Thursday, November 20, 2014

Everyday Math, Cognitive Development, and "Explain..."

"Explain how your number model matches the number story."

At the beginning of the year, my middle school students often sunk into a state of mental anguish and defeat at the mere sight of the word explain on a math assessment.   They often were not nearly as intimidated by mere math problems as they were math problems that involved writing.  It was like a double-whammy of academic anxieties.  First you want me to do math, and now you want me to write... at the same time.

Even more interesting is to see the reactions of first graders when they are asked to "explain" how they solved a problem.  There are so many questions to consider when we ask young students to explain their thinking.  Are they cognitively ready to explain their thinking?  Do they understand what we mean when we ask them to explain their thinking?  Do WE understand what we mean when we ask that question?  Should we give them a low score if they are not yet able to write the words down on the paper?

The increased emphasis on self-explanation, critical thinking, and concept-based instruction in teaching math is more present than ever in Everyday Math and has some parents asking a lot of questions.  They wonder: What ever happened to memorizing facts and learning how to add and subtract?  Math will always be math... Why has teaching math changed so much?

Are they cognitively ready to explain their thinking? 

There is increasing evidence coming from years of research that seems to indicate children benefit greatly from concept-based learning and self-explanation (being able to explain one's thinking to another) as they mature.  In other words, the earlier students become accustomed to showing and sharing their thinking, i.e. being able to not just solve the problem, but also to show a partner exactly how they solved the problem, the better they become at both procedural and conceptual learning as teenagers and adults.  They become better problem solvers.  So some students are more ready than others for this kind of thinking, but the consensus among the pedagogical researchers is that starting early has a positive impact.  Here are some links that help explain this... The first is a video, then the other pieces are articles that cite research.

http://developingchild.harvard.edu/resources/multimedia/videos/inbrief_series/inbrief_executive_function/

http://www.parentingscience.com/kids-learn-math-and-science.html

http://www.parentingscience.com/teaching-critical-thinking.html

http://news.uchicago.edu/article/2013/01/23/children-s-complex-thinking-skills-begin-forming-they-go-school

It is also important for us to remember that as we see a great variety of abilities to explain how they solve problems, that we are not asking them to analyze their thinking in grades 1 and 2 as much as were are simply asking them to show their thinking.  This is the precursor to the kind of critical thinking they will be asked to demonstrate later.  Being able to show their thinking is a very important step in the development of analytical thinking and being able to self-critique and revise their thinking.  Still, it is often difficult for children to put into words their mathematical thinking.  It often amounts to a language and writing lesson.

"What did I do?  First I read the number story. Then I wrote down 5 + 1 = 6 because 5 tickets and 1 more ticket equals 6 tickets in all."

I am amazed at the experience and skills veteran teachers demonstrate when they read and make sense of student answers!  Likely there is inventive spelling and lack of punctuation, digit and letter reversals, and interesting drawings to interpret.  First grade teachers become real experts at making clear sense out of beginning writers' answers.  Sometimes, it is appropriate for the teacher to write the words down as the student dictates his or her thinking.

Scoring questions such as these is not often a fast and simple process.  Careful analysis is required, and sometimes challenging interpretation.  I know some teachers get together to score assessments so they can consult with one another on such matters, and when it is possible, that is a terrific idea.

In closing, I should emphasize that this kind of learning experience is coming from scientific research, which influences the mathematicians, teachers and pedagogical experts who wrote the Common Core Content Standards for Mathematics and the Standards for Mathematical Practice, and those standards have influenced the writers and publishers of every major math curriculum in America right now.  The simple answer to the befuddled parent is this:  The trend today in mathematics instruction is teaching more for understanding; we are not abandoning fact practice and procedural learning, but we are introducing more ways to teach for greater understanding of math concepts.  Research shows that greater understanding of math concepts comes from giving children more opportunities to think for themselves, to construct and deconstruct mathematical problems, and to be able to explain their problem solving strategies to others.


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