So! I'm SO EAGER for your reactions to this book! Once I started highlighting, I couldn't stop! And that was just in the introduction!
What do we do when the United States is lagging behind in mathematics education? We visit with the country that is doing the BEST job and learn from them! Singapore continues to lead the pack. We can summarize Hazekamp's introduction with the following key points:
1. C-P-A (Concrete-Pictorial-Abstract), or the P can be R for representational
2. Gradual Release
3. Teacher Think alouds (Modeling is important!)
4. Manipulatives are necessary!
5. Place Value is CRUCIAL!
These are simply good instructional practices. Let's move on to the addition chapter, where the real excitement begins!
Chapter 1 - Addition (pages 11-35)
When I first heard the term "number bonds", I pictured a new type of glue stick hitting the market. It wasn't a term I was familiar with at the time and the bubbles looked kind of weird to me, but I realized it wasn't much different than part-part-whole. The neat thing about number bonds is that they grow as the students grow. Kindergarten students will be using number bonds to compose and decompose numbers to 20 and 5th graders can use a number bond to compose and decompose fractions. I like the examples below because I did not see an example in the book of number bonds with more than two parts but it is certainly possible.
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| What do you think this number bond is showing and for what grade level? |
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| 5th Grade Number Bond (decomposing fractions to add) |
Did you notice that the traditional, or standard, algorithm was listed as the very last thing a student does? Prior to that, students need to master the conceptual understanding of the mathematical concept and that means lots and lots of hands-on experiences in the classroom.
Reflection Questions:
What other applications of number bonds did you find interesting and think you might try with your students?
What was the toughest concept for you to think about trying with students?
How does your current math textbook present addition to your students? For example, if you are teaching 5th graders to add fractions or 2nd grade students to add two-digit numbers, how is it presented to students at first? Concretely, pictorially, or abstractly?
First, let me start by thanking you Carrie for organizing this book study. I always have good intentions, but not always the best follow through. By doing this book study with a group I will be held accountable and I can share my math geekiness with others who can appreciate it, rather than roll their eyes and think ~oh great, here she goes again talking about math and school.
ReplyDeleteFirst Image* I'm thinking prime factorization used/taught in 5th grade. What an asset it will be if students already see numbers in this way prior to embarking on this topic! The ground work will have already been laid and students will be able to focus their comprehension skills on the purpose of factoring rather than how to decompose the number 8.
Second Image* It never occurred to me to teach improper fractions this way. I teach students to add the numerators (parts of the whole), keep the denominator the same (whole), and then convert the improper fraction to a mixed number if needed. With this strategy the steps are simplified all while maintaining the distinction between the whole parts. An improper fraction is difficult for students to visualize. With this example the improper fraction does not even muddy the waters.
Reflection Questions
1. I find teaching time and measurement to be a difficult concept to teach for two reasons.
One – there are just so many units. Years, months, weeks, days, hours, minutes, seconds, and don’t even get me started on measurement!
Two – the amount of time allotted to teach the concept
I will definitely use number bonds to help better illustrate this difficult concept. Students can “bond” the numbers in units of measurement, rather than base ten.
2. The toughest concept for me to try will probably be working with fractions with unlike denominators. Students will have to really have a firm grasp of fraction equivalencies.
3. Well…….I don’t exactly follow the textbook when teaching new concepts. I probably use more hands on activities than most 4th grade teachers, but I still find myself talking about the algorithm first (abstract) and working my way backwards towards concrete to “prove” how it works by using pictures and manipulatives.
For the 1st image, that was a good guess! I didn't see it that way. It's actually a 2-3rd grade number bond showing part-part whole (or in this case, part-part-part-part-whole). In 3rd grade it can be used to show repeated addition leading up to multiplication facts.
DeleteHow do you think working from concrete to abstract (manipulatives to algorithms) will work for you this coming year?
I was so excited when we were doing the math curriculum map training last week with you and you started talking about the CPA approach. I am really loving this book so far. I like that it breaks down the steps of what you need to do and gives it to you in an example. It’s all right there for you. It was interesting to see the order in which to teach the kids the strategies as well. I’ve always taught the kids multiply ways to solve a problem. Then after one student shares their answer and how they got it, I ask who got it a different way and ask them to share.
ReplyDeleteThe first picture with the number bond I think could be a first grade problem (for your higher kiddos). It shows another way to make 8 (a different way of thinking than the basic 4 + 4, 2 +6, etc).
I am extremely excited to use number bonds with the kids. (Already found a bunch of ideas on pinterest). I will definitely be using them with addition and figuring out the missing addend.
The toughest concept will be the left to right addition for both myself and the kids. A good foundation in number bonds will help with that though. It is hard for the kids to think deeper and it will take a lot of practice, time, and modeling.
Our current math textbook presents addition a few different ways. There are some making ten, ten plus 1, and ten plus 2 strategies. I could see how that was related to making ten in the book. I think it is presented concretely with manipulatives, then pictorially, and finally abstractly. Not sure there is a whole lot of practice in each step though, especially the first one where there needs to be lots of manipulatives.
I think the making 10 to add/subtract is going to work great with the kids. If we start concretely, which primary teachers are especially good at doing, and build towards mental strategies, the students are going to be math superstars.
ReplyDeleteI love that this book reinforces what we are presenting at the trainings this summer with the new maps! Hoping that teachers will see the benefits.. Can't wait to continue reading!!
ReplyDeleteOne of the reasons the book was picked derived from the need to give further support. I love it even more that some of our paras have joined this talk o our RtI can be even stronger!
DeleteIs this working now?
ReplyDeleteYup! Welcome!
DeleteAfter the Stages of Addition (pg. 12), I wondered if anyone gave themselves a test question for practice. Then I wondered if any one of the stages was trickier to learn (to teach) than another? I think, depending on the grade level being taught, that Number bonds would be the trickiest to learn how to instruct. In K-2, adding with them is easy but you have tried making a 10 to subtract with them?
ReplyDeleteThe students, prior to subtracting, would have been making 10 to add numbers. This is a place value strategy. They would move on to making 10 to subtract.
Consider the following: 23 - 9
(20) (3) - 9
20 - 9 = 11
3+11 = 14
So the answer is 14.
Want to try? Hammer out 32 - 8.
Hammer time: (30 - 8) + 2 = 24
ReplyDeleteI wasn't familiar with the language, number bonds, but I am familiar with the concept; a number bond shows how a number (whole) is made up of parts. I thought it was really cool to use this method with fractions. When Hazekamp decomposed the equation, using number bonds, I thought it was brilliant. The equation was
3/5 + 4/5 and she manipulated the numbers to make it easier to compute. (3/5 + 2/5) + 2/5= 1 2/5
I kept thinking to myself...why didn't I think of that??? It only makes sense to create a new equation that is compatible and easier to compute. I honestly think that Number bonds will need to be taught from day one and used every day to really get the concept, but it will be a lot of fun. If we start this method from day one we are going to have the students thinking deeper. Before you know it, they'll begin seeing the compatibility of the numbers for themselves and will begin moving numbers around in all different kinds of mathematical computations. This actually thrills me! I am a total geek, and things like this really stokes me to the max!!
I totally agree with the C-P-A method; you cannot possibly expect students to understand the abstract until they've gone through the concrete and pictorial first. Oftentimes you can combine the concrete/pictorial with interactive technology and move to the abstract. I think our book/digital lessons attempts to do this, but it's really not enough; the students are too accustomed to it and tend to zone out if this is all you use. You really have to mix it up with a variety of methods, and this book will definitely help us add more tools to our teacher tool box. I think that is amazing!
Please let me know if this works.
ReplyDeleteYou are a go!
DeleteI think I am in😊
ReplyDeleteI think the biggest aha for me was using number bonds for fractions and measurement conversion. Using the model that is so easy when adding in general, really helps with composing and decomposing fractions as well as mixed numbers. Using the bonds to help student understand how many grams, for example are in a kilogram is just brilliant!
Taking the concept of fractions to create and understand equivalency can be daunting but I'm so excited by the new strategies that nothing feels tough. I'm sure that will change. Envisions spends most of its time using the standard algorithm despite it not being part of the standard or only a pice of the standard. It often speaks to using other strategies or manipulatives but the lessons don't teach them.
ReplyDeleteI agree! It also does not offer enough concrete experiences to students prior to the abstract tinking it wants them to master. I tink ,ost textbooks operate this way.
Delete