Teaching to Big Ideas

On Big Mathematical Ideas, Cathy Fosnot writes:


Underlying these strategies are big ideas. Big ideas are “the central, organizing ideas of mathematics—principles that define mathematical order”(Schifter and Fosnot 1993, 35). Big ideas are deeply connected to the structures of mathematics. They are also characteristic of shifts in learners’ reasoning—shifts in perspective, in logic, in the mathematical relationships they set up. As such, they are connected to part-whole relations—to the structure of thought in general (Piaget 1977). In fact, that is why they are connected to the structures of mathematics. Through the centuries and across cultures as mathematical big ideas developed, the advances were often characterized by paradigmatic shifts in reasoning. That is because these structural shifts in thought characterize the learning process in general. Thus, these ideas are “big” because they are critical ideas in mathematics itself and because they are big leaps in the development of the structure of children’s reasoning. Some of the big ideas you will see children constructing as they work with the materials in this package are unitizing, compensation, and equivalence.


There are two pieces of evidence from cognitive science that support teaching to Big Ideas instead of teaching 180 discrete and perhaps haphazardly connected lessons.


1. It is much easier to remember ideas that are connected together in more complex schema.


This is why it takes many repetitions to remember meaningless information (like a string of randomly chosen numbers) and far fewer repetitions to remember meaningful information (like a poem).

Acronyms like SOHCAHTOA are far easier to remember than the three equations this acronym represents because the acronym provides some structure to the information to remember. The acronym allows us to chunk the information to remember into smaller, easier to manage pieces. The same principle applies to anything we want to remember.

I see Big Ideas as recursively being formed of many smaller ideas and that a Big Idea is a way of linking the smaller ideas together in the same way that the SOHCAHTOA acronym links together the words sine, cosine, tangent with opposite, adjacent, and hypotenuse.

Discrete ideas each day vs collections of Big Ideas


2. It is much easier to remember information that we keep coming back to and are asked to repeatedly recall.


If I teach to Big Ideas, which may last more than a single session, the odds are greatly increased that the smaller ideas from which the Big Ideas are formed will be repeated across different learning sessions. This is critical because our brains are designed to forget information we don’t re-use and to remember information that is repeatedly helpful.

Repeating small ideas within big ideas

Suppose I want kids to remember Big Idea A, which is formed of smaller ideas A1, A2, A3, etc… and I teach this Big Idea over the course of a week. As I teach, I might ask kids to use idea A1 when working on idea A2, and then use ideas A1 and A2 while working on idea A3, all while asking kids to periodically attend to the relationships between A1, A2, and A3 as they are part of the Big Idea A itself. This means that during the course of a week, students may need to study idea A1 once, and recall idea A1 many times as they make connections to the other ideas of the week.

As the image also indicates, when teaching to Big Ideas means we can deliberately and explicitly make links between different Big Ideas, which means that across different weeks of instruction, the small ideas that make up Big Ideas can be referenced and repeated many times during the year. In our Algebra I course in our curriculum project, we don’t have a unit on graphing functions as graphing functions comes up in all seven units of Algebra I.


Here are some consequences of choosing to teach to Big Ideas instead of discrete small ideas:

  1. You have to name both the Big Ideas and the small ideas from which they are formed.
  2. Your curriculum can no longer be a collection of your favourite individual tasks as task selection and sequencing is far more critical.
  3. Your lessons have to be designed to make the connections between big ideas explicit for children rather than implicit. A rich schema is unhelpful if it is woven invisibly into your curriculum.



A Task is Not a Lesson

Does this image represent a lesson or a task?       I’ve noticed that opinions are split on this question with some people calling the image above a task and others calling it a lesson. In my opinion, unless an image like this includes a description of how the teachers and students will interact […]

Why Inquiry Fails

Here are at least six problems that often make inquiry-based lessons fail.     Some Problems Students have too much information to process when attempting to solve a problem which can quickly overwhelm their working memory. When given a new problem type, students do not always have all of the prerequisite knowledge necessary to approach […]

Teaching Two Courses on Instructional Routines

This spring I’m going to be co-teaching two courses with Kaitlin Ruggiero on instructional routines.   Learn more here Learn more here The first course will focus on using the instructional routine, Contemplate then Calculate, as a tool for learning how to use the 5 Practices for Orchestrating Productive Mathematical Discussions. The second course will […]

Five traps of technology

“Mathematics is not done with a computer. Mathematics is not done with pencil and paper. Mathematics is done with the brain.” ~ An anonymous participant of the Computer Based Mathematics Summit, London, 2011     The heart of mathematics education is ensuring that students develop both knowledge of mathematics (here is a definition of mathematics) […]

Just Google It

In an age where fake news is beginning to dominate the media consumed by millions of people around the world and Google’s results are being gamed by racist organizations, claiming that students don’t need to know anything because “they can just Google it” is irresponsible at best and negligent at worst. Students (and adults) are […]

Is Teacher Marking Necessary?

Teachers do a lot of marking of student work. But is it necessary?     In this comprehensive review of the literature on feedback, corrective feedback (example shown below) without mechanisms for correcting that feedback were found, unsurprisingly, to have little impact on student learning in most cases.   Unfortunately, there is also good evidence […]

Moving beyond CUBES and keywords

It is well known that children often struggle to solve word problems in mathematics. One strategy that is used to support students with having access to word problems is called CUBES. Another is to have students identify all of the keywords in the problem. (Update: Margie Pearse wrote a longer response to these same two […]

Writing Curriculum

An experiment     Let’s try a little experiment. Take a look at the following network graphs and think about what is different for each graph and what is the same for each graph.     Now look at this matrices associated with these network graphs.     Which network graph do you think goes […]

The difference between performance and learning

When my son was initially learning about fraction notation, he told me the following, right at the end of a class.   My son: Daddy, 1/3 is the same as 3/4. Me: Why is that? My son: The 3 in the denominator tells you how many quarters there are in the fraction.     Strictly […]