Why Maths Students Need To Understand The Concept
Before We Teach Them The Procedure
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Jody, a friend who was a nurse, told me once that she’d been hopeless at school maths. In particular, she didn’t understand ratios. This caused her constant stress due to her fear of making mistakes with drip mixtures - way back in the days when drips were mixed manually from ratios.
Even back then - decades ago - I was passionate about the effectiveness of presenting mathematics conceptually, about giving students a chance to understand maths from the get-go. So I made Jody an offer. I said ‘Give me 20 minutes of your time to play around with ratios, and let's see what happens’. She readily agreed. Of course, I didn’t show her a procedure to calculate the quantities of drugs needed to make a drip mixture. Jody knew the procedure. She’s been using it for years. What bothered her was that she was ‘blind’ when following the procedure because she had no understanding of what she was doing. She was genuinely fearful of, one day, accidentally killing a patient. Obviously, the approach I used was conceptual. I drew diagrams. I asked leading questions - questions that required her to draw on her own thinking; questions that she couldn’t apply a procedure to. I began with simple examples and gradually built complexity. In other words, I immersed Jody in the concept of ratio. |
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This allowed her to make sense of ratios and to make sense of the procedure she’d been using.
Poignantly, at one point during this process, Jody looked at me incredulously, and said “You’re kidding me! That’s it? It’s that simple?”
“I’m afraid so”, I replied.
“But this is easy. Why didn’t I understand this at school?”, she asked.
Poignantly, at one point during this process, Jody looked at me incredulously, and said “You’re kidding me! That’s it? It’s that simple?”
“I’m afraid so”, I replied.
“But this is easy. Why didn’t I understand this at school?”, she asked.
Why didn’t I understand this at school?
There are two answers to this question.
1. A need to learn
The first is that as a nurse who was rightfully fearful of killing someone via an incorrect drip mixture, Jody's need to understand ratios was high when I presented her with the opportunity. Whereas, when Jody was at school, she would have seen little or no relevance in learning ratios. Or any mathematics, for that matter. At school, her need to learn was low.
And having a need to learn is super-critical to effective learning. But that’s for another article.
1. A need to learn
The first is that as a nurse who was rightfully fearful of killing someone via an incorrect drip mixture, Jody's need to understand ratios was high when I presented her with the opportunity. Whereas, when Jody was at school, she would have seen little or no relevance in learning ratios. Or any mathematics, for that matter. At school, her need to learn was low.
And having a need to learn is super-critical to effective learning. But that’s for another article.
2. Understanding
The second reason why Jody didn’t understand ratios at school leads us to the core of this article.
Clearly, I wasn’t with Jody when she was at school. But I’ll bet you my mortgage and throw in the power tools that Jody was taught procedure after procedure re how to calculate various quantities using ratios. And that during this process Jody had no real understanding of what the heck she was doing as she replicated the procedures written for her on the board. And I’ll bet my car and throw in the trailer that all the while Jody would have had little if any, genuine understanding of the (very simple) part-vs-whole concept underpinning ratios.
And fifteen years later she still had zero understanding. Until I took 20 minutes to present ratios (and fractions) to her in a way that immersed her in the concepts, requiring her to think in a way that allowed her conceptual understanding of ratio to be born from within.
Note:
Rather, I presented her with diagrams, used a couple of bits of equipment and asked her some leading questions. This caused her to use her thinking process rather than her shakey memory of procedures she didn't understand.
The second reason why Jody didn’t understand ratios at school leads us to the core of this article.
Clearly, I wasn’t with Jody when she was at school. But I’ll bet you my mortgage and throw in the power tools that Jody was taught procedure after procedure re how to calculate various quantities using ratios. And that during this process Jody had no real understanding of what the heck she was doing as she replicated the procedures written for her on the board. And I’ll bet my car and throw in the trailer that all the while Jody would have had little if any, genuine understanding of the (very simple) part-vs-whole concept underpinning ratios.
And fifteen years later she still had zero understanding. Until I took 20 minutes to present ratios (and fractions) to her in a way that immersed her in the concepts, requiring her to think in a way that allowed her conceptual understanding of ratio to be born from within.
Note:
- I didn’t EXPLAIN any concepts to her.
- I did no TEACHING-BY-TELLING of concepts.
- I didn’t attempt to directly TRANSMIT my understanding of ratio to her.
- I didn’t do any of these things because explaining, direct-teaching and attempting to transmit are ineffective ways of enabling people to understand concepts.
Rather, I presented her with diagrams, used a couple of bits of equipment and asked her some leading questions. This caused her to use her thinking process rather than her shakey memory of procedures she didn't understand.
An analogy of school mathematics
There is a strong parallel between Jody’s story and that of the average school student. I can’t prove it, but have a strong sense it's there.
Jody represents a majority of school students who struggle trying to remember procedures without possessing the minimum required understanding by which to make sense of them.
The inference here is that WAY too many students spend WAY too much time in WAY too many maths lessons not understanding the work they have been given. (Check out 70% of Capable Students Are Failing At Mathematics)
Understand, I'm not attacking mathematics teachers here. Heck, maths teachers have the toughest gig on the planet. But I am challenging the dominant pedagogy.
The main problem is NOT that students don’t remember procedures.
The main problem is that (many) students don’t understand the CONCEPTS upon which the procedures are based - and, as a result, cannot make sense of the procedures, let alone remember them.
To expect students to remember procedures when they don’t understand the related concepts is a ridiculous expectation. And yet, arguably, this has been integral to 200 years of mathematics education.
Heck, I tortured my students this way for years!
Jody represents a majority of school students who struggle trying to remember procedures without possessing the minimum required understanding by which to make sense of them.
The inference here is that WAY too many students spend WAY too much time in WAY too many maths lessons not understanding the work they have been given. (Check out 70% of Capable Students Are Failing At Mathematics)
Understand, I'm not attacking mathematics teachers here. Heck, maths teachers have the toughest gig on the planet. But I am challenging the dominant pedagogy.
The main problem is NOT that students don’t remember procedures.
The main problem is that (many) students don’t understand the CONCEPTS upon which the procedures are based - and, as a result, cannot make sense of the procedures, let alone remember them.
To expect students to remember procedures when they don’t understand the related concepts is a ridiculous expectation. And yet, arguably, this has been integral to 200 years of mathematics education.
Heck, I tortured my students this way for years!
But … but … but ...
I expect some serious push-back to this line of argument. I expect some readers will be protesting “But I always present exceptional theoretical explanations before teaching any new procedure!”
Awesome! Exceptional theoretical explanations are ... exceptional. They have their place.
But here’s the thing … presenting exceptional theoretical explanations is not an effective way to transfer conceptual understanding. In fact, as I’ve already suggested, conceptual understanding cannot be effectively imparted at all. It grows from within the learner once the student has explored the concept by being prompted to thinking it through.
If we want students to gain conceptual understanding, we need them to engage with the concepts, collaborate with their peers, explore, inquire, ask questions, and most of all, use their own thinking.
Conceptual understanding is rarely if ever gained through being one-step-removed as a passive observer, taking notes and replicating procedures.
I’m not suggesting we stop teaching procedures (although some research suggests we should!)
What I am suggesting is that when introducing any mathematics that is new to students, where possible, present students with tasks that encourage them to think and engage with the underlying concept. THEN, present the related procedure. Or, better still, ask the students to determine the procedure themselves.
In summary, we need students to engage with concepts by using their own thinking before they deal with the related procedures. This is because engaging with concepts and having students draw on their own thinking breeds conceptual understanding. And this leads to more students spending more time in more lessons mastering the work they have been given; more students thinking “I can do this”.
Importantly, having students acquire conceptual understanding via well-structured activities that foster thinking leads to less students hating mathematics because they have no real clue about what the work that is in front of them.
For more detail on bringing conceptual understanding to mathematics students, check out Why We Need An Understanding-first, Procedures-second Approach.
For those who are convinced that having students engaging in concepts is a time-intensive process, I encourage you to read Three Ways Teaching Conceptually Can Save You Time in the Maths Classroom.
Awesome! Exceptional theoretical explanations are ... exceptional. They have their place.
But here’s the thing … presenting exceptional theoretical explanations is not an effective way to transfer conceptual understanding. In fact, as I’ve already suggested, conceptual understanding cannot be effectively imparted at all. It grows from within the learner once the student has explored the concept by being prompted to thinking it through.
If we want students to gain conceptual understanding, we need them to engage with the concepts, collaborate with their peers, explore, inquire, ask questions, and most of all, use their own thinking.
Conceptual understanding is rarely if ever gained through being one-step-removed as a passive observer, taking notes and replicating procedures.
I’m not suggesting we stop teaching procedures (although some research suggests we should!)
What I am suggesting is that when introducing any mathematics that is new to students, where possible, present students with tasks that encourage them to think and engage with the underlying concept. THEN, present the related procedure. Or, better still, ask the students to determine the procedure themselves.
In summary, we need students to engage with concepts by using their own thinking before they deal with the related procedures. This is because engaging with concepts and having students draw on their own thinking breeds conceptual understanding. And this leads to more students spending more time in more lessons mastering the work they have been given; more students thinking “I can do this”.
Importantly, having students acquire conceptual understanding via well-structured activities that foster thinking leads to less students hating mathematics because they have no real clue about what the work that is in front of them.
For more detail on bringing conceptual understanding to mathematics students, check out Why We Need An Understanding-first, Procedures-second Approach.
For those who are convinced that having students engaging in concepts is a time-intensive process, I encourage you to read Three Ways Teaching Conceptually Can Save You Time in the Maths Classroom.
Related Articles
Let's stop forcing our mathematics students to play the memory game! - here
Why We Need An Understanding-first, Procedures-second Mindset When Teaching Mathematics - here
Too Many Kids Hate Maths - here
The Understanding-first, Procedures-second Approach In Action - here
Why Compartmentalising Is A Bad Idea When Teaching Mathematics - here
Could These Four Aspects Of Mathematical Understanding Change The Way We Teach Maths? - here
70% Of Capable Students Are Failing Mathematics. What Can We do? - here
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