Abstraction vs Concreteness

Some concepts are extremely concrete while some are very abstract.

Example of Concrete

Polar bear's camouflage

Example of Abstract

The force of electrostatic attraction


An analogy is meant to act as additional support for students. Therefore, it is important to consider the following things that, if done badly, can make analogies less than helpful:

  1. the source

  2. the target

The source is is the idea that is supposed to be more familiar to the student. The target is, therefore, the less familiar idea. Analogies allows students to make inferences from the source to help understand the target. If knowledge of the sources is weak, then analogies fail.

Moreover, analogies have limits. Sources and targets don't map perfectly onto each other. There will be a point where an analogy stops working. Do not push analogies past this limit.

Grade 5 class learning about cells: The analogy of cells as little factories is only useful if students have a good understanding of factories first.

Moreover, this analogy has obvious limits as the factory is not filled with cytoplasm nor are cells populated by conscious beings.

Blocked vs Interleaved Practice

Block practice set only feature similar problems that relate to only one area of study. For example, a set of practice questions about potential and kinetic energy equations only.

This is opposition to interleaving where practice questions from other topics/units are introduced.

A pragmatic use of interleaved practice is to break up long sequences of practice by peppering in questions from related topics or by asking students to retrieve knowledge from other units. This discourages students from going on 'autopilot' mode (see Thinking and Participation Ratio).

Blocked practice given concepts A, B and C

Interleaved practice given concepts A, B and C

Challenge Equation

The difficulty of a task is related to the following 4 factors.

It is directly proportional to the number of interacting parts and abstraction as well as inversely proportional to domain knowledge and support.

This means the more a student has domain knowledge or receives support, the lower the challenge on a given task. However, the greater the number of interacting parts and the greater the level of abstraction, the greater the challenge.

The equation on the right captures this idea.

Check and Consolidate

A check and consolidate is similar to the check for understanding but it includes, for the student, an opportunity to begin consolidating their newly acquired knowledge. An easy way to do this is to have students actively use the new information in dialogue in writing. Mini whiteboards can be used, for example, to have students answer a series of questions. Best done after teacher explains a new concept.

Using MWB to answer questions

Check for Understanding

These are done during and after a teacher-lead explanation. We learn from Nuthall that students may appear to understand what we are telling them but in fact they often do not. Until the new knowledge has been consolidated, students may have a sense of understanding and would even self-report that they do, but the knowledge is ephemeral and very context dependent. Checking for understanding purposefully and regularly is important. Do not simply ask the class if they "get it". As I said, self-reports are highly unreliable. Instead, ask probing questions and attack the concept from multiple angles to truly check for understanding. The list to the right shows a few ways to check for understanding.

How to check for understanding:

  • ask probing questions

  • cold call students

  • use mini whiteboards

  • chorus responses

  • finger voting

  • coloured cards (red, green, blue and yellow voting)

  • hinge point questions

Cold Calling

This technique comes from Doug Lemov's Teach Like a Champion series. The idea here is that by selecting students whose hands are up to answer our questions, we are introducing a bias to our understanding of our class' ability: students whose hands go up are always the same 5 and they are either very confident of have the right answer - this unfortunately has 2 negative side effects:

  1. students tend to wait for the "smart kid" to answer the question and effectively opt out of thinking in your class

  2. you get misleading information about your class' ability

An important caveat in the way you ask the questions: always ask the question first, pause for thinking then call a person's name to answer (calling a name first tells students that they can stop thinking unless it was their name called).


The process of securing new knowledge in long-term memory and strengthening its connections to other knowledge (developing the schema).

Despite sometimes feeling as though we understand what is being presented to us, new knowledge is very ephemeral. Until we practice it enough and it finds its place within a schema, new knowledge is at risk of disappearing.

The image to the right shows the evolution of new knowledge as it begins isolated and becomes increasingly more embedded in our schemas. The image also hints to what teachers should do throughout this process to ensure the proper and permanent acquisition of this new knowledge.

Image taken from Adam Boxer's Teaching Secondary Science

Declarative Knowledge

Knowledge of facts, ideas and concepts. For example, the law of conservation of mass.

Typically, it is difficult to write practice questions for declarative knowledge as there are only so many ways to ask for it to be represented. This is in contrast to procedural knowledge (e.g. math equations, or other algorithmic sequences of steps) which is easier to write practice for as varying numbers or starting conditions is easy to do.

Consider using generative learning strategies to practice declarative knowledge.


This acronym is a tool for supporting students when solving mathematical problems. This strategy is far better than the all too popular triangle formulas (see here). Triangle formulas are meant to help students remember how to write and apply a formula but do the opposite of their intended purpose. Students who tend to be cognitively over loaded by new equations now need to remember the equation, its meaning, when to use it and its triangular representation (lots of interacting parts see challenge equation). Other similar strategies exist. See here for FIFA.

Feedback: Directive vs Non-directive

Feedback exists on a spectrum from directive to non-directive. Directive feedback is best given to novice students who are new to a set of ideas or concepts. On the other hand, non-directive feedback is better given to non-novice students. This relates to the amount of support we give to our students (see challenge equation).

Directive feedback example:

Start by writing the equation for kinetic energy [points to example on the board]. Then insert the known values.

Non-directive feedback example:

There's a mistake in one of these questions, look it over again.

Generative learning Strategies

These are strategies that follow the Select-Organize-Integrate (SOI) model. Each of these three actions have students actively use knowledge.

The poster on the right summarize 8 such strategies.

These strategies are especially good ways of practicing declarative knowledge

Generative+Learning+poster (1).pdf

Interacting Elements/Parts

In any given task, the number of steps or the number of necessary ideas that are required to solve the problem are considered to be interacting parts.

Examples Low Interaction


This simple multiplication has few "parts" - recognizing the symbols for 2 and 3 and multiplication. As such, it is fairly simple to do in ones head.

Examples High Interaction


This multiplication has many more parts - the symbols 1, 2, 6, 3, 3 and 9, the multiplication symbol, understanding of place values as well as the multiplication algorithm. Even if we've learned this in school, the number of things we need to hold in our heads to solves this is too high for most people to do without some kind of aid (e.g. paper+pencil or calculator).

Lethal Mutation

Dylan Wiliam uses the term to describe a strategy that has changed so much from its initial conception or one that is used in the wrong context and now either does not help or may cause harm.

Monosemic vs Polysemic Questions

A monosemic question has only one correct response while a polysemic question can have more than one way to correctly answer it.

This is an important concept to think about when questioning the class (typically for prior/prerequisite knowledge checks or checks for understanding or when correcting quizzes with the whole class). Teachers don't have enough time to go over all the possible correct answers of polysemic questions which means that students who have different (but possibly correct) answers to questions may not get correct feedback about their work if the teacher only uses one example of a correct solution. On the other hand, monosemic questions are easy to correct as a class (i.g. "that's correct" or "no, that's not quite right...").

Example of a polysemic question:

What is an atom?

  1. the smallest part of an element

  2. a tiny particle

  3. a tiny particle made of protons, neutrons and electrons

  4. the fundamental building block of matter

Students hearing only #3 as the correct answer will not get the necessary feedback about the correctness and perhaps incompleteness of their work. Here, it is best to share multiple correct answers while pointing to the best answer to avoid confusion.

Example of a monosemic question:

What is the atomic number of Sodium?

Here, because there is only one answer, students will not develop misconceptions when hearing the solution to this question.

Performance and Learning

Often, we convince ourselves that because our students are capable of performing well on a task, they have learned something. The unfortunate reality is that while performance and learning are related, they are not the same thing. Many teachers will tell stories of students performing really well on a task and the next day, "it's like they didn't learn anything."

A consequence of this is when checking for understanding, rather than trying to identify what your students know, try looking for what your students don't yet know (you can be sure of what they don't know but not of what they do know). That said, don't ask impossible or very complex 'got you' questions as it becomes very hard to give good feedback if the teacher cannot identify the source of the problem.

Graham Nuthall

In his book 'The Hidden Lives of Learners', Nuthall points out that a strong predictor of learning is the number of times a student experiences a concept (especially in novel ways). He describes that experiments showed that if a student had experienced a concept at three different times, they are 80% more likely to have learned it than those with fewer than three encounters.

Daniel Willingham

Willingham, in this article, describes the difference between rote, inflexible and flexible knowledge and how practice as well as access to many varied examples are necessary in developing flexible knowledge.

Prior/Prerequisite Knowledge Check

These are very similar to typical checks for understanding (because yesterday's check for understanding becomes today's prior knowledge check...), but come at the start of a lesson to elicit both what our students already know about the lesson's topic and also what they ought to know. If the teacher discovers that prerequisite knowledge is missing, they ought to quickly supply it so the students may participate in the lesson.

Examples of prerequisite knowledge checks:

Grade 10 lesson on chemical changes - photosynthesis specifically. In any given lesson, the teacher will try to relate new knowledge to prior knowledge but also try to make connections to 'authentic' situations (within reason - and without cognitively overloading students). That said, the teacher will like to know if students understand some of the items in the 'not prerequisite' column but needs to know that the class understands items in the prerequisite column.

Ratio: Thinking and Participation

Two very good indicators of learning include time on task and level of cognitive effort. Therefore, it would make sense to increase your class' participation and thinking ratio (i.e. the relative number of students thinking hard and participating). See the adjacent PDF for examples.

This is a concept taken from Doug Lemov's Teach Like a Champion.

Examples of Ratio:

Participation and thinking ratio activity (2).pdf

Worked Example

Worked examples are the "in between" step after teachers model on the board and students do independent practice. They are especially good for novices who are still developing fluency as it lowers the cognitive load. Providing these alongside problems to form a "worked example-problem pair" helps reduce the split attention effect and therefore cognitive load.

Reflections worked examples

Working Memory

This is the site of your consciousness. This is where ideas are held in the mind as you solve problems or think creatively. There is, however, a small problem. The working memory is limited - it acts as a bottleneck of information. The evidence suggests that people can juggle anywhere between 1 and 7 ideas in their working memory at any given time - most people around 3-4 ideas. This means when presenting information to students, steps must be put into place to mitigate this limitation of the human mind. Failing to do so result in cognitive overload whereby no learning occurs.

Not optimal