Monday, July 28, 2014

Book Study Monday: Teaching Numeracy, Critical Habit 7



It's Monday, so that must mean it's time for great math discussions! If you're just joining us, we're reading and discussing Teaching Numeracy, 9 Critical Habits to Ignite Mathematical Thinking, by Margie Pearse and K. M. Walton.  

Reading Schedule

June 30
July 7
July 14
July 21
July 28
Aug 4
Critical Habits 8 & 9
Aug 11
Essential Components 1, 2, & 3
Aug 18
Essential Components 4 & 5

Habit 7: Summarize, Determine Importance, Synthesize Using Note Taking and Journaling

"Writing shifts the responsibility for learning away from the teacher and toward the students by encouraging personal learning"

As the authors note early on in the chapter (p 79), the three skills discussed in this chapter--summarizing, determining importance, and synthesizing--have "the power to force students to own their mathematical thinking and be able to take it with them into the real world."  Of course these are not skills that only support math learning.  In fact, they might be more often thought of as language arts skills.

Although we don't use textbooks for math instruction on my campus, I have used the textbook scavenger hunt idea with science texts in the past, and I think it's a great way to introduce students to the features of a nonfiction text.

Likewise, because we are an elementary campus, we don't do much note taking.  When I've seen it in the classroom, it has most often taken the form of the Interactive Cloze System described on page 86.  The teacher prepares an anchor chart prior to the day's mini-lesson, with blanks in place of key words and phrases. Students have a printed copy of the same anchor chart.  Students interact with the teacher during the mini-lesson to fill in the missing words. After the mini-lesson, students glue their completed work into their math journals.  I'm curious to hear what other elementary teachers say about note taking.

From note taking, the chapter moved on to journaling.  Journaling is huge, in my opinion, and should be an ongoing part of math instruction at every grade level.  I particularly liked when the authors referred to the concrete, representational, abstract sequence of instruction (p 92) and the way that math journals can be used to move students from the concrete to the representational and abstract.  It's so important that those stages overlap, and journaling is the perfect way to connect them.

The authors addressed and dismissed the elephant in the room--the time required for journaling--by citing research that journaling can help reduce the need for review and reteaching (p 93).  I think that underscores the fact that when students put their thoughts in writing, they "clarify their thinking, and as they write, they expand their knowledge of the math." (p 92).  In other words, learn it deeply the first time around and you won't need to review and reteach.

I loved the sentence starters on page 96, because students often just don't know where to start to put their thoughts in writing. And, I think, teachers can fall into a routine and overuse certain prompts.  I knew Margie wouldn't mind, so I took the liberty of creating a poster with some of the prompts.  Like the Math Fix Up Tools from Habit 1, I also made a smaller, B&W version students can glue in their journals.  Grab the writing prompts poster here.

If you are not using exit tickets in your classroom, be sure to read pages 96-97 and give them a try.  They are a quick, easy, and extremely powerful formative assessment tool.  Once you have the stack of exit tickets, you can easily sort them to create your small groups for the next day's instruction.  You don't need fancy forms for the exit tickets--plain index cards work great.

While there were many practical activity suggestions offered on pages 98-101, my favorites were the 3-2-1 Journal Entry (p 98) and having students Compose Problems (p 99).

I have used the 3-2-1 format with adult learners in professional development sessions, but I like the idea of using it with students. I also like the suggestion provided for changing up how to use the 3, 2, and 1 (examples of learning, nonexamples, ways to tell the difference).

An activity I have used extensively is what I call You Write the Story, which is a fancy name for having students compose problems.  You give the students an equation, such as 24 x 6 = £, and they must write a story problem that can be solved using the equation.  This is a VERY difficult activity for some students and takes a great deal of modeling, but it really does shine a spotlight on their understanding of the operations.  You can easily differentiate the activity by using larger or smaller numbers; different structures, such as 68 + £ = 133; and even two-step equations, such as (24 x 6) - 32 = £.

Your turn to sound off!



Sunday, July 27, 2014

The ABCs of Number Sense


Number sense.  A phrase that we hear a lot in the math world. Some kids have it and some don't. Many adults lack number sense! But what do we mean by number sense, and how do we help our students develop a strong sense of number?

I think it is important to have a working definition of number sense, and I like this one:
“…a person's general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems” (Burton, 1993; Reys, 1991)
—from NCTM’s Illuminations website
Notice how it incorporates the ideas of understanding number and operations, flexibility, mathematical judgments, and problem solving.  It also makes clear that we learn mathematics to use it.

Children come to school with a sense of quantity and number (Sousa, 2007), and just as the primary grades lay the foundation for strong literacy skills, they also set the stage for number sense and numeracy.

A is for Accountable Math Talk

What does your math classroom sound like?  Who does most of the talking during your math instruction--you or the students?  Most of us experienced quiet math classrooms growing up with little student collaboration, but all that is changing.  Wonderful books like Number Talks (Parrish), Classroom Discussions in Math (Chapin), and Intentional Talk: How to Structure and Lead Productive Mathematical Discussions (Kazemi & Hintz) expose us to the importance of student discourse and provide guidance on developing a classroom that thrives on rich mathematical discussions.  In her book Math Sense: The Look, Sound, and Feel of Effective Instruction, Christine Moynihan says it this way:
"I want noise and plenty of it--productive, purposeful, and meaningful noise--from everyone, students and teachers alike."
So how does this look in your classroom?  Simply put, it looks like you talking less and your students talking more!  From the first day of school, you want to create a culture in your classroom that supports respectful and productive discussions.  As you plan your lessons, look for opportunities for students to talk with each other. If you find yourself talking too much during a lesson, have the students do a quick turn and talk.  Develop rich questions that engage and challenge students and guide them toward the learning, rather than handing it to them on a silver platter.

Relinquishing the role of sage on the stage will not be easy.  But you will be amazed at the rich conversations your students will have and the deep learning that will result when you step back and let them do the talking.

B is for Bruner's Concrete, Pictorial, Abstract Approach

Ask a class of 5th graders this question, and you might be surprised by how many students answer one-eighth.  One reason for this misconception is that many students lack a mental image for these fractions. If our students are to have strong number sense, it is essential that our mathematics instruction provides students with the experiences necessary to develop deep, conceptual understanding.

The concrete, pictorial, abstract (CPA) approach to learning originated in the 1960's as a result of the work of psychologist Jerome Bruner.  It is often referred to as CRA, with the word representational taking the place of pictorial.  Bruner (and countless researchers since) determined that mathematical content was best learned when it followed a progression from concrete, hands-on experiences, to pictorial representations linked to the concrete learning, before finally reaching the abstract, or symbolic, stage.  To illustrate the sequence using a fraction example, students would first have numerous hands-on experiences with concrete models of fractions, such as Cuisenaire rods, fraction strips or tiles, fraction circles, etc.  At some point, students would begin drawing representations of the concrete materials they are using.  Notice how the concrete and representational overlap. Finally, the abstract, or symbolic representation would be introduced, again overlapping with the previous stages.

Now think about how math is taught in many classrooms--with textbooks and worksheets.  Do you see the problem?  Not very concrete, right?  That's not to say that we need to throw out all of our textbooks and worksheets.  We just need to make sure that we are consciously and consistently providing concrete experiences for our students, and that's actually pretty easy to do.  When topics are presented in abstract form on a worksheet or in a textbook, provide manipulatives for students to use and then have them represent their solution pictorially.  So students model 2 + 3 (abstract) with counters (concrete) and then draw their solution in their math journal (pictorial).

C is for Composing and Decomposing Numbers

With the introduction of the Common Core State Standards for Mathematics, the phrase compose and decompose numbers positively exploded onto the mathematical scene.  Don't believe me?  A Google search on the phrase 'compose and decompose numbers' now returns over a quarter of a million hits!

In plain terms, composing and decomposing numbers is basically the understanding that numbers can be combined (composed) to make bigger numbers and bigger numbers can be broken (decomposed) into smaller numbers.  In other words 2 and 3 can be combined to make 5 (composing) and 5 can be broken into 2 and 3 (decomposing).

While Kathy Richardson notes that "If basic facts are to be foundational, they must be based on an understanding of the composition and decomposition of numbers", the power of part/part/whole thinking goes way beyond fact fluency.  Consider the following examples showing how the ability to fluently decompose and compose numbers allows for flexible computation with not only basic facts, but also with multi-digit computation and even measurement concepts.


Kindergarten and 1st grade are critical for developing this deep understanding of composing and decomposing numbers.  Activities should be differentiated so that each child is working on his or her own "number" as determined by some variation of a "hiding assessment". For example, show a student a train of 5 linking cubes, then put them behind your back and break some off.  Show the partial train to the student and ask "How many more to make 5?"  If the student knows all the combinations for 5 with automaticity, try the combinations for 6.  The number that a student stumbles on becomes their number.  Use a variety of activities such as number bracelets, dot cards, and Shake and Spill to differentiate composing and decomposing practice.


Starting a new school year is both exhilarating and stressful. There is so much great research about instructional best practices that we often feel overwhelmed about where to start.  As you head back to school this fall, just remember your ABCs!
Head back to school with these great products for building number sense: Decimals and Fractions, A Year of Shake and Spill, Apple Math, and Twenty Apples

Saturday, July 26, 2014

#TrendingInMath



As the calendar creeps toward August, many of us have Back to School on our minds!  Check out these great ideas for getting your year off to a great start!

From the Blogosphere...

Back to school is around the corner for many of us, and Jen from Runde's Room compiled a wonderful list of first day tips!  There's also a FREE bio poem template you can download.
A life-sized number bracelet?  Check out this awesome DIY project using pool noodles and hula hoops from Reagan at Tunstall's Teaching Tidbits!
Looking for an engaging B2S gift for your kiddos?  Check out this glowing idea Laureen from Teaching with Laughter shared on the collaborative blog The Teaching Tribune.  You can download the labels for PreK-5th for FREE!!
Found on Pinterest...

math flowers. Give each kid a different number, make a whole garden! could adapt to upper elementary grades, too.
LOVE this number flower!  Download a FREE template.
Blog posts with math workshop ideas
This pin was a real find!  Be sure to follow the link over to the blog and lots of great posts on math workshop!


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