Getting Started with CGI, by Tanya Blais

My main math goal at the beginning of the school is to build my students’ perceptions of themselves as mathematicians who are capable of solving problems using their own strategies. 

To build students’ perceptions as of themselves as mathematicians, I catch them in the act of doing mathematics.  Beginning in the first week of school, I pose a daily word problem to students.  When I pose the problem, my only purpose is to support comprehension of the word problem.  No strategies are suggested or taught, even if the word problem involves mathematics that students have not yet learned.  Students are then asked to try to solve the problem in their math journals or with manipulatives. While they are working on the word problem, I get down at eye level with them and watch what they do.  My nonverbal communication is important here.  A kind, curious face respectful of their attempts speaks volumes at the beginning of the year.  They know that I am interested in what they are doing and start to believe that their ideas matter. Other times, I ask them to tell me or show me again what they did.  I can not emphasize enough how critical it is to be genuinely curious about how students are thinking and really listen to them.  Initially, I hold back from many of the other questions I will ask later in the year to extend their thinking (e.g. “What would your thinking look like with an equation?”) or encourage efficiency (eg. “Is there a strategy you could use that would be more efficient?”).  Although these questions can be highly effective, I don’t want the students to glean from my questions that I am judging their thinking or that I have preferred strategies that I want them to use.  Kids are smart.  Early on, I think they try to feel me out and see if I REALLY mean it when I say that I want them to do their own thinking.  

I continue to build students’ perceptions of themselves as mathematicians when students are sharing their strategies with me, a partner, or each other in a class discussion.  At the beginning of the year, it’s not unusual for students to believe that they are good at math if they get correct answers to math problems.  During our first several class discussions of student strategies, often only half of the students have arrived at an answer and sometimes fewer than half of those answers are accurate!  I’m completely fine with this because incomplete solutions and wrong answers gives me an opportunity to model that mathematicians work on challenging problems and persevere to figure them out.  In the first few weeks of the school year I deemphasize “the answer” so that I can emphasize the importance of the strategy. After students have an opportunity to try to solve a problem, I have a few students share their strategy with the class by recreating their strategy on the dry erase board with markers or my magnetic manipulatives.  Showing is much easier than explaining.  Even students who don’t have the words to explain their thinking, either because of limited experiences or because they are Emerging Bilingual, are able to show what they did to their classmates. My more verbal students can often add explanations to other students’ strategies.  Other times, I model the explanation to match what the students did.  Over time, the students who initially struggle to explain their strategies move from pointing to their strategy and stating what tool they used, to truly giving an explanation that details their thinking or the thinking of another student. The process of sharing strategies with each other allows me to catch students in the act of being mathematicians who are focused on the strategy rather than the answer.

Another critical goal I have in my classroom at the beginning of the school year is to ensure that students are listening to each other during the discussions of student strategies.  I’m explicit with students about the importance of listening to other students.  I tell students if they didn’t have a way to think through the problem, this is their opportunity to learn a strategy or if they already have a strategy for this problem, this is an opportunity to learn a new strategy.  One of the most beneficial things I do that reinforces the importance of listening is to have students immediately use what they learned from their classmates to solve a follow up problem.  After the class discussion I will give students the same word problem with new numbers and encourage them to try a strategy that they just learned from another student.  I love this because it is often at this point in the lesson where students experience success at the same time as they learn the importance of listening to each other. Encouraging students to use each other’s strategies is really different than supporting students to use strategies that I demonstrate because students come to see that if their classmates can generate strategies they can as well.

I have a favorite variation of posing a follow up problem that I’ve named Manipulative Rotations.  The students are split into several small groups.  Each small group is given a different type of manipulative.  Some of the manipulatives that I’ve used are base ten blocks, linking cubes, and hundreds charts.  I’ve also provided dry erase boards as the “manipulative”.  Lastly, one of the “manipulatives” isn’t a manipulative at all, but is simply use your brain only (and I guess you always have your fingers!).   For the first rotation, I pose a word problem or equation to the students and ask them to solve the problem with the manipulative at their small group table.  I usually allow students to select the tool that makes sense to them to solve a problem but sometimes I find that supporting students to use different manipulatives can lead to greater flexibility. A few students share how they solved the problem using the manipulative at their table with the class.  Then the students rotate to the next small group table with a different manipulative.  I pose the same word problem or equation, but with different numbers.  The sharing of strategies and rotations are repeated until students have been through all of the rotations.  With this Manipulative Rotation activity, I find many benefits.  Students build confidence in themselves as mathematicians with having the same problem repeated multiple times.  Some students who struggle to understand the problem in the first few rotations begin to understand it and develop a strategy.  Students are also highly curious and listening as students explain their thinking, because they know that they will be rotating to that manipulative later on and will want ideas of how to use it.  Lastly, it exposes student to a greater variety of manipulatives that they can choose to use in the future.  Sometimes upper grades students are reluctant to use manipulatives; this activity can also help students to use manipulatives.

One final goal I have for students related to building their perceptions as mathematicians is to build their flexibility with solving math problems and equations in different ways.  Not only is this flexibility important, but each different strategy that the students use to solve the problem involves different mathematical concepts.  For example, if a student always chooses to solve multiplication problems with a counting strategy (either repeated addition or skip counting), the student continually uses the concept of understanding the relationship between multiplication and addition.  However, without developing flexibility to solve multiplication problems with other strategies such as relational thinking (ie. derived facts), the student may never engage with the distributive property or associative property of multiplication.  Although I care about the accuracy of their answers, I care more about their understanding of mathematical concepts which happens when students’ strategies develop and grow.   

Some students resist solving a word problem or equation another way, even though I routinely ask them to solve the problem using multiple strategies.  One technique that works for these students is to have students draw lines across their paper, splitting it into multiple sections.  They are then asked to solve the problem a different way in each section.  Although the expectation is exactly the same, something about having the paper sectioned really motivates the resistant students to work towards this flexibility.  It also gives an additional reason to listen carefully and ask questions when other students are explaining their strategies in class discussions.  

The beginning of the school year is such an exciting time and important time. I feel a lot of pressure to get it right so that my students can develop a perception of themselves as mathematicians who are capable of generating their own strategies for solving problems. Over the years, I’ve experimented with numerous ways to start my math time and have found the ideas I have shared in this blog as tried and true ways to start the important work of developing students’ mathematical thinking throughout the school year.