Radical constructivism is an exciting theory of how best to teach mathematics. This lesson provides an overview of what radical constructivism is and describes how it might be applied in a contemporary classroom.

## What is Radical Constructivism?

**Constructivism** is an approach to learning that argues that learners will not understand knowledge if they are simply taught facts as pre-existing entities. Rather, each learner must not only come to knowledge on his or her own terms; he or she must actually create the knowledge from scratch. Every learner constructs a knowledge base that he or she then builds on as part of moving through the world.

**Radical constructivism** is the idea that all learning must be constructed, and there is no utility or meaning in instruction that is teacher or textbook driven. Radical constructivism is often referred to in reference to mathematics, but it can be difficult to understand and enact. In this lesson, math coach, Ms. Novelo, will share how she helps the teachers she works with to understand radical constructivism in math.

## Beginning With Concepts

Ms. Novelo believes that the main idea underpinning radical constructivism in math is that concepts are much more important than facts or algorithms. Therefore, when young students are learning addition, it is not important for them to memorize their doubles or ways to make ten.

It is equally unimportant for them to learn about carrying or regrouping. Instead, it is important for students to understand the concepts underlying addition. Some questions that Ms.

Novelo encourages teachers to pose over time are:

- What happens when you get more of something?
- What happens when two groups of people come together?
- How can you tell that you have more now than you did before?

Concepts are just as important in more sophisticated mathematical concepts. Ms.

Novelo does not advocate, for example, that high school math teachers require students to memorize formulae for calculating and graphing averages. Instead, when they are studying statistics, Ms. Novelo recommends beginning with contexts that are meaningful to students and asking them questions like:

- What might you want to know about this data?
- How can you figure out what you need to know in order to tell a story about this data?
- Which operations might be effective at helping you make a meaningful graph, and why?

Students who lack a conceptual understanding of fundamental math concepts will struggle with higher order thinking, and they will also lack a sense of ownership of the ideas and processes they are working with.

## Real-World Application

Ms.

Novelo thinks that one of the cornerstones of radical constructivism is the constant awareness of students’ use of math in their real lives. She encourages the teachers she works with to imagine a day in their students’ lives and write down all of the different mathematical ideas and skills that come up over the course of the day. Ms.

Novelo reminds teachers that this might look very different depending on cultural, socioeconomic and geographic concepts, and that is okay. Part of the beauty of radical constructivism is that it is inherently culturally sensitive, since students are constructing frameworks of knowledge that makes sense to them.

Teachers that Ms. Novelo works with then design conceptual investigations, questions, and activities that allow students to explore the mathematical ideas most relevant to their daily lives. They work with numbers, but also shapes and conceptual questions. Students often design their own story problems and tackle questions involving sophisticated algebraic reasoning from a very early age.

## Releasing Time Constraints

One of the biggest challenges Ms. Novelo sees her teachers facing is the idea that mathematical skills must be mastered within a certain time frame. Radical constructivism essentially asks that teachers release these time constraints and trust that students, over time, will construct the concepts and skills they are ready for. This is challenging, since many contemporary curricula and sets of standards ask, for instance, that multiplication be mastered by the end of third grade. In true radical constructivism, these temporal frameworks would not exist.

However, Ms. Novelo works with her teachers to strike a balance between encouraging students to develop their own understandings but still prepare them to meet the demands of a standards-based education system.

## Openness to New Ideas

Most of all, Ms. Novelo asks that her teachers remain **open**, meaning, in this case, that they should be willing to look at the world from perspectives different from how they were raised or taught. Students might initially construct knowledge in ways that seem inefficient and wrong, and it is all too tempting for teachers to simply correct them.

Radical constructivism means being open to accepting that students might actually have something to offer the world of mathematical understanding even from a very young age. And what may seem to be mistakes need not always be corrected.

## Lesson Summary

Radical constructivism is a theory of teaching and learning that encourages students to construct knowledge and build on a conceptual understanding of mathematical knowledge. In radical constructivism, concepts and real-world application are key, as is an openness to new ideas and a willingness to be flexible with how learning is timed and measured.