Sunday, March 1, 2015

Rebranding "Show Your Work"



Marketers know that a product's brand is everything.  If a product is not selling to its potential, one solution is often to tweak the brand, or "rebrand" the product.  Kentucky Fried Chicken changed their branding to KFC to de-emphasize 'fried chicken" and appear to be a more healthy option.  If it works in business, shouldn't it work in education?

In math classrooms across the world, students are told on a regular basis to "show their work".  I wish I had a nickel for every time those words came out of my mouth during my educational career.  It is certainly done with good intentions--it is critical that students are able to communicate mathematically. Not to mention the valuable feedback teachers receive when they analyze the "work" a student has shown.

My problem is not with the process, it's with the words, so I have been experimenting with rebranding "show your work".

As I see it, there are two major problems with asking students to show their work. First, the words hold a very negative connotation in the minds of students.  It's something they have to do.  Furthermore, the words are often delivered in a way that is not conducive to cooperation.  "John, if I've told you once I've told you a million times, you've GOT to show your work."  "Valerie, I'm not taking this paper until you show your work!"

Second, many students don't show their work because they don't know what the heck it means!  My favorite is the student who circles the multiple choice answer he thinks is correct and then x's out the other choices.  If you ask him, he is "showing his work".

For my suggestions, I will address the second problem first.  Students have to be specifically taught what it means to show mathematical thinking (see how I'm rebranding it?).  This happens through a great deal of modeling and coaching.  I'm blessed to work with small groups exclusively, so I am able to coach my students one-on-one, helping them to understand how to comprehend and make sense of each and every sentence in a math problem.

Now, to overcome the negative connotation of the words "show your work", we have to stop using them.  Think about it, when you are solving a problem do you think to yourself, "Hmmm, I've got to show my work."  I don't think so.  What I DO do, is make notes to myself as I interact with the problem.  Those are now my go-to words when working with the students--I document my mathematical thinking by making notes as I interact with the problem.

I have been using this approach with my students for about a month now, and I am very pleased with the results.  They seem more open to the process and, through coaching, they are learning how to take notes on their own and determine important information.  It's a thinking process, not a rote procedure. In a follow-up blog post, I'll discuss more about how I help them make sense of problems.

I'd love to hear your thoughts!!



Saturday, February 21, 2015

Fly on the Math Teacher's Wall: Fractions



Welcome to the Fly on the Math Teacher's Wall blog hop!  In this recurring blog hop series, a great group of mathematics bloggers, covering all grade levels, band together to squash mathematical misconceptions.  This time around, we're tackling fractions.

The misconception I am discussing is that a larger denominator means a larger fraction. Ask a handful of third graders (or 4th graders or 5th graders...) which fraction is greater, 1/8 or 1/4, and most are likely to quickly tell you 1/8.  With big, proud smiles on their faces.  You're nodding your heads out there--I see you!  You've been there.

There's a reason this misconception is so widespread.  Up to this point in their educational career, bigger numbers always meant bigger values.  Eight is greater than four.  When students begin to learn about fractions, they erroneously apply whole number reasoning to fractions.  One-eighth must be greater than 1/4, because 8 is greater than 4.

So how do we address this misconception?  First and foremost, students must have tons of experience with a variety of concrete and pictorial models of fractions.  Use fraction tiles, fraction circles, Cuisenaire rods, number lines, and cut paper strips.  It's pretty hard to look at models of 1/8 and 1/4 and not see that 1/4 is greater.  The idea that fractions should be explored using manipulatives and models is very apparent from the wording of the 3rd grade Texas TEKS, but not so much in the CCSS.  I cannot overemphasize, however, do NOT rush to abstract symbols.

The other way we can overcome this faulty reasoning is to help students truly understand the meaning of the denominator.  The more parts an object is divided into, whether that object is a pizza or a number line, the smaller the parts.

Both the CCSS and new Texas TEKS address the issue of comparing fractions in a way that will help students deeply understand the denominator. The 3rd grade standards are really well written, once you get past understanding all the 1/b and a/b references, and will definitely result in better fraction number sense for our students.

Here are the standards for comparing fractions for both the CCSS and the TEKS:
Notice a few key points...

  • Students are only comparing two fractions, not ordering more than two
  • The denominators are limited to 2, 3, 4, 6, and 8
  • Students only compare fractions with either the same numerator or the same denominator
  • The TEKS specifically mention words, objects, and pictorial models along with symbols
  • Both include verbiage about reasoning about their size and justifying the conclusion
  • The CCSS states that students must understand that the reasoning only works when referring to the same whole

Let's first consider comparing fractions with the same numerator.  When two fractions have the same numerator, it emphasizes that a larger denominator means smaller parts.  Look, for example, at the representations below of 1/8 and 1/4.  When you look at one piece of each pizza, the idea that eighths are smaller than fourths is pretty clear. It works anytime the numerators are the same.  For example 2/6 and 2/4, or 3/8 and 3/6.
Third grade students are also required to compare fractions with the same denominator.  This emphasizes that the denominator describes how many parts the whole has been partitioned into (thereby influencing the size of the parts) and the numerator describes the number of those equal parts you have.
 

That's it!  Those are the only two types of comparison 3rd grade students need to do. And if they truly understand and can explain these comparisons, the next generation of students we send up won't leave us scratching our head when they say 1/8 is greater than 1/4.

Do you like the cards pictured above?  Well, guess what?  They are my gift to you tonight.  This freebie includes 16 cards--8 comparing equal numerators and 8 comparing equal denominators--along with a a recording sheet students can use to document their thinking in both words and symbols. Click here to grab your freebie!

Finally, who doesn't love a giveaway?  In the spirit of a fraction blog hop, one lucky winner will receive my Fraction Bundle and five winners will select a fraction product of their choice! But hurry, the giveaway isn't around for long!

a Rafflecopter giveaway
Ready for the next stop along the hop?  Head on over to Adventures in Guided Math!


Monday, February 16, 2015

What I'm Reading...Intentional Talk



"When we press beyond procedural explanations into explanations that include reasoning, we are supporting students in justifying their ideas." Intentional Talk (Kazemi/Hintz)
When you are planning instruction, how often do you consider the sound of your mathematics instruction and the conversations you want your students to engage in?  If you answered not often, then Intentional Talk by Elham Kazemi and Allison Hintz might radically change the way you plan.

Math is no longer a spectator sport.  We know that to truly develop mathematical thinking, students need to be active participants--they should be doing math and talking about math.  This dramatically changes the role of both student and teacher. The teacher becomes a facilitator, rather than a giver of knowledge, while students drive the work and the conversations.  Intentional Talk provides a road-map for that change.

If you have tried incorporating accountable talk in your classroom, then you know it is easier said than done.  To focus your efforts, the authors outline four guiding principles of classroom discussions in the Introduction and differentiate between Open Strategy Sharing and five targeted structures, each with its own goal and talk moves.

  • Compare and Connect
  • Why? Let's Justify
  • What's Best and Why?
  • Define and Clarify
  • Troubleshoot and Revise
Also included in the Introduction is a classroom example of a teacher using Open Strategy Sharing and two targeted follow-up structures with her class.  Following the Introduction, each type of talk structure receives its own chapter.

The chapters are thoughtfully organized to provide all the tools you need to implement each structure.   Each chapter contains information about the strategy; planning considerations, including completed planning templates (the Appendix contains blank templates); and both primary and intermediate vignettes.  The vignettes are the true power of the book, because you feel as if you are actually in the classroom observing a master teacher at work. References to research and the CCSS Mathematical Practices are sprinkled throughout the book, but do not distract or become tedious.

This is a book you will use as well as read.  You can't read a chapter and not have a new strategy to try in the classroom tomorrow!    



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