There is a story often told about maths teaching. It is a story of how, in olden times, children were taught rote mathematical procedures. They were never taught conceptual understanding of the principles involved. These days, we have computers to perform mere procedures for us and so, instead, we should focus on conceptual understanding.
This is flawed logic.
Take the principle of equivalence. This is an idea that is often investigated in educational psychology experiments as an example of conceptual understanding. When children first meet mathematical equations, they are of the form 2 + 3 = ?. This means that they reasonably, but incorrectly, infer that an equals sign (=) is a command to write a correct answer. In fact, an equals sign means ‘the same as’ and a failure to grasp this may cause problems later when students have to solve problems of the form 2 + ? = 5.
The principle of equivalence is also important in solving simultaneous equations by elimination. The basic logic is as follows:
If A = B and C = D then A – C = B – D
I remember being quite mystified by this seemingly magical move at school. This may be because nobody ever pointed out the logic and how it utilises the principle of equivalence, or it maybe that they did do this but I have forgotten.
That fact is that a lot of these revered items of conceptual understanding are straightforward items of declarative knowledge. It is the application of these principles that is important and difficult to learn and that’s why maths lessons traditionally devote the majority of time to this.
In contrast, I could train a parrot to give me the correct dictionary definition of what the equals sign means. Does the parrot understand? Maths teachers who focus on conceptual understanding don’t tend to ask students to recite dictionary definitions, but they may as well do so. Making posters or having discussions about a relatively simple item of declarative knowledge is not much different.
The power of mathematics is the ability to move from simple, straightforward axioms, step by step towards something that is not at all obvious but incredibly useful to know.
Yes, it is important to make these axioms clear, and I am quite prepared to accept that maths teachers do not always do this or do not return to them often enough to reinforce them. But to focus on these axioms at the expense of procedures is to completely misunderstand what mathematics is and why it is so powerful.
Filling the pail 
Great post Greg.
What irks me here is how vague a term “conceptual understanding” is. It takes a fair bit of experience with math to be comfortable with the ideas of axioms and proofs. In the example you give does it mean truly knowing what a graduate mathematician does about transitivity of equivalence for any algebra or does it just mean comfort with counting objects or measuring distances on a number line?
When people call for conceptual understanding they are often just being vague about what they are talking about. Of course as you point out if they are explicit with a description then it is easy to teach it to a parrot.
But someone who has solid knowledge of the properties of integers and has a good intuitive feel for how they work could enjoy playing the 24 game for an hour and explain which properties could be used to translate any solution to another.
https://www.24game.com
Goodness! That has just made a lot of maths much clearer to me. I don’t think it is being taught at all. I certainly wasn’t but I’ve just had a lightbulb moment.
I’ve always struggled with maths mostly as I was of the age when the ideas of not teaching tables by rote became fashionable. I felt like piggy-in-the-middle with the schools saying “no you don’t need to learn it off by heart” and my parents saying “yes you do”. Sadly, neither option worked for me and I pretty much “gave up” in school around age 8 particularly after one lesson where we did basic subtraction. I knew I had all ten sums correct as the “star” pupil had just come back with hers all marked right and I found my answers were the same. Off I trotted and had every single one marked wrong. I now know why but at the time I was just very confused. School had taught us a wierd way of working out subtractions. My mother finding me confused at home showed me her way which made sense (I still use it now). I had used that method. It was a salutary lesson to 8 year old me and I simply gave up at that age.
So its not that I can’t do maths – when I have to and get to practice, I surprise myself by eventually being able to do the process in my head. A very good example was percentages. I never managed to figure these out in school. Then in my first job, it became a major part of my work. I found a good tutorial and learnt it and practiced and within 2 months was doing the calculations in my head. Sadly, I always had to back up using a calculator as years of not being “good” at maths has meant I don’t trust myself.
Your story sounds very familiar, I’m sorry to say!
Greg,
Would you care to comment on “conceptual understanding” as it relates to social studies? I have a colleague who talks about it constantly. I’ll pin him down as to what exactly he means next week when we go back to school here in Texas.
I can’t articulate exactly why, but I find it difficult to agree with him. I will be quoting you on a poster in my room. “Critical thinking largely emerges from sufficient knowledge about a subject.” Just to tweak my principal…
Enjoy your blog.
Thanks
Yes, exactly right. The fetish over conceptual understanding is an attempt to thwart a mischaracterization of traditional math teaching as “rote memorization without understanding”, The result, as you point out, is often “rote understanding”.
Regarding the linear equations. when I teach simultaneous equations I’ll exhibit something like x + y = 2, and ask if we could add, say, 2 to both sides. The answer comes back from the students as “yes”. When I ask why, they tell me that adding equal amounts to both sides of an equation results in equality on both sides. “Suppose I have an equation of x – y = -8. Are the amounts equal on both sides of the equation?” Students will agree. “So can I add x-y = -8 to the first equation?” Students will agree because they see I am adding equals to equals. After that I ask why I would want to do a thing like that? They will usually see that it’s because doing so eliminates the “y’s” in the equations. Nothing mysterious, and we are working with prior knowledge.
Reblogged this on traditional math and commented:
Greg Ashman also writes about conceptual understanding in math and does a good job of it.
I am not sure that I can comment on how this applies to maths education. But from a physics point of view, all physics education research points to results that show that an increase in conceptual understanding leads to an increase in problem solving ability. However the reverse isnt always true.
There may be a difference in maths, but in physics students bring into the classroom their own mental models (preconceptions) about how physics works, so we need to spend appropriate time building the correct scientific models.
Furthermore science has a very precise language, and unfortunately many of the words that have a very structured meaning in science (force/speed/acceleration/momentum) are prevelant in everyday language, and mean very different, or very general things.
Because of this, a conceptual understanding in physics is vital. Otherwise students will continue to believe that textbook physics problems and “real-life” are two very different systems…
“But from a physics point of view, all physics education research points to results that show that an increase in conceptual understanding leads to an increase in problem solving ability. However the reverse isnt always true.”
Could you please point me to this research? I am interested in how they define and assess conceptual understanding.
Sure Greg. Anything with the “Force Concept Inventory” is good. Its the go-to test in Physics Education Research, so there are heaps of research using it, and articles about it.
Eric Mazur is an entertaining entry into PER https://youtu.be/WwslBPj8GgI
Mestre has a great early overview of the field, but I cant find an online free access to the article. Its called “Learning and Instruction in Pre‐College Physical Science”
Finally Beichner has a contempary review of Physics Education Research: https://www.researchgate.net/publication/237064083_An_Introduction_to_Physics_Education_Research
Enjoy. Physics Education Research is very well established (but that was a surprise to me!) And as far as I can tell, within the research there doesnt seem to be this two sided debate that there is in maths? I could be wrong, but thats my current understanding.
My observation is that most things are a feedback loop between theory and application. A failed application is negative feedback that some aspect of theory is not understood. The loop continues until the errors of application are eliminated.
Physics education generally means college level, in which domain knowledge of sufficient size as to allow for greater understanding of what is behind various procedures. Nevertheless, even college level students tend to lean toward mastery of procedures first, unless the concepts are part and parcel to the procedures. Sometimes they are–sometimes they are not.
Reblogged this on The Echo Chamber.
Here’s a post for you:
Click to access basicskillvsconcpt.pdf
from 1999, H. Wu
Hey Greg,
This post of yours came back at me like really good/bad reflux after a few days of watching other peoples’ first forays into Chat GPT. They would always try to “stump” it with what they felt were esoteric or “hard to answer” questions, like, “What is love?” or “What is the meaning of life?” or “Define creativity”. It finds these prompts absolutely the EASIEST to reply to. Ask it if Michaela follows the evidence around student wellbeing and it’s hilariouslyt patchy, inconsistent and equivocal. It strikes me as the same misunderstanding you talk about here: Assuming sophisticated declarative knowledge is the same as actual thought. Jordan Peterson said it well (and I paraphrase), the key limitation of Chat GPT is that it doesn’t have to interface with reality, just words. It has no mechanism by which to test its assumptions other than consensus on the internet.
This mistake (between declarative knowledge and thinking) has been an issue in English assessment for a long time now I realize.
This year I managed to get only band 5 and 6s for advanced English and only band e4s and one e3 for extension English – For other NSW HSC English teachers this is a fairly odd occurance, especially in my context of a disadvanted school with no native English speaking students. Those results absolutely shot up the ranking of my this small school, because English scales other subjects. And I attribute a large part of this success to knowing how flawed the assumptions of markers and the HSC format are. Now, I am not one of these “abandon tests” nutjobs. But Chat GPT really does show the lie that posing a different exam question every year but allowing students to memorize a few other key quotes and concepts is a good test of thinking in English.
Over the years I have absolutely abandoned the idea of reading the texts themselves. Because the assessment will only ask for a handful of quotes from it. What is taken as “critical thinking” or “personal voice” is linking the text to general/historical knowledge related to the text on a conceptual level – essentially declarative knowledge, as you say. It’s like a maths test that just asks you to parrot “conceptual definitions” without doing much maths. So I teach this and the format of essay writing for at least 50% of my coursework (often more).
I do not apologize for teaching writing more than I teach literature, I think most students are too young for some of the texts we throw at them and have been poorly served in reading and writing instruction (and they respond much better to early to mid 1900s texts of cultural significance than either recent, more postmodern ones, or much older ones where the form of English is drastically changed).
But if the assumption is engaging with literature is good for you, then the assessment format does a lot to conspire against us having time to do that. And Chat GPT is only going to make it worse, beause it can do this kind of “elegant declarative knowledge over substantial thought” better than 95% of students. Even if we switch to all paper assessments, students will have Chat GPT write something at home and more or less memorize it. That won’t mean they understand NOTHING of what they say, but it won’t be optimal. As many have pointed out, Chat GPT will state a falsehood just as confidently as a fact, and the only way to know the difference is KNOW LOTS OF STUFF.
Anyway, I think this blog post is ripe for an update in relation to Chat GPT.