In contrast, conceptual progression (or epistemic ascent), means that concepts already understood by students are brought into new relations of abstraction and generality as further concepts are acquired and integrated into stu- dents’ understanding.

Conceptual understanding in mathematics means that students understand which ideas are key (by being helped to draw inferences about those ideas) and that they grasp the heuristic value of those ideas.

Teaching Conceptually

1. Start with a concept.
2. Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept.
3. Create an opportunity for students to explore that concrete example.
4. Check for understanding by having them write a concept statement.
5. Reflect on their thinking and decide next steps.

Declarative (Conceptual) Knowledge Examples of concepts: square, square root, function, area, division, linear equation, derivative, polyhedron. By definition, conceptual knowledge cannot be learned by rote. It must be learned by thoughtful, reflective mental activity.

Let’s go over them one by one:

1. Learn the same material using different medium.
2. Read through the headings, sub-headings & illustrations of the topic, first.
3. Create conceptual chunks of information.
4. Pause, recall & reflect.
5. Use analogies or comparisons to make the concepts memorable.
6. Space your studies and your practice.

They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention.

Procedural involves working out a procedure, but the students may not understand the reasoning behind a procedure. Conceptual knowledge is understanding the concepts in order to solve problems (so students may use any procedure). A great example is with fractions.

Conceptual Knowledge refers to the knowledge of, or understanding of concepts, principles, theories, models, classifications, etc. We learn conceptual knowledge through reading, viewing, listening, experiencing, or thoughtful, reflective mental activity.

The term conceptual fluency initially referred to several studies in which semantically related concepts are used. Reder [7] demonstrated that presenting words that are conceptually connected with the correct response have a positive effect on the rate of correct responses.

How to build fluency in your classroom

1. Use regular sessions to build fluency. Give learners a chance to practice their skills with separate fluency sessions that happen at a different time to the daily maths lesson.
2. Make every session count. Time with our learners is precious.
3. Try games and quizzes.

When knowledge is based on concepts that drive factual pieces of information from the world around us, it is called conceptual knowledge and focuses on regrouping big understandings and corresponding relationships among them. Conceptual knowledge highlights connections between the concepts themselves.

“Developing procedural fluency is a critical part of instruction to ensure that students are adequately prepared for their futures.” Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving.

Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. Connected with procedural fluency is knowledge of ways to estimate the result of a procedure.

Procedural information consists of actions, conditions for actions, and results from actions. This information is characterized by action verbs and imperatives, relatively short action sentences, step by step presentation of items, direct style, and if … then constructions.

It’s basically “how” you know to do something. The classic example of procedural knowledge is riding a bicycle. When someone was teaching you how to ride a bicycle, no matter what they said, you probably struggled to grasp it until you’d actually done it a few times.

We are using the term procedural knowledge to refer to those understandings, of whatever kind, which inform actions in response to practical investigation tasks in science in general; and contrasting this with the declarative understanding of the science domain which happens to be the setting for a particular task.

Contextual knowledge includes all kinds of things that give a background of the text, for example why it was written, what belief system it is situated in, what historical events influenced the writer.

The definition of contextual is depending on the context, or surrounding words, phrases, and paragraphs, of the writing. An example of contextual is how the word “read” can have two different meanings depending upon what words are around it. adjective.