The first session I attended at NCTM was "Teaching for Reasoning and Sense Making: How Does it Work?" and Fred Dillon, Jenny Salls, and Christine Thomas were the presenters. I have seen Fred Dillon before and that was part of the reason I chose the session. The other reason I chose it was because the title intrigued me. The pdf of the slides for this session can be found here.

Basically what reasoning and sense making does is puts the mathematical skills into practice. It fits right in with Race to the Top and the Common Core practices. I think I remember Fred saying that this really fits in with the first one on the list.

We proceeded to work through a problem about fish dying off in a lake and the lake being restocked each year and we were to figure out how many fish were in the lake at the start of the 2nd, 3rd, and 10th years and when it reached zero. Fred had taken a "typical" problem from a textbook and modified it to this. The problems that Fred talked about creating (like this one) have multiple entry points and can be done arithmetically and have multiple representations. There were five he listed but I didn't get them all written before the slide went off the screen. Three of them are Algebraic Formula, Graph, and Table.

The biggest problem that we have as teachers is when we tell students we will help them with the first step or tell them how to figure it out step by step. We need to let the students struggle, for we are not really helping them by telling them what to do. (see more on this in my next post on my 2nd session).

As students are working through the problem and asking questions, things to think about include "How do we handle this? Does it matter?" (at the time it was in reference to decimal values). The most important thing to do is try it and see - encourage students to explore and not to worry about mistakes.

Fred also mentioned that other similar type problems to this one could include compound interest, exponential decay, and half-life of medicines. Once students have completed this type of problem once, you can consider these related problems to see how they can apply their reasoning to similar, but differing situations.

Some other thoughts from what Fred had to say include:

Reasoning and sense making should occur every day. I can see this - it is the main part of math (non-content wise) that can help our students do well beyond schooling.

Reasoning and sense making is NOT a list of topics to be covered.

I have a note about "productive ways of thinking that have become customary" but I have no idea why that is there.

Things that should happen as a part of Reasoning and Sense Making:

**Good Questioning Techniques

**Adequate wait time (and we never give enough of it - we think it's 30 seconds and it's only been 10)

**Resist the urge to tell students everything

How can you get started?

**Recast the material as questions. The opening problem was a typical textbook example.

The other two speakers walked through examples with finding x- and y- intercepts, graphing linear equations, and difference of squares (factoring).

Here's where I am at with this sesssion -

Great topic. I needed to hear this. I need help in how to make it happen in my classroom. I picked up 3 of the Reasoning and Sense Making books from NCTM (25% discount yay!) and they move to my reading list.

But I need to learn how to be less helpful. I know we've talked about this in the math blogosphere and I probably need to search through Dan Meyer's blog to find it. Anyone else have suggestions on how to be less helpful and how to do these types of problems?

Here's the other thing - I still don't get how I am supposed to handle covering the content I am supposed to in my class. When does the teaching go on? Or do the kids figure it out from the problems? How does this all work? I guess that's my biggest issue on "rich problems." I think they're great, but how do I incorporate them into class and still get the content covered that I'm supposed to? Do any of my readers have any insight into this or know of someone who has blogged about it? Any and all help is appreciated.

## 3 comments:

As far as wait time, what has really helped me is using my timer. I've started giving them at least 2 minutes to read and think on their own before we even discuss. Sometimes I then have them share with a partner or we discuss as a whole. My instructional coach calls it 'individual think time'. Another thing I do is when I ask a question, I try real hard to accept more than one answer. I will ask for other opinions before saying anything is right or wrong and ask the students themselves to agree or disagree. I hope that by asking more than one person at a time that I am extending my wait time to give more students a chance to think and respond as well as taking the pressure off of only one person getting the right answer.

As far as rich problems, I think a good place to start is with an old concept. Give them a word problem that applies something they already know so that they are struggling with the setup but not the concept overall. An example for me is using word problems in systems of equations. My students knew they had to have variables and two equations so the productive struggling was in setting up the equations. Also, a rich problem could be an old concept related to a new one. Another example for me is in our circles unit, I presented them with a word problem about the endpoints of the diameter of a circle on a coordinate plane and finding the center. They were actually reviewing the midpoint formula, an old concept, but the struggle was applying it in a new setting.

And the most important piece of advice is to just start. Find a great problem and just throw it out there. From there it will be easier to make adjustments and get more creative with how to use them.

Thanks for the comments - the timer idea was mentioned in the session yesterday and I think I am going to have to use that more often. I can certainly use the one on the SMARTBoard.

Loved what else you shared - thanks!

--LMH

Thanks for an idea, you sparked at thought from a angle I hadn’t given thoguht to yet. Now lets see if I can do something with it.

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