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!    

Saturday, February 14, 2015

Remediation...Who Needs It?

I was a classroom teacher for many years, an instructional coach for several more, and I currently serve as a K-5 math interventionist. Each role has allowed me to look at remediation through a different lens. Now that I am intimately involved in the Response to Intervention (RTI) process, I am seeing patterns emerge in how we identify and service students deemed at-risk in math.  It can't all be about classroom grades or standardized test scores.  We have to look at the underlying reasons for a student's struggles.  And if we are identifying large chunks of our student population as Tier II or Tier III, we have to look deeper for systemic reasons.

Students with Behavioral Issues

Let's just take this one off the table.  If a child is failing math because of a behavior issue, I can't help.  Sure, I can forge a relationship with the student and coax the math out of him, but you can do that as easily as me.  It does no good to put a student with behavior issues in a math remediation group with students who truly need remediation.

Students Who Lack Current Grade Level Skills

Face it, students do not all learn at the same pace.  Some learn more quickly and some more slowly. If a teacher tries to teach all students at the same pace, some will fail. That doesn't mean those students need to be pulled out for remediation.  It probably means that the teacher should reflect on her instructional strategies to determine if they are meeting the needs of all students.  A teacher who underutilizes small group instruction will likely have a higher percentage of students not mastering grade level skills, because whole group instruction will not adequately meet the differing needs of students.

Students Who Lack Number Sense

These students are probably good candidates for RTI.  If a student does not understand how to compose and decompose numbers, see the relationships between the operations, or lacks a basic understanding of place value concepts, they will undoubtedly fall further and further behind in math until those foundational areas are addressed.  That said, this is the primary learning that is going on in K-2, so if a student lacks number sense in K-2, it could be argued that they just lack current grade level skills.  So then we go back to the last conversation--is the teacher differentiating instruction and working with students in small groups at their level?  

Students Who Can't Apply Mathematics 

"They can multiply, but when it's in a word problem they bomb it!"  These are often the students who fail standardized tests. Students need to always see mathematics in context.  Sure, they need to learn computation skills and how to generate equivalent fractions, but if they learn those skills in isolation, they never see the application of the skills in a real-world setting.  Just as we have to teach students mathematical concepts, we have to teach them how to dissect and solve word problems.  This should not involve tricks and key words, it must happen through modeling and strategic instruction, including reading comprehension strategies.  In Texas, process standards are embedded into 75% of the items on our state assessment.  If we don't embed process standards into 75% of our classroom instruction, our students will not be successful.

Students with High Mobility Rates

My heart goes out to these kiddos.  Their families can't stay put in one place long enough for them to learn anything!  They often come to us with huge gaps, because as they move around they miss big chunks of learning.  These students definitely benefit from intervention, which can close the gaps and get them back on track.

Students with Learning Difficulties

There are students who, despite our best instructional practices and efforts, can't seem to overcome their struggles.  Additional testing is often required to determine if these students require the specialized talents of a special education teacher.

I hope this list gives you some food for thought.  I'd love to hear your comments and personal stories about intervention!

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