Tuesday, July 14, 2015

Sing it Out!

In my building I work in an area called "The Primary Pod". This means that we have two clusters of 4 classrooms with no walls between them. We have rolling cubbies between them, shelves, crates, but no walls.

Can you imagine teaching without walls? It's not easy to keep my small groups focused! Now, because we have no walls, I can hear every little thing that happens next door which includes a whole lot of singing about numbers, letters and sounds. And let me tell you, those kinders know their numbers, letters and sounds inside out and upside down. Songs really helped them to solidify these concepts.

One favorite of the class next door is "5 Little Ducks". You know the one. The mother starts with 5 ducks but one by one they swim away reinforcing the concept of "one less" for numbers for 5 down to 0. No worries though, at the end of the song ALL of the little ducks come back :) 

It got me thinking that there are tons of songs and poems that count up and count back reinforcing the add on and take away properties of addition and subtraction but not so many songs about the part part whole relationship of addition and subtraction.

So I wrote one :)

Perhaps I was inspired by the 5 Little Ducks but this song is called the 10 Penguins and is to the tune of "Little White Duck". It looks like there are a lot of words but it's really just the one verse repeated with different numbers!

As you sing, I would create an anchor chart and move the penguins as each moves into the pool so the students can see and keep track of the number of penguins on the ice and in the pool as the song progresses. I would also have my students hold out their hands to show the two parts and clap them together as they get to the end of the verse when they say "__ and __ makes 10!". After singing the song a few times I may even ask students to show a number bond or a number sentence that demonstrates what is happening in the song.

So there you have it, a song, some movement and visual cues all wrapped into one quick and easy activity! Click HERE to grab a copy of the song and lyrics to use with your kids! 

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Sunday, July 12, 2015

Chapter 11: Teaching Student-Centered Mathematics

Early ideas about place value... it's a big concept with so much nuance that it can be very easy to be a bit to narrow minded and miss an opportunity for a connection with your students. Think about the number 52 for a minute. What is it, how do you know? It could be 52 ones, a set of 5 tens and 2 ones. And in that case what is a ten? "One ten is ten ones". And if we wrote the digits 5 and 2 but switched the order it makes 25 which is an entirely different amount. Makes perfect sense... right?

WRONG! For our youngest students, forming ideas about place value means reconciling what they know about counting individual objects and applying that principal to counting sets of 10 objects to form a new unit. This topic is SO big in fact, that I am going to focus my discussion of this chapter on early place value concepts in terms of developing a concept of ten and tools/models for representing these concepts.

The text states that there are three stages a student goes through when understanding the idea of ten.

  1. Ten meaning ten ones. 
  2. "Ten" as the name of a set of ten ones with a model. 
  3. "Ten" as the name of a set of ten ones without the need for a model.  
I saw this transition so clearly in my students last year. A group of my tier 3 first graders did not have a concept of teen numbers come winter. In fact, they could only count up to about 12 before errors would occur. We played games daily counting around a circle, playing dice games on a number line, etc. and their rote counting to 20 improved dramatically. One game we played included a set of two ten frames and dinosaur eggs. Each egg had a number between ten and twenty. My kids would choose an egg and build that number on the ten frames with individual counters, they would then put the number into a bucket and build the next number. The first few days, I had kids who would clear the whole board in between numbers. We talked about how it took a long time to get out enough counters for the next numbers and they then transitioned to thinking about how to change the counters to make the next number. This activity was great for number relationships but I still had students who would count EVERY. SINGLE. COUNTER. to be sure they had the right number. I continued to prompt and ask the students how many counters were in the first ten frame "How many are there up here?" and without fail they knew that there were ten. From there, they began to use their knowledge of ten to count on when building a number or even to think about the relationship between teen numbers. For example, if I have the number 15 and I need to make 16, I know I can add just one more and that will make 16- I don't need to recount. A week and a half to 2 weeks in to this center, a few students started to notice the pattern in tens and ones. I started to hear comments like "14 is just a ten and a 4". I questioned about how they knew and how they could prove it counting by ones and counting on from 10. We "tested" the theory out with each number and found that their rule worked! At this point, we were still in stage 1 with my students recognizing a ten as ten ones and were dabbling in the idea of counting the first ten as "ten" but it was really only as a construct of counting on. 

The next time I made them math centers we moved from dinosaurs to fish and the novel new idea was that instead of counting out ALL of the counters to make the teen number, they could choose between individual counters and a connected ten frame that says the word "ten" on it. At this point, my students were becoming much more comfortable with the idea of calling something "a ten" as opposed to "ten" meaning ten ones. 

Even typing that sentence was confusing.... no wonder this is such a big concept for our little friends!! I'll fast forward to the end of the year quickly to tell you that my students did get to that third stage. I know because they were able to add and subtract ten from any number 10 to 99 in their heads without a visual representation and without the problem written out. They had a strong place value understanding that allowed them to understand two digit numbers and manipulate them! 

Now, rest assured that we did not spend 3 weeks just building dinosaur and fish numbers :) These were just a few activities that we did that represented their overall understanding of teen numbers. What I did not want to do was rush to the second stage of calling ten ones a "ten" before they were ready or really understood why we would want to use tens because they would have been able to rotely build these numbers for me but would not have understand how or why the math we were doing worked. 

So what else did we do during those three weeks? We used many, many, many representations of tens and ones and experimented with these tools. The text goes on to talk about models that represent place value concepts. These include: 
  1. Groupable base ten models
  2. Pregrouped base ten models
  3. Nonproportional models
Groupable models include items like straws, snapping cubes, beans, linking chains or any item that can literally be grouped together. A student can take a set of ten, connect or group them and that item is now called a "ten". This step is SO important for students. Without this step, kids will rotely use a tool called a "ten" but may have no idea of how it is made and what it means. Once students are comfortable with a groupable model, they can move to a pregrouped model. This is a tool such as strips and squares, ten frame cards or base ten blocks. They are great for convenience sake once a student has moved past the need to see ones grouped into tens and are less cumbersome to work with so a student can spend more time focusing on the MATH and less time focusing on sticking cubes together :) Again though, this tool is only useful once students understand what it is and how it was made. Used to early, these models can be confusing or students may use them rotely but lack understanding of the meaning behind the tool. When students fully understand the concept of how a ten is made and have an understanding of magnitude, you can move on to non proportional models such as coins, dollars or place value disks.

As I have said in most previous posts, this book is SO dense with information I really have only barely begun discussing this chapter. Really. Truly. So... be sure to check out the other blogs on the link-up below hosted by Adventures in Guided Math and/or grab a copy of the book for yourself!

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Thursday, July 9, 2015

Chapter 10: Teaching Student-Centered Mathematics

Chapter 10

Chapter 10 was all about helping students to master basic addition and subtraction facts meaning facts where both addends are below 10. The philosophy behind this chapter speaks a lot to my personal philosophy on math intervention in general. Learning basic facts is not meant to be a rote activity. If a student doesn't know, for example, 4 + 5 you could teach them that it equals 9 but, at the end of the day, all they would know is that 4 + 5 = 9. There would still be 99 other facts for numbers 0-9 that the student does not know. That is a LOT of facts to learn and memorize! If, instead, you take a more systematic and strategy based approach based on number relationships, your students will have the skills to efficiently reason through the facts that they don't know until they are known from memory.

Three Phases of Basic Facts

  1. Students will intially use counting strategies in order to solve addition and subtraction facts. This will include counting all and then counting on. When counting all, a student has an object to represent each and every item. These may be manipulatives or fingers. A student will then move towards counting on. This may originally look like stating the first amount and counting on the second amount but students will then move to the more efficient strategy of counting on from the larger number. 
  2. Students move next to reasoning strategies which cover a lot of ground. Reasoning strategies include the properties of addition and subtraction (adding and subtracting 0 and 1) along with the commutative property. This also includes using related facts. This includes doubles +/- 1, make a ten, finding five, finding a double, etc.) When teaching reasoning strategies, you can provide a story context that lends itself to a particular reasoning strategy or you can explicitly teach a strategy and then ask students to apply this strategy to a variety of other problems. In order to use reasoning strategies, students need to have a strong concept of how to decompose the numbers to 10. For example, if adding 9 + 3 using the make a ten strategy, a student would need to know that nine and one more makes ten. They also need to know that 3 is made of 1 and 2. Without a strong foundation and understanding of how to decompose numbers to 10, the reasoning strategies will be little more than a rote application of counting activities. 
  3. As students become proficient with reasoning strategies, students will move from reasoning strategies to knowing their facts from memory. To know a fact from memory is defined as a students knowing a fact within 3 seconds without using any overt inefficient strategies. 
A last note, the chapter stressed that timed tests are NOT a strategy for teaching facts! There is no reason why students should publicly display the facts that they currently know. This encourages quick rote memorization with flash cards and a move to quick recall too quickly so that students are not focused on the relationships between numbers. Instead, encourage a student to compete only against themselves and their ability to be more efficient than they were the last time they tested their fact mastery. 

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