There are 8 Standards for Mathematical Practices in the CCSSM. They are the same for Kindergarteners as they are for 12th graders, as they are for adults and mathematicians. I love the very idea that, in theory, being mathy in kindergarten is essentially the same as being mathy in grad school.
One of the exercises we do is to make sense of the Math Practices in "Kid Friendly" language. It makes sense, right? The first few words of every practice is, "Mathematically proficient students..." so doesn't it make sense that they need to understand the practices? They are the ones who have to do the behavior. They need to own it. Unfortunately, the kid friendly language for this one can sometimes come across as cheerleading...You can do it! Never give up! Always try your hardest!
I mean, yes, these are necessary attributes of the practice. But they aren't the
mathematical attributes of the practice. If you look at the details of the practice, we begin to see things like, "They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt." Let's be honest...there isn't a teacher alive on the planet who hasn't watched a kid try to add 4+112+2 to solve this problem: "A family of 4 drives 112 miles on 2 tanks of gas. About how many miles did they drive on each tank of gas?"
I just finished a week long summer institute, and it never fails that teachers inspire me to want to be a better teacher. So I started thinking about the things that I do to promote this practice in my students. Let's
examine three "get unstuck" strategies from the practice. Along the
way, we'll put them in a more kid-friendly language, think about the
classroom
experience, and think about out what we can say to students to help push their
thinking
and behaviors around this practice.
From the practice: consider analogous problems
Kid Friendly Language: "Can I think of another problem like this one?"
From the classroom: We want students to categorize problem types as they make sense of them. So once they know that "I have 8 stickers and my friend has 7 stickers, how many stickers do we have altogether?" is a problem where we are putting things together, they can begin to think about all the other problems they've done where we put things together, and they can think about and use some of the strategies that they used to solve
those problems before.
What I say: Start open ended... "Can you think of another problem that we did that is like this one?" Add an optional more focusing question... "Is this problem like the puppy problem? Or is it like the pumpkin problem?" Or how about a downright leading question... "Is this problem like the puppy problem where we joined them
together? Or is it like the pumpkin problem, where we took them away?"
From the practice: They monitor and evaluate their progress and change course if necessary.
Kid Friendly Language: "Can I act it out? Can I try smaller numbers? Can I make a story? Can I draw a picture?"
From the classroom: Part of persevering is having strategies for getting "unstuck". There is certainly an art to abandoning a current strategy if it isn't working, and knowing how to start on a new, possibly more fruitful, strategy. Kids don't often do this gracefully. They
do, however, lick paper, roll on the floor, throw cubes at each other, take out a book, doodle, etc. So when we see students get off track this way in math class, it's a good bet that I'm looking at somebody who is stuck.
What I say: Always start with a question..."Can you show me how you've tried to solve the problem?" My next questions are usually based on what they reveal, but I'll usually refer them to the list of "Can I..." statements to see if there isn't a new way to think about the problem. If a student is just working through pencil/paper strategies and nowhere near a correct interpretation of the problem or accurate answer, it's not unusual for me to give them an answer. "So, if I told you that the answer is 15, could you show me with your blocks why that works?" or "So, if I told you that the answer is 15, could you draw a picture to explain how the problem works?"
From the practice: Mathematically proficient students check their answers to problems using
a different method, and they continually ask themselves, “Does this
make sense?”
Kid Friendly Language: "Do I get the same answer if I try it in another way? Can I convince a friend?"
From the classroom: You always have quick finishers, and students tend to over rely on a few strategies, whether they are efficient/accurate...or not. Suggesting that they try another strategy, or hooking them up with a partner who got the same answer to compare strategies, can be a useful differentiation technique.
I've also hooked up kids with very different answers to try to convince each other that each is right. Good times, good times.
What I say: "Andrea, Isaac just got a very different answer than you...can you two get together and see if you can understand what each other did, and come to an agreement on what's happening in this problem?" And when students ask me if they got it right I say, "Well, you've convinced me, but you also need to convince somebody else."
I made this little poster to use with my students. You can get a copy of it by download it for free by clicking
Math Practice 1 - FREE (PDF). Telling students "never give up!" and "keep trying even if it's hard!" does little if we don't also help them develop the tools that get them unstuck.
One final note: Never underestimate the power of the "take a break strategy".
George Polya said, "A problem isn't a problem, if it can be solved in 24 hours." The end goal of any one problem solving experience is not necessarily "an answer". It's to develop strategies and mathematical understandings. That's not completed in a 50 minute math lesson. So go home, sleep on it, play soccer, do something else. Fresh eyes are the problem solver's strongest tool at times.
.JPG)
