Are You Open Middled?

If you’re not open minded about your math instruction, then you’ll probably never be open middled either.

You’ve heard of open ended questions and problems, but have you heard of open middle?

Me neither . . . until now!  I didn’t really get it at first, but I always like to try new things that might benefit my students, so . . .

Back in December I used this “Sum of Fractions” problem from the Open Middle website with my students:

It was surprisingly easy for most of the students.  Only the students who are working below grade level struggled with it much, and even then, with a fraction circle kit they were able to figure it out with out too much difficulty.

Here’s how another student solved it in my afternoon class:

Did you notice what I added to the instructions to try to open it up a little more?

This might seem like a good thing, and in one sense it was.  I was glad to see that so many of my students had developed their understanding of fractions to a level that made this “challenge” not much of one.  However, this meant that the problem was with the problem, and I’d have to do some more work to add some more complexity.

As we shared different ways to solve this problem almost every student came up with two fractions that equaled 1/2.  A couple students in my afternoon class did come up with different ways to make 1 whole without using just halves, so the extra prompt I added did bear some fruit.

So next we tried this:

This provided a little more challenge, but many of the students thought about cutting the half into fourths and finding fractions that would be equivalent.  The extra question prompting the students to examine other possible solutions produced some students thinking about combinations of thirds, sixths, and ninths.

The best way I know how to describe what “open middle” really means is that there is a common entry point for the problem (beginning) and there are actual solutions (ending), but the path or process students use (middle), is open!

Overall it provided some great discussion about fractions and constraints in a problem situation.  It was also good for students who normally know the answer to math problems without having to think much.  They at least had to try out a few different combinations before they found something that worked, and then they got to search out other possible solutions.

I will definitely use the open middle concept again! The challenge for me is to design the problem with just the right amount of challenge, but I guess that is really the point of any good lesson.  In the constant quest to assess our students’ thinking this is a great way to get some insight into what they really know and understand.   I also love how it supplies opportunity for different ways to reach a solution.

Here’s the website if you want to look at some more examples:

http://www.openmiddle.com/

 

 

You Number Talkin’ to Me?

128 divided by 8. I chose this problem because my class was in a unit on division and I had already done a few number talks where I was hoping students would use multiples of 10 times the divisor to help them solve it.  You know, problems like 186 divided by 6 where it’s just begging you to think about 6 x 30.  They were getting more proficient at that way of thinking, so I wanted to use a dividend that wasn’t so close to a multiple of ten of the divisor.

Here’s what my students came up with:

There’s almost always someone who wants to do the standard algorithm, so instead of fight it, I just slap it up on the board as they walk me through it and then move on by saying, “Did anyone think about it another way?”

Every now and then, there is also someone who arrives at the right answer, but their strategy doesn’t make sense mathematically and wouldn’t work if tried with other numbers.  (See the one in red at the bottom) I ask the students what they think and if they have any questions for the one who explained the strategy.  They asked things like: “Why did you round to 100?” and “Why did you add 4 to 96?” Sometimes they are not sure if it would work with other numbers and want to try it out.  I love that they are genuinely thinking about each other’s strategies and trying to follow the line of thinking of a classmate!

I totally dig the two strategies in the middle of the board!  (The ones written in black and green) These students were thinking multiplication and using facts they are comfortable with to build up to 128.  What is written in red on the two middle strategies came out of me asking clarifying questions so the rest of the class could more easily see the thinking that led to the quotient.

I want to give them more experience with these types of problems for a few more days, but then I want to move into two digit divisors and problems with remainders to see what they think up!

I am crazy if I number talk to myself?

Do you ever number talk to yourself?

 

About 400 Words on Estimation

“So then after I multiplied the two numbers, I rounded the answer to the nearest hundred to get the estimate.”

This is a common explanation I’ve heard over the years from students as they wrestle with the purpose of estimation.  One of my follow-up questions has usually been something like, “If you already know the exact amount, why would you need to estimate?”  Usually these students are just trying to get the “right” answer, or they are students who are still afraid to make a mistake so they calculate the exact answer and then work backwards to get the answer they think I’m looking for.

Hopefully this year I can use Estimation 180 to give my students a wider variety of real world estimation opportunities!  Each question on the website is accompanied by a picture (you could easily bring in many of these actual objects to your classroom for students to see and touch) and prompts to think about what estimate would be too high and too low before students settle on their answer.  I also like how many of the challenges build on what they discovered the day before.  It lets students consider prior knowledge and comparison as they try to make an accurate approximation.  The other feature that is convenient and exciting is that there are situations that deal with length, time, weight, volume, money, Legos, bacon, and much more!  Really, this site had me at Legos and bacon.

I’m not sure Estimation 180 will address the aspect of estimating that involves rounding numbers to make the calculations easier to do mentally.  Perhaps as my students and I engage in these questions they will at least become more comfortable with being wrong about an estimate, and working off of past mistakes to try to be more accurate in the future.

When I reflect on when I actually approximate values in my day-to-day, it normally occurs in conversation with someone else.  The exact amount is not necessary, but I’m just trying to communicate about how much time something will take, or about how much something will cost, or about how big or small something is.  I probably need to regularly present my students with scenarios where no paper or pencil is involved, because I feel like that has more relevance to why and when we estimate.  We estimate when we have neither the ability or necessity or time to calculate the exact value in our current situation.

A Brief History of Twitter

I’ve only been on Twitter for about a year, and even though I’m only in my mid-thirties, I’m usually behind the times when it comes to technology and social media trends.  I never had a computer in my house growing up, and because of that, I’m usually fine getting along without the latest phone, tablet, laptop, app, beeper . . . wait, forget that last one.  Originally, I started my Twitter account so I could follow my friends who are storm chasers @tornadotrackers.  After all, I was chasing with them in Kansas and Nebraska and I wanted to see how the amazing footage and pics we were getting were being shared with the world! (check out this video I helped shoot: https://www.youtube.com/watch?v=pMPc1EEyDxY&feature=youtu.be

A part of me still agrees with Texas head coach Charlie Strong when he said that social media will be”the downfall of society.”  I mean we can share our most idiotic, heat-of-the-moment thoughts with thousands, maybe millions, of people without really thinking it through first.  That’s scary to me.  So I haven’t done much tweeting over the past year.

However, this summer I must admit that I have become somewhat of a twitter junkie.  I’ve started following some old friends, as well as my favorite author @ericmetaxas. I love being connected with people I admire. The exchange of ideas and resources that occurs so quickly is continuous.  There’s always something new to read, see, or be informed about on Twitter.

Thankfully, because of the Math Rocks Community I’m also learning how Twitter can be an amazing resource for teaching!  I started following some math Tweeps like @mathminds and @mathcoachcorner and am already starting to get overwhelmed (in a good way!) with ideas and resources.  I’m also looking forward to participating in #elemmathchat and #mathphoto15 at some point.

Lastly, one of the most important things I’ve realized about Twitter is that I’m not just following, but I have the opportunity to be a leader.

Math Adventure Time

One of the things I want most for my students this year is for math to be an adventure!  Adventures are fun and challenging and rewarding and memorable and . . . I guess when I think of an adventure what comes to mind is a journey that has purpose.  It is not always easy, but in the end a feeling of accomplishment and victory is experienced.  That’s what I want for ALL my students each day this year, and over the course of the whole school year.

So . . . how to make this happen?  I need to present them with adventurous math experiences.  I need to help them learn to have a growth mindset, so when it gets challenging the adventure doesn’t screech to a halt.  I like the idea of building in number sense routines like “Which One doesn’t Belong?” and “Always, Sometimes, Never.”  On a very practical note, in order for these things to actually happen regularly I will probably try to find four or five different ones and then do a different one each day of the week.  Like Monday might end up being “WODB” day in my plans.

I also love the purposefulness of Intentional Talk! (hence the title) These are things that I have been trying to do, with varying levels of success, for years in my math discussion times.  I just didn’t have names for different types of discussions and didn’t really plan for different types of discussions. They just kind of happened when the subject matter or students dictated it.  I also want to learn more about “talking moves.”  Again, something I’ve tried to kind of make up and teach my students when having a discussion, but it sounds like someone has refined the details and tested the procedures for us, so I am excited about trying it out.  The more intentional I am able to be with planning our math discussions, the more my students will get out of the experience.

I think pacing of math units will be one of my biggest obstacles in this adventure.  I might feel the pressure to take a short cut to finish “on time” instead of really letting students wade through the waters of their own thinking and understanding.  I’m hoping that in the Math Rocks community we can continue to encourage each other to persevere, knowing that the reward for our students will be well worth the struggle we endure in the midst of the adventure.

Longhorns, Legos, Littles

I’m a huge Texas Longhorn fan.  Especially football, but that’s probably redundant.

Sometimes I wish I was a kid again, just so I could spend all my time building with Legos.

I’m a Dad.  Of small children. Someday I’ll be a Dad of large children. Lord willing.

Oh yeah, and I teach math.