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?

Math Monday Blog Hop #25

What are you doing this week? Share lessons, books, and resources that you're using in math.








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Math Monday Blog Hop #24

So excited to get back to math with my kiddos!!  What are you doing this week? Share lessons, books, and resources that you're using in math.








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Review: Math Dice Jr.

As of TODAY, my continuing education math workshops for teachers are done for the season. I'm very excited to get started with the school year a bit closer to home. Oh wait...in our home!

Last year, my youngest attended preschool two days a week. This year, at 5.5yo, we'll be doing some minimal kindergarten work. Recently, we were excited to receive a game that we can use as he begins learning more about MATH.


MathDice Jr. is designed to be used with ages 6 and up. At 5, my son is on the young side. To play, the rules say to roll the 12-sided Target Die and the five 6-sided Scoring Dice. All are large, chunky and bright...fun playing pieces for young children.

The Scoring Dice can be combined to make addition or subtraction problems that equal the target number. So, for example, in the picture below, you could use 6 + 2 to reach the target of 8. Since my son isn't ready for subtraction, we added another element...  If one of the colored Scoring Dice exactly equals the Target Dice, that dice can also be used to achieve a match. My son spent a lot of time counting the dots on each dice to see which matched the target. It was nice for counting practice.

The original instructions call for players to move one space on the track for each die used. Again, we played an easier version (our invention) to meet my son's developmental level. Player #1 (whomever rolled the dice that game), got the first opportunity to make the number, either through addition of several dice or by one-to-one correspondence, "matching" a dice to the target number. If he found a way to make the number, he called "Math Dice," showed his results, "captured" the dice he used (removing them from the play area) and moved one space on the track. Then, the next player in the circle could try to reach the target number from whatever dice remain. If he can match the target number, he also calls "Math Dice" and gets to move one space on the track. We continue around the circle until no possible combinations for the target number remain. Then the player who did not roll the last time gets to roll. Using this method, the play stays pretty balanced; no one is able to get far ahead of others on the track.

The first few times we played, I tried to let my 9yo son play using any operation (add, subtract, multiply, divide) to reach the target number. This left my younger son at a significant disadvantage. So I'd "help" him...again using any operation. But this meant my younger son got very little out of it. So I told everyone that for now, we'd only do one-to-one correspondence or addition. As my youngest learns more math, we'll add more operations...which will make it more of a family game. Right now it's mostly a counting game for the youngest.

The game itself comes with a nice little canvas game board and all pieces fit into a convenient drawstring bag. It was a nice compact game to carry along on a campout. No game markers come with the game but my boys have enjoyed picking out buttons (note the long button in the middle photo!) to use. On the MathDice webpage, you can pick up a pdf for a game board and some pretty cool looking game markers.

My opinion in a nutshell? This is an inexpensive, fun, educational game that could be used by teachers (a great math work station!) or by homeschoolers (a great math workbox!)


Disclosure: I received a free Math Dice Jr. game in exchange for a frank and unbiased review. Thank you to ThinkFun for the opportunity! I was not paid for this review, nor was I required to write a positive review. Frankly, I love the ThinkFun products I've had the opportunity to try...most of which I've purchased myself. :) If you purchase this product through my Amazon link at right, all commissions go toward foster care through Grace and Hope at no additional cost to you. I do not keep any money myself; I am hoping to be able to sponsor an additional child in foster care through commissions on this site.

Math Monday Blog Hop #23

First, the giveaway winner for a new children's book, How Big is a Million, is Kim @ the Learning Hypothesis. Congratulations!

Join in the Math Monday Blog Hop! We'd love to see what you're up to this week in MATH! :)







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What's a.... Purpllinker ?????

When I initially read about Purpllinker on Timberdoodle, I was intrigued.

What's a Purpllinker?
It's purple. It has 7 sections of 5" plastic "arms" that are jointed together at each end. Extended, it creates a purple "stick" that is about 2.5' long. But bend the joints and you can create any number of creations: letters, numbers, pictures, shapes, and more.

I asked to review it, thinking it would be great for my 5.5yo to use to explore letters. After I opened the package, we did that for about 5 minutes. Then, other family members started coming into the room. My 9yo grabbed it to make pictures. My teenagers took it and started twisting to see what shapes they could make. My husband snatched it and tried to make fancier letters. In essence, it made its way around the room, and the family, with each age level intrigued in different ways.

Always looking for quiet church toys, I stuck it in my bag on Sunday. My youngest made his own inventions with it, forming letters in his name as well as a variety of pictures. It was a great quiet toy. I just had to encourage him to make creations below the top of the bench so other congregants weren't distracted by a bobbing purple wand. :)

Over the last few days, Purpllinker has been sitting on my desk, waiting for me to review it. When my 5yo comes in and starts climbing on my back while I'm typing (not an unusual occurrence!) he'll often see the toy, pick it up, and start creating.

Thoughts:

1. I'm not sure that I'd use this for teaching the alphabet. If you look at the suggested formation of letters inside the cardboard packaging, many of them look quite different than what a child would be taught. You simply cannot make every traditional letter--at least "normal" looking letters--using this tool. However,...

2. If a child was familiar with many letters (as my son is), it is a nice challenge to get him to try to come up with a way to form any given letter. You might say, for instance, "Can you come up with a way to make the letter S?"

3. I would use it for creative play. Ask the child to invent his own picture. It's a nice quiet toy that folds very compactly to take along to appointments or long car rides.

4. It remains to be seen if this is what I call a "long lasting" toy/learning tool. I'm going to put it with our school tools and use it for a preschool workbox item; I hope it will continue to intrigue my 5yo.

5. I loved the minimal packaging. It came in one small plastic bubble with a small cardboard overlay. I was surprised, however, that the reference chart for how to make the shapes was on the inside surface of the cardboard packaging which didn't come off cleanly...at least for me it didn't. In our house, that chart isn't likely to last long. Since we won't likely use the number/letter visual models much, it's a minor issue, but if families buy it expecting their children to frequently refer to the chart, they may be disappointed in its ability to last over time.

6. I'm most excited about using this as a tool to explore geometric shapes. Ask the child to make a square. Then ask him to tell you about the properties of the square as he touches the model. "It has four corners. Four sides." Or a more advanced child might say..."It has 4 right angles." The teacher could do higher-level challenges such as, "Can you make a model of a hexagon?" For this purpose, it seems like it could be used in endless math extensions in a homeschool or public school classroom...at a variety of levels/grades.

I asked my boys to rate it. My 9yo gave it a 3 out of 5. My 5yo gave it a 10 out of 5. Maybe we need to do some number work with the Purpllinker. :) I'd give it a 4 out of 5. It definitely fills a unique nitch.

Get your Free Timberdoodle Homeschool Catalog to view the Purpllinker and other fun educational items.

Disclosure: As a member of Timberdoodle's Blogger Review Team I received a free Purpllinker in exchange for a frank and unbiased review.

Math Monday Blog Hop #22

It's Math Monday! What math-y things are you considering on this Labor Day? Also, don't forget to enter the giveaway for a new children's book, How Big is a Million.





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Review: Math Dictionary for Kids

Whenever possible, I integrate the use of mathematical terms in daily lessons. This year, however, I intend to expand student exploration of terms through Math Notebooking. (That post will come soon!) I was happy to receive a review copy of Math Dictionary for Kids as it appears to complement my plans.

The first section of the dictionary is organized by mathematical strand:

• Whole Number and Operations
• Measurement
• Algebraic Ideas
• Geometry
• Decimals, Fractions, Percents, and Ratios
• Statistics and Probability

Along with textual definitions, examples are often given. I especially appreciate the colorful visuals models that are frequently used. In some instances, multiple strategies are presented. For example, in "division strategies," the reader sees examples of strategies using multiples tables with larger numbers (if you've never seen this, it's often a very efficient strategy to use as an alternative to traditional long division), repeated subtraction, and the use of manipulatives. With the increasing amount of information that people need to process, it's essential that children learn to use a variety of strategies, considering which is most efficient for the problem at hand. For adults (and children) who've never been exposed to multiple strategies, the "Quick Reference Guide," gives a brief overview.

I'm especially partial to a section of the book called "Learning with Manipulatives." Again, for students or adults new to manipulatives--or for those wanting a review--this information clearly shows a variety of uses for manipulatives in mathematics. The visual models look very familiar after using similar models in my favorite curriculum, Bridges in Mathematics. I frequently lead workshops for teachers on how to use base ten pieces and grid paper for teaching multiplication and division, which leaves me feeling like the content is very up-to-date. Since most adults didn't have the opportunity to use manipulatives in their own math education, it's not unusual to see manipulatives set aside in classrooms (both at home and at school) or used as little more than playthings. This guide would be very instructive in helping adults and kids to see the value in using such tools to solve problems. After learning strategies for multi-digit multiplication with manipulatives, I can now do many problems in my head that used to take me significantly more time on paper.

Any negatives? Not really. I found one error. (p. 139 the array shows one dimension as 22 but in the problem it shows 21.) If anything, I'd love to see a book that pushed the exploration of strategies even further, encouraging people to look beyond the standard algorithm for the methods that are truly the most efficient. But this is a very nice start.

I wouldn't hesitate to use this book with students. Or to keep it nearby as a good reference for the teacher. The cover says it's for grades 4-9, but I'd use it with 3rd graders as well.

P.S. At the moment, this book is a good deal on Amazon.


Disclaimer: I received a review copy of Math Dictionary for Kids. I received no other compensation and this review is my opinion...good, bad, or otherwise. :) If you purchase this book through my Amazon link, all commissions go toward foster care through Grace and Hope at no additional cost to you. I do not keep any money myself; I am hoping to be able to sponsor an additional child in foster care through commissions on this site.