Friday I finally felt a normal amount of behind, so I started going through my notes and writing them up. Portland State is very generous and they are offering two grad credits to anyone who attended all three days of the conference and writes about it. There were some specific prompts to write to and I learned that my writing voice for papers is different from my blogging voice. At least it feels different. NWMC was great but not so life changing that I sound different, apparently I just write differently when typing into Word than when I compose in Blogger.

On Thursday I attended two workshops:

**What makes algebra hard to learn?**

**Steve Rhine**

This session was full of resources and links to even more
resources! I look forward to thoroughly exploring the AlgebraicThinking.org
website, especially their database of problems used in research. Upon return to
my classroom I was pleased to discover some of the apps loaded on our school
iPads are from Steve Rhine. Hearing him discuss the intentionality behind these
programs makes me much more likely to use them with my students. One in
particular is Point Plotter. I never would have understood the goal of this app
if I just tried it – a common misconception students hold is that there are a
limited number of points between two points on a line (ex: only lattice points
count) – this app pushes students to find as many points as they can between
two points. This addresses a misconception while simultaneously encouraging
students to use the equation of the line and/or definition of slope to
calculate points.

I am teaching Algebra 1 for the first time in five years and
it’s been an eye (re)opening experience. I had forgotten how challenging it is
to explain the basics; I wish I’d attended this session (as well as many
others) during the summer so I wouldn’t have made some of the mistakes I
already have with my students. Luckily it is still early in the year so I have
time to address their misconceptions. One of those is the difference between an
expression and an equation. As a mathematically proficient adult, it is quite
obvious to me that equations and expressions are very different objects, but
when I asked my students about it this week they struggled to differentiate between
the two. We have all seen the mistakes where students try to ‘solve’ an
expression using inverse operations – Steve Rhine shared that this can be due
to students feeling “lack of closure” when their answer is an expression rather
than a number as it has been for the rest of their mathematical experience.

Another place where students struggle is in understanding
variables. The idea that a variable changes with different contexts but is
constant within a context isn’t one I’d wrestled with, let alone helped
students wrestle with. The x in problem three has no relation to the x in
problem four, however, if problem five is 4x + 4 = 5x students must realize
that the two x’s in that equation are the same. As a group we decided that a
variable is an unknown

**quantity**because that definition encompasses x=5 as well as x=y+3. Steve emphasized the importance of a variable representing a quantity and not an object. H can represent the height of Harry, but H doesn’t equal Harry. Instead we should refer to variables as containers. It worked out nicely that I had started equations with this pennies lesson because now I can refer to variables as cups of pennies – a mystery quantity.
Our final topic of the morning was graphing. Students
struggle to understand the difference between discrete and continuous graphs.
One suggestion was to ask students who connect discrete data points what the
midpoint represents. For example, if they connect Sally’s height to Ted’s
height, is the midpoint meaningful on that graph? Lastly, lines having constant
slope is a big idea, but one that students don’t often wrestle with. If we only
give students tables where the patterns jump out at them (ex: x values always
increase by one) then they don’t get the opportunity to engage with the
concept. We need to make sure to give students that challenge them to think and that bring out misconceptions so we can address them!

**Fostering Algebraic Thinking and the CC Math Practices**

**Irving Lubliner**

I thought the last session was full of resources, and then I
got to this one – we received a bound notebook of activities! I have yet to go
through the entire thing and I can’t wait to have a chance to do so. Irving
Lubliner mixed teaching practices with content throughout the workshop. He made
his teaching moves explicit so we could reflect on those as much as the
activities. When someone gives a wrong answer he finds the question that their
response correctly answers, in other words, he finds something right about
their solution. We practiced for a few responses and it was a fun challenge
that reminded me of My Favorite
No, an activity I’ve used weekly. It would be great to infuse that spirit
throughout the course. He used tickets as rewards for participation and great
ideas. I am generally hesitant to give students extrinsic rewards but my school
is using the PBIS model so I need to consider it. I appreciate that he rewarded
bravery rather than only correct answers.

Just like my morning session, we spent time talking about
expressions vs. equations. My notes say *Spend time on this!* so I had better
do just that. One example in this session was about language precision – you can’t
double an equation, you double each expression. When evaluating expressions he
gave a great tip for getting students to remember to use the order of operations –
look at the expression in chunks. This process will help students think about
terms which is helpful for expressions with variables as well. Underline the
expression until you see a + or -, then write “later.” Repeat until you reach
the end.

__200*2__+__(3+4)*5__–__3*6__. Next evaluate each term (in whatever order makes you happy). Then rewrite each subtraction as adding the opposite. Finally, evaluate the expression (in whatever order makes you happy). This method pushes students to think about the structure of the expression, and also allows them to use their number sense – if the expression has 15 and -15 use those opposites instead of going from left to right.
We played with a great model for solving equations. I’ve
never seen the utility of a function machine model until this one:

Earlier this year I tried to use a representation where we evaluated at a
specific value of x to show the ‘forward’ steps, then worked back up to determine
the ‘backward’ steps. This diagram is much more intuitive and shows the ‘socks
and shoes principle’ clearly. During the session he showed an example with 7 or
8 steps, I wish that example had made it into the book – it made a very
complicated equation seem easy. I will have to search for or recreate it since
my students still need more practice solving equations.

Update 10/20/2014: The internet is awesome. Someone from NWMC saw this post and sent it to Irv and he responded with the image I was wishing for!

Update 10/20/2014: The internet is awesome. Someone from NWMC saw this post and sent it to Irv and he responded with the image I was wishing for!

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