Real Algebra, Blog #2
In the previous blog I talked about the importance of providing a reliable route to success in math, and specifically in algebra, for all students. Such a route does not currently exist.
There are three fundamental flaws in existing algebra curricula:
• They do not effectively get across to students the value of learning each topic.
• They do not develop conceptual understanding.
• They do not facilitate true mastery of skills and abilities.
I’ve listed these in order of importance as well as the sequence in which they should be accomplished for each topic. Despite the fact that having a reason to learn whatever you’re learning always comes first, in this article we’ll talk about conceptual understanding.
Fundamental to the development of conceptual understanding is recognition of the difference between a word or symbol and the concept that it represents. This idea goes back to the Greeks, and very likely earlier than that, but it doesn’t take a philosopher to see the truth of it. If a concept were not different from a word or symbol, then we wouldn’t be able to communicate about the same concept in different languages, and by using different symbols. The fact that we can do so demonstrates that concepts are mental ideas that are represented by words and symbols.
This may seem a little abstract, but it is essential in the development of any curriculum resource that seeks to guide and help students to achieve true understanding, because for many students the development of conceptual understanding goes beyond providing students explanations involving words and symbols. It requires experience.
The question to ask ourselves as curriculum developers, instructional designers and teachers is this:
How can we provide students with experiences that develop their comprehension of a concept?
As we do this, we need to keep in mind that when you try to explain a concept with words and symbols that the student doesn’t already understand (that is words and symbols which are not connected in the student’s mind to comprehended concepts), you make that concept inaccessible rather than accessible.
The usual process for teaching a concept to a student is to start by explaining it with language that the student doesn’t understand, and then have them take some action to engage with the concept. That’s backwards.
The correct sequence is to first develop understanding of the concept, and only then introduce the language (nomenclature and/or symbols) associated with the concept. In that way the student can easily attach the language to the concept that already exists in their minds.
This is not theoretical. It’s been the basis of my curriculum development work in many different contexts for decades.
As an example, here’s how Real Algebra treats the concept of slope (to continue reading, go to the full blog, here).
Stage 1: Get across the value of learning the concept.
Using language that is likely to already be part of the student’s working vocabulary, describe with plenty of visual support how controlling the steepness of a path or line is used to achieve goals in engineering, art, and finance. (Here are examples of short videos we’ve created for that purpose.)
Stage 2: Establish conceptual understanding through experience.
Student a) identifies and creates lines which have greater or lesser slopes than given lines, b) identifies lines which have positive versus negative slopes, and © extends lines without changing their slope.
Stage 3: Develop mastery of mathematical language and actions related to the concept.
Demonstrate to the student how slope of a straight line in a coordinate system can be measured as a ratio of the differences of vertical coordinates and horizontal coordinates for two points on the line. Provide a ladder of exercises of gradually increasing difficulty and diminishing scaffolding that result in mastery of the ability to calculate the slope of a line in a coordinate system or create a line of given slope.
You might think that this concept-first approach takes more time, but in truth it’s faster over time, because each conceptual understanding adds to the foundation on which further conceptual understandings can be constructed. Understanding of language and symbols accumulates right along with that. And once an element of language is closely associated with a concept in the student’s mind, that language will forevermore summon that concept, which means it can be used in the development of conceptual understanding in more advanced topics.