# Building Realizations: Understanding Multiplication through Exploration

#### 1 Classroom Norms

As annotated from Boaler (2015), Cohen & Lotan (2014) and original work:

To be able to teach in a classroom that navigates both the content and the mathematical mindsets I have found, corroborated with other research, that situating and the *living* the norms of a collective classroom space is vital. Here are the norms that situation the collective environments that I design as a mathematics educator:

1. We believe that each of you can accomplish all that you wish to accomplish with your education. There is no such thing as innate mathematical ability or genetics that make you good at math.

2. We do not believe that being an exceptional mathematics learner is more important than the many other intelligences and abilities that you may possess.

3. We have high expectations for you, and expect you to achieve at the highest level.

4. We love mistakes.

5. We do not care if you are slow at mathematical computation, we care if you are thinking and learning.

6. We want you to question everything.

7. Mathematics is about connections, communication, and creativity.

8. Our work together is about learning, not about performance. Mathematics is a subject of growth that takes time, effort, and struggle no matter how smart society thinks you are. It takes time.

Students will read through the norms provided in digital, poster, or paper form. If this is a one-to-one environment, I would suggest having the norms as something that is embedded in your LMS (moodle, canvas, blackboard). The discussion will be a explicit part of every collective space created around the content of this course. ***I have found that creating collective spaces, that build a culture is linked directly to those that create the space. So, all evolved must make an explicit effort to abide and navigate according to these norms. Any contradictions should be discussed immediately when found.

#### 2 Emphasis of Realization: Building an evolving understanding of Multiplication

First we build a realization to answer the following 2 questions:

1. What is a number?

2. What is multiplication?

A realization is fundamentally different than a definition (Berkopes, 2014; Renert & Davis, 2013):

Realizations contain all of the following:

1. Formal basal definitions

2. Images associated with the concept

3. Algorithms associated with the concept

4. Metaphors associated with the concept that aid learning the concept

5. Applications of the concept in the "real world"

6. Debate; is the concept an invention or a discovery?

Build using (whiteboard, googledoc) or any other sort of sharing platform:

A realizations chart that discusses all of the 1-6 facets (for both number and multiplication) in a way can be communicated with the rest of the class. The key here is that there are no wrong answer to realizations charts, they are meant to evolve daily in a classroom as the students mathematical perceptions evolve.

#### 3 Entailments: Building on what we think we know about Multiplication

Here you can use any version of a basic line multiplication algorithm. I have chosen a png that will provide a good initial discussion topic. The goal here is to discuss and see if this is a viable way of operating. Have students record carefully their thoughts so that a broader discussion can be promoted within the class. Use the video only if the classes discussions have stagnated and cannot be remediated through scaffolding.

Students interrogate the line multiplication picture with the following questions in mind:

1. Take a look at the following, what is happening here?

2. Does this picture give you a correct answer to the question?

3. What parts of mathematics is happening here?

4. Does this *always* work?

#### 4 Wrap-up and Realization Evolve

Teachers should now discuss the line multiplication in a way that links how students defined multiplication and number in their realizations chart and see if there is a need for annotating or further discussing their realizations charts. The evolution of these charts is critical to helping students understand the living norms mentioned in the first part of the lesson.

Assess and evolve the realizations chart based upon the work done with the line multiplication activity. Discuss further what is multiplication, what is number.

Answer the following:

1. What does it mean that our conceptions of these matheamtical concepts have changed?

2. Is there just one mathematics?