## “I have no idea why they would do this”

In the midst of a very good post, Julie writes:

They also distributed the exponent to both numbers inside the parenthesis.  I have no idea why they would do this, because I don’t even teach them the power to a power exponent rule.  I have them expand any set of parenthesis with an exponent.  They do love the distributive property, but we-e have never, ever, ever, distributed an exponent.  Sigh.  I failed.

There are a lot of wonderful reflective moments in that post, but I find the above fascinating. Kids are distributing exponents inside the parentheses.  You might be tempted to explain this mistake as confusion between the power rule for exponents and the distributive property, but seeing as the kids haven’t yet learned the power rule that explanation seems false, at least for Julie’s kids.

Distributing exponents is tempting for kids. Why?

Julie has a strategy for helping her kids move past these sorts of mistakes:

So, starting Monday, I am going to have one problem of the day for both classes posted on the board.  It will look like a variation of this.  Evaluate  -3x^2 – 2x + 5 when x = -2.  I’ll throw in fractions, decimals, and any other basic, easily forgettable concept.  This should help them quickly practice evaluating algebraic expressions, exponents, and the order of operations, EVERYDAY.  And we will go over it, together, everyday.

Will repetition work? Will she need something else? Here’s hoping that there’s a follow up post from Julie letting us know how her experiment goes. The question for me is, what are the returns on any given genre of problem? Do they diminish with repetition? How different does a problem need to be in order to promote real learning?

## “An excuse to make my students do algebra”

Last year, at any given time, about one-third of my tests were old skills ‘wrapped’ in a new geometry context. .. the assumption was always: These problems are an excuse to make my students do algebra because they still need to learn it and these problems will force them to do so.

Reflection 1: This is fundamentally dishonest – I’m ‘tricking’ my students into learning algebra by making it reappear throughout the whole year.

Reflection 2: This practice kept the cognitive demand of my classroom at a continually low level.

Another great post from Dan Schneider over at MathyMcMatherson. Visit the post for interesting questions about the relationship between teaching and assessment. I think that I would challenge his take on curriculum — he writes that curriculum is “the order that I present mathematical ideas” — but he’s got great thoughts about the ways that assessment decisions can drive daily decision-making. If you’re part of the SBG crowd, you’ll want to check this post out.

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## GCFs and Productive Struggle

Cross Posted from Not an Ordinary Day.

So when Tina (@crstn85) shared the idea of #FailFriday, I never intended on taking her literally. Unfortunately, yesterday was definitely a Fail Friday.

In my Algebra 1 class, we began looking at Greatest Common Factors on Thursday. It seemed straight forward enough and by the end of the block, the majority of the students seemed to understand prime factorization and how to determine the GCF of 2 monomials.

When Friday started, I should have known when I went over the homework from Thursday that they really hadn’t internalized what they had learned. So I spent more time on the homework than I intended (which is a good thing) and then started with factoring by grouping. What I thought was a fairly straightforward process that used the skills they had learned the previous day and we had just painstakingly reviewed, became a situation that I didn’t know how to proceed with other than continuing to try to reframe the material in a different manner.  Nothing I did seemed to work and I became increasingly frustrated.

Thankfully, since I am student teaching, my mentor teacher came to the rescue with a different approach.  Afterwards, when I thanked her for jumping in and saving my bacon, she admitted that the idea had just come to her and that the technique she used she had learned for a completely different context. She was also complementary on the lesson I had taught Thursday saying it was better than she would have taught it.  She reiterated that I would learn these things over time and not to get too frustrated. Easier said than done for me!

The biggest thing I still struggle with is that despite one career behind me, I am a new teacher, plain and simple, and I’m not going to have the perfect, engaging lesson every day. Second, I need to not be so critical of myself and that I need to learn from the mistakes and move on. Most importantly, I love manipulatives and visualizing math (my introduction to algebra tiles was mind-blowing). So, although many of the teachers I am working with currently think manipulatives invite problems in a HS classroom, I need to remember that I can still draw visuals on the board and that when I have my own classroom I can try all the manipulatives I like.

Bottom line, I need to continue to productively struggle as I learn to be the best teacher I can be.

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