Visualizing Numbers

"1,2,3, Look at me.  Tell me the number that you see."

This was today's refrain as we worked on Quick Images.  The Investigations curriculum makes extensive use of this type of activity.  Students are briefly shown an arrangement of dots, and they are asked to quickly determine how many dots they were shown.  They are not given enough time to count the dots one by one and so they must adopt other strategies, combining small groups, etc., to determine the amount.  These activities are supposed to recur regularly throughout the school year with dot arrangements that become more complicated over time. 
 
I have sometimes wondered about the usefulness of this activity.  It isn't an activity that occurs in traditional math curriculums, and I didn't spy anything in the common core standards about visualizing numbers.  Then I read Subitizing: What Is It? Why Teach It? by Douglas H. Clements which details the various theories and research supporting Quick Images activities and makes a convincing case for providing regular opportunities for students to practice recognizing and visualizing number groups.  It would seem that recognizing number groups is a skill underlying many of the skills required in the common core. 
 
There are a few suggestions that Clements makes that I'd like to work on applying in the classroom.  The first is to include quick recognition of auditory or kinesthetic numbers... how many taps?  how many dings?  The second is to work on recognizing numbers in different formats like the "Tens Frame."  I've been using the tens frame pretty extensively since it is prominently featured in the Scott Foresman curriculum, but today I introduced it to students for the first time using Quick Images. 
 
I worked with small groups at the round table on quick images-- youngers were identifying numbers on a tens frame; olders were identifying addition sums on a tens frame.  Meanwhile a new math choice activity was working with dominoes, matching numbers to create a domino path. 
 
 
 

War and Double War

Early on in the year I like to introduce the card games "War" and "Double War."  In the Investigations In Number, Data, and Space curriculum (Dale Seymour Publications, 1995) these games are taught to first and second graders as "Compare" and "Double Compare."  I used to use this terminology, but the kids all recognized the game as one they'd learned at home as "War," so now I just go with their name for it. 

You probably learned to play "War" yourself as a child, but in case you've forgotten... The deck of cards in dealt out evenly between two (or three) players.  Each player slaps a card down and the player who's card is the greatest number collects all the cards.  If there is a tie, the players have a "war," slapping down three cards upside down and one card facing up.  Again, the player with the highest card takes all. 

"Double War" is a great adaptation for older students.  This time around, instead of slapping down one card only, each player slaps down two cards, adds the cards, and totals are compared.  The player whose cards total the greatest number takes all cards.  

To simplify things a bit, I have removed all the face cards from the decks used in my classroom.  This helps with some of the other card games we play as well. 

As the year goes on, these games serve as great filler activities and regular opportunities to practice math fact fluency.  Often I ask students to play a card game like this with a partner as they arrive in the morning because students can usually play without a lot of teacher monitoring.  On this first introduction, however, I am sure to circulate around the room, observing carefully how students approach the task.  Some students will need to count the pictures on the cards in order to compare the numbers, while others will quickly recognize totals.  It is possible to learn a lot about students' mathematical understandings by observing them at play. 

Dot-to-Dots

Since I am still working with small groups, I added in a new math menu choice for children working independently.  Since it is still very early in the school year, I need a choice that children will feel confident taking on without a lot of instruction, but one that gets them practicing skills.  Dot-to-dot puzzles meet this criteria beautifully.  Younger students do puzzles with numbers 1-20.  Olders do puzzles with numbers 1-100.  Sometimes I've included puzzles that count by 2, 5, or 10.  Here everyone is working on ordering numbers, but doing a task that they consider fun, and dare I say, un-school-like.
Someday I will probably have to break down and buy a good dot-to-dot book, but for now, I make do with puzzles I've collected online.  The french site http://www.pointapoint.com/ is my main source.

Counting Books

Worked on making our own counting books today. Above you can see samples from a first grade book with digits and pictures and a second grade book with pictures as well as number words in complete sentences. (One note about the first grade picture-- the child was working on the number five page as I snapped the shot. He does know his numbers, he just isn't finished yet!)

It was really interesting to listen to the children chat as they worked. One of my second grade boys spent a lot of time explaining to the first grade pal sitting beside him the difference between the first and second grade assignments.

The Winning Number Game

The winning number game is one of the simplest examples of a game that can be made more or less challenging for different levels of students.

Today first graders who are working on writing numbers and number recognition played the game with one dice while second graders who are working on basic fact fluency played the game adding the total of two dice. Later in the year, first graders will play the game with two dice and second graders who will then be working on adding number strings strategically will graduate to adding three dice.

Big Collections and Small Collections

Counting and estimating are tasks that both first and second graders need to work on. To differentiate this work, I offer "big collections" and "small collections" for students to estimate and count. Today's math menu included the choice to work with these collections--just odd batches of items kept in ziplock bags--old keys, spoons, plastic shapes and doodads. The "small collections" for younger children number between five and twelve items. The "big collections" for olders number between fifteen and fifty. The worksheet accompanying the tasks asks students to record "what I had" and an estimate and an actual count. A few of the younger students who were anxious about recording "what I had" in words, drew a picture of the item in that space instead.