NCTM Essential Mathematics Teaching Practice #4—Facilitate Meaningful Mathematical Discourse

Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments. 

For years I have listened to the cries of those who hate mathematics.  The most common thread of complaint is that math doesn’t make any sense and in school they fail to learn the rules.  I remember days in the student seat, feeling lost, trying to follow the teacher’s train of thought as she jotted on the blackboard.  I often wonder how these same folks would feel about mathematics if they were fortunate enough to sit in a math classroom with a deep focus on sense-making.  At the core of these powerful classrooms is a community of learners who contribute to the intellectual capacity of the room.  The strength of these communities stems directly from the power of great classroom discourse.

Leading a great classroom conversation around mathematics is a complicated task.  It is far more than a mathematical show and tell.  Teachers have to consider the mathematics that they want students to learn and select student work and questions that will help move the class’ understanding of the concept forward.

Here are some things to keep in mind to help make your next classroom conversation high-leverage:

1. Plan ahead.  Start with a rich task that promotes problem-solving and reasoning.  Selection of the right task is important.  If there is only one obvious solution or strategy you may not get the diverse ideas that make for great classroom debate.  Discussion designed around steps in a procedure are not likely to produce the deep conceptual understanding you’re hoping for.                      
2. Anticipate student responses.  If possible have a team teacher solve the problem.  The better prepared you are beforehand, the better equipped you will be to guide students to the mathematics from the student strategies.                                                                                                 
3. Of course there will always be the one unanticipated strategy.  The art is to honor student thinking while ensuring that the underlying mathematical ideas are the common thread in the conversation.                                                                                                                                             
4. As students are working, circulate and make note of students’ strategies.  You should be planning who you would like to have share.  Think about which students’ work makes clear connections to the mathematical ideas central to your lesson.  Keeping that in mind, you want the samples that you select to have variations to provide talking points for students to contrast the strategies.                                                                                                                                           
5. Select the order in which you would like students to share.  Plan to go from the least to most sophisticated strategy.                                                                                                                               
6. While students share, think of what questions you will ask to help the class connect the strategies to each other as well as to the mathematical ideas.  Guide students to ask questions.  Students should be encouraged to defend their thinking, question one another, and critique the reasoning of others.  This purposeful, deep conversation develops higher-level thinking skills as students make sense of a variety of strategies.

Having students reflect on their learning is a great way to end a lesson focused around discourse.  It can also be an effective formative assessment.  If you use learning targets you may wish to wait until the students end their conversation to share the target and allow time for students to connect their discussion to the target. 

If you haven’t used this type of instruction in your classroom before it can be a bit of a shift.  To think that one or two tasks can elicit as much learning as 50 practice problems is a major change in thinking.  The power of the learning comes from the ideas that students present and the connections they make, not solely on answer getting.  For students who are proud of quick answers, taking time to debate the process can be frustrating at first.  Through these rich experiences students will come to see the value in exploring mathematical concepts deeper.  Often teachers who are shifting to this level of discourse will comment that they saw a deeper level of insight or thoughtful work from particular students that they did not typically see.  This just validates that this focus on sense-making is so powerful in student learning and has great potential to assist our struggling learners.

Already implementing mathematical discourse in your classroom?  Take your class to the next level by facilitating student to student classroom conversations.  Use of accountable talking stems or other statement/question starters help empower students to communicate clearly with peers to critique the reasoning of others and defend their thinking. 

Want tools to help you get started?  Number Talks by Sherry Parrish is an excellent resource which includes a DVD with sample lessons.  These 5-10 minute classroom discussions can help you perfect your technique and build your confidence for larger discussions.  When you’re ready to implement more challenging tasks, start with some from Illustrative Mathematics or from the Coherence Map at AchievetheCore.org.

We'd love to hear how you are implementing classroom discourse in your classrooms!  Share with us on Twitter (@KYCenterForMath) using #KCMTalks or on Facebook at Kentucky Center for Mathematics.

NCTM Essential Mathematics Teaching Practice #3—Use and Connect Mathematical Representations

Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

It was within my first week of professional learning as a math interventionist that I realized what I had done.  Well not what I had done, but had neglected to do, that was very likely the root of my students’ mathematical struggles over the years.  I wasn’t alone as you could hear the murmur of teachers across the state remorsefully apologizing to the years of students that they had wronged.  

What, you may ask, can move a competent teacher leader to want to fire herself from her charge?  The 3 aspects of number.  Such a tiny piece of information and yet so powerful.  As I learned through Math Recovery, behind every number are 3 dimensions, verbal (what we say “three”), quantity (the amount that we see “* * *”), and the symbol (“3”).  To be proficient, math students need a complete understanding of the numbers around them.  For many that are struggling, one or more of these dimensions are missing or flawed.  As an educator, I had failed to provide my students with a balance of these 3 aspects in my instructional program and their shallow interpretations were showing in peculiar ways.

I could still recall my Melanie who would write the strangest answers on her math assignments.  One day while students were “making 10” to add, she came to me with her paper.  “Mrs. Rowland, I just don’t get it.”  On her paper was written 9 + 6.  I explained, “All you have to do is take one from the six and give it to the nine.  That makes it ten plus five.  You know that one.” She stood there a few moments longer before informing me that there was no one in the problem, just a nine and a six.

Truly, Melanie wasn’t the only student that suffered.  Even my quick, eager learners were limited in their ability to think deeply about problems.  Through my greatest efforts of timed tests and bare number tasks I had created a league of the fastest counters known to man.  These students, though quick, were still counting by one for the majority of the problems they encountered.  I had done very little to help them think deeply about the mathematics they were learning.

According to NCTM, having students use mathematical representations (which have significant overlap with the 3 aspects of number), and helping them make connections between them, is critical in getting students to think deeply about mathematics.  These representations include visual, symbolic, verbal, contextual, and physical, each one providing a new dimension in which to view mathematical concepts. The diagram shown here illustrates conceptual networks developed when using various representations (NCTM, 2014, p. 25).

As students move between these representations their understanding of math concepts deepen.  These representations provide opportunities for students to look for structure, analyze essential elements of mathematical ideas, and communicate their thinking.  As noted by the NCTM (2014, p. 25): According to the National Research Council (2001), “Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.”  When students have strong connections to these representations, their ability to solve problems is greater as they can move with ease between them to make sense of the problem.  Connectedness of representations is not only good for our students, it’s essential!  

In my intervention classroom, the opportunities that students had to move between these representations helped them to understand mathematics in an authentic way.  The more responsibility for making those connections that I could give to the students, the stronger and more lasting their understanding.  We used mathematical tools to explore ideas and visual models to record our thinking.  These visual models provided opportunities for students to compare, defend, and debate strategies and ideas.  Intervention students benefited as well from the additional time and space to look deeper at the algorithms that they had been taught either in the classroom or at home.  These connections allowed them to explore the validity of these algorithms and reflect on their understanding of the process giving students the perfect climate to be active in their own learning.  

Ready to give your lessons a make-over?  Here are some of my top tips for creating connections in the classroom!

1. Teach concepts in context.  Let students ponder mathematical ideas in the real world.  Traditional textbook series start with the math then end with the application.  Don’t be afraid to flip it!  Teachers in middle and high school grades can explore some high interest projects at the website matheliscious.com.  Students can engage in mathematics around how to become more popular on Instagram or to determine exactly how much your pizza is costing you.  For younger grades I recommend Marilyn Burns Math Solutions as an excellent resource for teaching in context.  Here is a link to her blog with tips on getting started: http://marilynburnsmathblog.com/wordpress/using-childrens-literature-to-teach-math/

2. Know the representations that fit best with the concept that you are about to teach.  Some great places to look are John Van DeWalle’s Teaching Student Centered Mathematics, illustrativemathematics.com, and the Kentucky Numeracy Project.  You can also get a deeper understanding of the intent of the standards and their representations from the coherence map at Achievethecore.org.  Be prepared to allow students to explore the mathematical ideas using these representations, not just introduce them and move on.  If you teach from a textbook series you may want to ask yourself who is reasoning with the models.  If the answer is you or the book you may need to modify the lesson to put the responsibility on the students.

Be aware that physical models and manipulatives are great for allowing students to reason through mathematical ideas at all grade levels.  Make sure to allow students to use the concrete representations for exploration and mathematical reasoning.  If the teacher is simply modeling how one would use the model to get an answer, students may interpret that the learning goal is just that, copy the teacher.  

3. As your students share their ideas and strategies in classroom discussion, model their thinking on the board using the best representation that matches their thinking.  This allows a natural place for students to engage in the model.  Allow students time to discuss and debate using the model or representation as the backdrop.  Be prepared to shift models if the strategy shifts.  You can help prepare students by having them map out their thinking prior to sharing.  Encourage students to record their thinking, including labeling their work, so that they are able to share their thoughts with a peer.  

4. Make “Math Talk” a regular part of daily math instruction.  Provide opportunities for students to talk through the math they are learning.  Math talk goes beyond whole group conversation.  Think about natural opportunities in the lesson for students to clarify their thinking with a peer.  Many teachers use math workstations or guided math centers for this purpose.  Whatever your structure, you want to maximize engagement!  Think Pair Share allows many students to engage in math talk at a time.  

5. Be prepared to model your thinking for students, sharing what you notice and what you wonder about.  Allow students time in inquiry mode.  Bring them together to discuss strategies and ideas.  Your job is to formalize the math for your students.  Keep in mind your big ideas and goals for your unit.

6. Review your lesson plans for opportunities to connect the 5 types of representations, thinking about how to ensure students will interact with multiple representations.  This is a great job for math PLCs to work on together!

7. Give your independent practice work a check-up!  How many connections do students need to practice to complete their independent work?  There is more power for our students to complete 1 carefully chosen task in multiple ways, using various representations, than lengthy assignments with multiple problems and limited connections.  Less can be more!  Better yet, allow students to select their problem from a generated set of tasks and demonstrate their thinking in multiple ways!  

8. Use these representations to provide feedback that moves students deeper into the concept!   Ask students questions to illustrate a new connection.  If the task was bare number ask, “Can you write a story problem that would match this number sentence?”  

Nothing can replace the power of supporting students to make connections using various representations.  We would love to hear from you!  What great ideas do you have for using representations and connecting mathematical ideas?  Tell us what you think!  Share with us on Twitter @kycenterformath #KCMTalks or on Facebook at Kentucky Center for Mathematics.    

REFERENCE:

National Council of Teachers of Mathematics. (2014). Principles to actions; ensuring mathematical success for all. NCTM, Reston, VA.

NCTM Essential Mathematics Teaching Practice #2—Implement Tasks that Promote Reasoning and Problem-solving

Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies.

Last week, I had the opportunity to lead an enrichment class for 6^th grade math.  The students had been studying operations with decimals.  I introduced them to the online game Kahoot.  If you’ve never experienced this online hit you should give it a whirl at your next social gathering.  This resource houses hundreds of quizzes ranging from educational to trivia.  (You should always preview the questions and verify accuracy of answers before using in class.)  Students join from their personal devices using only the game pin number on the screen, no usernames or passwords to memorize.  Once students have locked in their answers the game shows a bar graph of responses and highlights the correct and incorrect responses.  The students loved the interactive format and that scores are shared at the end of each question.  Once they were familiar with the game, I informed them that had to create their Kahoot this week by designing their own questions.  We began with the target on the board.  We had a rich discussion of the vocabulary.  What are operations and examples of operations?  What is a decimal?  What are contexts that we see decimals in?  Initially, they told me they had only seen decimals with money.  The students had to create 2 bare number tasks and 2 story problems.  To up the challenge, they were required to solve their problem to provide the correct response, but they also had to generate 3 incorrect responses.  In order to help them succeed in the game, they needed to make those incorrect responses reflect the common errors and misconceptions they believed their classmates might make when solving the problem.

You could tell that some students were unsure of what to do at first.  There is something comfortable in traditional “find the correct answer” tasks.  I was requiring much more from them.  The responsibility for making connections was now theirs.  As class came to a close, the students were excited to share their questions with me and anxious to see if their questions were to be chosen.  As I poured through them I could see classmates’ names and a celebrity or two mixed into their story problems.  I also received a few blank cards from a student who shared that she was too scared to write story problems.

The next day as we played the game, students announced the problems that they had created after we solved them.  They were excited to see when they had tricked a classmate with a misconception.  One student explained that he created the problem “9.999 + 444.4” because he knew students would just add the digits without thinking about the decimal.  He pointed out that 9.999 is actually much smaller than 444.4 when you think about the decimal.  We also discovered that students had found many contexts for decimals other than money.  At the end of the game, I was able to view a spreadsheet of each student’s response as a formative assessment piece.

    

 We know that student learning is greatest in classrooms where the tasks consistently encourage high level student thinking and reasoning and least in classrooms where the tasks are routinely procedural in nature.  Implementing tasks that promote reasoning and problem solving allows students to engage in their learning using the math practice standards in an authentic way.   In this activity, the students had ownership of their learning and accountability to their peers.  From a teacher’s perspective it required a great level of trust in my students’ ability to engage in and make meaning from the task. 

Having students generate possible questions for an answer is an easier reasoning task to create.  This is similar to a routine that many teachers use called “Number of the Day.”  In this routine students generate equivalent representations or expressions for a number.  It’s important to note however that once a task such as this becomes routine we lose some complexity of thought.  Be prepared to replace old routines to keep the rigor high in your classroom.  I have seen intermediate classrooms begin the year with number of the day and replace it with fraction of the day or decimal of the day.  Keep in mind as well, that to be successful at such open ended tasks students have to develop a comfort level with multiple solutions.  They must learn to accept answers from their peers that may not reflect the same depth of thought as their own.  Students who struggle with this, such as my friend with 2 blank question cards, should be monitored and encouraged until they develop a deeper capacity to work with such ideas.

Children’s literature is a great launching point for high-level tasks.  One of my favorite tasks pairs nicely with The Napping House.  After reading the book we play “Who Lives in My House?”  In this task, students tell the number of feet living in their house.  The class sets to work creating all the possibilities of human and pet configurations that could be in the student’s home.  This activity allows for modeling, multiplication, and even writing expressions with variables.

Not all tasks must be real-world problems.  Students can investigate mathematical concepts in the context of materials as well.  With primary students I love to explore odd and even numbers with the game “Spill and Compare”.  In this game, partner pairs are given a number of two-sided counters in a labeled cup.  They take turns spilling the counters and comparing the number of red and yellow counters.  Students record their results on paper by tallying under the headings “More Red”, “More Yellow”, and “Same”.    As students engage in this game, pairs with odd numbers will begin to notice that they never get to mark “Same”.  As the complaints roll in I usually give a “permission to cheat” rule.  If you haven’t gotten “same” you have permission to move the manipulative yourself to create the same.  This is where I will allow my most vocal student to be the spokesperson for the odd numbered pairs.  This student gets to inform the class that while they have tried some of the numbers are not able to have the same number of reds and yellows.  As a class we collect the data from each partner pair on chart paper and identify which numbers worked nicely for the game and which did not.  Then we name them “even” and “odd”.  Because students had this authentic experience with even and odd before being introduced to the words, it gave the words meaning.  They have a much deeper understanding of odd and even than they would have received from traditional methods. 

High-level tasks do not necessarily require a great deal of time to create, but they do require great thought.  It’s important to keep in mind the mathematics that you would like students to engage in.  Identifying the big ideas first helps you to find connections between math concepts.  Select a context that is relevant to your students.  If you need to create your own tasks, search mathematical tasks or try some of the resources listed at the end of this post.  Professional learning communities are a great place to collaborate on high level tasks.  After trying a few with your class you may find that you better understand how to create them.

Once you have created or selected your task, you will need to be mindful in how you present the task to the class.  Launch the problem, without giving hints.  Support students as they work.  This is a time for you to gather evidence of student learning and strategies.  You may consider asking guiding questions.  Keep in mind that after the investigation you will bring the class together to summarize the learning.  The student work time is when you will be preparing for this discussion.  Plan to allow students to share their strategies from the simplest to the most complex.  If students are working in cooperative groups, listen as they work.  Make a note of students who may have made important connections to the learning you had hoped to address.  When the task time has ended, bring the students together to share their thinking and noticings.  Use the students’ experiences when you can to summarize the learning of the day.  These authentic experiences create rich connections that ensure lasting learning will occur.

This is also a time to monitor your own behaviors and thoughts.  It is part of our human nature to want to help.  This is why many of us became educators.  But the temptation it very real to help our students to the point of giving them our own thoughts.  When students make authentic connections the learning is lasting.  Preparing yourself in advance for these feelings will help you suppress the urge to think for the students.  If you see a student off track in their thinking encourage them to discuss it with a peer or ask some guiding questions, but refrain from just telling.

Some teachers have concerns with such tasks because of the time required.  With strict adherence to a pacing guide or textbook series teachers feel pressure to move to the next lesson or skill.  In many cases, these rich, open-ended tasks allow students to engage in multiple standards and can expose math that we hadn’t intended to introduce just yet.  For example, in a fourth grade classroom students were exploring equivalency of fractions in a game setting.  The teacher had planned to spend the next week on addition and multiplication of fractions, however the students had made the connection during the task.  The teacher was able to follow up with some practice and continue through the curriculum.

Want to find more tasks to promote reasoning and problem-solving?  Try these great resources!

http://www.nctm.org/tcm-blog/ is NCTM’s blog for Teaching Children Mathematics.  If you don’t already follow this blog, you should!

http://www.nctm.org/Classroom-Resources/Problems/ is NCTMs problem database.

http://www.k-5mathteachingresources.com/ has games and tasks by grade-level and strand.  I especially like their literature links.

http://www.insidemathematics.org/problems-of-the-month/download-problems-of-the-month, problem-solving tasks from Inside Mathematics.

Have a great resource of idea about reasoning and problem solving tasks?  We want to hear from you!  Tell us what you think!  Share with us on Twitter @kycenterformath #KCMTalks or on Facebook at Kentucky Center for Mathematics.