I worked with young children, mainly pr-k-5. When anything was being explored in math the question was always "How did You do it?" All the different ways were collected on the board for all the children to see. Young children like to see their work up on the board so they worked hard to think of a new or different way. Each volunteer explained how they thought about the work. Some methods were often more efficient and some more interesting. Children were then asked to choose someone else's method to use to solve the problem. At the end of the lesson kids were asked if they found a new way of working that they liked better than their own. Some did, some didn't.
Ah, this! Breaking the notion that there's only one way to do mathematics correctly takes intentionality, and this sort of "Write Around" activity is also one I love! (more here: https://www.mathycathy.com/blog/2012/11/revamping-a-writing-strategy-for-math/) Several others I like that promote discourse and not necessarily a direct route to the "right answer" are Stand & Talks (thank you Sara VanDerWerf... more here https://www.mathycathy.com/blog/2019/03/using-apple-classroom-for-stand-talks/ and here https://www.mathycathy.com/blog/2019/05/lead-learn-with-stand-talks-across-content-areas/) and "Find 'n Fix", which can be as simple as creating a template with 4 blocks, and a problem worked from start to finish in each block. Students are asked to "grade" the paper together, providing constructive feedback to the imaginary student for any errors... the fun part is, not EVERY worked problem contains errors! Again, promoting discourse regularly that considers multiple methods helps coach students away from the "one way" they may think they "MUST" do math. Finally, during lessons, I try to throw in exaggerated examples that make a silly, emotional impact. "Could I add 1,000,017 to both sides of this equation as my next step? Could I subtract 19.458 from both sides?" Guiding students to understand that these are "legal" algebraic moves that may not be HELPFUL, but they're also not WRONG, can encourage students who may later solve a problem less efficiently than a peer, but both got there... one just took the highway, and the other, the scenic route!
Additionally I believe that math frustrates students because they don’t like being wrong. We discuss how the don’t get upset in English because everything is a “draft,” not right or wrong. And so I have tried to change my language to “that’s a good draft,” “let’s revise this draft.”
Sure! When working w more than 1 student, "any particular routines or activities you use to help students deconstruct the idea that there is only one correct way to do math" are to try to involve the whole group/class into the discussion, somehow. If discussion w a single student, trying to get the student to lead w an example as a jumping off point to work out the problem.
I used to teach Decision Theory in the UK for A level. The head examiner told me you could 100% without a correct answer. Only interested in the method used. I found that my “brightest” student so struggled with this topic. They just could not cope with the uncertainty. As there were multiple correct answers there were no answers at the back of the book. Looking at what someone else had done did not help as there were multiple ways of doing it! Proved to me that to pass the traditional topic exams you did not need to be good at maths but good at remembering and using algorithms.
To say the same thing: school math is taught as a matter of performance. Teaching it is management of the resulting alienation. Making mistakes in understanding a world that interests me is not alienating.
I like the idea of different paths to solving a maths problem as that way the student can claim the knowledge as their own rather than passively accepting the one path.
I’m (to put it mildly) a mature student but am teaching myself so that means a mix of revision and new knowledge. What I do is work with a book or usually several books (plus internet sites) until I hit a roadblock which tells me that I need to learn something more basic before I can progress so I jump back to where I think I need to restart or revise and then jump back further if I have to or move forward until I can deal with the roadblock.
This is quite easy with hyperlinks on the internet and some maths sites already do a version of this. As do some books but to my knowledge none of the above have an organised way of doing this where right from the start the student chooses the path they want to travel which is formed by their chosen outcome.
The advantage of this method of learning is that the path is very personalised and the motivation is strong as at any given point I’m studying something that is directly relevant to my goal at the time.
Also in my case it helps to have some good strong reasons for wanting to learn. For instance one of my main drivers is I want to at least have some grasp of Quantitive Analysis as I feel it’s an underused mathematical tool for looking at connections between things.
Another driver of mine is to investigate science methodology and for that Statistics is one of the major subjects so there my path is determined by by that driver or goal.
The disadvantage of this method is that it can’t be used in the classroom although I suppose it could be modified as the article and some comments suggest.
> When all of those different paths converge on the same destination
This language made me think of mathematics as a physical place and how math operations allow you to travel in that place. I am also imagining how different travel directions look in a mapping app on my phone with multiple options to get from a start point to an end point. Each route may take different amounts of time and distance, the “leaf” route is more energy efficient, some routes may have tolls, and the options are very different when driving, walking, biking, or taking transit.
I think it is a very valuable thing to know that there are multiple mathematical paths that can lead to the same place of understanding and while the most direct routes may be faster, you also may discover things you didn’t expect when taking a meandering walk.
Oo yeah nice. Like what are you optimizing for. Getting a number correct at the end of a problem is a valuable objective. But a meandering path can result in more learning than a direct one, provided someone helps you tune into your surroundings.
I worked with young children, mainly pr-k-5. When anything was being explored in math the question was always "How did You do it?" All the different ways were collected on the board for all the children to see. Young children like to see their work up on the board so they worked hard to think of a new or different way. Each volunteer explained how they thought about the work. Some methods were often more efficient and some more interesting. Children were then asked to choose someone else's method to use to solve the problem. At the end of the lesson kids were asked if they found a new way of working that they liked better than their own. Some did, some didn't.
There are like ten different special moves in this comment. Thanks, Paula.
Ah, this! Breaking the notion that there's only one way to do mathematics correctly takes intentionality, and this sort of "Write Around" activity is also one I love! (more here: https://www.mathycathy.com/blog/2012/11/revamping-a-writing-strategy-for-math/) Several others I like that promote discourse and not necessarily a direct route to the "right answer" are Stand & Talks (thank you Sara VanDerWerf... more here https://www.mathycathy.com/blog/2019/03/using-apple-classroom-for-stand-talks/ and here https://www.mathycathy.com/blog/2019/05/lead-learn-with-stand-talks-across-content-areas/) and "Find 'n Fix", which can be as simple as creating a template with 4 blocks, and a problem worked from start to finish in each block. Students are asked to "grade" the paper together, providing constructive feedback to the imaginary student for any errors... the fun part is, not EVERY worked problem contains errors! Again, promoting discourse regularly that considers multiple methods helps coach students away from the "one way" they may think they "MUST" do math. Finally, during lessons, I try to throw in exaggerated examples that make a silly, emotional impact. "Could I add 1,000,017 to both sides of this equation as my next step? Could I subtract 19.458 from both sides?" Guiding students to understand that these are "legal" algebraic moves that may not be HELPFUL, but they're also not WRONG, can encourage students who may later solve a problem less efficiently than a peer, but both got there... one just took the highway, and the other, the scenic route!
Love all of this, Cathy. Many thanks.
Additionally I believe that math frustrates students because they don’t like being wrong. We discuss how the don’t get upset in English because everything is a “draft,” not right or wrong. And so I have tried to change my language to “that’s a good draft,” “let’s revise this draft.”
The drafting language is really powerful and could benefit math a lot. Adding a link to Jansen's book here for an extra treat:
https://stenhouse.com/products/9781625312068_rough-draft-math
Sure! When working w more than 1 student, "any particular routines or activities you use to help students deconstruct the idea that there is only one correct way to do math" are to try to involve the whole group/class into the discussion, somehow. If discussion w a single student, trying to get the student to lead w an example as a jumping off point to work out the problem.
I used to teach Decision Theory in the UK for A level. The head examiner told me you could 100% without a correct answer. Only interested in the method used. I found that my “brightest” student so struggled with this topic. They just could not cope with the uncertainty. As there were multiple correct answers there were no answers at the back of the book. Looking at what someone else had done did not help as there were multiple ways of doing it! Proved to me that to pass the traditional topic exams you did not need to be good at maths but good at remembering and using algorithms.
To say the same thing: school math is taught as a matter of performance. Teaching it is management of the resulting alienation. Making mistakes in understanding a world that interests me is not alienating.
"Teaching as the management of alienation" is the sort of phrase that'll stick with a person.
I like the idea of different paths to solving a maths problem as that way the student can claim the knowledge as their own rather than passively accepting the one path.
I’m (to put it mildly) a mature student but am teaching myself so that means a mix of revision and new knowledge. What I do is work with a book or usually several books (plus internet sites) until I hit a roadblock which tells me that I need to learn something more basic before I can progress so I jump back to where I think I need to restart or revise and then jump back further if I have to or move forward until I can deal with the roadblock.
This is quite easy with hyperlinks on the internet and some maths sites already do a version of this. As do some books but to my knowledge none of the above have an organised way of doing this where right from the start the student chooses the path they want to travel which is formed by their chosen outcome.
The advantage of this method of learning is that the path is very personalised and the motivation is strong as at any given point I’m studying something that is directly relevant to my goal at the time.
Also in my case it helps to have some good strong reasons for wanting to learn. For instance one of my main drivers is I want to at least have some grasp of Quantitive Analysis as I feel it’s an underused mathematical tool for looking at connections between things.
Another driver of mine is to investigate science methodology and for that Statistics is one of the major subjects so there my path is determined by by that driver or goal.
The disadvantage of this method is that it can’t be used in the classroom although I suppose it could be modified as the article and some comments suggest.
> When all of those different paths converge on the same destination
This language made me think of mathematics as a physical place and how math operations allow you to travel in that place. I am also imagining how different travel directions look in a mapping app on my phone with multiple options to get from a start point to an end point. Each route may take different amounts of time and distance, the “leaf” route is more energy efficient, some routes may have tolls, and the options are very different when driving, walking, biking, or taking transit.
I think it is a very valuable thing to know that there are multiple mathematical paths that can lead to the same place of understanding and while the most direct routes may be faster, you also may discover things you didn’t expect when taking a meandering walk.
Oo yeah nice. Like what are you optimizing for. Getting a number correct at the end of a problem is a valuable objective. But a meandering path can result in more learning than a direct one, provided someone helps you tune into your surroundings.