Happy Holidays!

From the Elementary Math Team ~
Wishing you and your families a very "Mathy" Holiday! 

As we take the next couple of weeks off to spend time with our families, friends and loved ones, we hope you will continue to find the math in all your Holiday FUN!!

            Yummy 4x3 array of
ham and cheese holiday sandwiches!

Holiday cake balls and pretzels!
What array(s) do you see?
How many treats are on this tray?

We would love for you to tweet us your favorite holiday math photos @lisdelemmath

Best wishes for a fabulous NEW YEAR! 

Problem-Solving Sharing: Whole Group Area

This week we are excited to add another element to the Math Classroom Challenge on Problem-Solving Sharing. Thinking of classroom space, it is essential to designate a large area of the classroom where students can come together and become mathematicians in a real mathematical community. Take a look at these pictures of Sandy Handrick's 5th grade classroom:

Sandy has a large area for these young mathematicians to come share ideas and strategies that emerge as students do the math and teachers facilitate discussions. Notice that she even has ample space on her white board for student sharing.

What is problem solving?
Problem solving is a process of inquiry where students are investigating mathematical and real-word situations. This could be in the form of a specific problem type in which careful number choices are selected to bring out mathematical relationships, or it could be a problem in which students are applying their mathematical thinking to an authentic, real-life situation such as problem-based learning or performance tasks. Problem solving is NOT giving the students a problem and telling them a certain operation to use and how to solve it step by step. If that's happening, who's doing the problem solving?

How often should students be problem solving?
Every day! Students should have the opportunity to solve a variety of problem types and real-world problems in which they can explore and investigate mathematical relationships. Problem solving every day is critical for a successful math community. For example, if students only have the opportunity to problem solve once a week such as every Friday, that is less than 36 times a school year that they will problem solve. That's not enough! Think of mathematical thinking like a muscle that students must exercise. It takes time and practice for that muscle to get stronger. Making a connection with reading, do you only have your students read once a week and expect fluent readers?

I have so many TEKS to teach, how do I fit in problem solving every day?
One of the biggest misconceptions in mathematics is that skills must be taught in isolation. This is just not true! Our math standards (TEKS) are connected in so many ways. When we only focus on one TEKS or skill in a day or week, this creates a misunderstanding in our students that skills are not connected. One good problem can encompass many TEKS.

How should students share their thinking?
This can be done in many ways! Here are a few examples:

1. Select 2 or 3 students to share to the whole group: We've all had the whole group sharing time where students aren't paying attention. Usually, this is because we select students to randomly share, students don't understand the learning goals, and therefore, they do not make any connections between the student who just shared their strategy and their own ideas about math. Good whole group sharing should only take 5-10 minutes and be focused around a clear instructional goal. Ask yourself, "What is my purpose for these students sharing?" Do you want focus on a big idea, connections between different solutions and strategies, or moving students from less efficient to more efficient strategies?

2. Partner Share: Partner students based on their levels of understanding. Partner a low with a medium and a medium with a high. Be flexible! As students grow in their understanding toward different concepts, partners will need to be changed. The main point here, is if you want to get the most out of sharing time, students should not be partnered randomly.

3. Gallery Walk: Have about half of the students post their strategies so they are visible. This could be done by writing on a chart paper, simply leaving their journal open, or using technology such as the "Show Me" app on the iPad. These students stand by their strategy as the other students walk around and talk, ask questions, collaborate, prove and communicate their thinking to one another. The next day, students can switch roles.

Where do I get problems?
Luckily, LISD's math curriculum is rich with examples of problem types and performance tasks to support this area of instruction. We have several problems and a performance task for each Unit of Study. Please remember that these are problem suggestions and we encourage you to tailor them to fit your student's interest and appropriate number choice.

We would love to hear your questions about this topic! Please share with us your thoughts! Happy problem solving!

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. 

Mathematizing the World

As educators, we often get asked, "How can I help my child at home?"

One way of supporting children's mathematical development is to help them "mathematize" their world by seeing math all around.  Start by being curious about your child's ideas and what they are pondering.  Notice everyday experiences and ask questions that help build on their mathematical thinking. 

For example, when visiting the Texas State Fair last week, math was EVERYWHERE! 
It was easy to capture some digital images with the use of a smart phone.

How many fair tickets did I purchase?

If each ticket costs $2, how much money did I spend on tickets?

It takes 5 tickets to ride the bumper cars.  How many times can I ride?

How many ducks in the pond?
How could you group the ducks to easily count them? 
How many would there be if you added 1, 10 or 100?
What if you took one away?

Estimate the height of the Ferris wheel.
How many buckets are there?
If a bucket holds 3 people, how many people will the Ferris wheel hold?
If takes 45 minutes for one rotation on the Ferris wheel, how many minutes would 3 rotations take?

Look at the balloon array!
How many rows of balloons are there?
How many columns?
What is the product? 
How many 6x4 arrays can you find?
What fractional part is blue?

How many inches tall are you?
How many feet tall must you be to ride?  
If you are not tall enough, how many more inches must you grow?  
What is the difference between 36 inches and 42 inches?













Howdy BIG TEX!
If Big Tex is 52 feet tall, how many feet taller is he than you? 
How many inches tall is he?  
If Big Tex turned 60 years old in 2012, what year was he built?

When our children are thinking like mathematicians, they make sense of numbers, learn to persevere, are able to reason, and use multiple strategies to solve problems.  All of these things help our children become passionate about math in the world we live. 

I challenge you to "mathematize the world" and have some rich conversations with your child.  

 

Math Word Walls

As part of the Math Classroom Challenge, today's post is about how you can use math word walls to support students in communicating with mathematical language.

Word walls are relatively common in the Language Arts realm, but why might we want to use them in math? How can we set up word walls in a way that give students the opportunity to actually make meaning of the words?

What exactly is a math word wall?
A math word wall is an area of the classroom in which key content words are displayed on a list or chart in order to help students learn mathematical vocabulary. Word walls aid in encouraging students to spell these words correctly and use them when they are speaking, reading and writing about math. You could also have other word walls for science, social studies, etc. It's a great tool!

At River Place Elementary, 1st grade has math word walls right on a large piece of chart paper!

Here's another example. This is simple and eye level for the students to see!

Here is another example from Becky Vaughn's 4th grade classroom at Westside.

Heidi Dominguez, teacher at Westside, set up a spot in her math area for her word wall right at the beginning of the year. It is nice and low for her class of 1st graders!

How often do I change out the words on a word wall?
Math word walls are constantly changing based on student level of understanding, unit of study, and/or real-world problem they are solving. Once students understand the words, you can take them down and add new words.

I like this word wall because of its simplicity. Grab a piece of chart paper, select about 10 core words from your unit of study, gather the students around and hang it up in the math area of your classroom. Add to it as the students learn. Don't buy something that is pre-made! It makes more sense to involve students for a couple of reasons:
1. It supports the LISD curriculum. Teacher stores usually don't have the most rigorous words.
2. Different students understand different words and that will change every year based on your class.
3. If students are involved in the creation of the word wall it sends a message to them that you care what they think and you care that they learn the words. Spotters ready...let's learn these words together!

What is the most important thing to remember about a word wall?
As my language arts friend Diana says, "They are dynamic and interactive!" Students need opportunities to make meanings of words on the word wall so they can communicate. Here are a few ideas:

I Say, You Say!
1. Point to the word on the word wall and have the students pronounce them with you.
2. Point to the words in order and mix them up when students become more fluent with the pronunciations.
3. Have students work together in partners with one partner pointing at the words and the other partner saying the words. Then, the partners switch and do the same thing.

Word Connection
1. Have students pair up.
2. Pick two words from the word wall and write them down.
3. Have students talk about how the words connect.
4. Share answers with another pair or as a class.

Vocabulary Cards
1. Have students use the Frayer model to create a vocabulary card for each content word on an index card.
2. Now the students have their own set of word wall cards!


Ready, Set, Redo!
1. Take down the words from the wall and have students sort them in a new organized way.
2. If students have their own set of cards (like mentioned above with the Frayer Model) they can just use their own.
3. Record the sort in their math journal.

Those are just a few basic strategies that I have found most helpful in getting students to communicate and learn math language, but we know that there are multiple paths to the same solution! What are some ideas that you have to make math word walls interactive and meaningful? Comment below!

THE CRAZY MATH LADY!

Look who showed up at Reagan Elementary to promote mental math, number talks, and fact strategies! 

The CRAZY MATH LADY!  (I've heard she has also been spotted at Giddens Elementary before.)

The CRAZY MATH LADY dresses up to bring warm-up activities into the classrooms. She enters with clappers, yelling "WooHoo," startling many teachers and students.  She introduces herself as the CRAZY MATH LADY because she is CRAZY about math. (Aren't we all?)   

She always begins her mini-lesson with a chant, "When I say Math, you say Rocks! Math....Rocks! Math....Rocks!" Then she gathers the students on the carpet for a quick number activity.  

On this day, she handed each student a hundreds chart. The hundreds chart is a great model for seeing patterns in numbers. Most children decided to use the hundreds chart, while some challenged themselves to try it mentally and turned their charts over. 

The activity was a series of problems to calculate quickly.  It went something like this: Put your finger on the sum of 10 and 25, add 50, subtract 4. Is the sum of the digits 9? 

Wow, she went CRAZY fast but it was fun and engaging.

Hopefully after a few more weekly visits from the CRAZY MATH LADY, students will become more confident in their mental math ability and less dependent on using their hundreds chart. 

The next day students were still wondering . . . who is the CRAZY MATH LADY???  
One never knows where and when she will show up next!!   

The lesson idea is from Mary Alice Hatchett at http://www.mahatchett.com./
  

Traditional Algorithms VS Invented Strategies

There are significant differences between traditional algorithms and invented strategies.

The traditional algorithms are based on performing the operation on one place value at a time with transitions to the next position. They involve trades, regrouping, "borrows," "carries" or Dead Monkeys Smell Bad!  The procedures are rigid.

Traditional algorithms take the understanding out of place value. They are "digit oriented." They rely on procedures or steps that must be done in a specific order and usually start in the ones place. 

Look at theses examples of traditional algorithms for addition and subtraction. They follow a procedure.

Think about what is being said as you solve this problem: 7 plus 8 is 15. I put down the 5 and carry the 1.  One plus 5 plus 3 is 9. Three plus 2 is 5. The answer is 595.

Think about what is being said as you solve this problem: 3 minus 7, you can't do it so you have to go next door and borrow 1. Since they don't have any to give you they go next door and borrow one. The 3 becomes a 2 and the zero becomes a ten. Now they have one you can borrow so the ten becomes a 9 and the 3 becomes a 13. Now you can subtract! 13 minus 7 is 6, 9 minus 6 is 3 and 2 minus 1 is 1. The answer is 136.

Invented or flexible strategies develop a good understanding of the operations especially the commutative property and the distributive property of multiplication. Students start to see the relationships of addition to subtraction, addition to multiplication, and multiplication to division. What an important concept! 

Invented strategies involve taking apart and combining numbers in a wide variety of ways. They are "number oriented." Most of the partitions of numbers are based on place value and start in the largest place. 

Now look at the examples of the same problems. Although these may look complicated many invented strategies can be done mentally and just recorded on paper.

Think about what is being said as you solve this problem: 300 plus 200 is 500, 50 plus 30 is 80 and 7 plus 8 is 15. I add them together and 500 plus 80 equals 580 plus 10 is 590 plus 5 more is 595. The answer is 595.

Think about what is being said as you solve this problem: 300 minus 100 is 200. Since I know 67 + 33 gets me to 100 then 200 - 67 is 133. But I still have the 3 ones in 303 that I need to give back. 133 plus 3 is 136. The answer is 136.







Your Turn: 
Think about the traditional algorithms for multiplication and division. Are they rigid procedures? Are they "digit oriented?" Do they build understanding of number relationships? 

Be careful: Don't turn an invented strategy into a "procedure." 

Students must be allowed to develop their own strategies based on their own understandings!

Math Anxiety

Imagine that you are in a restaurant, seated with many people at a large table. In the course of conversation the person sitting next to you laughingly remarks, “I’ve never been good at reading. Can you read the menu for me?” 

What would you say? Would you laugh along and confess that you never really learned to read either? What would others say at the table?

         

Now imagine the same scene, only this time the person next to you says, “How much do I owe? Can you add it up for me?  I’ve never been good at math.” 


What happens this time? You can expect other people at the table to chime in cheerfully, "I've never been good at math either."

When did it become socially acceptable to say, "I've never been good at math?"

Math anxiety or the fear of math is not uncommon. This emotional reaction to mathematics is based on an unpleasant experience from the past which harms future learning. 

OUCH: Those past learning experiences could be from my math class!!!

So how do I avoid creating math anxiety in my students? 

1. Be aware of your own attitude towards math. 
Students pick up on your math passion or lack of it. Avoid saying, "I wasn't good at math." Positive attitudes comes from quality teaching for understanding. 

2. Ditch the idea that "some" people are just good at math. 
Anyone can be good at anything when they have perseverance. 

3. Give help early. 
When you see a student struggling take action immediately! Work for understanding, and skip the "tricks." Understanding the math is what it is all about.

4. Help your students learn how to "shake off" mistakes. 
Develop a classroom environment where students can take risks, share their thinking, and celebrate mistakes! Everyone can learn from mistakes. Why do you think pencils have erasers?

5. Make math engaging and relevant. 

Look at all of the work you are giving your students. From warm-up to independent practice, is it engaging? Is it relevant? Does it have just the right amount of challenge? Would you want to do it? If not, fix it!

6. Dispel the myth that there is only one answer or one way to get the answer. 
There are a variety of ways and tools to solve problems. Allow the students to create their own strategies that build upon their own understanding. Don't make them solve it "your" way! 

Let's do our part to put an end to Math Anxiety!