Monday, September 15, 2014

Are 6 x 5 and 5 x 6 the Same?

9/18 Update: This post, originally titled 6 x 5 and 5 x 6 are NOT the Same! has generated the most amazing conversations!  I have left the post as written, but I just felt that I needed to change the title. Enjoy the dialogue! :)

Greetings from the long lost Math Coach!  I am now officially a math interventionist and lovin' life! My 4th graders have been representing multiplication with square tiles and writing equations to match pictures of arrays (like the ones shown above).  These investigations have uncovered a misconception in their thinking that I wanted to share with you tonight.  While 6 x 5 and 5 x 6 have the same product, they are not the same thing.  The multiplication sign really means "groups of", so I encourage students to read 6 x 5 as 6 groups of 5. This practice helps students make sense of multiplication.

I have two soap boxes to stand on tonight, so please humor me.  First up is my CRA soap box. Making the switch from additive thinking (6 + 5) to multiplicative thinking (6 x 5) is a huge transition for students, and it must begin with LOTS of concrete (hands on) experiences. Over time, you need to overlap the concrete, representational, and abstract stages of learning to help students smoothly bridge the distance between concrete and abstract.  Look, for example, at the cards below. After selecting these two cards, students can use counters to make 3 groups of 5 (concrete), draw 3 circles with 5 stars in each circle (representational), and write the equation 3 x 5 = 15 (abstract).  All three stages of learning in one activity.
Next soap box.  As teachers, let's be sure we are using precise language.  If you are describing a multiplication equation, be sure that you are calling the numbers being multiplied factors and the answer to the multiplication problem the product.  I've also found that students are easily confused about rows and columns.  A simple anchor chart showing this vocabulary is a great reminder for both you and the students.
The images shown above are from my There's Nothing Alien About Multiplication unit.  It includes 4 different games and a themed multiplication division chart.  Click here to download one of the games for free!

Tuesday, August 26, 2014

The Q Factor: Adding Quality to Our Questions

One of the neat things about being an educational blogger is that you get to connect with educators across the country and even the world.  This past summer I attended the Teachers Pay Teachers conference in Las Vegas, and I had the extreme pleasure of meeting Greg Coleman, aka Mr. Elementary Math.  Naturally, we hit it off, and I knew that collaboration would be in our future.  Tonight, I present my very first guest post, written by Greg.  There's even a freebie at the end!

The Q Factor

Do you find yourself asking questions like, “What place does the underlined digit hold in the number 256?”  Many of us ask questions like this during our math instruction. 

As educators, we find ourselves looking for ways to ask questions that will both challenge and grow our students’ understanding.  It is important that we strategically plan lessons to incorporate higher level, open ended questioning.

Take a look at the questions below and consider how they differ.  Which question drives to the heart of student understanding?

After looking at the questions above, you may be thinking  “How can I easily create questions that will help drive to the heart of student understanding?”  Good Questions for Math Teaching, by Peter Sullivan and Pat Lilburn, is an excellent book that supports educators in asking good questions.  According to the authors, good questions contain three characteristics:
  1. They require more than remembering a fact or reproducing a skill.
  2. They build students’ learning through answering questions, and the teachers’ learning about their students from their attempts.
  3. There may be several acceptable answers.

You may be asking yourself, “How can I gather or create questions that fit the 3 characteristics?”  There are 2 methods that are used by Sullivan and Lilburn to address this concern.

Method 1: Working Backward
·         Step 1 – Identify a topic
·         Step 2 – Think of a closed question and write down the answer.
·         Step 3 – Make up a question that includes (or addresses) the answer.

Using method 1 may look like this:
Now it is your turn! Use the example above to adapt a question with the answer in mind.

Method 2: Adapting a Standard Question
·         Step 1 – Identify a topic
·         Step 2 – Think of a standard question.
·         Step 3 – Adapt it to make a good question.

Using method 2 may look like this:

Now it is your turn! Use the example above to adapt a question with the answer in mind.

Take a look at some additional questions for elementary grade students that followed the methods above. Click on the images to download the Quality Questions template.

Though this process takes time, I challenge you to try it at least 2 – 3 times each week.  Notice the difference that great questioning techniques have on student thinking.  It is worth a couple of additional minutes to move our students in this direction as the Common Core Standards are requiring more from all of us. 

P.S. This is my VERY 1st guest blog post and I really want to thank Donna for extending this opportunity to me.   Donna is the best!!


Saturday, August 23, 2014

Math Books That Will Change Your Teaching Blog Hop

This week I am participating in a blog hop with a group of amazing educational bloggers, and we are sharing professional resources that have changed our teaching.  There are so many great books to choose from, and you will read about many of my favorites along the blog hop.  After reading my post, simply click on the link at the bottom of the page to follow the trail to the next stop.  Oh!  And did I mention that we are each giving away a copy of our book?  So be sure to visit each stop and enter the giveaway! 

Guided Math Conferences, by Laney Sammons

In 2009, Laney Sammons' book Guided Math was published and, quite honestly, it changed the way we think about teaching math. In this groundbreaking book, Sammons outlined a framework for teaching math that was based on tried-and-true literacy instructional practices.  I think the group impacted the most has been primary teachers (K-2), who often felt very confident teaching reading, but usually less so teaching math.  By connecting a process they were familiar with, Guided Reading, to math instruction, she empowered teachers in a whole new way.  As a math instructional coach working with K-2 teachers, I felt I was speaking their language when discussing Guided Math.  I saw their confidence grow, right along with the achievement and confidence of their young mathematicians.  It was, therefore, an absolute thrill to meet Laney at the NCTM Annual Meeting & Exposition in New Orleans this past April.  In addition to being an inspirational author, she is also a genuinely warm person.

Since publishing Guided Math, Laney has continued to write and publish additional books to support the Guided Math framework. In January of this year, she released Guided Math Conferences, and that is the book I am writing about today.  My current focus for personal professional development and research is the use of accountable talk in math instruction.  So when I saw that Laney had a new book on conferring, I snapped it up!  It did not disappoint.

Let me first say that even if you do not teach using the Guided Math or workshop approach, there is still much to be learned from this book.  Throughout the book, Sammons addresses the fact that not all teachers use the Guided Math structure and provides suggestions for implementing math conferences regardless of your instructional setting.

Buried deep in the book, I found this gem of a quote that sums up the value of conferring with students: "When teachers confer with students, student thinking becomes more visible--to teachers and to the students themselves."

In Guided Math Conferences, Sammons provides a compelling, research-based argument for making math conferences a priority in your classroom, citing the emphasis on increased rigor and critical thinking skills, the focus on the CCSSM Standards for Mathematical Practice, the value of formative assessment, and the importance of student learning goals, among other things.  As with all of her books, though, she addresses the practical aspects of implementation, with chapters that tackle the most common reasons why conferring does not take place (time and managing the other students during conferencing) and provide step-by-step instructions for managing conferences, including helpful forms for recording conference notes.

You may be thinking, "I do small-group instruction.  Why do I need conferencing, too?"  Sammons covers that early on in the book, comparing and contrasting the math conference with a math interview and small-group instruction.  Later in the book, she also describes the connection between conferring and small-group instruction and explains that the conferring process compliments small-group instruction by providing insight into the needs of students and data for grouping.

When you're ready to get started, she outlines the structure of a guided math conference and explains the different types of conferences.  That's right!  Conferences are not one-size-fits-all.  Snapshots from a variety of grade levels provide glimpses into the different types of conferences. 

Like all of my other Laney Sammons books, this one will soon be dog-eared and worn.  It is a book you will continue to use and reuse as your refine your teaching practices.

And now for the best news!  Laney has graciously agreed to provide a signed copy of her book to the winner of my giveaway!  My sincerest thanks to Laney and Shell Education for their support. Please be sure you fulfill the requirements for each entry--I will check the validity of entries before awarding the prize.

a Rafflecopter giveaway

Ready for the next great book?  Head on over to Beyond Traditional Math!

Beyond Traditional Math

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