Monday, January 25, 2016

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 6th 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! is NCTM’s blog for Teaching Children Mathematics.  If you don’t already follow this blog, you should! has games and tasks by grade-level and strand.  I especially like their literature links.

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.    

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