Yes!!!! I've been trying to find meaningful ways to work this into my curriculum. I have a sequence where I ask kids to... "Do the same thing to both sides of the equation such that..." with all types of non-standard goals. 2x + 5 = 12. Do the same thing to both sides of the equation until the left side is 10x + 10. Multiple paths, and a familiarity with the moves you can do to both sides of the equations. (And the practice, say, of multiplying both sides by 2, which doesn't come up a lot if you are just solving for x).
I used to get to completing the square (say, x^2 + 12 + 31= 2), and students would say... "Wait, how can you just add 5 to both sides of the equation? Where did the 5 come from?" Now students are more comfortable with the idea that they can do whatever they want as long as they do it to both sides.
I struggled with this just last week. I love the exploration of "wrong steps", I dont think my students loved it as much as me, afterwords mentor teacher said "no you cant divide by 5" and I said "well actually you can..."
I'm not sure I agree that picking a number is "incorrect"- I think you're saying it's mathematically incorrect if you choose a number greater than or equal to 3. But if I chose 1 then it is correct! And I'd argue it's even moderately useful in getting me closer to a solution. I think I'd argue that choosing a "wrong" number (like 3) is actually quite useful as well, when you're starting out at least.
Yeah, thanks for pushing my thinking here, Leandra. Why I called it incorrect is that the goal of the problem is to come up with a representation of ALL the solutions of the inequality and picking a number or numbers is only ever going to reveal SOME of the solutions. I may need to think of a better example of incorrect and useful here, though.
Yes!! I’m always saying “is this a legal move?? Sure. But is it helpful move??” When students can see the flexibility and the “art” behind math… great things can happen!
I would go one step further and refrain from calling a step as "useless." IMO this is too negative for students who are just working up the bravery to try something, even if they don't know if it will lead to the solution. I write about this here:
Also I agree with @Leandra below that "pick a number and try it out" is not incorrect. We call it "guess and check" and it can be very helpful in finding the solution(s).
hi Dan, This post has nothing to do with what you are talking about here. I wanted to leave a comment on your last Math Teacher's Lounge about math anxiety but saw no way to do that.
When your expert on math anxiety said he saw no way to get rid of math anxiety my bristles went up. The way to get rid of math anxiety is to teach math correctly from kindergarten onward and involve the parents while you are doing it. My K students know there are lots of tools they can use to approach problems to help them understand what to do and how to solve the problem. They also know that everything they do is connected in some way to to something they are experiencing or have experienced. Later in school kids get anxious when they feel they have no path whatsoever to follow or no tools they can use to solve or understand a problem. In my classrooms all the way through 6th grade the kids have access to a wide variety of tools and they understand that that using tools does not make them stupid just the opposite it makes them strong because they can then approach a problem from a lot of different angles.
When I have had new students enter my class at 6th grade they at first believe that math is something they must do in their heads and if they cannot do that then they are no good at math. At first the tools seem "babiesh" but then they see they give them power. When I give a quiz my student have access to anything they think will help them and they settle in to think and work their way through. Kids who never thought they could be good become excited in the prospect.
Teachers contribute to math anxiety beginning in K and continuing forward. When they don't fully understand a process they teach their students the processes must be memorized. They don't see the connections to learned work so they don't ask their kids to look for them. They fawn over those students that excel and shake their heads at those who don't or now, send them beginning at K to remedial help! Math Anxiety is a self fulfilling prophecy....we are all anxious when we don't understand and have no tools at out fingertips to help.
Sorry I took up the space here...didn't know what else to do so I used another tool to help me solve my problem.
“Transformation Golf” is an excellent activity to introduce this idea of legal and useless mathematics. I will ask students to try to get the most efficient solution (and oftentimes there are multiple!) or they can try to find the least efficient solution. I get to talk about coterminous angles because students like to do a few spins before they do the last transformation. I love when a problem can be solved with a 180 rotation, but then someone enters 3600+180 and calls it a trick shot😂.
Anyway, reflecting on this makes me want to take the same energy we get going from this lesson and apply it to solving equations/inequalities. I love Don Steward’s “algebra snakes and branches”:
And I think it could be made into a really fun activity much like transformation golf, where the goal is to isolate the variable, but you can make a snake that is as long or as short as you like. I’m going to give it a try, but hopefully one of the great activity builders @ Desmos sees this and likes the idea enough to build it :)
When a student proffers a "move" that the rest of the class thinks is the wrong move, I never challenge it, as long as he has phrased it as needing to be done to both sides. I just do it—even if it leads to some bizarre fractional coefficient—and ask someone for the next "move". If someone tries to say the previous move was "wrong", I inform them that anything they know how to do is permitted, except for dividing by zero. I think acting nonchalant about these "flubs" reduces anxiety about being willing to participate. What I have *not* done is use the term "legal" for this concept, but now I will.
> I just do it—even if it leads to some bizarre fractional coefficient—and ask someone for the next "move".
I love the kind of feedback you're offering here. We do a lot of "interpretive feedback" in our curriculum. The student contributes a thought to the computer and the computer, rather than evaluating it, simply interprets in a context for the student's reflection and revision. This is like that, but for thinking that is too sophisticated and unstructured for the computer to understand. The student contributes a thought to your discussion and rather than evaluating it, you're just interpreting it in the problem.
Yes!!!! I've been trying to find meaningful ways to work this into my curriculum. I have a sequence where I ask kids to... "Do the same thing to both sides of the equation such that..." with all types of non-standard goals. 2x + 5 = 12. Do the same thing to both sides of the equation until the left side is 10x + 10. Multiple paths, and a familiarity with the moves you can do to both sides of the equations. (And the practice, say, of multiplying both sides by 2, which doesn't come up a lot if you are just solving for x).
I used to get to completing the square (say, x^2 + 12 + 31= 2), and students would say... "Wait, how can you just add 5 to both sides of the equation? Where did the 5 come from?" Now students are more comfortable with the idea that they can do whatever they want as long as they do it to both sides.
That's a super helpful reference to completing the square, which includes a step that feels at first quite useless.
I struggled with this just last week. I love the exploration of "wrong steps", I dont think my students loved it as much as me, afterwords mentor teacher said "no you cant divide by 5" and I said "well actually you can..."
I'm not sure I agree that picking a number is "incorrect"- I think you're saying it's mathematically incorrect if you choose a number greater than or equal to 3. But if I chose 1 then it is correct! And I'd argue it's even moderately useful in getting me closer to a solution. I think I'd argue that choosing a "wrong" number (like 3) is actually quite useful as well, when you're starting out at least.
Yeah, thanks for pushing my thinking here, Leandra. Why I called it incorrect is that the goal of the problem is to come up with a representation of ALL the solutions of the inequality and picking a number or numbers is only ever going to reveal SOME of the solutions. I may need to think of a better example of incorrect and useful here, though.
Yes!! I’m always saying “is this a legal move?? Sure. But is it helpful move??” When students can see the flexibility and the “art” behind math… great things can happen!
I would go one step further and refrain from calling a step as "useless." IMO this is too negative for students who are just working up the bravery to try something, even if they don't know if it will lead to the solution. I write about this here:
https://mathproblemsolvingskills.wordpress.com/2021/02/07/mistakes-i-do-and-dont-correct/
Also I agree with @Leandra below that "pick a number and try it out" is not incorrect. We call it "guess and check" and it can be very helpful in finding the solution(s).
"As long as it’s legal, we can try it." That's wonderful.
hi Dan, This post has nothing to do with what you are talking about here. I wanted to leave a comment on your last Math Teacher's Lounge about math anxiety but saw no way to do that.
When your expert on math anxiety said he saw no way to get rid of math anxiety my bristles went up. The way to get rid of math anxiety is to teach math correctly from kindergarten onward and involve the parents while you are doing it. My K students know there are lots of tools they can use to approach problems to help them understand what to do and how to solve the problem. They also know that everything they do is connected in some way to to something they are experiencing or have experienced. Later in school kids get anxious when they feel they have no path whatsoever to follow or no tools they can use to solve or understand a problem. In my classrooms all the way through 6th grade the kids have access to a wide variety of tools and they understand that that using tools does not make them stupid just the opposite it makes them strong because they can then approach a problem from a lot of different angles.
When I have had new students enter my class at 6th grade they at first believe that math is something they must do in their heads and if they cannot do that then they are no good at math. At first the tools seem "babiesh" but then they see they give them power. When I give a quiz my student have access to anything they think will help them and they settle in to think and work their way through. Kids who never thought they could be good become excited in the prospect.
Teachers contribute to math anxiety beginning in K and continuing forward. When they don't fully understand a process they teach their students the processes must be memorized. They don't see the connections to learned work so they don't ask their kids to look for them. They fawn over those students that excel and shake their heads at those who don't or now, send them beginning at K to remedial help! Math Anxiety is a self fulfilling prophecy....we are all anxious when we don't understand and have no tools at out fingertips to help.
Sorry I took up the space here...didn't know what else to do so I used another tool to help me solve my problem.
“Transformation Golf” is an excellent activity to introduce this idea of legal and useless mathematics. I will ask students to try to get the most efficient solution (and oftentimes there are multiple!) or they can try to find the least efficient solution. I get to talk about coterminous angles because students like to do a few spins before they do the last transformation. I love when a problem can be solved with a 180 rotation, but then someone enters 3600+180 and calls it a trick shot😂.
Anyway, reflecting on this makes me want to take the same energy we get going from this lesson and apply it to solving equations/inequalities. I love Don Steward’s “algebra snakes and branches”:
https://donsteward.blogspot.com/2018/01/algebra-snakes-and-branches.html?m=1
And I think it could be made into a really fun activity much like transformation golf, where the goal is to isolate the variable, but you can make a snake that is as long or as short as you like. I’m going to give it a try, but hopefully one of the great activity builders @ Desmos sees this and likes the idea enough to build it :)
Thanks for this inspiration!
"I love when a problem can be solved with a 180 rotation, but then someone enters 3600+180 and calls it a trick shot😂." Extremely fun!
“Coterminal” whoops
When a student proffers a "move" that the rest of the class thinks is the wrong move, I never challenge it, as long as he has phrased it as needing to be done to both sides. I just do it—even if it leads to some bizarre fractional coefficient—and ask someone for the next "move". If someone tries to say the previous move was "wrong", I inform them that anything they know how to do is permitted, except for dividing by zero. I think acting nonchalant about these "flubs" reduces anxiety about being willing to participate. What I have *not* done is use the term "legal" for this concept, but now I will.
> I just do it—even if it leads to some bizarre fractional coefficient—and ask someone for the next "move".
I love the kind of feedback you're offering here. We do a lot of "interpretive feedback" in our curriculum. The student contributes a thought to the computer and the computer, rather than evaluating it, simply interprets in a context for the student's reflection and revision. This is like that, but for thinking that is too sophisticated and unstructured for the computer to understand. The student contributes a thought to your discussion and rather than evaluating it, you're just interpreting it in the problem.
how am I to understand "less than 0". What have you changed in the way that words I use work?