Why the ‘Conceptual Understanding’ Debate Is Arguing Over the Wrong Premise

Can We Actually Teach and Test Conceptual Understanding?

In my previous article - The Lost Blueprint of Learning, I walked through the taxonomy developed by Philip Tiemann and Susan Markle (1983). Their central insight is beautifully simple: not all learning tasks belong to the same class. Gaining a precise understanding of how different forms of knowledge are constructed fundamentally changed how I teach, how I co-plan and how I deliver professional learning for other teachers.

Recently, I was supporting a group of Kindergarten/Foundation teachers during a collaborative planning session. They were preparing a subtraction lesson.

The initial unit draft leaped straight into the procedure. The plan was to hand the five-year-olds a collection of counters, tell a brief story and show them how to physically take the blocks away to find how many were left.

I paused and asked the team a question: “Before we show them how to do the action of taking away, do they actually know what subtraction is?”

The initial planning was going to result in the students executing a mechanical routine, a counting algorithm. But to truly set them up for success, we had to start by teaching the concept of subtraction first. Subtraction isn’t just an action, it is a structural mathematical category where a part is separated from a whole. If we jump straight to the procedure of moving counters, students focus entirely on the physical steps of the task rather than learning to discriminate a subtraction scenario from any other mathematical relationship.

The Conceptual Understanding Debate

Dr. Greg Ashman has reopened a conversation that matters deeply to how we spend our days in the classroom. In his recent posts on Filling the Pail (Conceptual understanding is a myth and The curse of conceptual understanding), he revisits the persistent question of what we truly mean by ‘conceptual understanding’.

Bill McCallum, writing at Mathematical Musings, takes a different tack, arguing the term can still guide us if we look closely enough. While they arrive at different conclusions, both are trying to solve the exact same high-stakes classroom puzzle from different angles. Ultimately, they are both chasing a way to ensure that what we teach sticks, transfers and empowers a child to think mathematically.

I share Ashman’s unease with the word “understanding.” As a cognitive label it is hard to observe and harder still to measure. When a student repeats our favourite diagram or explanation, we are often too quick to label that performance as understanding. It might be, but the evidence is thin.

I agree with both impulses regarding transfer, but the missing piece is clarity about the kind of transfer we want. True transfer is not about running a different procedure, rather it is about carrying a structural concept into unfamiliar examples and still recognising the category.

Most of us were never taught to think about instruction through the lens of Tiemann and Markle. I know I was never told about it. Their work sits quietly in the background of the principles of Direct Instruction (DI), behaviour analysis and instructional design. Yet when we adopt their categories, the fog around “understanding” begins to lift.

Introducing their categories here isn’t about using a theoretical weapon to prove anyone wrong or declare a winner in the maths wars. Instead, their work offers a practical bridge. Concepts become teachable. Assessments gain sharper edges. The framework repays the effort it takes to learn it.

The Hidden Anatomy of a Problem

To see why this distinction matters so early on, we need to turn to the framework developed by Tiemann and Markle (1983). Their work provides the exact toolkit we need to rescue concepts from educational mysticism.

When we teach a student to add two-digit numbers on an empty number line or the order of operations or the standard vertical algorithm, we are teaching a sequence. Tiemann and Markle call this simple cognitive learning. It is choreography. It answers the “how.”

A concept is different. A concept is a mental category, a class of objects, symbols or events that share common, defining characteristics (Layng, 2019). Concept learning has occurred when a student can look at a completely novel, unseen instance and correctly identify whether it belongs in that category or not based on its attributes (Johnson & Bulla, 2021).

Learners demonstrate it when they meet a brand new instance and decide, with confidence, “In or out?” That is discrimination and generalisation working together. If we want conceptual understanding, we want students who can do this even when the surface features keep changing.

Unpacking Ashman’s Example: (3x = 18)

Ashman offers a familiar algebra question:

“If a student sees (3x = 18) and writes (x = 6) what kind of knowledge is that?”

Tiemann and Markle give us a structured answer. Solving (3x = 18) involves three kinds of learning:

1. Paired Associates (Basic Memory): Recognising “3” and “18” and recalling instantly that (18 ÷ 3 = 6) so working memory stays free. If this memory is not automated, their working memory stalls before they even begin.

2. Algorithms (Procedural Sequence): The student executes a step-by-step routine. They look at the expression, determine the operation connecting the number and the variable, apply the inverse operation to both sides and calculate the final value.

3. Concepts (Complex Cognitive Category): This is where true conceptual understanding lives. To solve this equation with actual insight, a student must master the concept of a variable (x) as a placeholder for an unknown value and the concept of the equals sign (=) as a structural state of balance, rather than a command that means ‘find the answer’.

Teach only the routine and students still succeed on (3x = 18). Present (18 = x∙3) or embed the relationship in a story problem and many hesitate. Students who hold the concept recognise the balance regardless of format. That is conceptual transfer.

Defining a Concept: Critical and Variable Attributes

To teach a concept explicitly so that students can navigate these variations, we cannot just recite a textbook definition and hope for the best (Layng, 2019). This is where the structural design of Direct Instruction intersects perfectly with content analysis. In DI, we focus on identifying the generative relations within a content domain to ensure learners can do more with less extensive instruction (Slocum & Rolf, 2021).

We bring this upfront analysis to life by breaking the concept down into its Critical Attributes (the features an instance must have to belong to the category, or ‘must-have’ features) and its Variable Attributes (the features that can change without altering the concept, or ‘can-have’ features) (Tiemann & Markle, 1983; Twyman & Hockman, 2021; Williams et al., 2025).

This plays out across subjects.

Prisms

• The “Must-Haves” (Critical): A 3D solid with two identical (congruent) faces opposite each other, connected by flat parallelograms.

• The “Can-Haves” (Variable): The shape of the base (triangle vs. hexagon), the size (scale), or whether it’s standing up or lying on its side.

• The Test: Show a cylinder. It has the identical opposite faces, but the surface is curved. It’s “out.”

  

Prime Numbers

• The “Must-Haves” (Critical): A whole number greater than 1 with exactly two distinct factors: 1 and itself.

• The “Can-Haves” (Variable): The number of digits, parity, whether it is near an even number, or how “dense” it is compared to other primes.

• The Test: Show the number 1. It has only one factor. It’s “out.” Show the number 9. It’s odd, but it has three factors (1, 3, 9). It’s “out.”

Mammals

• The “Must-Haves” (Critical): Warm-blooded vertebrates that possess hair/fur and mammary glands.

• The “Can-Haves” (Variable): Size, habitat (land vs. water), or means of movement (walking, swimming, or flying).

• The Test: Show a platypus. It lays eggs (a massive variable shift), but it has hair and mammary glands. It’s “in.”

Persuasive Writing

• The “Must-Haves” (Critical): A debatable claim, evidence tied directly to that claim, and language intended to influence an audience.

• The “Can-Haves” (Variable): The tone (angry vs. logical), the length, the format (an essay vs. a tweet), or the publication medium.

• The Test: Show an Instruction Manual. It uses professional language and has a clear goal, but it lacks a debatable claim. It’s “out.”

Students who internalise the concept protect those critical attributes. Remove one and they reject the example. Shift the variables and they stay steady.

The Danger of ‘Proceduralising’ Concepts

This is where good intentions can backfire. The common error in progressive pedagogy is trying to teach a concept by turning the explanation into a mandatory, step-by-step procedure. I call this the ‘proceduralisation’ of concepts. I know this because for much of my career, I’ve been there to!

We often turn teaching a concept into a step-by-step procedure. Photo by Denys Sudilkovsky on Unsplash

If you force a student to draw a specific bar model, a rectangle or a fraction wall every single time they solve a problem, you have not taught the concept. You have simply taught a drawing algorithm. The student follows a new mechanical routine: Step 1: Draw a box. Step 2: Cut it into sections. Step 3: Shade the parts. The visual model, which was originally intended to illuminate the concept, shifts into just another mindless sequence to execute.

Direct Instruction avoids this confusion by aiming for faultless communication, structuring lessons so that they communicate one and only one, logical interpretation (Twyman, 2021). True conceptual instruction achieves this by presenting a ‘rational set’ of juxtaposed examples and close-in non-examples (Johnson & Bulla, 2021; Tiemann & Markle, 1983; Twyman & Hockman, 2021).

Following DI’s core sequencing rules, such as the Difference Principle and the Sameness Principle, we can systematically map out the exact boundaries of a concept for our students (Spencer, 2021; Twyman, 2021). We hold variable attributes steady while stripping away one critical attribute at a time to show students exactly where the boundary of the category lies.

Then we vary the variables. We rotate prisms. We shift persuasive writing from a speech to a social media post. We swap mammals from dolphins to bats to platypuses. Students who still see the concept are demonstrating the transfer Ashman and McCallum both want to find. We ensure their understanding is tied to the structural concept, not the specific visual tool we used during Tuesday morning’s lesson.

Locating Concepts in the Instructional Hierarchy

To understand why this matters for daily practice, we can place this structural analysis directly into the Instructional Hierarchy framework developed by Norris Haring and Marie Eaton (1978). This model reminds us that all learning progresses systematically through four distinct stages: Acquisition, Fluency, Generalisation, and Adaptation (Ardoin & Daly, 2007; Daly et al., 1996).

The current educational debate completely scrambles these stages. Progressive practices frequently demand Generalisation or Adaptation that forces students to solve complex word problems or write elaborate explanations before the student has achieved basic accuracy and Fluency with the underlying facts and algorithms. On the flip side, a purely mechanical approach traps students permanently in the Fluency stage of a single, highly specific procedure.

True concept learning bridges the vital gap between Acquisition and Generalisation. When we use a rational set of examples and non-examples, we are explicitly training the student how to generalise the concept to any valid scenario in the real world, while discriminating it from non-examples (Haring, 1988). Once a concept is fully generalised, and the procedural algorithms are fluent, the student naturally reaches the stage of Adaptation. They can apply their knowledge to solve novel, complex mathematical problems because their working memory is protected from collapse (Daly et al., 1996).

Assessing Conceptual Understanding

This brings us back to Greg’s core point: how do we measure this without the test turning into rote mimicry?

We do not test a concept by asking students to define the concept or to reproduce a familiar diagram they saw on the whiteboard. Those are simply verbal or motor algorithms that can be memorised and regurgitated on cue (Tiemann & Markle, 1983).

Ashman rightly questions assessments that value a student’s ability to act out an explanation over their actual mathematical capability. McCallum wants evidence that understanding exists. Both concerns are valid. We can respect them by designing tasks that require students responding to new instances.

If you want to know if a Year 6 student understands prime numbers, you do not ask them to list the primes up to 20. Instead, you hand them an unfamiliar number, such as 51, and ask them to determine if it belongs in the category, requiring them to justify their decision by testing for hidden factors.

Offer a Year 5 student a solid with matching triangular faces and curved sides. The response “Not a prism. The sides must be flat parallelograms” signals the concept is intact.

Ask a Year 8 science class whether a platypus remains a mammal despite laying eggs. Students who cite the critical attributes of warm blood, hair and mammary glands have transferred the concept. In English, hand students an unfamiliar article and ask if it fits the persuasive genre. Those who locate the claim, the evidence and the audience moves are applying the concept, not reciting a checklist.

Moving Beyond the Procedural vs. Conceptual Debate

Unpacking the exact architecture of a concept does more than just settle a debate about procedures. When we equip ourselves with a comprehensive taxonomy of learning, we gain utility that transforms how we plan everyday classroom interactions. The current debate gets stuck in a simplistic binary: procedural versus conceptual. Because of that, teachers frequently select the wrong instructional tools for the job.

Consider how often we try to force ‘conceptual understanding’ onto tasks where it doesn’t belong, such as basic maths facts. A maths fact is a basic memory task, what Tiemann and Markle (1983) call a Paired Associate. The instructional goal here is absolute automaticity, an instant stimulus-response relation (6 x 7 = 42). Yet, educators will often display a complex visual array or an area model at the exact same time they introduce the fact, hoping to anchor it conceptually.

This creates a massive split-attention effect. Instead of building a clean, automated memory association, the teacher inadvertently jams the student’s working memory with an analytical task. Visual arrays are fantastic tools for teaching the concept of multiplication during acquisition, but using them during a fluency drive on maths facts fundamentally misinterprets the category of learning required.

We see the exact opposite mistake when teachers introduce mental strategies. A mental strategy (e.g. choosing whether to round and compensate, jump along an empty number line, or use a double plus one) is supposed to be a Complex Cognitive Strategy (Tiemann & Markle 1983). It is a heuristic for navigating an ill structured problem.

Yet, because the system lacks a precise vocabulary, we routinely ‘proceduralise’ these strategies. We turn them into a rigid, robotic set of steps: Step 1: Look at the units. Step 2: Round to the nearest ten. Step 3: Subtract the difference. The moment we script the strategy into an invariant sequence, we transform it into a Simple Cognitive Algorithm. The student hasn’t learned a strategic problem-solving repertoire at all. They have just memorised a brand new, highly fragile procedure that falls apart the second a problem doesn’t fit the script perfectly.

Moving Forward Together

How can we move forward together? Photo by Sdf Rahbar on Unsplash

None of this diminishes the thoughtful critiques and defences that Ashman and McCallum bring to the table. Their writing highlights genuine problems we see every day. Empty rhetoric about understanding and the longing for something richer than rote routines. The Tiemann and Markle framework gives us tools to meet those problems head on. It clarifies what a concept is. It shows us how to teach it. It points to assessments that honour the goal without rewarding mimicry.

Procedural fluency and conceptual understanding are different cognitive architectures. Both matter. Both require instructional designs built for their specific demands. Algorithms flourish with explicit modelling, faded scaffolds and deliberate frequency building to minimise cognitive load (Johnson & Layng, 1992). Concepts flourish with sequences of examples and non-examples that etch the category boundaries in students’ minds (Twyman & Hockman, 2021).

Grounded in instructional design frameworks like Tiemann and Markle alongside the precise sequencing logic of Direct Instruction, we can confidently move past the old, cyclical ideological stalemates. We can define exactly what a concept is, teach it with laser precision and test it with complete objectivity.

If the term “understanding” continues to blur our thinking, we can set it aside and speak directly about concepts, procedures and the behaviours that demonstrate each. Once we do, the debate becomes far less mysterious and the path to better teaching becomes far easier to see.

Next time you introduce a new idea, ask yourself:

Am I teaching them the ‘how-to’ algorithm or do I want them to know what something is and isn’t? Try one ‘close-in’ non-example and see if the fog lifts for your students, too.

References

• Ardoin, S.P. and Daly, III, E.J., 2007. ‘Close encounters of the instructional kind—How the instructional hierarchy is shaping instructional research 30 years later’, Journal of Behavioral Education, 16(1), pp. 1–6.

• Daly, III, E.J., Lentz, F.E. and Boyer, J., 1996. ‘The instructional hierarchy: A conceptual model for understanding the effective components of reading interventions’, School Psychology Quarterly, 11(4), pp. 369–386.

• Haring, N.G. ed., 1988. Generalization for students with severe handicaps: Strategies and solutions. Seattle: University of Washington Press.

• Haring, N.G. and Eaton, M.D., 1978. ‘Systematic procedures: An instructional hierarchy’, in N.G. Haring, T.C. Lovitt, M.D. Eaton and C.L. Hansen, eds. The fourth R: Research in the classroom. Columbus: Charles E. Merrill Publishing Company, pp. 23–40.

• Johnson, K. and Bulla, A.J., 2021. ‘Creating the components for teaching concepts’, Behavior Analysis in Practice, 14(3), pp. 785–792.

• Johnson, K.R. and Layng, T.V.J., 1992. ‘Breaking the structuralist barrier: Literacy and numeracy with fluency’, American Psychologist, 47(11), pp. 1475–1490.

• Layng, T.V.J., 2019. ‘Tutorial: Understanding concepts: Implications for behavior analysts and educators’, Perspectives on Behavior Science, 42(2), pp. 345–363.

• Markle, S.M., 1991. Designs for instructional designers. Seattle: Morningside Press.

• Slocum, T.A. and Rolf, K.R., 2021. ‘Features of direct instruction: Content analysis’, Behavior Analysis in Practice, 14(3), pp. 775–784.

• Spencer, T.D., 2021. ‘Ten instructional design efforts to help behavior analysts take up the torch of direct instruction’, Behavior Analysis in Practice, 14(3), pp. 816–830.

• Tiemann, P.W. and Markle, S.M., 1983. Analyzing instructional content: A guide to instruction and evaluation. Champaign: Stipes Publishing Company.

• Twyman, J.S., 2021. ‘Faultless communication: The heart and soul of DI’, Perspectives on Behavior Science, 44(2), pp. 225-241.

• Twyman, J.S. and Hockman, A., 2021. ‘You have the big idea, concept, and some examples... Now what?’, Behavior Analysis in Practice, 14(3), pp. 793–804.

• Williams, C.L., St. Peter, C.C., Perone, M., Aguilar, M., Cederberg, B.A., Gregersen, D.J. and Richardson, E.J., 2025. ‘Using must-have and can-have features to improve conceptual learning’, Journal of the Experimental Analysis of Behavior, 124(1), e7003.