Conceptual & Procedural Knowledge

Measurement of Constructs

I really tried my best.

I surf the net like Cotty catching a barrel off the coast of Nazaré.

The algorithms of LinkedIn, BlueSky, Facebook all are programmed well – they cycle through hot takes on math instruction. I have engaged in strong self-regulation. I have avoided commenting on posts I have seen across platforms. However, the recent debate has cycled back to one of the old comforts – conceptual knowledge versus procedural knowledge.

Brief Warning

I really try to only write a Substack post when I have an idea for something that, in my opinion at least, is centered around meaningful and practical information [also when I have time….hence the LOOOOONG layoff]. I try to always keep Arthur Glenn Dowdy IV in mind because he’ll ask, “What does all this jawn mean?” (i.e., why is this important?).

This post will deviate from that goal. I promise it will end with me sharing less than helpful information and me asking questions I have no answers too! So if you want actionable advice that can be used tomorrow, I’d recommend hitting that tiny x on this tab. However, if you want to read some things I have been thinking and talking about with other people keep going.

Brief Descriptions

I want to provide a brief description of the terms we will be discussing today while also sharing a couple resources for readers. It is VERY hypocritical for me to provide definitions for these terms given what I will share later – but alas I am human.

• Conceptual Knowledge. The National Council for Research in their Adding it Up report defined conceptual knowledge as “comprehension of mathematical concepts, operations, and relations” (Kilpatrick et al., 2001, p. 5). The Common Core State Standards for Mathematics explicitly reference this definition. Rittle-Johnson et al., 2015 provided a similar description, knowledge of concepts, which are abstract and general principles. Rittle-Johnson did provide an additional descriptor, conceptual knowledge can be implicit or explicit, and thus does not have to be verbalizable.

• Procedural Knowledge. The National Council for Research in their Adding it Up defined procedural knowledge as “skill in carrying out procedures flexibly, accurately, efficiently and appropriately” (Kilpatrick et al., 2001, p. 5). Rittle-Johnson et al. 2015 provided the following description:

  • procedural knowledge is often defined as knowledge of procedures (e.g., Byrnes and Wasik 1991; Canobi 2009; Rittle-Johnson et al. 2001). A procedure is a series of steps, or actions, done to accomplish a goal. This knowledge often develops through problem-solving practice, and thus is tied to particular problem types.

Controversy

People argue over the order and weighting of importance. I am only going to explain the two extremes of this debate because this is not the focus of the post but I want readers unfamiliar with the debate to have a little background knowledge.

• Conceptual Knowledge RULES. This side argues that conceptual knowledge is the more important knowledge domain. Because this is viewed as the more important of the two, when planning an instructional sequence they would recommend always prioritizing instruction that focused on conceptual knowledge building at the beginning of an instructional unit before transition to procedural knowledge building.

  • Conceptual —> Procedural

• Procedural Knowledge RULES. This side argues that procedural knowledge is the more important knowledge domain. Because this is viewed as the more important of the two, when planning an instructional sequence they would recommend always prioritizing instruction that focused on procedural knowledge building at the beginning of an instructional unit before transition to conceptual knowledge building.

  • Procedural —> Conceptual

Please do not submit me for self-plagiarism! I purposefully copy-pasted the structure for both sides to highlight how it can often read as just polar opposites in arguments for both sides.

• Example. Prioritizing conceptual knowledge first will make learning procedures more efficient. Prioritizing procedureal knowledge first will make learning concept more efficient.

Verbal Theories versus Formal Theories

I know my readers all love Art Dowdy now and his questions on, “What does all this jawn mean?” However, I do have more than 1 friend! Benjamin Heddy works at the University of Oklahoma with me, so I actually see him more frequently than Arthur. Ben is a THEORIST and I mean that lovingly. When he teaches classes he oftentimes has students using little sticky notes or index cards derive their own new adaptations or theories from other prominent ones. For example, this past spring he taught a course on conceptual change theory and one of the PhD studies in special education discussed how theory development enhanced her conceptualization of her dissertation topic idea (I would share it here but haven’t talked with her and don’t want to play spoiler - but it is a SWEEEET idea). That was a lot of words to say Ben likes theories.

I don’t want to put words in Ben’s mouth, but I believe on one of our trips to Couch he was articulating that theory development supports thinking deeply about phenomena but theories must be tested.

Quick story time.

SCIPIE is an educational psychology organization. Benjamin was the president in 2025 and the president typically holds the conference in the town/city of their university of close by. So SCIPIE was on OUs campus.

Jason Chow is a special education faculty member but honestly he kind of is a chameleon because he collaborates across so many different disciplines! Jason came to SCIPIE and presented on a panel geared towards graduate students. On the panel Jason said, theories are pointless….

okay, I am being more dramatic than I need to be and honestly pulling out of context the quote! But the heart of what Jason was communicating was that being theory-aware is good (i.e., knowledge about the theoretical work people are pulling from) but ultimately the important part is theory application and data collection to align whether theoretical models hold and to provide meaningful information on how theoretical models can inform educators.

Last thought here on theories and data. A recent post on LinkedIn by Daniel Quintana stopped me in my tracks - I sent it to Benjamin right away! The basic premise is that when we stay in the verbal theory space (i.e., description of theory in word space) we can never be proven wrong because we can always add more words to have the theory still be accurate (with caveats). The post goes on to document a study the researcher carried out where they tested the verbal theory through research (with data) and identified that 6 of the 8 hypotheses held – 2 did not. The author highlights testing the theory with data actually provided great specificity of the theory and caught something that might have gone unnoticed if relying only on verbal theory descriptions.

What is my point of including all of this? Words can be tricky beasts.

Toothbrush Problem

I have lost many toothbrushes to children because I cannot get over the idea of sharing a toothbrush with someone else. Luckily, everyone in our house now has electric toothbrushes besides me so that mistake doesn’t happy anymore! Okay - let’s get back to the topic - what are we talking about toothbrushes?

In Psychology, researchers have created the term toothbrush problem. Simply put, “Psychologists treat other peoples’ theories like toothbrushes — no self-respecting person wants to use anyone else’s” (Mischel, 2008). New theory building has the potential to enhance our understanding about the world while also undermining our knowledge of a phenomenon because there is less cumulative science occurring.

This phenomenon can also be traced to measurement (Elson et al., 2023)! Researchers consistently create new measures for research projects despite multiple measures being used in prior studies. For example, a systematic review and meta-analysis was published on the relation between teacher self-efficacy and student academic achievement. The meta-analysis included 71 studies, I counted 37 (if my one-to-one correspondence is good) unique instruments used to measure teachers’ self-efficacy (Ma et al., 2025; see Multimedia Component 2 if you want to double check me).

Words

Please stay with me! Last subheading before we cycle back to conceptual and procedural knowledge. I was talking with my colleague Mike Crowson about a recent paper I had read focused on a grit (Credé et al., 2017) and Mike pointed out that beyond consciousness the construct resilience may also share extensive overlap with the construct.

Mike went on to describe the idea that we use words to attempt to explain the world around us but it is always an imperfect description of whatever phenomenon we are aiming to capture. And these words can be interpreted differently by different people based on their prior learning histories and/or the current context in which the word or phrases of words are being used. One real life example of this is a survey all employees at the University of Oklahoma are asked to fill out, an item asking something to the effect of “Do you have a close friend at work…..” The different interpretations of friend colleagues could take have been a point of reiteration each year prior to answering the survey.

So where does this leave us?

1. People like creating their own theories/ideas of why something happens

2. People like creating their own instruments to measure things

3. Words are tricky to use and have be interpreted similarly across people

Conceptual & Procedural Knowledge

We made it back to the beginning. Which is more important? Which should be prioritized in order of instruction?

I feel like we are stuck arguing about these two elements in the BIG BROAD abstract sense as if we can measure holistic conceptual knowledge and holistic procedural knowledge. Yet, we just discussed that:

1. Different theories. People are going to approach conceptual and procedural knowledge from different theoretical leanings.

2. Different measures. People will create their own measures for conceptual and procedural knowledge.

So where am I at? I feel the need to do a lot more reading across more expansive literature to wrap my head around the usefulness of categories math learning as conceptual or procedural. Because in my current, pessimistic mood - it seems to cause an argument and I am struggling to see the applied utility. If you have some thoughts on anything shared please pass them along - I’d love to hear them. I hope to be a little more productive in writing on here during the summer break!