I’ve posted before about my efforts to correct student’s conceptual mistakes regarding integer operations. Jason Dyer dropped by and made some great comments about a student’s inability to visualize, which really helped me pinpoint ways that I could improve a student’s conceptual understanding without going back to procedural reteaching – taking the time to make students imagine the number line has made a measurable difference in my students comfort with integers and acceptance of this thing called the ‘number line’.

So, that’s one hurdle I’m starting to get a handle on. Next is: algebra. In particular, the concept that my students do not understand at all is equality. What does it mean when we put two quantities on opposite sides of an equal sign? Why can I combine things regularly when they’re on the same side, but I need to do the opposite thing when they’re on opposite sides? What do you mean by ‘same side’ and ‘opposite side’? What do you mean ‘get the x by itself’, or ‘move this to the other side’? I have students who will make algebra mistakes and end up with expressions like “15 = 7″ and not realize how fundamentally incorrect this is. Or, students who will lose the equal sign in the middle of an algebra procedure, not realizing that the equal sign is the most important part of the equation (the second-most important thing being the variable, but this whole sentence is just my opinion). Again, the rules and procedures they’ve memorized are so jumbled up that they’ve lost sight of what they’re really trying to accomplish.

So… how do I fix this one? I’ve mostly been hammering in rules and correct procedures because I’ve been struggling to come up with a conceptual foundation that I can build from. With integer operations, there’s the number line, which is (1) intuitive, (2) somewhat concrete, and (3) easy to manipulate. With algebra, there’s… I don’t know. Algebra tiles is the only concrete thing that comes to mind, but (1) I don’t have any (although this is a cheap excuse because I could get some if I really wanted to), and (2) I’m worried about how my students will respond to going back to something as simple as algebra tiles (I can just imagine “We’re in high school – not preschool” and them shutting down). I’m also not sure how to teach with algebra tiles and have that transition into the symbol-manipulation mistakes they keep making. I’ve tried using money and concrete word problems to help (“I go to a casino and double my money, then tip the dealer 10 dollars. I leave the casino with \$200. How much money did I start with?”), but that also doesn’t help with the procedural mistakes they keep making. Posing those word problems just reinforces the ‘I’ll just keep guessing numbers until I get the right one’ mentality (which isn’t necessarily bad, it’s just not the thing I need to ‘fix’).

Anyway – I’ve been thinking about Jason Dyer’s ideas on Visual Algebra – in particular, the representation of an equation as a bunch of boxes that you’re trying to match up. I tried this with some students, using dots to represent the equation “3 + x + 7 = 12″. Below is (essentially) what I had them do.

Start by representing the numbers as dots with the x as an unknown quantity of dots. Then pattern-match the dots on one side with the dots on the other – I had them do this by drawing boxes around each of the dots so they matched up. Then, whatever dots are remaining, that’s your x. I like this because, if my students buy into it, it makes a connection between algebra and something tangible like dots or boxes. They also immediately began to understand phrases like ‘same side’ and ‘opposite side’, and saw the merit in combining like terms before ‘moving’ across the equal sign. I also like that it uses the intuition of matching a pattern, which is something visual and (in my experience) natural. However, for this example at least, it doesn’t scale so well when you have a coefficient in front of the x; ie: 2x + 3 = 13.

Jason’s method of solving this type of problem involves imagining the “2x” as an unknown number of rows with 2 squares in each column – which I think is brilliant. He’s essentially tapping into the idea that multiplying a times b is the rectangle you get when you have a units on bottom and b units on the side (I probably like this even more because I teach geometry, so this representation has multiple connections).  He then goes through the motions to demonstrate the connections between the pattern matching aspect of the visual and the algebraic manipulations – every time you match the two sides of the equation, you can get rid of it. What follows is mostly ripped off from Jason, but I made a change to how I end the problem (his version of solving a similar problem is here):

In step 1, we represent the problem pretty much exactly how Jason has it set up. However, notice that I’ve labeled the rectangle such that the x is on the side (representing our unknown number of rows) and the 2 is on top (representing our known number of columns).
In step 2, we match up the constant dots on both sides
In step 3, we rearrange the remaining dots on the right side into a rectangle so they begin to match the endless rectangle on the left side. In this case, we want 2 columns and it just so happens we end up with 5 rows
In step 4, we pattern match again. I need 5 rows on the left so it will match my 5 rows on the right.  This means that my number of rows (aka: x) must be 5. Therefore, x = 5.

Anyway – here’s the whole point of this post. I’m trying to find the right ‘thing’ to complete this analogy: Integer Operations is to the Number Line as Algebra is to _______________________. (In writing this analogy, I wonder if ‘Algebra’ is too big of a word. Maybe ‘solving linear equations’? Or ‘symbol manipulation’?). Right now, the only thing I’ve got is this idea of Visual Algebra, which I like but am curious if anyone else has any other ideas.

UPDATE 11/10/11: The comments are worth reading – got some advice from some fellow teachers, than added a long reflection on how I’ve continued to try and teach this. Something I’ve realized that came out in my comment: “I like the number line because when I ask a student ‘why?’, they can justify it with the number line. I guess I’m looking for something similar to that with algebra – if I asked a student ‘why?’, what would they point to and say ‘because of what I did here’. I think this is why I was first drawn to the visual algebra – it gives them a way to justify their operations in a way that isn’t completely procedural”. Read more in the comments. Also, check out Tina C’s very timely and similar post about finding the right way to get students to ‘get’ variables: http://crstn85.blogspot.com/2011/11/variety-of-variables.html

UPDATE 11/13/11: One thing I’m discovering and enjoying about the math blogging community is that several teachers all tend to have the same problems all at once. Jason, Tina, and I are all having problems with fixing misconceptions – and now, Bowman Dickson is having algebra misconception problems too. I wonder if the strategies he uses to tackle to problem will be different from mine, given that he’s teaching calculus to seniors while I’m teaching geometry to sophomores. I don’t really have any answers for him, but again – I like that several teachers across several levels all felt the need to post about the same type of problem around the same time.

From → Curriculum, Math

1. Algebra is definitely too big of a word here. I think symbol manipulation is too. If their problem is with the meaning of “=” and solving linear equations, then focus on that.

The typical analogy for solving linear equations in one variable is the balance beam, isn’t it? That’s why with x + 3 = 12, you can take 3 off of both sides (a.k.a. subtract) and maintain the balance. Or if you have 2x = 10, you can take half of each side off. (I’ve mostly seen algebra tiles used for factoring– I’m not sure that they make the meaning of “=” really clear.) Balance beams may be even harder to come by than algebra tiles, but if they’re just not getting equality, it would probably help. And they transition well into Jason’s Visual Algebra.

In any case, I would avoid saying things like “move the seven to the other side” when you mean “subtract seven from both sides.” Have a conversation about why some people say “move -blah- to the other side” and what they really mean. Emphasize the difference between talking loosely, and speaking precisely. It’s not that “move it to the other side” is wrong, it’s just informal, and can be confusing if your listener doesn’t know exactly what’s going on. You should do your best to model precise language. I always tell my students they’re allowed to speak loosely, but only if they can demonstrate that they can also speak precisely when it’s called for. But I make myself speak precisely.

2. The “official definition” of algebra is generally along the lines of “Algebra is a generalization of arithmetic.”

What I’m doing in my book (actually my one for adults which I have yet to post about but is easier to write and doesn’t require interactive stuff) is spending the first quarter simply on arithmetic, playing around with the absolutely concrete stuff (and using small, simple numbers) before moving on to the alegbra. I am not worrying about being insulting here because even something as simple as counting can have deep things to say about it.

I would say the puzzles I have which involve the tree structure directly are the strongest fit for your analogy. I have yet to even start adapting those into a full curriculum yet, though; they might just make things more rather than less confusing.

All I understood was “I am an awesome teacher.”

4. It’s been a few days since I posted this initially, which means I’ve had a few days to try and teach this with all of your comments in mind. Here’s what I’ve learned:

(1) Kate made me realize that my vocabulary, by default, is not formal enough for what I want my students to do. She’s right – saying things like ‘add to both sides’ is much better than ‘move ___ to the other side’ – unfortunately, I’m too used to speaking informally about algebra, which may be why my students are having a harder time understanding it when I try to tutor them. I’ve been trying to make a conscious effort to keep my vocabulary as systematic and consistent as possible.

(2) I tried using the balance-beam analogy for a few students and it’s been working wonderfully. I drew a picture and everything. I compared it to hanging scales that they see in super-markets. It also helps me with the vocabulary issue mentioned above – with the scale analogy, it’s hard not to say ‘add to both sides’ or ‘subtract from both sides’. The only thing that sort of breaks the analogy is when you have 0 or negative numbers on one side – ‘how can I take away 4 from both sides if one side doesn’t have any?’. I had to blur the lines between the concrete analogy and the algebraic manipulation. Other than that, this will probably become by starting point with reteaching algebra.

(3) My students _still_ do not want to combine like terms when they’re on the same side of the equal sign. I let a student evaluate ’11 – 6 = x’ by adding 6 to both sides, getting ’11 = x + 6′, then subtracting 6 from both sides, getting ‘5 = x’. I think the scales will help with this, or the visual algebra.

(4) I tried the visual algebra trick with a few students. One thing I like about it is it gives students a place to start playing around, just like you can on the number line. If you’re stuck with an integer operation problem, you draw the number line, then follow the few rules you’ve memorized. I watched one student get stuck with an algebra problem, draw it out with blocks, then start pattern-matching and moving forward with it. I’m still not sure if this is better/worse than the scales, but I think it tackles a different kind of problem – giving students a place to start from and experiment with an algebra problem.

Writing this helped me find the words to express something else: I like the number line because when I ask a student ‘why?’, they can justify it with the number line. I guess I’m looking for something similar to that with algebra – if I asked a student ‘why?’, what would they point to and say ‘because of what I did here’. I think this is why I was first drawn to the visual algebra – it gives them a way to justify their operations in a way that isn’t completely procedural.

(5) Jason’s definition of algebra, along with this very timely post from Tina C (http://crstn85.blogspot.com/2011/11/variety-of-variables.html), made me realize that there are actually a few conceptual blocks I’m trying to fix. Algebra is definitely too big of a word. There’s the symbolic manipulation problem, which I think the scales/visual algebra helps with. There’s the notion of equality, which I think the scales definitely help with. But then there’s also this conceptual issue with the very nature of a variable and the whole ‘goal’ of an algebra problem. I think this is connected to the mistake I saw in (3) – if they knew that the goal was to get x = ___, then they should have known that adding something to both sides is the _opposite_ of what they want to do. I’m not sure how to tackle this conceptually. Reading Tina’s post and Jason’s definition of Algebra, I think the key is some sort of inquisitive ‘aha’ moment when they realize that variables are a tool that let them generalize patterns they see in arithmetic. So maybe I need to find a way to reteach that as well. If that makes sense.

(6) Jason – I’m not gonna lie – I don’t fully understand your tree puzzles, even with the explanation. I guess I don’t know how to read what the solid and dotted lines mean. Am I supposed to work backwards somehow? I think I’m missing something – it might be one of those things that, if it was explained to me in person, I’d understand. Sorry =/.

(7) No Jen – _you’re_ an awesome teacher.

5. I’m so glad you linked to my blog because then I backtracked to yours and got to read this excellent post! I’m excited to try the blocks and dots method. These are big questions that don’t have easy answers, but if we keep working at it somehow the students are sure to learn something!