The Powerful Array

How many meatballs do you see?

How many do you see now?

See how arrays are so powerful?

An array is a systematic arrangement of objects. This model is as important to multiplication and division as the number line model is to addition and subtraction. The visual representation of rows and columns helps students as they develop their proportional reasoning. Like the part/whole box for addition and subtraction, the array identifies the parts (factors) and the whole (total area of product) and can be used to demonstrate and prove student strategies. (Number Talks, Helping Children Build Mental Math and Computation Strategies)

As student move on beyond the basic multiplication and division facts, the array can model and prove partial products which is also called the distributive property. Students build arrays with color tiles, base-ten materials, or on grid paper.

Look at the array below. How many different ways can you find the answer to 8 x 25?

Here are just a few different ways that can be proved by cutting or folding the array.

Halving and Doubling:
(4 x 25) x 2
Four 25s equals 100 (think money) and double it you get 200.

Partial Products:
(8 x 20) + (8 x 5)
Eight times 20 is 160 and 8 times 5 is 40. 160 + 40 = 200

Using Multiples of 10:
(8 x 10) + (8 x 10) + (8 x 5)
80 + 80 + 40 equals 200

Here is an example of how base-ten materials can be used to build a 24 x 36 array. How would you count this?

Click here for a GREAT lesson:
So Many Ways to Separate Arrays

Now, share how you use arrays!

Guest Blog: Transforming Math Homework

About a week or so ago, Rayla Rucker, a fourth grade teacher at Pleasant Hill Elementary, asked me what I thought about some math homework she created for her class. As I opened the Google Doc, my eyes got huge and my heart jumped out of my chest. It was the best homework I have ever seen in math! So, I asked and Rayla agreed that she would share her thoughts on homework as our guest blogger this week! Thank you, Rayla, for inspiring us all as you do what's best for your students.

Homework.
There isn’t another word in the English language that elicits more dread.  Kids don’t want it, parents either want more or less of it, and teachers (myself at least) don’t have time to grade it…so, usually it is a meaningless endeavor that no one is happy about.  I mean, sure, there will always be student’s that have it in your hand before it’s due, but there will also be kids that never, EVER turn it in no matter how many times you call or email.  

So, how did I get all my kids and a lot of my parents (I can only speak for the ones that have tweeted or emailed about it) excited about math homework???

It all started with a tweet.

If you talk to Mark Koller, my principal, about me,  he’ll probably joke and say...”Be careful, she’ll tweet about you!”  because he knows I have gone a little twitter crazy.  The above tweet was one of my first with a real life problem solving idea.  I thought, how can I make kids and parents see that math isn’t just something you do for 75 minutes a day 5 days a week?

And then this happened,

and I thought, “Well that’s pretty awesome! Maybe I’m on the right track.”

This was just the beginning of a shift in thinking about what math is, or can be, or maybe should be.

I have a student that just can’t stand math.  He rushes through worksheet assignments and turns them in without a second glance with an abysmal degree of success. One day he was done, and I was meeting with a group and I just could not get him to settle.  I was irritated.  So, I called him up, handed him one of those catalogues we all get, full of junk to buy for our  classrooms. I said, “Hey, Johnny (names changed to protect the innocent),  I have $50 to spend on the classroom store.  Can you take this catalogue and see how much we can get?  I want the most and best for the money. Write down the items, how many we’ll receive, and the total spent and give it back to me.”  Well imagine my shock when he spent a couple of days quietly studying,  adding, subtracting, multiplying, and dividing whole numbers and decimals.  He was doing a whole lot of math without even realizing it.   A reluctant student was all over it, AND the math was correct.  The difference in real life, she’s going to order these things for our classroom, let me help...and ugh, another boring worksheet, was definitely apparent.  

Obviously, that made me realize that maybe, just maybe math worksheets weren’t the way to go.  Honestly, I have never been a fan of worksheets, but in math especially, I saw the benefit of extra practice.  It’s weird because in Language Arts I have always been a stickler about keeping everything authentic, so why the disconnect in math.  Can math be authentic?  What would that look like? That’s when I met Beth Chinderle.  

Beth sat down with my team, answered our questions, and helped me understand what authentic math could look like.  The best part was she gave me permission to cut out the worksheets for homework.  I don’t know why I needed that permission, but I did.  From there, my brain was really on fire with the possibilities.  One night, I was sitting at the computer and it came to me.  I created a choice board revolving around what families and kids actually do in their everyday lives that will allow them to practice the specific skills we are exploring and learning in class.  They can see from these choices that math is a skill, like reading that they will do all of their lives, and can be enjoyable.  

So, I tweeted about it, sent an email about the change, and nervously brought it to my students.  I guess the nervousness came from my excitement, I mean, what if they hated it?  They gave me that familiar, what are you up to now glance and looked it over.  I went through the choices with them and they were totally digging it.  

Now, I have been teaching long enough to know that the initial excitement can dwindle when the task has to actually be accomplished, but to my surprise the next day when I asked how many of them had done their homework, over half raised their hands.  On Friday, homework was due, and every single kiddo was excited to share what they had done.  Many of them had completed 2, 3, or 4 of the tasks.  They were only asked to do one for the week.  Several even asked if they could submit ideas for the next time!  

This change in homework is part of an even bigger, and much scarier transformation of my math class.  Scarier because it’s not what I am used to, but maybe it’s just what I, and more importantly my students need.  So, I am happy, proud of my kiddos, and a little proud of myself, too.  To me, students are like play dough, they roll and bend with the changes easily.  Most teachers, though, are more like that really hard modeling clay.  We can and will change, but it takes a lot more work to get us warmed up enough to shift in a new direction.

Check out the homework here: Rayla's Real Life Math Homework

Thank you again, Rayla, for sharing your journey! :) I've also added this post to the Math Classroom Series as an important element of the math classroom.

Number Talks

I was worried about my children not understanding Number Talks, but it was amazing how quickly they wrapped their brains around different mental strategies.

Christi Muto

3rd Grade, Parkside Elementary

Number Talks are classroom conversations around purposefully crafted computation problems solved mentally. What a great way to "warm-up" students' brains, but more than that, they are a fun way to build number relationships.

Key Components of Number Talks

1. Classroom environment and community

• Number talks build a cohesive math community. It is so important that this is a risk-free environment.  Designate a place in your room where the students sit altogether on the floor. Have a place to write because in Number Talks the teacher does the recording!

2. Classroom Discussion

• Since the computation is done mentally, provide plenty of time. Use a signal such as thumbs up on their chest to show they have the answer. Students are given the opportunity to share their strategies and justifications with their peers. The benefits are students clarify their own thinking, test other strategies to see if they are logical, apply number relationships, and build a repertoire of efficient strategies.

3.  The Teacher's Role

• Since the heart of Number Talks is classroom conversations, the teacher becomes the facilitator. The teacher writes down all the students' answers. Then the students can "justify" their answers by sharing their strategies. While the student is explaining a strategy the teacher is recording the strategy on the board. What a great way to model recording strategies!
• Teacher poses questions to the students to lead the conversation. By changing the question from "What answer did you get?" to "How did you solve this problem?" the teacher is able to understand how the students are making sense of mathematics.
• Don't be afraid to share incorrect solutions. Wrong answers can lead to great classroom discussion and point out misconceptions a student may have.

4. The Role of Mental Math

• Number Talks help the students focus on number relationships and use these relationships to solve problems. When students approach problems without paper and pencil, they are encouraged to rely on what they know and understand about the numbers and how they are interrelated.

5. Purposeful Computation Problems

• Careful planning before a number talk is necessary to design the problem that is "just right." The learning target should determine the numbers and operations that are chosen.

Here is what Leander teachers are saying about Number Talks:

I begin class several times a week with number talks. This is a great way to get kids thinking about math concepts and not just memorizing "how the teacher said to do it". It also models how to communicate your math thinking. I see students using the same symbols that I modeled sometimes in their problem solving. We have a common format for communicating. By having the students defend their answers, it also helps them understand "justifying" when communicating solutions. Number talks are a great way to start a lesson.

JoJo Fentress 

2nd grade, Naumann Elementary

I use them almost everyday! We talk about numbers (odd/even, what comes before after on the number line, how many more/less to get to 5 or 10, different ways to represent a number - numeral, tallies, doubles, an octopus for 8, triangle for 3, twins for 2 and anything else we can thing of that relates to the number) and I use the dot cards. The kids love the dot cards!  One day I had a Watch Dog in my class during number talks. He owns his own computer gaming software company and is a really smart guy. He came over to me later and told me how impressed he was with the dot cards and what we were doing with them - especially since it was a kindergarten class. I have a pretty high math class overall, so I am also including partial number sentences when the kids tell me how they saw the dots. So for the number 8 we had things like:

8

5+3

3+3+2

4+2+2

6+2

2+1+5

I also have all the kids put up fingers as the person tells us how he saw the dots. That way they are actively participating and making the number. The biggest success is that kids don't just think of 8 as 8 and counting out 8 objects - they understand that it can be represented different ways and can be taken apart and put together many different ways.   

I also talked about the dot cards with parents at our parent teacher conferences. I wanted them to understand what we are doing with them. I had one mom tell me that they were what her second grader needed because she was struggling with her math facts. I told her that when her second grader was in kindergarten we did not have them.  :o(

Does it sound like I like number talks? You bet! It's always one of the most fun and engaging times of our day.

Colleen Welliver

Kindergarten

Steiner Ranch Elementary

After doing Number Talks with my students, one of my students came in the next day and asked me very excitedly, "Can we do "Math Speaks" again?

Ann Hutton

1st Grade, Mason Elementary

You can learn more about Number Talks in the book:
Number Talks: Helping Children Build Mental Math and Computation Strategies by Sherry Parrish

What are your thoughts about Number Talks? Please share! Post your comments below.