Student Notebook Strategy Posters

This quick, easy idea is one that works well for student notebooks. When I work with students on strategies, I often create a classroom anchor chart for the wall. I like to record the name of the student who used the strategy, along with a title that clearly describes the strategy. Kids love to see their own names in print and when they're asked to name what happens in the strategy, they often delve into rich mathematical thinking and discussion to define exactly what it is that they've done.

To give students greater ownership in the process, I invite students to make their own posters to go in their math notebooks or journals. I give each child an 11" x 17" paper, folded near (but not on) the halfway mark and 3-hole punched on the left. This way, the poster can be folded and added to their math notebooks as permanent reference.

Today, we made posters for Addition Strategies. If you click on the photos, you can see that we depict and name a variety of strategies. You'll also notice that this exercise is appealing to the artists in the crowd.

Multiplication Strategies: x2, x4, x8

Over the past week, my son and I have made tremendous progress with multiplication. (Intro post here.) Each day, we add to his fluency toolbox by looking at specific strategies. The (related) strategies for 2s, 4s, and 8s, have been especially fruitful. Let's look at why...

2s...Dare to DOUBLE!
Twos are easy-peasy. Just a matter of doubling. We can see an example in this array.

If you multiply something by 2, you only need to double. Instead of 1 group of 6, you have 2 groups of 6; you just double 6.












4s...Double-Double
In Bridges in Mathematics, the strategy for multiplying by 4s is called Double-Double. It's easy to see why.

We already doubled when we multiplied by 2. To go from 2x a number to 4x a number, we double. So we double, then double again.

6 is doubled to 12 (x2)
12 is doubled to 24 (x4)

I bet you can guess what's coming next!





8s...Double-Double-Double
We call the strategy for 8s Double-Double-Double.

For 8x, we double 3 times:

6 is doubled to 12 (x2)
12 is doubled to 24 (x4)
24 is doubled to 48 (x8)

Can you see it in the model?

This is not a multiplication "trick" but rather a strategy with meaning behind it. Children need to see the visual model and understand what "Double-Double-Double" means. Once they understand the concept, they can apply it in wonderful ways.

I asked my son (just developing fluency with single digit multiplication) to consider these problems.

8 x 15 = ?

He doubled 15 and got 30. He doubled 30 (60). And doubled once more to get 120. So 8 x 15 = 120.

4 x 13 = ?

Double 13 to get 26. Double 26 to get 52. (Of course he then wanted to keep going and figure out 8 x 13. Double 52 and get 104!)

25 x 8 = ?

50, 100, 200, done! This problem was also a great opportunity to talk about another strategy. Do you know what it is? Leave your ideas in the comments below to start a RICH exchange.

Hope you're having a double dose of summer fun!

Credits:
Little Girl Graphic from: www.mycutegraphics.com
Number Frames (free app) from: http://www.mathlearningcenter.org/web-apps/number-frames/

Multiplication Fluency: Summer Practice

Ready for a start-to-summer quiz?

Question: What happens when the math coach's child begins the summer by taking a multiplication fluency assessment in which he answers 20 problems in 4.5 minutes when the fluency guideline is 20 problems in 1 minute?

Answer: Summer math!  (Don't you wish you lived at my house?)

In case anyone else is in a similar predicament, here are a few resources to get you started...

First, "fluency" does not equate memorization. If you're interested in the difference between "by memory" and "memorization," check out this article. Fluency means accurate, efficient, and flexible mathematical thinking. Think about reading fluency. A fluent reader is not just fast. 120 words-per-mind counts for nothing without comprehension. Fluent readers AND mathematicians are  accurate, efficient, and flexible.

Although every child needs to master all three areas, he may demonstrate challenges in one area over the others. In our case, flexibility is an issue. Although my child knows some strategies for working with multiplication, it doesn't appear to be something that's been emphasized in his education. To that end, we are working to increase his strategy toolbox.

I pulled the Multiplication & Division Discussion Cards from Opening Eyes to Mathematics. (Cards are located on pp. 32-35 in this pdf, free from The Math Learning Center.) We flip through several cards a day and talk about what strategies could be used to solve a problem. For example:

What multiplication expression is represented here? (8 x 8)

How could we look at pieces of this array to help us solve the problem? Maybe we could see it as two parts: 8 x 5 and 8 x 3.

So 8 x 8 = (8 x 5) + (8 x 3) = 40 + 24 = 64

Or maybe you see it as two groups of 4 x 8:

What about this one? Could you use what you already know about 10 x 7 to help you figure out
9 x 7?

Strategies become critical when you get to larger multiplication problems, so this is definitely something we want to work on now.

Although fluency doesn't equate speed, it is generally expected that students be able to complete 20 problems in 1 minute to meet fluency standards. With the strategies in hand, I plan to assess his progress using the free online program, Xtra Math. I would not use this in isolation as I don't want to overemphasize speed, but when combined with visual models and strategies, it's a reasonable way for  both of us to track his progress.

Multiplication Strategies with Number Pieces App

Earlier this week, we started "Rethinking Multiplication Fact Memorization." Several people have asked me for examples of strategies that could be used to develop fact fluency. Here's a quick video using the online version of the new Number Pieces App (free) to get you thinking. Please leave a comment if you're interested in additional videos on multiplication strategies.

Rethinking Multiplication Fact Memorization

My friend Tamara at Teaching with TLC asked a really great question. Specifically, she asked about a condition called dyscalculia, but more generally she questioned what to do if a child doesn't seem to be able to memorize multiplication facts, no matter what you try. To answer that, I'm going to tell a story...

I used to conduct math (teacher) trainings with "Julie." She was (is!) a phenomenal math teacher. But one day, while working with a group of 4th/5th grade teachers, the topic of multiplication table memorization came up. After a number of folks shared their ideas and perspectives--including their frustration with kids who just don't get it, Julie spoke, close to tears. She told us how she struggled throughout childhood with the fact that no matter what she did, she could not memorize multiplication facts. It became a huge point of anxiety. Yet mathematical thinking and strategies came naturally to her. She explained how at 40something, she still does not know her facts. BUT, her knowledge of math strategies is so great that she can compute quickly and easily.

As she spoke, I reflected on my own experience with memorizing facts. As a 4th grader, I was scholastically competitive. Surprising, I know. When the class multiplication chart went up, the race was on! I was eager to conquer the chart (and beat my classmates.) I memorized it quickly. To this day, if I need to recall a fact, I do this sing-songy thing in my head to find it. ("Nine times nine is eighty-one.") But here's the irony...  I didn't have a clue what I was doing--what multiplication meant. I was great at memorizing, but it was really no different than memorizing a random list of words, letters, or numbers. It carried absolutely no meaning. Yeah, I could use them to do higher level multiplication and division, but my knowledge of the gamut was pretty limited because I'd done nothing more than memorize, first the facts and later the larger algorithms.

My friend Julie, in the meantime, fully understood the mathematical thinking behind multiplication. She had a wide range of strategies to pull from in order to figure out facts. I meanwhile, had none.

Which brings us to the concept of fluency. The definition of fluency is "good command." I would wager that as kids, Julie had excellent command of multiplication facts while I had relatively little. Today, Julie uses strategies to compute facts just about as quickly as I can locate them with my little sing-songy memory.

When I teach multiplication, my goal is fluency, not instant recall. I want kids to command multiplication--by thoroughly understanding what it is and being able to express their understanding through a wide range of strategies.

That sing-songy voice occasionally fails me. Julie's strategies never do. Her toolbox is loaded; mine has one unreliable wrench. ;)

In Salman Khan's book, The One World School House: Education Reimagined, he quotes from Nobel Prize-winning Eric Kandel, a neuroscientist:

"For a memory to persist, the incoming information must be thoroughly and deeply processed. This is accomplished by attending to the information and associating it meaningfully and systematically with knowledge already well established in memory." 

In order for something to be memorized [well], it has to make contextual SENSE! 

For more thoughts on fluency, read "Teaching for Mastery of Multiplication."

P.S. I'm on the edge of my seat, waiting for the online version of Number Pieces to come out. When it's available, I'll show you some spectacular strategies for multiplication. (Even this 40something can add a few tools to the ole' toolbox!)

Subitizing - Making Sense of Numbers

I'm in love.

With subitizing. Rhymes with oobitizing. (I always thought it rhymed with rub-itizing.)

Know what it is? If you're teaching, it's a great tool to add to your toolbox!

Ever since I read about it in Teaching Children Mathematics, I've been on a mission to learn more. Basically, it's the ability to instantly recognize a quantity. An example that frequently comes up: dice. Most adults see a number on a die and immediately recognize the number, without counting. We define two different types of subitizing:

Perceptual Subitizing - a number is instantly recognized, usually with quantities of 5 or less.

Conceptual Subitizing - recognizing small quantities within a number and combining to find a total. For example, you might see 7 dots as a combination of 3 dots and 4 dots.

You can see how the ability to subitize leads to all kinds of math skills. If you can instantly recognize a quantity, you can count faster, add/subtract/multiply/divide more easily and efficiently...the list goes on. It's a no-brainer that we want our kids to master this.

So I've been trying.

I started by placing objects on a peg board, showing a quick glance, and having the child replicate it on his own board. I also tried flashing quantities on a DIY counting rope.

More recently, I've been flashing quantities on a five frame. We used ducks after our lesson with Little Quack's Hide and Seek.

I've only done subitizing a handful of times (probably less than 5) this year. My little student showed a good grasp of numbers up to 5, but anything beyond that was pushing it. I thought. Shows how much I know.

Yesterday, I printed off some subitizing cards. (Went to a TON of effort. Smirk.) Googled "subitizing cards" and printed out cards from the first Google link. (Apparently, they go to a private school site, but I found the same ones linked here titled "subitizingcards.pdf.") They are FANTASTIC. Mostly because I didn't have to make them. But also because they fit neatly into a cardboard flap. I literally grabbed a piece of cardboard out of the recycling box and had a flap made in seconds. You'll see how it allows you to flash the dot pattern and then reveal the number after the child states an answer.

I apologize that the child's volume isn't more clear. I need to invest in an external mic. Listen carefully, however, and you'll hear the reasoning that went into his answers. You might think about when he used conceptual subitizing...or when he was able to instantly see smaller quantities within the number and add them together to get a total.



Although you certainly don't need to ask a child how he arrived at an answer, I found it very helpful to help assess my student's abilities. I was surprised that he knew so many number combinations. Formal addition is going to be an easy next step.

Now...go subitize! Here are some more links/resources to assist you:

Math Four
A terrific blog entry, Why Learning to Subitize is Important. (Thanks, to Tricia for the suggestion!)

Staff development with 1st grade teachers
In this YouTube Video, the facilitator guides teachers through steps that children might take in gaining number sense. It's 20 minutes but really gives you an education.

Math Coach's Corner
Donna has several posts about subitizing. She writes that subitizing is "foundational to the development of number sense." She links blacklines from John VanDeWalle's book with lots of subitizing cards. I'm printing these next. Also read her post describing different ways that kindergarten students saw a dot pattern.

Dr. Nicki's Guided Math Blog
Love Dr. Nicki's resources. She has several posts on using subitizing in the guided math setting.

Freebie! - Number Rack
Read about Math Learning Center's new free app, Number Rack. It could be easily used for subitizing, both online and on an iPad/iTouch. The entry also links you to a free activity/lesson book.

PDFs - Use with Subitizing
This site has a whole list of pdfs to print for subitizing with kids.

Subitizing: Counting Rope & Speedy Pictures Game

I've been intrigued with the concept of subitizing or "instantly seeing the quantity" ever since reading about it in a recent issues of Teaching Children Mathematics. I've started using our DIY counting rope to flash quantities in front of my 5yo. Want to join us? If you're a homeschooler, just make a counting rope and pull a small number of beads to one end. Then hold up the rope for a brief look (long enough to see the beads but not long enough to count each bead individually.) Teachers could put the counting rope under a document camera or lay game chips on an overhead for a similar experience.

Then, extend learning with this "Speedy Pictures" game.  You can choose to flash a visual representation of a number using fingers, dice, rekenrek, eggs, or number discs. The flash time may be increased or decreased, according to student needs. Students get a moment or two to see the quantity--but not count one by one--before being asked to enter a total. This activity would make a great computer-based Math Station.

Quick Image Math - Fluency in Basic Addition

Interested in helping your students (at home or school) become fluent in basic addition? I just read a fabulous article on the topic in the Sept. 2011 issue of the National Council of Teachers of Mathematics (NCTM) magazine, Teaching Children Mathematics.

In the article, author Gina Kling asks "what exactly does fluency mean, and how might fluency differ from having instant recall of each and every basic fact?" Great question. And one with which many teachers and parents struggle.

NCTM defines computational fluency as efficient, accurate and flexible ways of computing. Kling writes, "Traditionally, learning basic facts has focused on rote memorization of isolated facts, typically through the use of flash cards, repeated drilling, and timed testing. However, ...drill alone does not develop mastery of single-digit combinations."

The article goes on to describe strategies that fluent students use and suggests several ways that teachers might help students to become more fluent. She talks about ten frames and games. But the concept that jumped out at me was subitizing.

Subitizing is "instantly seeing the quantity." Kling suggests using quick images to show students a representation of a number "with the expectation that they will retain a mental picture of what they saw and then use that image in some way." The key? "...to flash the image quickly enough so that students [can] not rely on counting to determine their answer."

I decided to try it this morning. I put pegs into a mat, gave a 5yo a quick image, and asked him to then replicate the number of pegs in his own mat. The mathematical thinking is apparent in the video below as he decomposes the number four, seeing it as "these two and these two make four." As you can see in the video, he doesn't count the numbers. He knows that it is four because he recognizes 4 as a double of 2 or 2+2=4. Note that we've done no formal work with addition; I've never heard him say,"two plus two makes four."

In the representation at right I asked, "How do you know it's 5?"

"Because 2 of these  and 2 of these [he pointed at two on one side, then two on the opposite side] and 1 of this [pointed to middle] makes 5."

It's quite exciting to see his mathematical reasoning.

We're on our way to learning basic facts. I'd like to try other methods using subitizing. The author suggests one could flash a pattern of dots (think of 5 or 10 frames) or a certain construction of unifix cubes. Watching my boys this morning, it'd actually make a great partner game in a math station at school or in a workbox at home. Each student could make a quick image "puzzle", give their partner a quick peek, challenge the partner to replicate it and then check the work by explaining the mathematical thinking involved. For instance, each child could have a five frame, put pennies on it (anywhere from 0-5) and have the partner try to replicate it after a quick peek. What an engaging way to develop fluency! As I was jumping around the web I noticed that DreamBox Learning has a whole category of games/activities for quick images. (I registered--free--and could immediately access.) Also, while this 5 Frame exercise isn't presented as a quick image, it's also a great resource for fluency practice.

P.S. Want to check out something similar for bigger people? Check out "What's Your Number Sense?" on Mathwire and go take the Panamath Test and check your own Number Sense! It's fascinating!

P.P.S. Just found a couple links:
What is Subitizing and Why Teach It?
Where Will You Land? A Game Focusing on Subitizing
Subitizing: What Is It? Why Teach It?

Milk Cap Math: MADS Elimination

Here is another milk cap math game for basic +/-/x/division practice. My 15yo daughter thought of the name, MADS Elimination. (Thanks!)

The object?

M: multiply
A: add
D: divide
S: subtract

...in order to eliminate all milk caps on the playing field. (Before you go completely MADs!)

You'll need:

• two sets of 10 milk caps (any color), numbered 1-10
• two, 1-6 dice

To play the game, each player forms a line of milk caps, 1-10. Players take turns rolling the dice. When dice are rolled, the player may choose to multiply, add, divide, or subtract. So, for example:

Dice rolled: 3 and 5

I can:

M: 3 x 5 (won't work since my caps are 1-10)
A: 3 + 5 = 8
D: 5 / 3 = 2.5 (won't work)
S: 5 - 3 = 2

So I could eliminate either the 2 or the 8 cap. If the cap(s) has already been eliminated, I lose my turn. The first player to eliminate all caps, wins.

You can make this a little easier (preschool) by using 6 caps (1-6) and one die (1-6) and removing caps from the table as you roll each number.

You can make this a little harder by using bigger numbers (on the caps) or having the option of rolling multiple dice. For example, you could roll three, 1-6 dice, and make a number sentence using all three numbers. (You could also use dice with different numbers: 0-5, 4-9, etc.)

Example using 3 dice:

I roll: 1,3,4

I could do any number of combinations...I won't list them all, but for example:

3 - 1 + 4

1 x 4 - 3   (You could even talk about order of operations here.)

4 - 1 + 3

Lots and lots of possibilities for this game!

Running out of math caps yet?? ;)

This is another great math workbox game! ;)

Addition Fact Games--Make Ten

We've been working on materials from Bridges, Grade 3, of The Math Learning Center. I want to share some fun activities, designed to promote fluency with addition facts.

One card game, "Make the Sum," comes with the MLC materials. Prior to receiving the materials, we played "Make Ten," a variation based on the MLC game using a deck of regular cards.

Materials Needed:
deck of cards, A = 1, then 2-9 cards (remove all other face cards)

Object:
To collect combinations of cards that make 10. Player with the most cards at the end of the game wins.

To play:
Shuffle the deck and lay face down on the table. Draw top card and lay face up on the playing field. Player #1 looks at the card and says which card he would like to draw in order to make 10. He draws a card. If he can make 10, he tells the other player which cards add up to 10 and takes those cards. If not, he adds his card to the field.

Player #2 then says which cards he would like to draw in order to make 10 using the cards already on the field. He draws and then either makes 10 or, if not, lays his card on the field and play goes back to Player #1. Combinations of more than two cards are desirable as the players with the most cards at the end of the game wins.

Here's a video demonstrating how to play the MLC game.



My student gained more skills practice in "Make 10" by using an on-line game (also recommended by MLC) called "Math Lines." This game is unbelievably addictive. It's embarrassing to admit, but I made my little student stand next to me and watch while I played because I wanted to see how many levels I could master. ;) He loves the game as well.

Farm addition is another game that provides practice in addition facts. We found it just as we finished My Little Farm.