The Core of the Matter Issue #3
Why consolidation is essential to progressive mathematization…
My colleague, Isobel, has this saying that has always stuck with me, “It’s all about the debrief.” At this point I would probably go as far as to categorize it as a mantra. Even though she is referring to the work we do with adults, it rings every bit as true when talking about students. In fact, I contend the debriefing of learning, which goes by many names in math, such as consolidation (Liljedahl, 2021), math congress (Fosnot & Dolk, 2001), or Neriage (in Japan), is the most important part of any lesson. It allows for students’ informal thinking to be formalized, and for their learning to be solidified and encoded into long term memory. Peter Liljedahl (2024) refers to this process as going from, “meaning making to meaning made.”
However, the consolidation process, as powerful as it is in supporting student learning, is an extremely difficult thing for a teacher to do well, as it requires extensive expertise. In a math classroom, for example, a teacher must possess a strong understanding of standards, learning progressions, and the development of student thinking. Furthermore, they have to be skilled in facilitating student discourse and keeping all students engaged and thinking. On top of that they need to have a high level of tacit knowledge in order to best navigate all the decision points that will come up during the course of the discussion, what Deborah Ball (2018) calls discretionary spaces.
At this point I think it would be prudent to clarify some things about what a consolidation or congress is and is not. The first thing to clear up is that it is not a show and tell, or just a time to share. If students are simply providing a narrative of their solution process, or just sharing their answer, nothing is being consolidated. Rather, it is a much more intentional activity, where the teacher is actively selecting what pieces of student work to highlight during the discussion and in what order. The goal of a consolidation is focused on weaving together the key mathematical ideas that emerged throughout the lesson in order to help students make sense of those ideas, in a way that connects new learning to prior knowledge.
A second and related clarification point is that the consolidation is not just about discussing different ways students got to the answer. Again, simply hearing about a bunch of different solution strategies on its own will not lead to anything being consolidated. The teacher needs to press students to make connections between the various strategies presented, and then relate these similarities to the underlying mathematical ideas, properties, or principles from which they were derived. This idea of students starting with informal strategies and ideas, and then gradually moving towards more abstract and formal ways of thinking is referred to as Progressive Mathematization. The term stems from the work of Hans Freudenthal, a Dutch mathematician, who focused much of his career on researching how to best teach the subject of mathematics. Later, Adrian Treffers, a mathematics curriculum researcher and disciple of Freudenthal, made a further distinction within the construct of progressive mathematization by delineating two different types of mathematical activity, which he called horizontal and vertical mathematizing.
According to Treffers, horizontal mathematizing occurs when a learner comes up with strategies to organize and solve math problems embedded within a real world context. Freudenthal describes horizontal mathematizing as, “going from the world of life to the world of symbols.” Treffers goes on to explain that horizontal mathematization occurs when any of the following activities can be identified: identifying or describing specific mathematics in a general context, formalizing and visualizing a problem in different ways, recognizing relations and regularities, recognizing related aspects in different problems, and transferring a contextual problem into a mathematical problem. Treffers describes vertical mathematizing as going beyond what was constructed during the horizontal mathematizing phase. It is defined as the process of organizing within the mathematical system itself. Freudenthal described vertical mathematization as, “moving within the world of symbols.” Treffers adds vertical mathematizing occurs when the following activities can be identified: reorganizing within a mathematical system, representing a relation in a formula, proving regularities, refining and adjusting models, using different models, combining and integrating models, or formulating or generalizing a mathematical model.
Source: Drijvers, 2022
It is important to note that moving from horizontal to vertical mathematizing should not be seen as a linear process, but rather as dynamic and cyclical in nature. Additionally, Freudenthal cautioned educators not to place more importance on one over the other. Rather, he argued that both were important for students to experience as a part of progressive mathematization.
Classroom Connection
The construct of vertical and horizontal mathematizing is important for teachers to understand, specifically as it relates to the instructional goals during the consolidation phase of a lesson. A teacher must have a plan to select and sequence student work in such a way that allows for both horizontal and vertical mathematizing to occur.
For example, let’s suppose we are teaching a third grade multiplication unit. We have chosen a task from the Context for Learning Mathematics (CFLM) unit, The Big Dinner, where students work to find the price for different amounts of apples and carrots for a Thanksgiving dinner. While the class is working on the problem we notice that many students have made use of a partial products strategy to find the various prices. For example, some students added the price for 5 pounds to the price for 1 pound to get the price for six pounds, and others doubled the price of 5 pounds to get 10 pounds, etc. Our goal then during the consolidation could be to select student work that highlights the use of partial products (horizontal mathematizing) as a means to examine the conjecture that decomposing a factor and combining the subsequent partial products will work for all multiplication problems. Internalizing this idea leads to a generalized understanding of the distributive property (vertical mathematizing).
Let’s take another example. This time let’s imagine we are teaching a grade 7 class and students are working on the Lewis Carroll Cats and Rats Problem. The task asks students to figure out how many cats are needed to kill 100 rats in 50 minutes if we know that 6 cats can kill 6 rats in 6 minutes. There are a lot of different ways to approach this problem, and I will not go through all of them, but what they all have in common is that they make use of the relationship between the different variables in the problem. The goal for consolidation would be to select work that highlights strategies which leverage these relationships such as repeated addition, scaling in tandem, or doubling and halving (horizontal mathematizing), in order to push students to identify and generalize different types of proportional relationships (vertical mathematizing). For instance, you might select a student work sample that used a table to show that 1 cat can kill 1 rat in 6 minutes leading to the conclusion that 1 cat can kill 8 rats in 48 min, 2 cats can 16 rats in 48 min, and so on until they arrive at 12 cats killing 96 rats in 48 minutes. The work sample shows that from here they decide they need an extra cat in order to ensure all 100 rats are killed in 50 minutes. After discussing the utility of this strategy, the focus could shift to analyzing the proportional relationship between cats and rats. Specifically, that 1 cat can kill 1 rat in 6 minutes. This also allows for a discussion on the idea of the minutes being an invariant that mediates the relationship between cats and rats. In other words, the ratio between cats and rats is 1:1 given the minutes remain constant at 6. Next, you could show a solution where the students concluded that 6 cats can kill 1 rat in 1 minute and scaled up to figure out that 6 cats can kill 100 rats in 100 minutes. From here they may have figured out that in order to kill 100 rats in half the time they need to double the number of cats to 12. This opens the door to focus on the relationship between cats and minutes, which are inversely proportional to one another. An increase in cats will lead to a decrease in the minutes needed to kill a constant or invariant number of rats, and an increase in minutes will require a decrease in cats needed.
Related Reads
This chapter from the book, Current Studies in Educational Disciplines, provides a nice overview of the idea of progressive mathematization, including horizontal and vertical mathematizing. It also summarizes other tenets from the work of Freudenthal and his predecessors which has come to be known as Realistic Math Education (RME).
This article uses the vertical/horizontal mathematizing framework to analyze the thinking of middle school students reasoning about fractions.
This book, by Smith and Stein, outlines the 5 practices the authors believe are necessary for orchestrating productive mathematical discussions.
Peter Liljedahl’s book, Building Thinking Classrooms, has an excellent chapter on consolidation.
This article, by Akihiko Takashani, provides a digestible summary of the structure of a Japanese mathematics lesson, including the consolidation portion which is called Neriage.
This link takes you to a resource I wrote to assist teachers in planning and implementing an effective consolidation. It is part of a series of instructional recipes I authored which you can find on my website via this link.
This link takes you to a Substack post my colleague Isobel wrote where she explains the origin of the recipe metaphor and its role in the process of supporting teachers to be researchers of their own practice. While you're reading this post don’t forget to subscribe to Isobel’s Substack, The Coaching Letter, where she writes about leadership, organizational improvement, coaching, instruction, and everything else adjacent to those topics.