One of the Mathematics Teaching Practices from National Council of Teachers of Mathematics (NCTM, 2014) is:
Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses goals to guide instructional decisions.
Generalizations are summaries of thought and overriding ideas that we want our students to understand from their study. They are our mathematics goals to focus learning. Lynn Erickson (2007) first coined the term “generalizations” and in education circles generalizations are also known as enduring or essential understandings. It is a statement that connects two or more concepts in a sentence of relationship. Here are some examples of generalizations:
Students understand that:
- Linear functions show relationships that exhibit a constant rate of change
- A function represents a mapping (rule) that assigns each input (domain) with one output (range)
- For right-angled triangles, the area of the square drawn from the hypotenuse represents the sum of the areas of the squares drawn from the other sides
- Logarithm laws provide a means of changing multiplicative processes into additive processes and can be used to find inverses of exponential functions
If my students said any of the above statements I would be satisfied that they have fully understood the topic and actually have a deep understanding of the concepts involved.
To help craft quality generalizations Erickson (2007) suggests a scaffolding process.
Here is a specific example of how to scaffold a generalization in order to improve the quality of the generalization.
|Level 1||Students will understand that…..Logarithm laws affect logarithmic expressions|
|Level 2||How or why?Logarithm laws reduce logarithmic expressions utilizing the inverse process of exponential functions which represent continuous compounded growth.|
|Level 3||So what? What is the significance or effect?Logarithm laws provide a means of changing multiplicative processes into additive processes and this can provide the means to find inverses of exponential functions (continuous compounded growth).|
In summary here are the three levels for crafting generalizations:
|Level 1||Students will understand that…|
|Level 2||How or why? (Choose which question is most appropriate)|
|Level 3||So what? What is the significance or effect?|
It is important to note here that critical understanding in mathematics is usually achieved through a level 2 generalization therefore the target for curriculum development and instruction is level 2 type of generalizations. Level 3 generalizations are used if we wish to extend an idea or apply to a real life situation.
How do we draw out conceptual understandings from our students?
John Mason (1996) encapsulates the essence of math teaching in the following quote: “a lesson without the opportunity for learners to generalise is not a mathematics lesson”.
In a concept-based model of curriculum and instruction we do not tell students the generalizations at the beginning of a unit of work or at the beginning of a lesson. We guide students to communicate generalizations through a variety of strategies.
The following table illustrates some examples of how we can draw conceptual understandings from our students.
Wathall, J. T. H. (2016) Concept-based mathematics: Teaching for deep understanding in secondary classrooms. Thousand Oaks, CA: Corwin.
Crafting generalizations is an important part of unit planning and if teachers know what their end goals are for their students then the rest of the unit planning process is enhanced.