Algebra in 5th grade

[Not_a_Number]

2 hours ago, medawyn said:

They have problems unlearning that an equal sign does not mean "the answer is".  I am working with a second grader right now, and I have to remind him daily that = means "the same as" or I get many wrong answers from him.  I purposely write equations in a variety of ways to make him think through the problem, but if I don't prompt him and read the first 2-3 with him, guaranteed I'll get a slew of wrong answers because he sees an equal sign and immediately assumes the answer goes there.  A problem like 2+3 = 3 + x will automatically get a "5". He's quite obviously accustomed to a page of cookie cutter problems requiring little thought, but it's astounding how many bad habits that has taught him.

I'm shocked because I struggle with math and fret about teaching my own kids, and while I by no means feel sanguine about higher levels, I now feel pretty confident that I'm giving them a good base in early elementary math.

Oh, you are SO right. That is the one big problem I had teaching kids -- unteaching this habit. 

You know what's a really good scaffolding step for this? An equation grid! You cross out the wrong equations and then color them in, and you get something fun, like a letter. Here's an example from my class: 

 

Edited December 6, 2020 by Not_a_Number

[Not_a_Number]

49 minutes ago, EKS said:

The universities are also the only places where tracking occurs on a large scale.  First to get in and then people self sort after that.

Oh, I was tracked in high school, lol. I was in the gifted program, and one had to test into it. It wasn't nearly as good. 

[daijobu]

That equation puzzle is pretty brilliant and really drives the point home.  

[Shoeless]

On 12/4/2020 at 8:14 PM, JHLWTM said:

I’m going to put in a plug for Critical Thinking CO’s Balance Math and more. It uses a visual representation of a standard two arm balance to help kids understand equations in a very intuitive way. 

Similar idea, but we had one of these balance scales when DS12 was little: Algebra balance scale

We did a lot of experiments with putting weights on one side and then figuring out different combinations of weights to balance the scale. For example, I'd put a weight on "10", and then he'd figure out all the different combinations to make the other side balance. We had a regular pan balance that we played with, too. The goal was to teach that while yes, 2+2 = 4, you could also say that 2+2 = 3 +1. We did a lot of rearranging of blocks, coins, poker chips to find all the different ways of making the scales balance. 

[drjuliadc]

On 12/5/2020 at 11:24 AM, Not_a_Number said:

It's possible I hang out with more self-aware people, lol, but I know lots of professors who think that kids aren't prepared enough and aren't able to do hard math for that reason. 

I have a grand total of two professor patients. Only one is a math professor. She didn’t believe me when I told her there were people teaching things like number theory and counting and probability to kids even before high school. 

It could be the level of the only university here also. Old Dominion University is not MIT.

I just thought the topic of elementary and high school math inadequacy might come up among the hundreds of parents of “high performing” students, or even the many high school or elementary math teachers I’ve met throughout the years. It hasn’t and I’ve specifically asked about it a lot.

Edited December 7, 2020 by drjuliadc

[drjuliadc]

On 12/5/2020 at 2:54 PM, daijobu said:

Yeah, white people kinda suck.  They arrive at college unprepared for college level work.  So they drop out of engineering programs, but it was okay because they could pick up a liberal arts degree and still maintain a healthy social life and join a fraternity, then either go on to law school or go work in marketing at some big company.  

Now those back up white collar jobs are disappearing, and more unprepared students are unwilling to give up on the CS degree they hoped for.  In the 80s they would have quietly chosen another major.  Now they are complaining. This article is about poor kids, but I think it also applies to white middle class kids as well.  We may see pressure on professors to lower standards so  these kids can graduate with CS degrees, and those professors will have little incentive to make a stand for maintaining high standards, and follow the lead of their liberal arts colleagues.  

 

Yes, I am nothing but white and I was unprepared for college level calculus.  I don’t think it was my fault. I would have worked hard in high school in any math that was available. I was one of the very best math students in both of the two high schools I attended. 

Don’t take anything in my tone as if I am offended.  I am not in any way. I think my experience is described very well in the Richard R. video I mentioned and you, thankfully, reposted recently. The high school math I got wasn’t hard enough and it started too late, hence, this post. 

Btw, freshman calculus was the only thing I felt unprepared for.  My undergrad degree, Medical technology, required me to finish pre-med in 3 years instead of 4 (with the exception of two classes with labs), so maybe all of pre-med in 3.25-3.5 years. Required isn’t exactly correct. I’m sure they would have let me into the program if it took me longer, but that is the path they put you on and I just did it. I didn’t realize until years later that is what I did. 

Organic chemistry was up there in the challenging department too, but I am not sure what else could have prepared me for it. I recently read that that is what keeps a lot of wanna be doctors out of med school, their inability to pass organic chem.  I just got my Cs in one semester of calculus and organic chemistry, got A’s in everything else and moved on.

Pre-med doesn’t require calculus based physics either.  I never thought of premed or medical school as being as “hard” as engineering or computer science because the math is easier. Apparently a lot of engineers don’t think that way. The endless lists of straight memorization inherent in medical education intimidates them.

In my first week of chiropractic college I had to memorize 150 named prominences of the skull and spell them correctly. One of them was external occipital protuberance. 

I actually like your article. Yes, she was a little whiny, but I admire her hutzpa and her grit. If Stanford wants to accept people on an equal opportunity basis, they shouldn’t just throw them to the wolves once they are there. They shouldn’t lower their standards either.

So colleges have to teach high school now.  I don't feel bad for colleges. They are charging enough they should be able to do lots of things.

[Not_a_Number]

1 hour ago, drjuliadc said:

I have a grand total of two professor patients. Only one is a math professor. She didn’t believe me when I told her there were people teaching things like number theory and counting and probability to kids even before high school. 

It could be the level of the only university here also. Old Dominion University is not MIT.

I just thought the topic of elementary and high school math inadequacy might come up among the hundreds of parents of “high performing” students, or even the many high school or elementary math teachers I’ve met throughout the years. It hasn’t and I’ve specifically asked about it a lot.

Oh, that's very interesting. I definitely I know many people who are aware of it, but it's quite possible they aren't representative. A fair number are second generation immigrants or math professors at good schools, I guess. 

 

49 minutes ago, drjuliadc said:

I actually like your article. Yes, she was a little whiny, but I admire her hutzpa and her grit. If Stanford wants to accept people on an equal opportunity basis, they shouldn’t just throw them to the wolves once they are there. They shouldn’t lower their standards either.

Yeah, they've had a similar issue at DH's work -- they accepted "affirmative action" kinds of candidates to grad school, then didn't provide any support to them and watched them flounder. That simply doesn't work and is unfair to the kids... how are they supposed to pass the classes without the background?? 

As you say, if you're going to try accepting people on an equal opportunity basis, you have to provide structural support for them to succeed. Otherwise, it's an empty gesture. 

[wendyroo]

On 12/5/2020 at 2:59 PM, Not_a_Number said:

By the way, I tend to agree with the students on this one... most of my calculus kids who were deeply unprepared and unable to do well were really working hard. They were working very hard. The problem was that they had no idea HOW to work in such a way that got results. Some of my calculus kids spent HOURS rereading solutions to homework problems in the vain hope that they'd remember it. They didn't realize that this was ineffective and I didn't know how to convince them that it was. 

I watched this video recently and this discussion is reminding me of it: 

 

 

On 12/5/2020 at 3:24 PM, daijobu said:

Thanks, I hadn't seen this "meta" Veritasium video.  I love those videos precisely because he addresses the misconceptions directly.  I especially enjoyed the one about where all the carbon in a tree comes from.  The soil?  Then why isn't there a big whole surrounding a tree where the carbon used to be?  

I also wanted to model study skills to my kids, with an emphasis on reading skills.    I mentioned in another thread that I didn't have my kids use online learning until high school, mainly because I wanted them to be self sufficient with their study skills.  They couldn't count on good teaching at university, so they need to be able to make up the difference when needed.  

There's a way of learning challenging material from a textbook that many people seem to miss.  A student needs to be sitting upright, pen and paper in hand, writing what they see in the textbook, following along line by line, summarizing topics or copying equations as needed.  You don't sit back in bed with a book on your lap reading, like you would a celebrity memoir.  

I've been working my way through this thread, and came to these two posts. I liked the clip about the problems with videos like Khan Academy, so I decided to watch a Veritasium misconception video. I chose "Where Does The Sun Get Its Energy?" because I have been reading The Story of Earth to the kids.

I watched the video on my own, and then did a "pre-test" with my 7 year old, Spencer, by asking him how the sun makes energy. His answer was not verbose, but moderately accurate: "Fusion when hydrogen turns into helium". I wondered if the video would help clarify in his mind that one hydrogen atom does not "turn into" one helium, but rather the hydrogens are colliding and joining together. Also, I was hoping that the video's water-soaked sponge ball metaphor might help him expand his answer to include the idea that energy is given off in the process of the atoms joining.

This is not a happily ever after story!! Oh, my goodness, after watching the video, suddenly Spencer started talking about fire and burning on the sun and trying to integrate those into his "fusion, hydrogen turns into helium" model. He had latched onto a misconception that the people in the video had expressed! I guess I understand. Minute for minute, you do spend a high percentage of the video's runtime listening to wrong information, and when presented with incorrect information, the host does not label it as such in any way, rather he seems to accept it at face value.

I guess the message I take away from this is that addressing the wrong misconceptions can be just as damaging as not addressing any misconceptions. So, it is vital for me as a teacher to probe my student's understanding so I can help them eliminate incorrect mental models and replace them with stronger ones. But watching a video addressing misconceptions that the student doesn't actually share, even if the correct information is then taught, can pollute a correct mental model with incorrect information. We could call it the "You Can Never Go Down the Drain" problem - when Mr. Rogers sang that song, it helped kids who were having that misconception, but it planted that idea in the heads of kids who had never thought of it before.

[Not_a_Number]

27 minutes ago, wendyroo said:

 

I've been working my way through this thread, and came to these two posts. I liked the clip about the problems with videos like Khan Academy, so I decided to watch a Veritasium misconception video. I chose "Where Does The Sun Get Its Energy?" because I have been reading The Story of Earth to the kids.

I watched the video on my own, and then did a "pre-test" with my 7 year old, Spencer, by asking him how the sun makes energy. His answer was not verbose, but moderately accurate: "Fusion when hydrogen turns into helium". I wondered if the video would help clarify in his mind that one hydrogen atom does not "turn into" one helium, but rather the hydrogens are colliding and joining together. Also, I was hoping that the video's water-soaked sponge ball metaphor might help him expand his answer to include the idea that energy is given off in the process of the atoms joining.

This is not a happily ever after story!! Oh, my goodness, after watching the video, suddenly Spencer started talking about fire and burning on the sun and trying to integrate those into his "fusion, hydrogen turns into helium" model. He had latched onto a misconception that the people in the video had expressed! I guess I understand. Minute for minute, you do spend a high percentage of the video's runtime listening to wrong information, and when presented with incorrect information, the host does not label it as such in any way, rather he seems to accept it at face value.

I guess the message I take away from this is that addressing the wrong misconceptions can be just as damaging as not addressing any misconceptions. So, it is vital for me as a teacher to probe my student's understanding so I can help them eliminate incorrect mental models and replace them with stronger ones. But watching a video addressing misconceptions that the student doesn't actually share, even if the correct information is then taught, can pollute a correct mental model with incorrect information. We could call it the "You Can Never Go Down the Drain" problem - when Mr. Rogers sang that song, it helped kids who were having that misconception, but it planted that idea in the heads of kids who had never thought of it before.

Ooooh, that's super interesting!! Thanks for test driving those videos. Sorry that wasn't a very successful experiment! 

The idea that introducing NEW misconceptions is just as bad as not addressing OLD misconceptions makes a lot of sense!! I guess I've never tried to do that... when I address my kids' misconceptions, it's practically always one-on-one. 

In one of my AoPS classes, I did have an example where kids would have TWO different misconceptions at the same time. But I mostly handled those by whispering to the kids individually, as opposed to having a canned script for it. (I did have a canned script for slightly earlier, when there was one overwhelmingly popular misconception.) 

I think this comes back to the role that a well-designed computer program could play in education... first, the program could spot the specific misconception, and then it could play the correct video? (Or give the correct exercise/problem/game.) 

[Not_a_Number]

On 12/5/2020 at 10:39 PM, daijobu said:

That equation puzzle is pretty brilliant and really drives the point home.  

Thank you! It was the scaffolding step I thought of for the kids in my classes that really couldn't handle questions like 

3 + 4 = x + 5

from the beginning. I got a serious "deer in the headlights" look from some of the kids who weren't used to non-straightforward math. So I thought I'd provide them with the equations myself, so they didn't even have to think about solving for anything, just about whether the equals sign is true or not. I like breaking ideas down into even smaller ideas, so the steps aren't cognitively taxing 🙂 .

By the way, the other thing you realize if you run puzzles like this is that MANY elementary school kids think the following is true: 

3 - 9 = 6.

That's because they don't have a robust model of subtracting as taking the second number from the first number. You can go through all of elementary school thinking that the definition is "take the smaller number from the bigger number" and not get much wrong 😉 . The other thing that tests for that misconception is questions like 

5 - x = 3,

of course. 

Edited December 7, 2020 by Not_a_Number

[wendyroo]

7 minutes ago, Not_a_Number said:

I think this comes back to the role that a well-designed computer program could play in education... first, the program could spot the specific misconception, and then it could play the correct video? (Or give the correct exercise/problem/game.) 

I agree. I have always thought that Prodigy (and many other math apps/programs) don't pay nearly enough attention to wrong answers.

I have many times watched one of my kiddos get a Prodigy question wrong that clearly demonstrates a common misconception, perhaps saying 47 is greater than 51. As a human teacher, my next step would be to backtrack to showing visual representations of two digit numbers and checking that the student can correctly name them. Then perhaps moving on to comparing single digit numbers to teen numbers showing the visual representations of each to emphasize how powerful that group of ten is compared to the ones. Next I would try comparisons of various 2 digit numbers, but continue to provide the visuals, and start with stark comparisons (27 vs 91) instead of close ones (47 vs 51). Only once that was mastered would I remove the visuals.

But I watch Prodigy and it just marks the problem as wrong and moves on to something completely different. The kids may learn that they got that particular problem wrong...but they have no clue why or how or how to fix their mental model.

[Not_a_Number]

14 minutes ago, wendyroo said:

But I watch Prodigy and it just marks the problem as wrong and moves on to something completely different. The kids may learn that they got that particular problem wrong...but they have no clue why or how or how to fix their mental model.

I have a long-term aspiration to make a mental model oriented math app. I've now experimented with lots of different mental models for different concepts and have some pretty good ideas how to elicit the correct model. (I've done some experimenting with this for AoPS precalc, where I worked really hard to establish the "unit circle" mental model for trig. It worked a lot better than what we were doing before, and kids can now USE trig in the second half of the class, which was by no means a given previously.) 

 

14 minutes ago, wendyroo said:

Then perhaps moving on to comparing single digit numbers to teen numbers showing the visual representations of each to emphasize how powerful that group of ten is compared to the ones. Next I would try comparisons of various 2 digit numbers, but continue to provide the visuals, and start with stark comparisons (27 vs 91) instead of close ones (47 vs 51). Only once that was mastered would I remove the visuals.

I've been shocked at how many people seem to think you can remove visuals early. No, you can't. You really can't. If you do so, you often get what I call "symbol shuffling" -- the rearrangement of symbols on a page according to half-remembered patterns, with no grounding in any actual mental model. You know, like "canceling the 2s" in the calculation below:

(3+2)/(4 + 2) = 3/4.

Why not, right? If the only thing you are interacting with is the symbols, and you remember that you can cancel 2s from top and bottom, why not do so when it's a plus and not a times? 

Edited December 7, 2020 by Not_a_Number

[wendyroo]

31 minutes ago, Not_a_Number said:

If you do so, you often get what I call "symbol shuffling" -- the rearrangement of symbols on a page according to half-remembered patterns, with no grounding in any actual mental model.

Yeah, we had a base case of symbol shuffling this weekend. Elliot has gone through the whole Hands on Equations word problem book using the manipulatives and now he is going through it again working out the problems symbolically on paper.

He had written an equation like: 18x = 12 - 6x. He kept adding and subtracting the 12 (correctly) from both sides of the equation, but wasn't making an progress because he couldn't see that he needed to add 6x to both sides...or subtract 18x from both sides, but either way, it had to involve an x term.

This a problem he would have no issues with using manipulatives, so I went to grab them to help him "see" what was going on. In the meantime, DH "helpfully" sat with him and told him to "just move the 6x". I about died! NO!! You cannot just "move" numbers around because you don't like where they are. That is a shortcut that masters can use and talk about because they understand all the processes that are packed into that "moving", but it is not helpful to interduce that idea to a beginner. He needs to see that we are not grabbing a 6x from one side of the equation and giving a -6x it to the other (and neither DH or DS even acknowledged that the term was in fact -6x, not positive 6x). We are simply doing the same thing to both sides of the equation and the result happens to look like we "moved" the 6x over and swapped its sign.

These types of incidences give me a lot of empathy for teachers who DO NOT want parents "helping". 

[Not_a_Number]

2 minutes ago, wendyroo said:

These types of incidences give me a lot of empathy for teachers who DO NOT want parents "helping". 

Oh, I absolutely do not want the parents of the kids in my homeschool classes "helping." I have a kiddo in the classes who has basically stopped making conceptual progress since I last saw her in March 😞 . I only saw them once a week, but we were definitely (slowly) moving forward. But now that we're doing Zoom classes, I sometimes see what the parents do to "help," and it's not helpful... just like the interaction you describe. 

Oh, and I had parents who had no clue how to explain "borrowing" except via the standard algorithm, so they taught their daughter that... the problem was that she didn't really get place value or trading in at all yet, so when I asked her to explain what this "borrowing a 1" thing meant, she had no clue. She went back to the poker chips, lol, and did fine with them. 

[daijobu]

4 hours ago, wendyroo said:

He had written an equation like: 18x = 12 - 6x. He kept adding and subtracting the 12 (correctly) from both sides of the equation, but wasn't making an progress because he couldn't see that he needed to add 6x to both sides...or subtract 18x from both sides, but either way, it had to involve an x term.

This a problem he would have no issues with using manipulatives, so I went to grab them to help him "see" what was going on. In the meantime, DH "helpfully" sat with him and told him to "just move the 6x". I about died! NO!! You cannot just "move" numbers around because you don't like where they are. That is a shortcut that masters can use and talk about because they understand all the processes that are packed into that "moving", but it is not helpful to interduce that idea to a beginner. He needs to see that we are not grabbing a 6x from one side of the equation and giving a -6x it to the other (and neither DH or DS even acknowledged that the term was in fact -6x, not positive 6x). We are simply doing the same thing to both sides of the equation and the result happens to look like we "moved" the 6x over and swapped its sign.

These types of incidences give me a lot of empathy for teachers who DO NOT want parents "helping". 

<Rolling up my sleeves and balling up my fists>  Give me a few minutes with your DH and I'll set him straight...

I love that you let your students add and subtract 12, even if it doesn't make progress in solving the problem.  As long as its legal, I always let them go down algebraic rabbit holes so they can see for themselves that it doesn't help them solve for x.  Also, I think this process encourages a sort of problem solving courage.  They can't see whether it will solve the problem or not, so they try it to see if it works.  Some students won't even try something without knowing whether it will work first.  

In that situation, if they tried adding and subtracting 12 a few times and getting stuck, I might say, "I like the approach of adding and subtracting.  I wonder if there is something else you could add or subtract from both sides?"  And then "I wonder if there is something else that you could add or subtract from both sides that will get rid of that pesky -6x?"    

[daijobu]

9 hours ago, drjuliadc said:

Yes, I am nothing but white and I was unprepared for college level calculus.  I don’t think it was my fault. I would have worked hard in high school in any math that was available. I was one of the very best math students in both of the two high schools I attended. 

 

True, true.  I sometimes think white people are strangers in their own country when it comes to math.  I get immigrant families asking me weirdly specific questions about AMC and MathCounts.  In contrast, white parents will ask me "What is MathCounts?  What is AMC?"  

[Shoeless]

7 hours ago, wendyroo said:

I guess the message I take away from this is that addressing the wrong misconceptions can be just as damaging as not addressing any misconceptions. So, it is vital for me as a teacher to probe my student's understanding so I can help them eliminate incorrect mental models and replace them with stronger ones. But watching a video addressing misconceptions that the student doesn't actually share, even if the correct information is then taught, can pollute a correct mental model with incorrect information. We could call it the "You Can Never Go Down the Drain" problem - when Mr. Rogers sang that song, it helped kids who were having that misconception, but it planted that idea in the heads of kids who had never thought of it before.

I ran into this issue a few years ago, while using Wordly Wise of all things! I don't remember the exact passage, but there was a statement in a reading passage to the effect of "Some people think boys are better than girls for xyz reasons", and then some follow up statements how that wasn't true. Kiddo BOMBED the reading comprehension question about the passage, and said "Well, the reading said that boys were better/smarter than girls, so it has to be true".  What?!  This wasn't an issue until this stupid reading passage.  I spent a lot of time undoing this message. (And yes, I did correct it; please don't think I'm raising a mini-sexist!). 

[Not_a_Number]

Just now, MissLemon said:

I ran into this issue a few years ago, while using Wordly Wise of all things! I don't remember the exact passage, but there was a statement in a reading passage to the effect of "Some people think boys are better than girls for xyz reasons", and then some follow up statements how that wasn't true. Kiddo BOMBED the reading comprehension question about the passage, and said "Well, the reading said that boys were better/smarter than girls, so it has to be true".  What?!  This wasn't an issue until this stupid reading passage.  I spent a lot of time undoing this message. (And yes, I did correct it; please don't think I'm raising a mini-sexist!). 

Aaaaaah!! That's a bad misconception to accidentally communicate, lol!! 

[Not_a_Number]

1 hour ago, daijobu said:

I love that you let your students add and subtract 12, even if it doesn't make progress in solving the problem.  As long as its legal, I always let them go down algebraic rabbit holes so they can see for themselves that it doesn't help them solve for x.  Also, I think this process encourages a sort of problem solving courage.  They can't see whether it will solve the problem or not, so they try it to see if it works.  Some students won't even try something without knowing whether it will work first.  

Yeah, I also make sure that DD8 feels empowered to do anything whatsoever, including guess the answer 😉 . Our first few years of equation work were all guess and check, and while she doesn't use it much now, she's very aware that it's a method. (Of course, nowadays, I'd probably congratulate her on finding a solution and then tell her that she'll only get credit if she shows me that's the ONLY solution, at which point she'd probably have to do algebraic manipulations anyway...) 

[Shoeless]

2 minutes ago, Not_a_Number said:

Aaaaaah!! That's a bad misconception to accidentally communicate, lol!! 

Right? We stopped using it after that, lol. I think I must have thrown the book away because it's not with the rest of the saved schoolwork. Yikes.

[daijobu]

40 minutes ago, MissLemon said:

I ran into this issue a few years ago, while using Wordly Wise of all things! I don't remember the exact passage, but there was a statement in a reading passage to the effect of "Some people think boys are better than girls for xyz reasons", and then some follow up statements how that wasn't true. Kiddo BOMBED the reading comprehension question about the passage, and said "Well, the reading said that boys were better/smarter than girls, so it has to be true".  What?!  This wasn't an issue until this stupid reading passage.  I spent a lot of time undoing this message. (And yes, I did correct it; please don't think I'm raising a mini-sexist!). 

There really is something to this.  I ran a MathCounts competition last year and we needed to collect demographic data on our students, so gender and race.  But we were specifically instructed to have the students complete the forms after the competition because even if data collection is ostensibly neutral, just asking will lower scores of minorities and girls.  

Sheesh.  I wish we humans were more resilient, lol.  

[daijobu]

I like Wordly Wise, but shame on them for not responding to recent research. 

[Not_a_Number]

4 minutes ago, daijobu said:

I like Wordly Wise, but shame on them for not responding to recent research. 

Hmmmm, this seems quite mixed, though? 

I wonder whether stereotype thread works for confident girls that KNOW they are good at this stuff? I would guess not, although certainly other psychological things do work. 

[vonfirmath]

On 12/4/2020 at 10:14 PM, JHLWTM said:

When I learned algebra, I was taught that to solve for the variable, you rearrange the terms until you isolate the variable on one side.  To do that, I was taught to move an addition or subtraction term to the other side, switching signs in the process. I was taught that when you move a multiplicand to the other side, you make it division (and vice versa). But I wasn’t told why. No one ever explained it to me as doing the same thing to both sides of the equation. 

 

No one ever showed that if you have x+y=7 and want to isolate the x you substract y from both sides? 

      so x+y-y=7-y

            x+0=7-y

             x=7-y

 

[Not_a_Number]

1 minute ago, vonfirmath said:

No one ever showed that if you have x+y=7 and want to isolate the x you substract y from both sides? 

      so x+y-y=7-y

            x+0=7-y

             x=7-y

What did they teach you, then? 😄 

I have to admit, I don't teach that explicitly. I just say "you can do the same thing to both sides," and "you want to get the x on its own" and I see what happens. (If a kid is them confused, I'll help out. But it's good to figure it out yourself.) 

[vonfirmath]

8 minutes ago, Not_a_Number said:

What did they teach you, then? 😄 

I have to admit, I don't teach that explicitly. I just say "you can do the same thing to both sides," and "you want to get the x on its own" and I see what happens. (If a kid is them confused, I'll help out. But it's good to figure it out yourself.) 

I had to get my son to write it out. He kept wanting to skip steps (Write as little as possible) -- which eventually is fine. But first I needed to see what he was getting vs not getting so yes, we wrote out extra steps to make sure it was clear.

[EKS]

7 minutes ago, Not_a_Number said:

I just say "you can do the same thing to both sides"...

I had a bonus student whose idea of fun was to multiply both sides by zero to make the problems "disappear."  

[Not_a_Number]

1 minute ago, EKS said:

I had a bonus student whose idea of fun was to multiply both sides by zero to make the problems "disappear."  

Hahaha. Well, it's valid, just not very informative! "Now we've convinced ourselves that 0 is 0! What does that tell us about x? Errr... nothing." 

[Not_a_Number]

2 minutes ago, vonfirmath said:

I had to get my son to write it out. He kept wanting to skip steps (Write as little as possible) -- which eventually is fine. But first I needed to see what he was getting vs not getting so yes, we wrote out extra steps to make sure it was clear.

Yeah, that makes sense to me. 

[vonfirmath]

18 hours ago, Not_a_Number said:

Yeah, I also make sure that DD8 feels empowered to do anything whatsoever, including guess the answer 😉 . Our first few years of equation work were all guess and check, and while she doesn't use it much now, she's very aware that it's a method. (Of course, nowadays, I'd probably congratulate her on finding a solution and then tell her that she'll only get credit if she shows me that's the ONLY solution, at which point she'd probably have to do algebraic manipulations anyway...) 

My son is amazingly good at guess and check. He seems to intrinsically be able to figure out the answer just by looking at problems... And the teachers were just great he got the right answer. Now that the problems have gotten difficult enough.... We're having to go back and relearn the steps, etc.

 

[Not_a_Number]

7 minutes ago, vonfirmath said:

My son is amazingly good at guess and check. He seems to intrinsically be able to figure out the answer just by looking at problems... And the teachers were just great he got the right answer. Now that the problems have gotten difficult enough.... We're having to go back and relearn the steps, etc.

It'll serve him well, though. Being good at guess and check means you really understand what you're doing when you solve an equation. No, it's not all you want, but it's useful cognitive scaffolding.