The Need For Conceptual Understanding - A Conversation With Ed Staples
Richard Andrew
? Maximising the Conversion of Ideas Into Long-Term Practice ? Collaborating with Schools, Consultants, Coaches & Subject-Matter Experts ? Raising Agency, Empowering Change ? Self-Reflective, Hybrid & Online CPD
Recently, I used the image-with-quote (above) as a LinkedIn post. I received the following reply from Ed Staples, a fellow Australian who has impacted the maths education landscape over many decades. Ed recalled a ‘wake-up moment’ for him early in his teaching career:
I recall once asking a student to solve a quadratic equation that I had written on the whiteboard.?He strode to the board and went through the procedure flawlessly arriving at the two solutions in the last line.?
When he’d finished , I asked him what the answers meant, but to my surprise he said he didn’t understand what I was asking.?So I rephrased - I asked him how the two answers related to the original equation.?
He confessed that he had no real idea. It was news to him that the two values of x “satisfied” the quadratic equation.?And it was a revelation to me that knowing a procedure doesn’t necessarily mean understanding a concept.?
It was concerning that a student could complete a procedural task correctly yet not understand what they were doing or why they were doing it.?I was glad it happened early in my career.
The above comment, I believe, sums up an all-too-common maths classroom scenario … students who get the correct answer but do not understand what they are ‘doing’.
This is only slightly different to the other common maths classroom scenario … students who get the WRONG answer and have no idea what they are ‘doing’ (!!)
Ed followed up by saying:
(This) goes to the very heart of how many students ‘see’ maths - a bizarre formulaically driven set of routines that make no real sense but are worth doing for some reason because it’s a core subject.?
Over the years, I would encounter students who’d say to me “I don’t need to understand it, I just want to know how to do it” After a while, I began to take that line as a great teaching opportunity to change their perceptions.
领英推荐
I mentioned how many teachers I work with tell that students commonly say, 'just show me the rule'. But can we blame the students? After all, isn’t this (being taught the rule) simply what they are used to? It’s up to us to change their mindset.
The reality is, getting students to construct their own understanding is a game changer. Why? Because it gives students a sense of control. And what springs from students having a sense of control? Understanding, authentic engagement and agency!
Some closing comments from Ed …?
Routines are learnt by rote, they’re tested and then forgotten. (The information) slides straight off. Nothing is really constructed by the student.?
(Whereas) collaboration between students with careful teacher guidance assists the student in their own construction of knowledge.?
Meaningfulness comes from constructing one's own knowledge albeit facilitated by the teacher. It’s powerful stuff!
Over to you …?
Have you encountered students who are disinterested in understanding, who demand the shortest path to the correct answer.
Do you believe it is possible to turn this attitude around? (It is)
Let us know in the comments.?
maths teacher | flipped learning expert | AI experimenter | content creator | teacher trainer
2 年The university entrance exam in my country, Turkey, consists of multiple choice questions and within one session, students have to solve at least three different tests among science (a combo of physics, chem and bio), maths, Turkish, social sciences and English. Calculators are not allowed. Some of the questions are quite advanced, like solving a quadratic equation, choosing the right solution and continue to reach the final answer, which requires one-page length of working in an IGCSE or iAL Maths Exam. But students have to solve such questions within a few minutes time and all they need is the right choice among the five provided with the question. In such a setting, procedural knowledge weighs much more than understanding, for all the students need is to reach the final answer in a particular type of question. However, I still think that understanding the concept may still help these students.
Educational Consultant Director at Raising Achievement
2 年I have a Year 12 student I am tutoring who came to me this week. In class, they were doing trigonometry and discussing sine and cosine. He said he understood the first two lessons and then he got lost. I asked him if he knows what cosines are about and why we use them. He said no so we started there. While some students may not want to know, one of my colleagues Mike Scaddan, said if you don't give knowledge an application at the very beginning of a lesson engagement, understanding, and memory for a skill will be impacted. Starting with a why and following with conceptual understanding is crucial.
Teacher of Mathematics/Department Chair at Millburn High School - Retired
2 年I just added the following additional post to the original interaction as a reply to another respondent: Wouldn't it be nice if all math teachers believed that having their students actually understand what they are learning was the focus of instruction? Wouldn't it be nice for all students to build a strong foundation for future study, to explore, to conjecture, to critique, to argue, to build consensus and to thereby remember what they learned because they really understood what it was that they were learning? Wouldn't it be so much more fun? Wouldn't it give instruction meaning? Wouldn't it be better than memorizing facts and procedures that will be forgotten as soon as the test is over, if they actually could hold onto it that long (and without needing a complete review of what they learned before the test?) Wouldn't it be nice if all of their studies of mathematics were actually related to one another and were not seen as isolated dips into unknown and unremembered pools of mathematical water? Wouldn't it be nice if the vast majority of educational professionals actually believed that learning was about thinking and not regurgitating disconnected facts? Why am I such an inordinate and unrelenting dreamer?