# Concept-based mathematics

##### Teaching for deep understanding in secondary schools

*Jennifer Wathall**, Head of Mathematics at Island School, Hong Kong suggests in a new book that the key to winning students to the subject lies in the explicit development of their conceptual mathematical understanding. *

**Maths anxiety**

A long-standing tradition sees mathematics as an elusive discipline and people often recall negative experiences when learning the subject. Timed tasks, and an over-emphasis on rote memorization of facts and algorithms with little conceptual understanding have created maths anxiety and fear.

**Concept based instruction**

Concept-based curriculum and instruction offers a real alternative. It was first introduced by Dr H. Lynn Erickson in her book “Concept-based Curriculum and Instruction for the Thinking Classroom” (2007) and is described as a three dimensional design model that organises the facts and skills with a third, conceptual layer. The concept-based model can overlay any curriculum and represents an approach to teaching and learning that focuses on creating a synergistic relationship between the facts, skills and conceptual understandings in order to encourage thinking and to develop intellect. This relationship is illustrated in figure 1.

Figure 1 Developing Intellect

**Applying the model to Mathematics**

Mathematics is inherently a language of conceptual relationships: it is important to highlight the concepts and conceptual understandings in any unit of study and develop critical thinking and problem solving skills.

“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)

(c) Jennifer Wathall. Illustration by Jordan Wathall

**Mathematical concepts and generalizations **

Concepts are organizing ideas or mental constructs that share common attributes. They can be abstract to varying degrees, universal and timeless. Mathematical examples in the topic of calculus include “derivative”, “rate of change”, “slope” and “gradient”.

Conceptual understandings can also be seen as enduring understandings or generalizations. Generalizations are two or more concepts whose relationship is described in a sentence. An example of a generalization for differentiation could be:

*The derivative may be exemplified physically as a rate of change and geometrically as the gradient or slope function.*

Notice in figure 1 the synergistic relationship between the facts, skills and conceptual understandings is supported by the inquiry process continuum. Inquiry based learning is a student-centred approach that develops independence and curiosity while allowing learners to construct their own meaning when learning.

**Developing conceptual understanding: the role of guiding questions**

When learning mathematics, students need to be given time and space to explore the concepts and to develop conceptual understandings. A concept-based model of learning promotes a thinking classroom, engages the intellect and develops critical skills necessary for the 21^{st} Century. This in turn depends on the effective use of guiding questions**, **which are a way to help draw out generalizations from a unit of study.

Different types of guiding questions with varying degrees of abstraction can be employed to develop conceptual understanding:

**Factual questions**: what we want students to know in purely fact terms. An example for the topic of differentiation is:

What is the product rule?

**Conceptual****questions**: these are questions that connect the factual content with the relevant concepts. For example

How do you explain the gradient function? Provide examples

**Debatable/ Provocative Questions**: this type of question overlaps with Theory of Knowledge and is designed to provoke deep abstract thought

Do limits apply to real life situations? Was calculus invented or discovered?

Using this type of question in a carefully planned way will draw out mathematical relationships, identify generalizations and deepen understanding, adding to the appeal of a subject that deserves to be enjoyed and not feared!

Jennifer Wathall.

Jennifer’s new book, *Concept-based Mathematics: Teaching for Deep Understanding*. Thousand Oaks, CA: Corwin, was published in February 2016.

It has been reviewed by Vicky Hill in International Teacher Magazine. See:

For more about Jennifer’s work, see http://www.jenniferwathall.com/

To order Jennifer’s book, use these links:

Amazon:

From the publishers direct:

For the USA:

For the rest of the world:

https://uk.sagepub.com/en-gb/eur/concept-based-mathematics/book246583

**Reference**

Erickson, H. L. (2007). *Concept based Curriculum and** Instruction for the Thinking Classroom*. Thousand Oaks, CA: Corwin.

Hello Jennifer,

I agree with your article and have employed similar methods.

Kind regards

Linda

In the elementary and middle grades, conceptual based learning generally involves the use of concrete materials, such as Hands-On Equations, fraction blocks or other manipulatives, including visual manipulatives. For example, instead of a 4th grade student learning a rule for transforming 2 1/3 into the improper fraction 7/3, the student can use the fraction blocks to see that each whole is composed of three parts of size 1/3. Hence, after working with the fraction blocks, the student can represent the process this way: 2 1/3 = 1 + 1 + 1/3 = 3/3 + 3/3 + 1/3 = 7/3. Eventually, the transformation from 2 1/3 is done mentally, but it is done cognitively, not using a rule.

Or consider the subtraction of fractions such as 3 1/6 – 1 5/6. Using the fraction blocks, the student clearly sees the need to rename the yellow block by 6 green blocks, that is, to rename one whole by 6/6, in order to have the required number of 1/6 parts to be able to carry out the subtraction. The transition to symbolic notation then becomes much simpler.

In essence, conceptual based learning means that the student has a sense that he understands what he is doing.