Wednesday, March 18, 2015

Math Practice #3 - Construct Viable Arguments and Critique the Reasoning of Others

Math Practice #3 is all about listening, analyzing, and responding to others viewpoints.  In a nutshell - this is authentic classroom discourse.  

If a student is proficient in Math Practice #3 they can:
  • Analyze problems and use stated mathematical assumptions, definitions, and established results in constructing arguments.
  • Justify conclusions with mathematical ideas.
  • Listen to the arguments of others and ask useful questions to determine if an argument makes sense.
  • Ask clarifying questions or suggest ideas to improve/revise the argument.
  • Compare two arguments and determine correct or flawed logic.
In the classroom, it is important for teachers to be able to prompt this thinking without giving it away.  The article Convince Me by Wendy Petti describes the implementation of this practice in the classroom where students need to take ownership of the problem.  

Below are some actions that should be seen in the classroom by students and teachers along with some examples of open ended questions that can be used for almost any scenario.  


  1. The K-2 teachers at Hatley have incorporated Number Talks into our daily routine. The primary goal of Number Talks is computational fluency, however, students are responsible to go beyond just sharing their answer. Students are asked to share their strategies for solving the problems with with each other. Students must learn to clarify and express their thinking with others. This verbal rehearsal serves them well when asked to write out their mathematical thinking throughout the day.
    Mrs. Umlauf- K
    Mrs. Novak- 1
    Mrs. Rowlands- 2

  2. ditto of above (third grade Rothschild)
    In addition to the number talks, we wanted our students to be exposed to kid friendly language for Math Practice #3, so we started using the primary poster and language found at the Jordan School District website. So, we have taught the process through number talks, but require them to use constructive arguments throughout the problem solving, group and partner work with in a lesson.

  3. Evergreen has been using Number Talks during WIN time. Students need a great deal of practice and modeling in using discussion techniques in math. Students need to become comfortable with citing evidence and being able to explain their thinking to their peers.

  4. Evergreen Kindergarten has been using math practice three in our sorting unit (topic 13). When one student shares their sorting rule, the other students are able to agree/disagree and share their thinking. We modeled how to respectfully disagree with someone as well as supporting your position.

  5. Rothschild Kindergarten
    We introduced our students to math practice three by playing a game and practicing with students what it looks like to listen to the math thinking of another student and how to respond. We provided students with math vocabulary to use (through the use of sentence strips), incorporating strategies that we have taught and been working on throughout the year.

    After we taught and modeled with a small group in front of students, they went into groups of two and played the game. After modeling the sentence starters/strips and having them visible in the classroom, we were able to have students reference the sentence strips which allowed for modeling and support without just giving them the answer. We noticed some of our students who struggled with the math concept, had a harder time keeping each other accountable. We will continue to work on the concept of respectfully arguing ones thinking.

  6. The first grade at Rothschild has implemented Math Practice #3 in many ways in our classroom. As a team we have all begun using math talks in our classroom. Each morning we have gather our classrooms and the students demonstrate different ways of getting to the solutions. The students share their ideas with their classmates. The first grade team will start using sentence starters such as I agree with __________, I solved it by ____________, and My thinking was different than ____ because, and the way that worked for me was. We are having the students use their own words to explain their thinking.

  7. Hatley 3-5

    It has been evident throughout our instruction that students often have a difficult time with Math Practice #3. In mathematics, students have been programmed to seek the "illustrious answer" but often fail to adequately justify their reasoning. Students know that if they give an answer in class, they must also be prepared to explain the "why" and "how" behind their answer. Peers are then asked to comment and add to their thinking. Students are eager and willing to critique the work of peers, but struggle to provide a detailed/justified explanation for their thinking. Continual reinforcement of this math practice, along with others will make Math Practice #3 become a habit rather than a rarity. We feel that it is crucial for students to respond to each others thoughts, thinking, and answers on a regular basis in all subject areas. Allowing students to do this will eliminate the "sage on the stage" in the classroom, and in turn, make your classroom a more engaging and student-focused environment.

  8. Rothschild 4th Grade
    We have started doing number talks in our classrooms and it is very empowering for students to share their strategies. It gives them a sense of ownership and a confidence boost and they have really been enjoying it! Students are using transitions when explaining their strategy and conversational moves when critiquing the strategies of others. As we work through explaining and critiquing, we encourage and help each other to use math vocabulary which strengthens what they are saying. The routine of it is also appealing to many students and after we were wrapping up for the day, a number of students asked if we could do just one more! We see this as a powerful method to share thinking and ideas.

  9. We have started using partners to discuss and check their daily work. After the mini-lesson, students are able to work at their own pace through the activity, perhaps the work mat. They meet with a partner to go over each section. Students are encouraged to discuss differences in their work using pictures, drawings, or charts on the work mats. Coaching talk has been modeled where students don’t just give the right answer but ask questions of their partner on how they came to their answer. Number talks have also become a regular part of our math lessons, where we model and encourage the students to use many of the sentence starters that were mentioned by other grade levels.

  10. Rothschild 5th Grade

    We really like Josh's (Hatley) answer above. We think that is very good.
    Math Practice #3 builds on #1 and since introducing #1, we've been building on #3 ever since that point. We've been working on explaining and justifying our answers to each other and then to the whole group. Each class we build on agreeing and disagreeing as a group/partner, and some kids have really taken off on this and done a wonderful job. Others have struggled and they are the kids who don't like to agree on much at all and like to argue about many things. It's nice to see the different thought process and their justifications can use a variety of methods to show understanding. We do encourage them to use math vocabulary and to be very precise when explaining things. We are continuing to grow with this, and it has taken us to new heights already it seems. The students seem to be very excited for math and can't wait to prove how they know their answer is correct.