One thing my district has been really giving a lot of attention to in recent years is number sense-especially the concept of "structuring" numbers. For those of you who aren't familiar with the term, I have shared below a small excerpt from the following article:
Structuring Numbers 1 to 20: Developing Facile Addition and Subtraction David Ellemor-Collins & Robert (Bob) Wright (Click HERE to read more!)
"In early addition and subtraction in the range 1 to 20, students can progress from using strategies involving counting by ones to using more facile strategies that do not involve counting. Researchers recognise this progression to facile addition and subtraction as critical mathematical learning, yet many low-attaining students do not make the progression successfully. There is a pressing need to understand how low-attaining students can progress to facile addition and subtraction, and to design instruction that facilitates such progress."
What we know to be true is that there is plenty of resource to support explicit teaching of "number"--starting with subitizing, then moving to structuring numbers to 5, 10, and eventually 20. What IS "structuring", you ask? Structuring numbers refers to the ability a student has to combine and partition numbers without counting. When you ask a child to list a "way to make 7", we want them to be able to instantly say, "6 and 1 or 5 and 2 or 4 and 3 or 7 and 0". If they need to think about it or count, they aren't "facile" with their structuring and this foundation needs to be strengthened.
This may seem like an odd thing to work on in the intermediate grades when our curriculum is filled with fractions and geometry and huge numbers and division...but this understanding that numbers can be broken up into parts and can be combined is the foundation of all of these! Don't we want our students to understand that 7/8 is the same as 2/8 and 5/8? Or that 1.1 is 1.0 + 0.1...and 0.9 + 0.2? Or that a right angle can be made from two angles that are 40 and 50 degrees? Or that 1,000 is a combination of 400 and 600--and 200 and 800..and so on? This is what is at the heart of structuring, and students who didn't "get it" with lower numbers are going to struggle with the more complex concepts. This idea of structuring is the foundation for addition and subtraction and, eventually, algebra concepts so we have to make sure they are securely grounded in the concept.
If you want to see how your students handle this, do a quick interview! Ask them the following questions...
"Tell me all the ways you know to make 5."
"Tell me all the ways you know to make 10."
"Tell me all the ways you know to make 13."
"Tell me all the ways you know to make 20."
What are you looking for? Can they do it quickly and do they sense the "pattern"? I love to notice which students are random with their answers and which students "get it"...that 5 is 5 and 0, 4 and 1, 3 and 2...some might even ask "Do you want me to do their 'other' fact?" or something similar (3 + 2 AND 2 + 3). It tells you a lot! Do they need their fingers? Are they obviously counting in their head? If they are, they may benefit from some intervention work with structuring. This is certainly not something we would expect ALL intermediate students to need--this is a concept that should be incorporated into math instruction in the primary grades, but as you and I both know--sometimes students don't get it when it is first presented...for a whole myriad of reasons.
There is WAY more to structuring than I am covering here...but I thought I'd give you a little something to think about--and a place to get started if you want to read more. If it looks like there is a great deal of interest, I can add in some more posts throughout the summer!
Using bead racks (rekenrek), linking/unifix cubes, or ten frames to show the "ten" and physically manipulating to make one more and one less, using base 10 blocks (usually down the road a bit), and other place value tools can be VERY helpful in letting students explore these ideas and really see and feel how numbers can be split and combined. If you don't have some of these tools in your intervention box, grab some! Hmmm...maybe I see another blog post coming up!
I thought I'd share a few games I use to help me when I'm either pulling students for an intervention group or want the students I have worked with to continue to practice. Students LOVE these games...and they are a great way to EASILY teach a concept and provide a way for practice. I will even send copies of the game home as replacement homework for these students. The first one I use with students who are working on structuring to 10...and then I have a second version that goes to 20 when they are ready. Make sure you don't rush it! The game is called "Block Out" and the goal is for them to roll dice and look for combinations of numbers to "block out" each number, 1 to 10. This forces them to really see how numbers can combine. I play it with them a LOT at the beginning to model my thinking. You might here me say things like, "I got a 1, 3, and 4. I know that 3 and 1 make four and 2 fours is 8." or "I got a 2, a 2, and a 6. I know my answer will even because all my choices are even." (I might even ask my "partner" to help me come up with ALL the possible answers for that one...like 1 + 2 is 4, 2 + 6 is 8, 2 + 2 + 6 is 10. This models for them how they need to be flexible with their thinking! Once they have really mastered "10", I bring out the manipulatives again and we explore the numbers between 10 and 20. When they are ready for "bare number" tasks--we play "Block out 20"!
Another great structuring game is "4 in a Row". This game is PERFECT to use with intervention groups or individual students who do not yet add 9, 10, or 11 fluently. It can also be available for math workshop or stations for students to build fluency once they are independent. Helping students “see” this is so important. Consider using base 10 blocks, linking cubes, bead racks, or other “hands on” manipulatives where students can really see how adding 10 impacts the tens place…and adding 11 impacts both the tens and the ones. Adding 9 is tricky for some students…showing them how
adding 10 and then “snapping off” a cube to show the “one less”. Using bare numbers (no visual model or manipulatives) should really only happen after students have used these hands on materials to build their understanding. Too often we ask students to “memorize” their facts before they understand the structuring concepts involved! Use this game to TEACH, then let students continue to play it to practice fluency!
So there is a little background on "structuring"--and I can certainly blog about some of the more "hands on" materials and even some "make your own" ideas if people are interested. Thanks for stopping by, and let me know if you would be interested in more information about this topic!