Today was the second edition of #mathsjournalclub. A bi-monthly twitter chat aimed at increasing engagement oaths teachers with the research on mathematical education. It was the first I’d managed to be involved with and I very such enjoyed it. As it happens, I’m working on the very early stages of my masters dissertation and have spent the day reading many articles about problem solving in maths and relational vs instrumental understanding, all of which links quite well to the featured article:
Mathematical ètudes: embedding opportunities for developing fluency within rich tasks Colin Foster (@colinfoster77) – article available here
I thoroughly enjoyed this article, it started by presenting the ideas of traditionalist teachers; those who feel repetition is the best route to fluency, if you don’t give opportunities to build fluency you are failing your students; and progressive teachers; those who feel mathematics should be all about exploration and never about drill and practice.
Colin’s discussion on these points was very insightful, suggesting that drill and practice may lead to embedding misconceptions and that learning by practice doesn’t always lead to understanding. This is something I can agree with, I’ve seen these misconceptions become embedded through drill routines and seen people who can follow a procedure without knowing why. I even remember a friend at uni who left with a first but only had a surface understanding of the procedures rather than a deeper relational understanding of the maths.
Colin also mentions the need for fluency, which I can also agree with. When number facts are known, and skills are developed to a fluent level, then that frees up more working memory to take on the more taxing tasks that come with a higher cognitive load.
Colin’s paper sought to provide an answer, to find a common ground that could satisfy both the traditional and progressive methodologies. That could build understanding and develop fluency. He looked for his answer in music, borrowing the idea of an ètude. Something designed to practice a skill, but also to be pleasing to an audience. His mathematical ètudes are designed to practice a skill set within a richer context which can also build a better set of problem solving skills and a deeper relational understanding.
He puts forward some excellent examples of what he means, examples of rich tasks that still have the elements of practice required for fluency, yet have self checking mechanisms built in so students won’t bed misconceptions. There is also easy enriching extentions for those who master something and there can be self differentiation, as the tasks are accessible to all but can be taken to many levels.
One example was in the carte sian coordinate system, instead of giving students tons of pairs of coordinates, give them a rule (ie the second is twice the first), this allows them to generate then plot their own pairs. The fact that the graph will show a pattern gives a great self checking mechanisms and also a prompt for the student to investigate further, the enrich and extend for the higher ability students could involve squaring or cubing. This simple task allows students of all levels to engage at the level best suited for them. I can’t wait to get back to school to trial some of these tasks.
cavmaths 
Learning Fluency within Rich tasks
Today was the second edition of #mathsjournalclub. A bi-monthly twitter chat aimed at increasing engagement oaths teachers with the research on mathematical education. It was the first I’d managed to be involved with and I very such enjoyed it. As it happens, I’m working on the very early stages of my masters dissertation and have spent the day reading many articles about problem solving in maths and relational vs instrumental understanding, all of which links quite well to the featured article:
Mathematical ètudes: embedding opportunities for developing fluency within rich tasks Colin Foster (@colinfoster77) – article available here
I thoroughly enjoyed this article, it started by presenting the ideas of traditionalist teachers; those who feel repetition is the best route to fluency, if you don’t give opportunities to build fluency you are failing your students; and progressive teachers; those who feel mathematics should be all about exploration and never about drill and practice.
Colin’s discussion on these points was very insightful, suggesting that drill and practice may lead to embedding misconceptions and that learning by practice doesn’t always lead to understanding. This is something I can agree with, I’ve seen these misconceptions become embedded through drill routines and seen people who can follow a procedure without knowing why. I even remember a friend at uni who left with a first but only had a surface understanding of the procedures rather than a deeper relational understanding of the maths.
Colin also mentions the need for fluency, which I can also agree with. When number facts are known, and skills are developed to a fluent level, then that frees up more working memory to take on the more taxing tasks that come with a higher cognitive load.
Colin’s paper sought to provide an answer, to find a common ground that could satisfy both the traditional and progressive methodologies. That could build understanding and develop fluency. He looked for his answer in music, borrowing the idea of an ètude. Something designed to practice a skill, but also to be pleasing to an audience. His mathematical ètudes are designed to practice a skill set within a richer context which can also build a better set of problem solving skills and a deeper relational understanding.
He puts forward some excellent examples of what he means, examples of rich tasks that still have the elements of practice required for fluency, yet have self checking mechanisms built in so students won’t bed misconceptions. There is also easy enriching extentions for those who master something and there can be self differentiation, as the tasks are accessible to all but can be taken to many levels.
One example was in the carte sian coordinate system, instead of giving students tons of pairs of coordinates, give them a rule (ie the second is twice the first), this allows them to generate then plot their own pairs. The fact that the graph will show a pattern gives a great self checking mechanisms and also a prompt for the student to investigate further, the enrich and extend for the higher ability students could involve squaring or cubing. This simple task allows students of all levels to engage at the level best suited for them. I can’t wait to get back to school to trial some of these tasks.
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