Simple Equations - Horizons' Style

Basic Equations with Variables, in Horizons Math (starting 2nd or 3rd Grade)

Horizons Math teaches Equations, and I think they start this in 2nd grade.  Many math curriculum teach equations with variables, but not quite like Horizons!  I'll give an example:

Most Math curriculum will teach something like this:
x + 6 = 11;
The next step, according to other Math programs is to turn the addition problem into a subtraction problem:
11 - 6 = x;
Then 6 is subtracted from 11 to get 5
5 = x

Horizons Math shows the algebraic step missing in other programs.  Here is Horizons' method:
x + 6 = 11;
Next step is to subtract 6 on both sides of the equation.  This cancels the 6 on the left side, and gets the 6 onto the other side.  It also gets the variable alone, which is what we want;
x + 6 = 11
    -6    -6
--------------
Here, the 6s cancel on the left, and you can subtract 6 from 11 on the right.
x       =  5

I would suspect that many kids have the same reaction as my child: why do it that way?  It doesn't make sense!  I can do it the first way intuitively, and I get the answer just fine.  Why subtract from both sides?  Today I showed it with blocks, and I think it made more sense.  See the video above to see how I explained it with the blocks.  The example problem is
n + 5 = 9

Hopefully, this is enough to get the pattern.  After I explained it this way, my daughter was able to apply the whole "subtract on both sides" idea to her math problems.  Thankfully, she automatically made the connection that she'd use the opposite sign when confronted with this kind of problem:
y - 2 = 8
It seemed intuitive to her to add 2 to both sides.  Thank goodness, because I was having a hard time figuring out how I'd show that with blocks!  Looking online, the explanations get into some pretty big concepts: negative numbers and making 0.  It's not that my daughter couldn't get those concepts, I just hate to pile on too many new concepts all at once:)
Want more?  Here's a Khan Academy video that shows the same thing (without the blocks):
http://www.khanacademy.org/video/solving-one-step-equations?playlist=Developmental%20Math

Distractions from learning pt 2: Math Manipulatives

Maria Miller, author of Math Mammoth (a homeschool math curriculum), wrote a blog entry called The Value of Manipulatives.  She explains how Math Manipulatives can actually become a learning distraction. In my last post, Distractions From Learning, I explain that sometimes, attempts to draw students into a lesson can misfire, and actually serve as a distraction.  Mrs. Miller's post is a great example of math manipulatives distracting from the lesson.  Is teaching with math manipulatives a bad teaching strategy?

Math Manipulatives - what do they do?
Manipulatives show numbers visually and concretely.  You cannot hold 2, but you can hold two apples.  Manipulatives present Place Value: a ones place, tens place, hundreds place.  Higher level math continues with place value, into the decimal / fraction places and into thousands, millions, billions.  Manipulatives give the student something to touch and move around (manipulate).  When shifts happen in place value, this can be taught concretely, by showing the real shift in numbers.  Place value blocks are a common concrete representation of abstract concepts. 

When do manipulatives distract?
Many of us have seen the hundreds blocks get stacked into a house, complete with a tens block fence, and ones place people.  Our little students are not thinking about place value when erecting a tall skyscraper out of manipulatives!  When students ignore the lesson and explore the blocks creatively, the manipulatives did not enhance the math lesson: the manipulatives distracted.

Another distraction: when the student does not make the transition from concrete to abstract.  At some point, 2 apples and 2 apples can be understood abstractly as 2 + 2.  This stage of moving from concrete to abstract repeats with each new concept.  Remaining in the concrete phase means that the student still doesn't understand the connection between those 2 physical items and its representation.  A student can fall into this trap even in higher math.  Manipulatives should be a short phase before moving to the next phase.  The natural next step is a visual presentation on the page, and lastly, being able to work the abstract numbers.

Sometimes, using manipulatives with incredibly large numbers serves as a distraction.  In Maria Miller's blog post, she has a video of a child working a math problem using a pictoral representation of the numbers.  The numbers are in the thousands.  Shifting and counting these numbers becomes problematic because the sheer size of the numbers gets in the way.  Working the abstract numbers is faster and more reliable.  In teaching the concept of large numbers, a manipulative or pictoral representation can help show the student "hey, this is what is really happening, behind the scenes"; but hopefully the student will catch on and be able to bring this understanding with her in the next step: traditional algorithms.