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Sunday, December 14, 2014

Children's Mathematics Book Study: Part 2

We are on to week 2 of our book study on Children's Mathematics.  I am so enjoying this book and this week's chapters really made me think about how I teach kids and what kind of problems I expose them to.  It also helped me see how far I have come in my teaching practice and how much of what I have done has been based on Cognitively Guided Instruciton.  

If you are just joining us, it is not to late to join our book study.  Grab a copy of the book and maybe a friend or two and jump in when you are ready.  Also, if this December is just to hectic for you, I will be starting another book study the second week in January.  Read more details about both book studies here.  

Here is the posting schedule for Children's Mathematics:
December 7: Chapters 1- 3
December 14: Chapters 4 & 5
December 21: Chapters 6 & 7
December 28: Chapters 8 - 10
January 4: Chapters 11 - 13

I will post each Sunday morning and share it on my Facebook page.  Please join in by leaving a comment on my blog post or Facebook page.  If you have your own blog and want to write a post about the book that works too!  Add your link in the comments section here.  Thank you to all who shared last week!

Chapter 4:  Multiplication and Division: Problem Types and Children's Solution Strategies 

3 Possible Unknowns

I loved the way the chapter started with the big ideas around multiplication and the 2 different types of division problems.  When I first learned about partitive versus measurement (sometimes called ) division, it completely changed my teaching practice.  Upon examining the problems in my math program and supplemental materials I was using with students, I came to the conclusion that most of the division problems were partitive or sharing problems.  When I first started asking different problem types, it was hard for my students who had not been exposed to these types of problems.  Now I make sure kids have experience with the 9 different problem types for multiplication and division whether we are working at the fact level, with double multiple digits or even with fraction and decimals.  

Primary Students Need to Solve Multiplication and Division Story Problems

Reading page 71 was a big wake up call for me.  I have been so focused on making sure my primary students are moving along with their additive reasoning and being exposed to the 12 problem types for addition and subtraction that solving multiplication and division types story problems with primary students as kind of fallen by the wayside.  Reading about how these problems support place value development and how "base-ten problems are essentially multiplication or division problems involving groups of 10" reminded me how important it is for primary students to solve more than addition and subtraction problems.  A resource I used a great deal when getting started with problem solving with little kids was the Kindergarten Kindergarten blog and it is definitely a resource I need to revisit.   

Further Reflection

I did take a peak at the questions for further reflection in the back of this chapter and LOVED number 2.  "Choose a book that you are reading with your students or a topic that you are studying.  Write one problem for each of the problem types that relate to the story or topic."  I think this will be a great way to ensure my students get experience with these types of problems.  I will be choosing a few books and doing this and will be sharing these with you as part of my Monday Math Literature posts.  Check back on Monday to see the results!

Chapter 5: Beginning to Use Cognitively Guided Instruction 

When I first started using Cognitively Guided Instruction it was hard.  I felt like I wasn't teaching.  If I didn't tell kids how to do it, how would they learn?  It definitely took some time and a few disastrous lessons before I realized that by allowing my students to construct their own learning was giving them a deeper, more meaningful understanding of math.  This chapter had some great ideas for getting started.  Here is what I think the 5 most important things are to remember if you are just starting out.

1) It is your job to stop "teaching" and start listening.  You REALLY need to listen to how your students are solving problems and not just assume you know what is going on.  
2) Make sure you are using problems of different types.
3) Be strategic in who you choose to share their thinking with the entire class.  Make sure you get a variety of ideas.  This is not the time to use call sticks or other random generators.  
4) Talk to your students about efficiency.  When multiple kids share strategies talk about how efficent each one was.
5) The hardest part can be starting!

I am looking forward to hearing your thoughts on this week's reading!  



  1. Chapter five looks like a lot of fun!

    1. It is a lot of fun! When you teach using CGI, it can be hard at first but it really is fun every day to see how kids solve problems and to see students (and teachers) learning from each other.

  2. Ch. 4 Multiplication & Division:

    As we learned in the addition and subtraction chapters, there are also problem types in multiplication & division and developmental solution strategies that children will typically choose as they progress in their understanding and sophistication in solving tasks.

    Children will initially solve multiplication & division problems by directly modeling the action and relationships described in the problems. In order to solve with direct modeling, children must recognize that they can simultaneously count groups of objects as well as individual objects. This understanding is indicative of the unitizing concept which is a major milestone for young learners and is the foundational basis for place value concepts.

    Children eventually replace direct modeling strategies with counting strategies, however, it is more difficult in multiplication & division since the counting will most likely involve skip-counting. Again, this involves a conceptual understanding of unitizing or the knowledge that groups of objects can be counted in the same way that single objects are counted. Eventually children use facts they know from memory to derive unknown facts. If we allow plenty of time for children to use, discuss, and reflect on derived fact strategies, without rushing the memorization of multiplication facts, there are several long-term benefits.
    1. Children develop an understanding of properties of multiplication and the relationship between multiplication & division as they develop the ability to accurately and efficiently solve multiplication & division problems.
    2. The focus on making sense of number facts also contributes to helping children see that mathematics can be understood and reasoned about rather than learned as a collections of unrelated facts and procedures.

    It is suggested that experiences solving multiplication & division problems in kindergarten & 1st grade can provide a foundational understanding of multiplication, division, and base -ten concepts in later grades. I agree with this statement but would caution that the emphasis should be on the development of initial grouping and partitioning activities. The authors specifically state that it is not necessary to introduce the symbols for multiplying and dividing in the early grades.

    Ch. 5 Beginning to Use Cognitively Guided Instruction:

    This chapter explained that in CGI classes, most student learning occurs as children engage in purposefully chosen problem-solving tasks. These problems might be posed as word problems (verbal or written) or symbolic equations but have been specifically selected to advance students’ understanding and sophistication in solving. Students develop and share their own problem-solving strategies, which become more efficient and abstract over time.

    The Number Talk book will be a great follow-up to this book and will be very helpful as it provides number sentence strings that encourage advancing to derived fact strategies. Your list of “the five most important things when starting out with CGI instruction” is “spot on” especially #1, the importance of listening and questioning, and #3, strategically choosing students to share strategies. In regard to #4, I am always very cautious about discussing efficient strategies because an efficient strategy for one student might be difficult for another student that is not at the same place developmentally. Efficiency is often equated with speed, so my personal preference is to focus on successful strategies that make sense to individual students, especially in the beginning. I thought the video clip 5.1 was a beautiful example of listening, questioning, and sharing strategies in a first grade CGI classroom!

    Thanks again for choosing this book and providing a forum to discuss it with other educators!

  3. Hi Lori! I am so glad you are participating and am thankful that you take the time to share your thoughts. "In order to solve with direct modeling, children must recognize that they can simultaneously count groups of objects as well as individual objects. This understanding is indicative of the unitizing concept which is a major milestone for young learners and is the foundational basis for place value concepts." Excellent point! Unitizing is really the key to so many things. I also like your point about not worrying about introducing the multiplication and division symbols in primary grades and just focusing on grouping and partitioning activities.

    I am very interested in others' takes on efficiency in problem solving. The reason I emphasis efficiency is because sometimes kids get so wrapped up in really cool ways to solve problems that they go out of their way to come up with this long and detailed solution. What looks efficient to a first grader is going to be very different than what looks efficient to an older student. The older my students get, the more I talk with them about efficiency.