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Developing Flexible Thinkers May Require a Less Flexible Approach

Over the past decade, emphasizing multiple solving strategies has permeated elementary mathematics instruction - a significant departure from the rote, procedural learning that previously characterized the subject. Flexible thinking has replaced rigid rules, unleashing student creativity in the process. This pedagogical shift has helped positively redefine the subject, which has led to better math teaching and learning.  Still, the practice is commonly mismanaged and overemphasized.

Elementary math teachers frequently provide questions that stimulate multiple solutions, and encourage students to share their solving strategies in front of the class.  In theory, this methodology creates a dynamic classroom experience that engages everyone.  In reality, however, it usually has the reverse effect, puzzling already confused children.  Many students initially pay attention, but their focus wanes with each passing explanation.  Learning becomes passive and the lesson more resembles a collection of oral reports than an instructive learning session.  By the time the last presenter shares, the majority of students have stopped listening and forgotten the earlier solving methods.

This lesson design is most dynamic and effective when teachers use the following approach. The mental math topic Add 98 to three-digit numbers will be used as a catalyst.

  • While lesson planning, the instructor selects a few strategies that they want their students to learn, e.g.

  1. Standard algorithm

  2. Add 100, subtract 2

  3. Make 100

  • At the beginning of the lesson, they present a problem such as 98 + 394 = and allow students an exploratory time to solve. As students try out different solving methods, the teacher probes the room, carefully selects which strategies will be presented, and the order in which they’re given.  If there is a valuable method that no child uses, the teacher will demonstrate it.

  • Strategies are shared from simplest to most complex.  When the latter is demonstrated before the former, struggling students often become confused, frustrated, and stop paying attention.

  • Presenters understand that they are providing instruction.  Therefore, the selected students need to articulate their strategies, or the teacher needs to efficiently paraphrase their mini-lesson.  Long-winded presentations lead to disengaged learners.

  • After a child presents, all of their classmates practice the strategy using analogous examples.  For the problem 98 + 394, three different presenters might demonstrate the following three solutions:

 
strategy1.png
 

After each student shares, the class practices the problems 98 + 496, 98 + 795, and 98 + 597, using methods I, II, and III, respectively.

  • Regardless of how brilliant or creative a strategy is, the teacher doesn’t select students, whose explanation will confuse their classmates.  This can be frustrating for both students and teachers, but resisting is critical to maximize learning.  Teachers can recognize and commend individual students at a different time of the day.

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Stimulating flexible thinking is also possible using a more traditional approach.  The teacher simply shares different strategies and requires their students to attempt each.  After students demonstrate proficiency in both, they use their preferred method.  If teachers don’t require their students to practice multiple strategies, then children will usually default to what initially seems easiest, never trying alternative solutions that might work more efficiently for them.  Below are two methods for solving a first, third, and fifth grade problem.

First Grade: Adding crossing the 10

 
strategy2.png
 

Third Grade: Multiply 3 by 9

 
strategy3.png
 

Fifth Grade: Prime Factorization

 
strategy4.png
 

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Several years ago, I met two second grade teachers who boasted that their students could perform Addition with Renaming (e.g. 467 + 285) in ten different ways.  They had just completed their sixth week of teaching the unit and possessed dozens of student work samples as proof.  But their claim was only half true. Fifty percent of their students hadn’t mastered one method and struggled daily as their teachers and classmates introduced new solving strategies.

I assume that if the teachers had set out to teach the standard algorithm and one or two other ways of solving, then 90% of the class would’ve learned the topic following a week or two of practice.  Those who mastered quickly could’ve worked with alternative algorithms and still wound up proficient in many strategies.  Lost in the discussion was that the teachers used an extra month on a topic that shouldn’t have lasted more than five to ten instructional days.

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I believe that teaching various solving strategies is a critical component of great elementary mathematics instruction. But - like any pedagogical technique - the practice can be overemphasized. Inundating students with various strategies often favors strong mathematicians at the expense of struggling le­­­­arners, because it creates illogical practice habits and mismanages class time. Therefore, flexible thinking shouldn’t be the aim of the subject, but instead the inevitable outcome of good teaching and learning.

The most successful math teachers I’ve seen understand:  If a strategy is worth teaching, then it’s also worth practicing, and too many methods can overwhelm students’ working memory, leading them to confusion.  They also know that teaching a repertoire of methods can be the same as opting to not teach other topics.  Weighing all of this, they encourage flexible thinking and praise creativity, while maintaining efficient pacing throughout lessons, chapters, and units.  In doing so, they nurture the brain and subject’s elasticity, while optimizing each students’ mathematical potential.