), in particular the ways in which he used story to teach concepts. One such story involves a dance that leads students to understand rules for integers, as dance partners arrive and leave. Again, I loved the idea of a dance hall full of dancing zeros and the addition and subtraction was very intuitive but the multiplication and division were not as immediately apparent. I began to think about the dancing zeros and how we might have 2 or 3 cars come and drop off equal numbers of dancers looking for partners or pick up equal numbers of dancers leaving partners behind. Eventually my dance hall became a bucket filled with zeros and I began inviting students to think about adding equal groups to the bucket or removing equal groups from the bucket. I also became very intentional about talking with my students about what mathematics symbols mean in the context of multiplying integers. This involved revisiting basic multiplication to consider the various representations we use in modelling multiplication. For the purpose of the bucket of zero, a set model is nice. For example, (+3) • (-2) explicitly means “Add 3 sets of -2” to the bucket of zero (Figure 7), (-4) • (+3) means “Remove 4 sets of +3” from the bucket of zero (Figure 8), and (-2) x (-5) means “Remove two sets of -5 from the bucket.” (Figure 9) What I liked about it right away is that it made explicit something that is often implicit, namely, we start with zero. Consider an elementary aged student being asked to do 3 jumps of 2 on a number line as an introduction to the process of multiplying. It is not correct to start anywhere on that line, we remind students to start at zero. The same is true with multiplying integers; we start at zero, a whole bucket of zeros.
When working with the bucket of zero, students easily see that we are adding to or taking from the bucket of zero, sets of equal size. The first number, or multiplier, tells us not only how many sets, but also tells us if those sets are added to the bucket or taken out of the bucket. The second number, or multiplicand, tells us the value of the set (magnitude and sign). So, multiplying (+a) • (+b) adds positives to zero, making the value of the bucket positive. Multiplying (+a) • (-b) adds negatives to the bucket making the bucket negative. Multiplying (-a) • (+b) removes positives from the bucket, leaving behind their negative partners, thus making the value of the bucket negative. Multiplying (-a) • (-b) removes negatives from the bucket, leaving their positive partners behind making the bucket positive. It is this last model that, I have found, really wins students over as they often, for the first time see a reason why multiplying two negatives makes a positive, a rule they likely have known for a long time. Once students know the reasons for the multiplication rules, division can easily follow with fact families or they can think about the bucket again.
For division, as is the case with whole numbers, there are two possible ways to make sense of this process – knowing the number of sets or knowing the size of the sets. For example (+10) ¸ (-2) can mean you make the value of the bucket worth +10 by removing two equal groups from the bucket. What is the size of each group? We would need to remove 2 groups of -5 from the bucket to make it equal +10 (Figure 10). We could also say that (+10) ¸ (-2) means using groups of -2 make the bucket worth +10. In this case I would need to remove 5 of these groups to leave the bucket with a value of +10. (Figure 11). I always want students to consider these two ways of thinking about division and then decide which one works best for their understanding. Again, this requires a revisiting of what division means and the various ways to model it. When students explore dividing a negative by a negative, these models help them to understand why the result is positive. So (-8) ¸ (-2) could mean take out two sets so that the bucket is left with a value of -8. I need to take out 2 sets of +4 and leave behind two sets of -4 in the bucket. (Figure 12) Or I could think using only sets of -2 make the bucket equal to -8. To make the bucket negative I need to add sets of -2, in fact I need to add 4 sets of -2. (Figure 13) The bucket of zero invokes a need for action of adding to or taking from, and this allows students to make sense of division of integers in a tactile and active way.
With my preservice teachers, I regularly invite them to explore the actions associated with multiplying and dividing integers using the bucket of zero before even discussing symbolic operations. This has frequently been met with surprise and amazement when their huh moments because aha moments for the first time and they see why these integer rules work the way they do.
Nat and His Undergrads Work with the Bucket
It was Lisa’s description of dividing by a negative number that, for the first time, brought the operation of integer division to life. By that I mean, that the action—the verbing—introduced the bucket of zero finally matched my concept of division as creating same-sized groups. Until this point, I needed to flip some of the counters before I could create groups, but the bucket employed the actions of filling up and removing from the bucket to handle the notion of negative groups, a concept that was inaccessible to a collection of two-colour counters.
After my experiences with developing a coherent model, with plenty of Lisa’s help, my goal for my pre-service teachers was to create a classroom experience along a similar trajectory. I assumed that, given the widespread use of two-colour counters, that my students would have fallen victim to the same inconsistency in the model that caused me such consternation and also sponsored this paper. Under that assumption, I wanted to elicit and solidify any rules for integer operations, work with models they might already be familiar with, and then closely examine the actions of those models in order to trigger similar concerns with those models that I experienced. In this way, the pedagogical trajectory I planned was half vindictive and half empathetic, and class was designed to provide opportunities for the pre-service teachers to make meaning with the model, and not simply to assume that because the model was now filled with meaning for me, that it comes ready-made and meaning-filled for another learner (Banting & Vashchyshyn, 2018; Greeno & Hall, 1997).
We started with a short introductory activity where I asked students to write a story problem that can be modeled, and then solved, with the following expression:
-2 + (-4)
Consider two stories written by one of the groups:
“I had two rotten apples and then four more apples went rotten. Then I made apple crisp.”
“I had two rotten apples and then someone gave me four more rotten apples to make apple crisp.”
In both these stories, the protagonist ends up with six rotten apples. Initially, discussion was centered on whether a rotten apple could, in fact, be considered a negative apple. Eventually we accepted this reality as true (and abandoned the notions of some sort of anti-matter apple), to tackle the larger conceptual issue: These stories feel like they model different actions, yet result in the exact same resolution. In other words, they represent different concepts despite resulting in identical, numerical results. The ensuing discussion exemplified the distinction between a calculative approach and a thinking and modelling approach to using mathematical models (Lesh & Doerr, 2003). In a calculative approach, the model is not considered a way to think about a problem, and the execution of the model becomes the problem in and of itself. If the number sentence model is considered strictly as a way to resolve the calculation, we might gloss over the debate of whether an apple going rotten is the same, conceptually, as being given a rotten apple. However, with a thinking a modelling approach, a model provides a way to work with a problem, and so discussing how the first number sentence feels more like subtracting ripe apples than adding rotten apples becomes a pertinent detail. This work with the story prompt was designed to invite students into a thinking and modelling mindset before I introduced the bucket of zero.
We talked about the rules and models that the pre-service teachers had encountered in their own schooling or in the course of their teacher preparation prior to that moment. As I had anticipated, many teachers recounted their calculative approach to a variety of models, which included collections of two-colour counters. I firmly embraced the role of devil’s advocate, repeatedly asking students to show me how to gather negative groups. Predictably, as I continually pointed out, they gathered positive groups of negative items, and the same debate broke out as was sponsored by rotten and ripe apples. Are these processes different? Why do we flip? Does the rule drive the model or the model drive the rules? When I was convinced that we were thinking and modelling our problems, as opposed to calculating them away, I explained the bucket of zero.
The bucket of zero was introduced through the same style of animation that appear in this article, with the exception that the timing of the actions was controlled through presentation software and was not pre-timed in a video format. We began by filling the bucket with zero in order to establish the concepts of zero pairs, multiple ways to create zero, and the mechanics of filling and emptying the bucket of zero. From there, the operations of addition, subtraction, multiplication, and division were modelled in that order, with particular attention paid to the cases that included negative groups. The sequence of animations was met with enthusiasm, and many of the pre-service teachers excitedly talked about the potential of the model for their classrooms. However, there were several moments where students asked critical questions of the model. For example, some debate arose around whether multiplication should be modelled by groups being added or subtracted one at a time (see Figure 14) or if the groups should all first be formed and then added or removed from the bucket all at once (see Figure 15). Some argued that the first treatment was not one of multiplication, but of addition. Others argued that they liked how that particular action of the model illustrated, for them, the connection between repeated addition and multiplication. The thinking and modelling approach reminds us that this particular problem, (-4) • (+3), is not, by its nature, a repeated addition problem or a multiplication problem, but dependent on how it is modelled.
The interrogation continued around the potential model for integer division. Unlike Lisa’s model, and corresponding question (animated above in Figure 10 & Figure 11), I presented a model for division (Figure 16) supported by a different question, “Two groups were removed from the bucket of zero, and now the bucket has a value of positive 10. How much was each group worth?” Students argued that they did not like the way the question was posed, because they were not doing the removing. Rather, it felt like they had arrived at the scene of a crime and were tasked with figuring out what mathematics had occurred, instead of doing the mathematics themselves. This insight strengthened the explanatory power of the bucket of zero as a mathematical model for all of us in the room—particularly myself. It would be simple to treat these challenges as negative events, as deficiencies in the clarity of the model or the method in which it was introduced. However, these queries illustrate, for me, critical evidence that students were not simply accepting a meaning-filled model, but rather interrogating how the new model afforded new mathematics.
Further evidence that the pre-service teachers had made personal meaning with the model was echoed in the year-end course reflections where several students mentioned the model explicitly when describing how the course helped them prepare for their future careers as mathematics educators. Their responses indicated an appreciation of the model as a tool for their future classrooms, but also an appreciation for the way they were encouraged to interrogate the model so it became a thinking tool and not just another brittle procedure, justified because the results matched the rules for symbolic computation. The bucket of zero was important for them, as it was for me, in reclaiming the act of doing mathematics.
Lisa’s Implications for Teaching and Learning Mathematics
The approach I use in my methods classes is typically to have students use two-colour counters at their desks to experience the modelling in a tactile way. I also make use of virtual manipulatives when discussing as a whole group so that students can see groups being added or removed in the ways we have shown with the animations. My pre-service teachers remark regularly that they always knew the rules for multiplying and dividing integers but never knew why they worked conceptually. Recently, a student declared that the bucket of zero conversation was “life-changing” for her. Making seemingly abstract concepts visible and tangible for students is an important part of the work we do as teachers and teacher educators. It is common to look for contextual situations to make meaning (Carpenter, Fennema, and Franke, 1996) but contexts for integer operations can feel very contrived for students – negative apples anyone? What can be more meaningful in a situation like this is to pay attention to the structure of operations, in particular what operations do to quantities. I draw from Mason, as described in Venkat, Askew, Watson and Mason (2019), who argued that “the notion of structure has an architectural quality, a spatial organization formed by specific relationships that place some element or elements in particular configurations with another element or elements, rather than in random arrangements” (p. 14). This description aligns with the idea of verbing where we use actions to form quantities in certain ways based on the actions manifested in the mathematical processes. My passion for verbing and spatializing math that involves just such a process – allowing students to play with structure, to build models that invite them to explore this specific structure, and to come to a more general understanding of the processes of mathematics. Only then do we name these processes. The bucket of zero lets us uncover the meaning of operations on integers by paying attention to the actions of these operations (multiplication and division) on quantities, in this case integers. Adding or removing sets of integers allows students to build models that show the meaning of multiplication and division of integers. From these specific examples, they can generalize how these processes work for any integers. I believe most, if not all, mathematics concepts can be developed in similar ways.
The story of our collaboration around the bucket of zero has asked us to play multiple roles. Our initial search for a model was inspired by our role of teacher and teacher educator searching for a model with explanatory power, but our process of refining the model placed us, quite firmly, in the position of student—tinkering with possibilities and organizing meaning. We feel that it was this (sometimes dizzying) experience of switching between teacher and learner that spoke meaning into the bucket, creating a mathematical model from a manipulative. Just like the story of its origins (which we have attempted to chronicle here), the overarching message of our work with the bucket of zero emerges as one with layers that, we feel, hold importance for our work as teachers and teacher educators.
First, the bucket of zero holds power, and it is our hope that you find the same power in your work with mathematical learners. For those students who have already encountered integer operations as a series of memorized rules, it provides a robust explanatory model to accompany the repetitious dictums and symbolic choreography of school mathematics, where the notation +3 – (-4) is met with the over-generalized chorus of “two negatives make a positive”. This story illustrates the potential of the bucket of zero to unlock meaning in learners who have had decades of unexamined participation in such a practice. The potential is far greater for learners who have yet to encounter the notion of negative numbers and might have their introductory experiences with the concept guided with meaningful actions and interactions. Due to our shared history as educators, we cannot gloss over the fact that the model is directly applicable to the work of teaching. In this light, we feel the bucket holds value because of its utility: It has potential to sponsor mathematical interactivity with the concept of integers.
Second, the bucket of zero forefronts the importance of learners-as-doers. The process of constructing meaning through experience is a powerful one, and this power continued to draw us back to the model. Through the course of our careers, we have heard and offered numerous metaphors for integer operations. Two of the more popular ones—temperature and finance—aim to tether the concept to the student’s existing operation in the world. Here the numbers are translated into objects (temperatures or currency), and then the operations are carried out with support from an analogous climate or bank account. Even setting aside a young learner’s understanding of accruing debt or the unnatural manipulation of dividing degrees Celsius aside, both of these metaphors have students reflect on a world that is forming outside of their control. The temperature and bank balance happen to them, not because of their action. Students recognize these contexts are contrived. Furthermore, the contrived contexts can serve to confuse rather than support sensemaking.
The bucket of zero has a different character, one that sacrifices the strained connection to the real world in favour of having students act in real ways. It illustrates the mathematical power that is gained by asking learners to play with abstract concepts (operations on integers) in non-abstract ways (adding, grouping, removing, stabilizing, etc. positive and negative counters). It is through this activity that mathematics maintains its active nature, at least long enough for the rules to maintain their meaning. In this way, the artefacts of integer operations—the often-recited rules—emerge through meaningful interaction with a model, and the learning trajectory begins with figuring and, once that figuring is complete, culminates in a naming. Ultimately, it is this trajectory that we feel is of utmost importance when teaching, learning, and doing mathematics.
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