# Teaching and Learning Corner: Mathcad and Constructivism (Part 1)

One of Mathcad’s key capabilities as a desktop tool is the documentation of intellectual property.  Many of our users leverage Mathcad’s mathematical, graphical, and text capabilities to capture the ideas behind their engineering designs. In design and engineering the importance of documenting the knowledge and beliefs behind one’s contributions is well established and critical to progress and success.

In education, modern theories of learning such as constructivism advocate for an approach to instruction that pays careful attention to the intellectual property of learners. In mathematics and science education, many studies support this approach. The research is so compelling that the first principle of learning advocated by the National Research Council for both math and science classrooms is Engaging Prior Understandings (Donovan and Bransford, 2005, How Students Learn, p. 4):

New understandings are constructed on a foundation of existing understandings and experiences.

It is evident that in order to engage prior understandings as source material for learning, a teacher must encourage students to express their knowledge and beliefs as their own. Since I think that Mathcad has capabilities that can support this approach to teaching, I intend to post several blogs in the next year addressing the topic of constructivism. Today’s blog will focus on the elementary level mathematics classroom.

The image below is an example of the type of problem that I have observed being used in an elementary school classroom.

## Validating the Intellectual Property of Learners

In an environment where students are going to learn by constructing new knowledge and beliefs on a foundation of existing experiential knowledge, it makes sense that teachers should encourage learners to “own” their ideas. In fact, many of the best teachers that I have known encourage this practice by labeling students’ problem strategies (“Carrie’s Strategy”) so that others can both employ and attribute it if they use it themselves.

An adult, non-educator may look at Carrie’s strategy and describe it as inefficient. There are clearly more efficient ways to evaluate the expression, such as the Check in the bottom of the image. A teacher who teaches from a constructivist perspective, however, may describe Carrie’s work differently:

1) Carrie is relying on her understanding of Base 10 numbers to group 10’s and 1’s and evaluate expression

2) Carrie is using the commutative and associative properties of addition well in her work

3) I wonder how Carrie would solve the following problems:

10 + 10 + 10 + 10

5 + 5 + 5 + 5 + 5

20 + 20 + 20 + 10

The same teacher might consider the 7 + 7 + 7 + 7 + 7 task as a good problem for her class because:

1) The task will give all of the students in the class an opportunity to show their understanding of the commutative and associative properties

2) The task may provide an opportunity for Carrie to be exposed to or to discover new strategies for evaluating expressions that will bring her closer to understanding how multiplication can be used.

3) The task will help to validate Carrie’s current ideas as valid intellectual property and thereby increase her confidence as a learner

## Managing the Intellectual Properties of Learners

In a constructivist classroom the learning of mathematics shares many characteristics with the practice of engineering design. The process of learning can be described as designing and understanding strategies for evaluating expressions and solving problems. In the example above, Carrie’s initial understanding (her way of seeing the expression) can be described as inefficient, but representative of her current knowledge and experience. Similarly, I think it is safe to say that initial engineering designs will be (a) based on existing knowledge and experience and (b) less efficient than final designs that have benefited from an iterative design and review process.

For teachers, like engineers, the extra effort required to foster the development of intellectual property is really hard work. In both settings, however, the evidence indicates that the process adds tremendous value. In both education and in engineering design, the process of developing intellectual property is worth the effort required to engage in effective project (or classroom) management.

I make this comparison because of I currently see and hear tremendous amount of interest in the improvement of science, technology, engineering, and math (STEM) education in the engineering community.  Based on the argument above, engineers may be very well positioned to contribute to the constructivist reform dialogue. The process of constructing knowledge in schools, even at the very earliest stages, benefits from the same patterns of social interaction (brainstorming, review, concept design, review,…) that engineers employ on design teams. Scientific research on learning tells us that this is true. Consequently, engineers may well be able to contribute to the improvement of STEM education by helping teachers learn to better manage and develop students’ intellectual property. This contribution should be considered complimentary to the mathematical and scientific expertise that engineers also bring to the classroom.

I encourage engineers with an interest or a role to play in STEM education to take the time necessary to prepare to work with teachers well as mentors. Fortunately, there are a number of excellent references available through documents prepared by the National Research Council, the National Council of Teachers of Mathematics (NCTM), and the National Science Teachers Association (NSTA). Below are some links to documents written to summarize research on teaching and learning for both teachers and the general public.

National Research Council

How People Learn

How Students Learn: Mathematics in the Classroom

How Students Learn: Science in the Classroom

NCTM

Principles and Standards for School Mathematics

NSTA

Next Generation Science Standards