In my last post I wrote about some materials we can use to engage students in puzzling through problems, the Contexts for Learning Mathematics units.  These materials present a wonderful model for giving students the opportunity to think through the meaning of a problem.  Then they engage students in applying their prior knowledge to find a variety of solution paths.

We can really help our students grow as mathematicians if we provide similar opportunities throughout the curriculum.  By presenting students with scenarios, problems, or puzzles that are unfamiliar, but for which they have sufficient prior knowledge to figure out a solution path, we give them the opportunity to think mathematically in constructing their understanding.  By engaging students in discussions about these scenarios, problems, or puzzles, we encourage them to think together about what makes sense and how they can apply what they know.  We are asking them to do math rather than just follow a procedure that has been explained to them.

So how do we do this?  Here are examples using the variety of types of problems in the table of common addition and subtraction situations (click here) and the table of common multiplication and division situations (click here).   At some point in the year, primary students might be comfortable with “add to” and “take from” situations where the result is unknown, but unfamiliar with “take from” situations where the change or start is unknown, or “put together” situations where one addend or both addends are unknown.  We can engage students in making sense of these unfamiliar situations by asking them what they notice about the new situations and what they wonder.  Rather then showing them what procedure to use, we can empower them to think mathematically by asking them to discuss the situation.  We can help them learn to use strategies that help all of us make sense of problems and situations: retelling, acting out (physically, with manipulatives, or with pictures), and visualizing.

Similarly, in the middle grades, when students are comfortable with “unknown product” situations with equal groups, we can have them discuss and apply sense making strategies to figure out “unknown product” situations where we are comparing (ex. “A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?”).  Rather than showing them what procedure to use, by discussing, acting out, and visualizing comparison situations,  students can work together to think about the meaning of these unfamiliar situations.  That will help them apply their knowledge about multiplication and make sense of the connections between these types of problems.

Talking about what they are doing and thinking, helps students learn to use  sense making strategies.  As they agree and disagree with each other, they learn to think through what makes sense and try out different ideas on a regular basis.  This helps them develop a “growth mindset”.

We need to provide regular opportunities for students to puzzle through math problems and ideas.  These ideas are right there in our curriculum.  When we work on new content standards, we can provide problems or situations for which students  have prior applicable knowledge.  We can use questions and prompts that engage them in modeling ideas and thinking about relevant concepts and skills.   Then our students can do the thinking and talking that will help them figure out how to apply that prior knowledge.  Let’s engage our students in puzzling and wondering throughout the content standards!

Discussing strategy with games is a great way to engage students in talking about their reasoning.  We often use games in math class. We can engage our students in  productive discussions if we have students work together to consider next best moves.

Nancy Anderson recently taught this simple but flexible game at an ATMIM  meeting in January.  Once we understood the game she had us think about the next best move for a particular scenario.

I am calling the game “Guess My Problem”.  It’s like Hangman, only it’s about making math equations.

Here is an example of how you could play it with 2 digit + 2 digit addition equations.

“Guess My Problem” (Like “Hangman”)

Try to use as few moves as you can.

Player 1 writes down the equation with blanks and the list of digits.

Ex. __ __ + __ __ = __ __

Player 1 records the equations with numbers in a hidden place.

As Player 2 guesses digits, player 1 fills in the blanks guessed correctly and keeps track of the numbers tried and the numbers eliminated.

So here is where we can start thinking about strategy:

After 3 moves, one game looked like this with two digits left to be placed:

1  __+­ ­­­­__ 1 = ­­­ ­44      Correct Guesses             Numbers Eliminated

1,  4                                           7

What is the best digit to guess next? Why?

 Challenge students to reason as a class about best moves.  Play it with larger numbers or different operations.  I’d love to see parents playing this with their children as they wait in a restaurant or for an appointment!

Thanks to Nancy Anderson, co-author of Classroom Discussions in Math,  for teaching this at ATMIM in January.

Getting students to discuss their math thinking is not always easy. Students may struggle with staying on topic, explaining their ideas, or listening and responding to others. I’ve found that role-playing, having students act out behaviors in a math discussion, gives students the chance to inhabit the roles of speaker and listener and improve their abilities within those roles.

I’ve helped teachers use role-playing to focus on student engagement during turn-and-talk. Because students communicate simultaneously in pairs during turn-and-talk, they all get opportunities to share ideas and practice communicating prior to sharing with a large group. But even with turn-and-talk, students can have difficulties. Teachers have expressed the following concerns to me:

• Only one of the partners talks and the other is mostly unresponsive
• One or both partners talk about something other than the given topic
• One partner gets frustrated and only has negative responses
• The partners have different ideas and don’t seem to think about each other’s ideas

Here are some scenarios we’ve used to help students reflect on their roles in turn-and-talk and boost their participation.

Scenarios and Roles

To introduce the role-plays, I told the class that we were going to act out different turn-and-talk situations and think about how our discussions can help use learn. We gave two people student roles and one person the role of the teacher. The roles were explained to each actor, but not to the students in the audience. The audience was asked to watch carefully and try to remember how the students were engaging in turn-and-talk. I often started with one role-play where students communicated effectively and one where they did not. This helped us list the kinds of behaviors that made turn-and-talk work well.

The role-play scenarios focused on math ideas students were pretty comfortable with. That allowed students to focus on the behaviors, not the math.

Examples I have used:

Scenario 1: Discuss how you could use one multiplication fact to help you find the answer to a harder multiplication fact (discuss addition for 1^st or 2^nd)

• Roles:
  • Teacher – presents the question: How can you use a multiplication fact you know to help you answer a harder multiplication fact?
  • Student A – only talks about how the multiplication facts were so hard to learn
  • Student B- tells which facts they used to help them figure out other facts
• Variation to show effective communication: To show a more effective turn-and-talk, Student A responds to Student B by asking how they used one of the facts to help them solve the other fact, and this developed into a discussion

Scenario 2: Discuss the meaning of an unfamiliar story problem

Roles:

• Teacher – presents the story problem to the students and asks them to talk about the meaning of the problem (I chose story problems recently used in class)
• Student A- talks about the action in the problem and how the numbers are related; then talks about what they need to figure out
• Student B- just gives one word responses; doesn’t show understanding

• Variation to show more effective communication: As above, but Student A asks Student B if he/she agrees or has a questions and gets Student B to identify what they don’t understand

Scenario 3: Discuss why you can use adding on or subtracting back strategies to solve a problem asking how many more are needed

• Roles:
  • Teacher- asks students to discuss why they could use adding on or subtracting back strategies to solve a particular problem.
  • Student A- talks about why they think adding on is best
  • Student B- talks about why they think subtracting back is best

Scenario 4: (Based on my book, Hatching Butterflies): Discuss the meaning of a given problem, ex. Last year’s class hatched 21 butterflies. This year’s butterflies have started hatching. So far we have 8 butterflies. How many more butterflies do we need to hatch to reach last year’s record?

• Roles:
  • Teacher: Presents the new problem and asks students to talk about what the problem is about and how they might get started with it
  • Student A: Explains that they know how many butterflies have hatched and need to know how many more need to hatch to get to 21.
  • Student B: Nods
  • Student A: Says maybe they could take away the ones that have hatched already.
  • Discussion continues with Student B explaining ideas while Student A is pretty quiet.
  • Teacher stops the turn-and-talk and asks Student A to tell the class about the ideas they shared as partners.

Mechanics

As a math coach in classrooms with two adults available to model a role-play, I first had one student play the role of the teacher and the classroom teacher and I took on the roles of students. If I were the only adult in the room, I would rehearse with two or three students first, so we could model role-playing for the class.

After the role-play, I asked the audience to describe what they saw each person doing. Students are very observant! They described the behaviors so accurately. Once we agreed on the roles of the actors, we would talk about how the behaviors shown seemed to help or hinder learning. Again, the audience was reflective and spot-on. Then I asked them to consider whether a scenario like this might actually happen in class and what suggestions they had for how the students in situations like this could use turn-and-talk more effectively. Here it was a bit harder for them to focus their recommendations on the problem shown in the role-play, but they had lots of good recommendations! We posted a list of the recommendations and referred to them frequently.

Your classrooms

I wonder if you have used role-playing in this way or if you’d like to try it. I ‘d love to hear about your experiences or other ways in which you boost participation in discussions.

If you want to try using role-playing in this way, you may want to focus on turn-and-talk, or other aspects of discussions like students restating and rephrasing ideas asked in class or students working in pairs. Occasionally repeating a role-play, perhaps altering it a bit, can remind students about how they can engage more effectively.

This post has a link to my webinar produced by SDE.  The webinar focuses on processes that help students make sense of story problems, specifically using reading strategies and discussing problems.  It uses examples from my Use Your Math Power books.  For a video clip showing another educator giving  tips about using reading strategies in solving story problems, click here.

Here is the link for my webinar, Use Your Math Power! (Gr. K – 3)

http://sde.com/Knowledge-Base/Complimentary-Webinars