Getting Kids to Organize and Record Their Math Thinking

Even though I teach gifted kids, I run into trouble getting kids to show their thinking as they work some of the demanding math tasks I set them. Notice I didn't say showing their work. I encourage mental math, so I don't need kids to write out their arithmetic, unless they're practicing a brand-new skill. Besides, that's not what these tasks are for.

It's not always easy to find problems that require these kids to stop and organize their thinking. I use problems from the Math Olympiad Contest Problems (Find it here.) to provide enough challenge to make my students stop and think about strategy. Monday through Thursday, one of the math centers is a challenge problem. The next day, we engage in a whole class Math Talk to look at answers and strategies. They drive my fourth graders (who are in the first year of the gifted program) nuts, particularly, because they're so focused on getting done and being first that accuracy can get kicked to the curb. It takes a while to learn that doing it twice is frustrating and relatively uncool and that taking time to organize their thinking before engaging in endless (and often inaccurate) guess-and-checking saves a bunch of time and trouble in the long run.  

Having gone through this process, my fifth graders have learned that stopping to look for patterns, math rules, and strategies is the way to go. They think the problems are fun and do a really good job with them. 

Students for both classes turn in a sheet on Friday with one of the week's problems documented. They are required to provide a visual representation along with the steps of their thinking and proof that they're correct. It looks like this:

Yep. It's pretty simple. My fifth graders are also pretty adept at choosing a visual representation that helps them organize their thinking. This is the anchor that's in the room:

So, in all, teaching kids to organize and record their math thinking takes:
1. Challenging problems. If the problems are too easy, the students have no real reason to bother with other representations and no authentic need to document their thinking.
2. Time. Time to work and wrestle with a challenging problem, time to discuss their ideas, and time to bump their butts and learn the hard way.
3. Opportunities to talk with the greater group about what they did and how they did it, which also gives them chances to see others' strategies.
4. Time. (See number 2.)
5. Knowing that at least one of the problems will be graded, but having some choice about which that is.
6. Consistency and, oh yeah, time.

I hope you'll find these tips useful! Don't forget to pin the anchor chart so you can make yourself one!

How Flipping my Classroom Rescued My Math Block

I'm tickled pink to be guest blogging this week over at Education to the Core! I've written about why and how I've flipped my math instruction, and the big-tme benefits of having done so.Curious? Click  the image to hop on over!

Are you arriving from Education to the Core? Welcome! I'm glad you're here! Click my Bloglovin' button at the top of the page to catch all of my freebies, teaching tips, and occasional shenanigans.

Growing Community, Critical Thinking, and Mental Muscle with Math Talks

One of my favorite routines in my classroom is Math Talks. For about 10 minutes each day, my kids discuss, debate, and confer, respectfully and focused on the topic at hand. They run them almost entirely on their own and when I listen to them, I feel like I might be doing a pretty good job, after all. 

What are math talks?
 In my classroom, a student, chosen randomly, places his or her solution to a complex word problem on the camera and, in steps, explains the process for solving it. Then, other students discuss the solution, adding their thinking, following some basic sentence starters that keep the conversation objective and on-task. 

How do you train your students?
At the beginning of the year, I lead as little as possible, but more than I will for long. We begin with an anchor chart:

 and I'll begin by modeling a talk. Something like this:
Jake has 163 Pokemon cards. Seth has 46 fewer cards than Jake, and Isiah has 84 more cards than Seth. How many more cards does Isiah have than Jake?

The presentation would sound something like this:
I found the numbers 163, which represents how many cards Jake has, 46, which represents how many fewer cards Seth has than Jake, and 84, which represents how many more cards Isiah has than Seth. My job is to find the difference between Isiah's cards and Jake's.

The word fewer tells me that there's a difference between how many cards Jake has and how many Seth has. Seth has 46 less than Jake. I'll subtract 46 from 163.

The words 'more' and 'than' tell me that there's another difference, this time between how many cards Seth has and how many cards Isiah has. Once I have Seth's number, I'll add 84 to it to find out how many cards Isiah has.

There's another 'more than' in the last part, which tells me that there's a third difference, but this one's between Isiah and Jake. I know how many Jake has - it's 163 - and once I find out how many Isiah has, I'll find that difference.

So, If Seth has 46 cards fewer than Jake's 163, he has 117, because 163-46 = 117 (and I'll show my arithmetic as I go). If Seth has 117, then Isiah has 201 cards because 117+84=201. Finally, to find how many more Isiah has than Jake, I'll subtract 163 from 201, which gives me 38 cards, my final answer.

Because the problem didn't give us a total for all three boys, it's tough to prove my answer, but I used a bar representation here and did the numbers again to show that my process and calculations are right. 

From here, the students may comment on my work. For the first several weeks, maybe even until the first report card, they are only allowed to begin their comments with "I agree with ________ because..." or "I disagree with ____________ because..." The because is imperative. It forces kids to reason out their responses. Until they're good at talking - and they may respond to each other's comments, but they still have to use the two available sentence starters - this is all they get.

Over time, once the routine and the language are down, I can step away from it. Sometimes, I'll need to step in to get some different voices involved, but that's about all.

How does this build community?
Respectful-but-lively discussion is good for everyone. Presenters are sometimes wrong, and that's where the "I disagree because..." comes into play. Without prompting, most kids are kind in their disagreements. I usually get, "I agree with _______ 's process, but disagree with her calculations because 7x8 is 56, not 54." If, in your classroom community, risk-taking and mistakes are a natural part of learning, then these talks are an extension of that.

How does this build math muscle?
 Every kid knows that he or she might be called up to present on any day, so there's that factor. I also structure my question sheets to make the presenting easier, with each step
in its own space, which encourages all kids to get through them. It's a little less intimidating if it's all laid out. 

I use 'CUBES' as a strategy for approaching story problems. The acronym stands for:
C - Circle the numbers. What does each represent?
U - Underline the actual question. In your own words, what are they asking?
B - Box the action words.
E - Evaluate. How many problems are in the problem? What operation(s) do you need?
S - Slash the trash (get rid of distractions) and Solve. How do you know that your process
      and calculations are correct? Did you get all of the possible combinations?
 Giving them this feels a little 'drill and kill', but it also gives my lower students a life ring when they feel like they're drowning. It also translates well to testing situations. 

The discussion, though, is the important part. Students are always encouraged to have more than one representation, which requires more creative thinking. It's not unusual, at the end of a talk, for a child to say, "I got the same answer, but did it totally differently. Can I share mine?" Whenever we can, the answer is 'Yes.' Seeing multiple ways of approaching a problem gives kids a variety of strategies, and, again, encourages risk-taking, which, in turn, values divergent thinking. Good for problem-solvers.  

Where can you find problems that work for this?
 What you'll have to decide first is if you want to do this whole- or small-group. When my class is largely at the same level, with a few on the fringes I do this whole-group. If, however, the kids are all over the place, we'll go small. If your kids are like mine, they'll be all-about-the-same on some standards and very different on others. It's OK to vary the routine. If you're doing this small-group, you'll have to simplify the steps of some questions and algebra-fy others to increase the challenge. There are a couple of really good resources available for this. I loved Minds on Mathematics and Good Questions: Great Ways to Differentiate Math Instruction

I invent a lot of my questions, but I also model some from quiz questions that kicked their butts. On Amazon, searching 'Challenging Math Problems' will get some hits, largely
based on or around Singapore Math. (Don't let the Singapore thing intimidate you. Most of the questions don't really require the bars.)

Or, you can head over to my TpT store, where I've whipped a few into shape. 4th grade whole number operations is ready. I'll be adding more to the collection as the year progresses:

Clicking the cover will take you there!

Thanks for stopping by. I hope you'll find that adding a Math Talks routine to your math instruction has the same kind of positive effects it had on mine!