How did you learn mathematics at school? What was your experience with mathematics? Unfortunately, besides math educators, many people report negative experiences in the classroom due to an overemphasis on rote memorization with little understanding of the underlying concepts of mathematics.

According to Dr H. Lynn Erickson:

“The reason why mathematics is structured differently from history is that mathematics is an inherently conceptual language of concepts, sub-concepts, and their relationships. Number, pattern, measurement, statistics, and so on are the broadest conceptual organizers.” Erickson (2007, p30)

**Concept-Based Learning: A Three-Dimensional Educational Design Model**

The Facts, Skills and Concepts in Math

Concept-based curriculum and instruction was first introduced by Dr H. Lynn Erickson in her book *Concept-based Curriculum and Instruction for the Thinking Classroom* (2007) and describes a three-dimensional instructional design mode which includes the third dimension: the conceptual layer; generalizations.

A traditional two-dimensional instructional design model focuses on lower order thinking skills, such as rote memorization of facts and skills, with little understanding. The figure below illustrates the two dimensional instructional design model versus the three dimensional instructional design model.

**Facts **in math include definitions, formulae in terms of symbols, and vocabulary. For example, in right-angled trigonometry students have to know the formulae for Sine, Cosine and Tangent a.k.a SOHCAHTOA. Knowing a fact does not necessarily imply understanding.

**Skills** are small operations that are embedded in more complex strategies. Skills underpin high order strategies. In right–angled trigonometry students have to able to rearrange the subject of the formula SOHCAHTOA to find missing angles or sides. Performing a skill does not necessarily imply understanding.

**Concepts** may be defined as organizing ideas or mental constructs, which can consist of one word or a phrase. Concepts have common attributes and are timeless, abstract to varying degrees, and universal. In mathematics all of our topics are actually concepts.

Some examples of concepts in the study of trigonometry are:

- ratio
- similar triangles
- angles

This table shows some examples of other concepts in mathematics.

Concepts in Mathematics |

independent/dependent
variables functional relationships domain/range patterns and sequences discriminant quadratic functions logarithmic & exponential functions trigonometric functions trigonometric identities gradient/ slope rate of change process of summation correlation scatter plots calculus standard deviation |

**Generalizations** are two or more concepts stated in a sentence of relationship. It is helpful for you to start off with the stem: “I will understand that…” when forming your generalizations.

For right-angled trigonometry a generalization could be:

“Students understand that *SOHCAHTOA connect ratios of sides of similar right-angled triangles, which help to solve real-life problems in surveying, architecture, and astronomy.*”

A learning experience that supports the understanding of this generalization is to ask students to measure a number of similar right-angled triangles and to record the different lengths of the sides and work out the ratios. Download an example of a recording sheet for students.

It is all three elements; facts, skills and concepts that constitute a three-dimensional instructional design model. I hope these three things will help provide you with a structure to support mathematics teaching and learning.

## Jennifer Chang Wathall / September 28, 2017

Yes I would be honored! Thank you!

## /

## Olivia Bartlett / July 6, 2016

Hello! This post was recommended for The Best of the Math Teacher Blogs 2016: a collection of people’s favorite blog posts of the year. We would like to publish an edited volume of the posts at the end of the year and use the money raised toward a scholarship for TMC. Please let us know by responding via http://goo.gl/forms/LLURZ4GOsQ whether or not you grant us permission to include your post. Thank you, Tina and Lani.

## /