Thinking about this a LOT now. The first part of my Algebra 2 class rebuilds the rules of rational expressions and polynomials. I'm trying to teach students that math is something to be created from choices and definitions (easier with exponent rules). This is much easier when we "create" new understandings from existing concepts, but trying to re-create things most people have seen makes high kids mad (I know the rule, why are you confusing me?) and indulges their bad habits. (No need to take notes, do practice, etc).
I tried getting around this by starting with matrices (no one has seen them); it worked really well. But the habits and mindsets didn't stick when we went back to things they'd seen before.
Hi Dan! Really admire you for sharing this and appreciate the conversations you create for everyone.
My wondering: In the video, you say "I asked you /how/ do you know?" but in the actual text of the question shared, you ask them to "explain /why/ it makes sense". Maybe I'm mistaken, but if that's the case, (and being a bit picky here), I think /how/ and /why/ can elicit two very different responses. "Why" may unintentionally add constraints, (i.e., what is the path to the correct answer?) but "how" can invite prior knowledge or one to produce an answer from their own lens or added narrative.
So, I'm wondering if you might end-up with different responses if the intial question was perhaps worded differently?
This is such a great and constantly top of mind question because at some point I let it go and often kick myself later for taking the easy way out. In Algebra I, I usually let the "a negative times a negative will give a positive" explanation fly when thinking about what value would minimize something like (x-5)^2 even if it is slightly superficial because we are building towards something else. I think in your case I might have given four or so expressions like -2*-4, -2+-4, -2+4, 2+-4, -2*-4 etc and asked pairs to decide which ones are negative. When debate/disagreement naturally occurred, I would have the class convert each expression into the language of floats and anchors to see when the mnemonic makes sense and when it might be misleading us. Thanks for this!
We were looking at just this the other day and started the class with sometimes, always, never statements.
It was great because it allowed the students to discover for themselves how brittle statements like a negative and a negative is a positive were.
Like you, the activity was a response to students confusion with addition and subtraction of directed numbers caused by mnemonics learnt outside of the classroom.
I haven't had much success trying to "fight" tricks like this. I find that when I try to convince students the trick isn't a good one, two things tend to happen: the students who really do understand the math behind it nod along because while they might use the trick they understand why it's a trick and they're actually in a good place. The students whose understanding is tenuous, who are relying on the trick because they don't have a lot else to go on, feel quietly frustrated because I am illegitimizing the only understanding they have.
My approach would be to focus on the limits of the trick. If it's FOIL, give students a few monomial/binomial multiplications mixed with monomial/monomial and monomial/trinomial. In this case, give a mix of multiplication, division, and addition with negatives. Provide context -- "I owe Jimmy 3 dollars, I owe Johnny 5 dollars, how much do I owe in total" has worked well for me in the contrasting case of -4 + -8. Craig Barton has a great idea of "same surface, different deep" problems where at first they look similar but require different underlying math. Maybe you give four problems with those same numbers of -4 and -8 and similar contexts, but each requires a different operation. Then the discussion focuses on the differences between the operations.
Here's a different way to say what I'm trying to get across: if my only response to a trick is to delegitimize it, I risk alienating students for whom that's all they have. Don't attack the trick -- see the trick as a valuable first step, find its limits, make clear why the trick doesn't apply beyond those limits, and use that discussion to help students understand the trick better.
Late to this, but I like the sometimes, always, never strategy. In the moment, I like the idea of sharing the four statements you pulled out, and saying that only one of them is true all of the time. Have students turn and talk to identify the always true statement, and explain why at least one other statement is sometimes false. After sharing out, I would specifically highlight the slipperiness of "two negatives make a positive" and give them examples of how that could be misinterpreted ("Does -2 - 4 make a positive?").
Once students are clear that "a negative times a negative equals a positive," I think they would be ready for your original push. This statement isn't a trick imo - it's a true statement that every student should know, but it still doesn't have the depth of understanding you're going for. Now the push is to explain why the statement is true, and students will have to use the floats and anchors to do so.
In general, when a trick is proposed I ask them to explain it, much like you did with the anchors and floats. And also to be very specific about what's happening. Ultimately, they will then know that ~ A negative "multiplied" by a negative = a positive.
Extremely relatable! I think in the exact moment I'd ask something like "But how do you know a negative times a negative is a positive? How could you check that it works in this scenario?" And then I think I'd plan a follow up with several examples like the one you mentioned: -4 * -3, -4 + -3, -4/-3, -3 - 4, -4 - 3, etc. Then I'd say- which of these are positive? Ya'll told me yesterday a negative and a negative makes a positive. (purposely using vague language there)
I would say, ask them to tell you what (-2)(-6) means in terms of the balloons and anchors, and not be afraid to be the one to point out to them the -2 is "taking away." Then re-ask the question about why it's positive, and then challenge them to generalize. I say this having not gone through the activity, and maybe you already did that scaffolding?
I love how you re-directed them to use the "vocabulary" of the floats and anchors in their explanation. If you had just let it slide, then they would NEVER make the connection. Yes, they will whine - - - but as we will all testify, they mess up when they just memorize a phrase.
In fact, in my "at risk algebra class" for kids repeating semester 1, we have solved equations with Crates and Rocks on a balance scale with our intuition and informal reasoning and they have done just fine. Moving that to the process of inverse operations and "formal" solving, we still frame the process with Crates on one side of the scale, Rocks on the other, and then share Rocks with the # of Crates. When they have an incorrect solution, I can see their work showing "always subtract from the left side" or "always divide by the number on the left side".
It's still disappointing to me that kids don't make the connection that multiplication is just repeated addition. But it seems like the floats/anchors seem to be reinforcing repeated addition/subtraction as the same as multiplication, which is TERRIFIC!
Definitely worth the effort and energy to have our kids use appropriate reasoning and contextual vocabulary instead of "Math Tricks", even though it sure would be easier to let it slide. ☺
I usually draw on a discourse which reduces the 'trick' to a bad thing. For example, with a trick like FOIL I usually say, "whoa, whoa, whoa... such strong language in math class. I'm not sure four letter F words have a place in this classroom" and then give that student a binomial multiplied by a trinomial to multiply and let them see if FOILing works. I think addressing those tricks in class and giving situations where they don't work forces the students to think beyond the tricks and seek to better understand what it is we are trying to steer them towards. Adding a bit of humour always helps reduce the position of power us teachers hold.
Thinking about this a LOT now. The first part of my Algebra 2 class rebuilds the rules of rational expressions and polynomials. I'm trying to teach students that math is something to be created from choices and definitions (easier with exponent rules). This is much easier when we "create" new understandings from existing concepts, but trying to re-create things most people have seen makes high kids mad (I know the rule, why are you confusing me?) and indulges their bad habits. (No need to take notes, do practice, etc).
I tried getting around this by starting with matrices (no one has seen them); it worked really well. But the habits and mindsets didn't stick when we went back to things they'd seen before.
Hi Dan! Really admire you for sharing this and appreciate the conversations you create for everyone.
My wondering: In the video, you say "I asked you /how/ do you know?" but in the actual text of the question shared, you ask them to "explain /why/ it makes sense". Maybe I'm mistaken, but if that's the case, (and being a bit picky here), I think /how/ and /why/ can elicit two very different responses. "Why" may unintentionally add constraints, (i.e., what is the path to the correct answer?) but "how" can invite prior knowledge or one to produce an answer from their own lens or added narrative.
So, I'm wondering if you might end-up with different responses if the intial question was perhaps worded differently?
Sharing here a nice reading outside the edu domain, but really got me to consider the difference between asking /why/ vs. /how/: https://www.oreilly.com/radar/the-infinite-hows/
This is such a great and constantly top of mind question because at some point I let it go and often kick myself later for taking the easy way out. In Algebra I, I usually let the "a negative times a negative will give a positive" explanation fly when thinking about what value would minimize something like (x-5)^2 even if it is slightly superficial because we are building towards something else. I think in your case I might have given four or so expressions like -2*-4, -2+-4, -2+4, 2+-4, -2*-4 etc and asked pairs to decide which ones are negative. When debate/disagreement naturally occurred, I would have the class convert each expression into the language of floats and anchors to see when the mnemonic makes sense and when it might be misleading us. Thanks for this!
We were looking at just this the other day and started the class with sometimes, always, never statements.
It was great because it allowed the students to discover for themselves how brittle statements like a negative and a negative is a positive were.
Like you, the activity was a response to students confusion with addition and subtraction of directed numbers caused by mnemonics learnt outside of the classroom.
I used this set of statements from mathspad.
https://www.mathspad.co.uk/teach/linkedDocuments/negatives/ASNnegatives.pdf
I haven't had much success trying to "fight" tricks like this. I find that when I try to convince students the trick isn't a good one, two things tend to happen: the students who really do understand the math behind it nod along because while they might use the trick they understand why it's a trick and they're actually in a good place. The students whose understanding is tenuous, who are relying on the trick because they don't have a lot else to go on, feel quietly frustrated because I am illegitimizing the only understanding they have.
My approach would be to focus on the limits of the trick. If it's FOIL, give students a few monomial/binomial multiplications mixed with monomial/monomial and monomial/trinomial. In this case, give a mix of multiplication, division, and addition with negatives. Provide context -- "I owe Jimmy 3 dollars, I owe Johnny 5 dollars, how much do I owe in total" has worked well for me in the contrasting case of -4 + -8. Craig Barton has a great idea of "same surface, different deep" problems where at first they look similar but require different underlying math. Maybe you give four problems with those same numbers of -4 and -8 and similar contexts, but each requires a different operation. Then the discussion focuses on the differences between the operations.
Here's a different way to say what I'm trying to get across: if my only response to a trick is to delegitimize it, I risk alienating students for whom that's all they have. Don't attack the trick -- see the trick as a valuable first step, find its limits, make clear why the trick doesn't apply beyond those limits, and use that discussion to help students understand the trick better.
Late to this, but I like the sometimes, always, never strategy. In the moment, I like the idea of sharing the four statements you pulled out, and saying that only one of them is true all of the time. Have students turn and talk to identify the always true statement, and explain why at least one other statement is sometimes false. After sharing out, I would specifically highlight the slipperiness of "two negatives make a positive" and give them examples of how that could be misinterpreted ("Does -2 - 4 make a positive?").
Once students are clear that "a negative times a negative equals a positive," I think they would be ready for your original push. This statement isn't a trick imo - it's a true statement that every student should know, but it still doesn't have the depth of understanding you're going for. Now the push is to explain why the statement is true, and students will have to use the floats and anchors to do so.
In general, when a trick is proposed I ask them to explain it, much like you did with the anchors and floats. And also to be very specific about what's happening. Ultimately, they will then know that ~ A negative "multiplied" by a negative = a positive.
Extremely relatable! I think in the exact moment I'd ask something like "But how do you know a negative times a negative is a positive? How could you check that it works in this scenario?" And then I think I'd plan a follow up with several examples like the one you mentioned: -4 * -3, -4 + -3, -4/-3, -3 - 4, -4 - 3, etc. Then I'd say- which of these are positive? Ya'll told me yesterday a negative and a negative makes a positive. (purposely using vague language there)
I would say, ask them to tell you what (-2)(-6) means in terms of the balloons and anchors, and not be afraid to be the one to point out to them the -2 is "taking away." Then re-ask the question about why it's positive, and then challenge them to generalize. I say this having not gone through the activity, and maybe you already did that scaffolding?
OMG, so glad students are the same EVERYWHERE!
I love how you re-directed them to use the "vocabulary" of the floats and anchors in their explanation. If you had just let it slide, then they would NEVER make the connection. Yes, they will whine - - - but as we will all testify, they mess up when they just memorize a phrase.
In fact, in my "at risk algebra class" for kids repeating semester 1, we have solved equations with Crates and Rocks on a balance scale with our intuition and informal reasoning and they have done just fine. Moving that to the process of inverse operations and "formal" solving, we still frame the process with Crates on one side of the scale, Rocks on the other, and then share Rocks with the # of Crates. When they have an incorrect solution, I can see their work showing "always subtract from the left side" or "always divide by the number on the left side".
It's still disappointing to me that kids don't make the connection that multiplication is just repeated addition. But it seems like the floats/anchors seem to be reinforcing repeated addition/subtraction as the same as multiplication, which is TERRIFIC!
Definitely worth the effort and energy to have our kids use appropriate reasoning and contextual vocabulary instead of "Math Tricks", even though it sure would be easier to let it slide. ☺
I usually draw on a discourse which reduces the 'trick' to a bad thing. For example, with a trick like FOIL I usually say, "whoa, whoa, whoa... such strong language in math class. I'm not sure four letter F words have a place in this classroom" and then give that student a binomial multiplied by a trinomial to multiply and let them see if FOILing works. I think addressing those tricks in class and giving situations where they don't work forces the students to think beyond the tricks and seek to better understand what it is we are trying to steer them towards. Adding a bit of humour always helps reduce the position of power us teachers hold.