Sorting People

We began our sorting and graphing work by sorting manipulable objects--buttons and lids.  Then we moved onto recording those sortings on paper, stepping from the concrete to the abstract.  Our next step today was to sort people and their preferences. 

Again, we used "Not Boxes," and moved from the concrete to the abstract.  I outlined large sorting boxes on the floor of our meeting area in masking tape.  Then sorted four or five children at a time into the boxes, first by looking at obvious physical traits (long pants/not long pants, wearing blue/not wearing blue), then by personal preferences (Do you like spaghetti/not like spaghetti?  Do you like swimming/do you not like swimming?)  After everyone had a turn, we repeated this process in a slightly more abstract manner by sorting name cards into the giant boxes.  Another step towards abstraction was writing children's names onto a set of boxes drawn on chart paper.  For homework tonight, children will sort the names of their family members into boxes after asking the question "Do you like pizza?" 

Was this lesson differentiated?  Not at first glance.  It was a whole group lesson without being particularly open-ended.  Tomorrow, as students do some of their own surveying, there will be more opportunity for students to modify the lesson to suit their own needs.  I'll have to think more about this one.  Perhaps there is something more that I could do with questioning throughout-- planning out questions ahead of time tiered for different levels.  Let me know if you have ideas!

NOT Box Recording

Graphing is all about taking the concrete--some actual things, actual events, actual people--and rendering a more abstract representation of a situation.  Children in the primary grades (and perhaps learners of any age) must work first with the concrete--the actual buttons, or lids, or blocks, or what-have-you-- before attempting the more abstract--the recording on paper.  Yesterday we just sorted the objects.  Some created very obvious sorts, some made the task more challenging by attending to unusual details.  Today we took the next step, recording these sortings on paper. 

Again, I began by modeling a sort and this time modelled recording my work on chart paper with pictures and words.  Then I created a sort and asked students to help me label my sorting rule.  Again, we had an open-ended workshop period with students creating as many sorts as they had time for, trading zipper baggies of buttons and lids for new ones when they were ready. 

NOT box sorting

Today as we launched our next math unit (sorting and graphing), I found myself wondering again about the validity of spending instructional time on what seem like simple skills.  The new common core standards  ask kindergarteners to "classify objects in categories," while first graders are supposed to "organize data in up to three categories" and second graders are supposed to create graphs.  I feel like these are fair goals for each grade level, but I also know that while classifying objects sounds simple, it can actually be incredibly complex.  When given a handful of buttons to sort, many first graders will create two piles:  the thin buttons and the buttons with holes.  When faced with thin buttons that also have holes, they may be completely stymied. 

Sorting buttons is an open-ended task that I find to be appropriate for both my first- and second-graders.  Up until now, sorting has been considered a preliminary skill for graphing work and again, is a task that will undoubtedly help them meet more complex goals.  I have decided to carry on with this work, even though the new standards would suggest it is more appropriate for younger students. 

Earlier this week, Organized Chaos posted about the complexity of counting to 100-- a seemingly simple task, but one that is not easy for kindergarteners to master.  She expresses frustration that more and more standards are being added to the list of kindegarten responsibilities, some of which are less than developmentally appropriate.  My frustration as we work with the new common core is that fewer standards sounds great, but cutting out simpler precursor skills only makes the learning more difficult.  Mastering fewer standards does not mean teaching fewer lessons.

Today's lesson is based on an Investigations lesson with sorting boxes called "Not Boxes."  Students need to identify an attribute and sort items into those that have the attribute and those that don't.  Any objects could be used for sorting, but today we used buttons and lids.  I began by modelling this sorting, then sorting again and challenging the class to figure out the rule I used for sorting.  Then I gave each child a paper sorting mat and asked them to sort a baggie full of buttons into a "Not Box" in any way that they saw fit.  As students sorted, I circulated around the room and tried to guess the sorting rules they were using.  Could I always guess?  Sometimes, yes, and sometimes, no.  Many times I needed to coach a re-sort, even for second graders. 

Staircase Patterns

There is a lesson in the first grade Investigations in Number, Data, and Space (Dale Seymour Publications, 1995) curriculum exploring the idea of a "staircase pattern," a pattern in which quantities grow.  This lesson is based on the Eric Carle book, Rooster's Off to See the World in which a rooster is joined in his travels by two cats, three frogs, four turtles, and five fish.  Carle includes a graphic on each page showing the number of animals in the group arranged in a staircase-type design.  The Investigations lesson suggests that the teacher begin reading-aloud the book, then stop partway through so that students can make predictions about what will come next by extending the pattern.   This is a nice example of an open-ended task which can accomodate a range of learners, but in my multi-age class, I generally need to stretch the task a bit more for the second-graders. 

   

Today we started with a read-aloud of Phyllis Root's One Duck Stuck. Obviously, Carle's book is the best book for this lesson since he includes the "staircase graphic" but many other books play with the same pattern.  Because I teach this lesson each year, and some of my students will have had the lesson the previous year, I alternate my read-aloud book.   One Duck Stuck is similar to Rooster's Off to See the World in that an increasing number of animals continue to join the group on each page.  

After a brief discussion of the staircase pattern in the book:

What might happen next?

Is this a pattern?

How is it a pattern?

What is the pattern unit?

Could we translate this pattern into letters? 

Could we translate this pattern into numbers?

Could we translate this pattern into pictures?

Could we translate this pattern into snap cubes?

I asked the class to make and record their own staircase patterns.  First graders generally made simple counting patterns, but I asked many of the second graders to try to make staircases that skip counted.   

During our share session at the end of the workshop period the group became very interested in the idea of making a really big snap cube staircase and worked together as a class to make the staircase you see above.  This led into counting how many cubes made up the total staircase. (We used 276 cubes!) Tomorrow I'd like to push students to investigate how many cubes make up the staircase patterns they created individually.

I also have thought that it would be neat to make some kind of large staircase pattern to display in the stairwell of our school, but I will need to think more about how to carry that out...

Pattern Necklaces

I have yet to meet the child that isn't drawn to jewelry.  Boys, girls, doesn't matter.  If it sparkles, they love it.  So today we made pattern necklaces.  This was an open-ended activity with some parameters set up to hold children accountable for math work. 

We made our necklaces out of very simple materials-- plastic pony beads and elastic cording.

 

Rules were as follows: 

All necklaces must have a pattern. 

Before you are allowed to touch the beads, you must make a plan.

The plan must include a colored drawing with the pattern unit circled and a translation into letters underneath. 

Second grade patterns needed to be more complex than a simple "AB" pattern. 

Where Did Patterns Go? An old curricular cornerstone is missing from the new common core standards.

So patterns don't actually appear to be part of the new common core standards. I'm loving the idea that the common core is more streamlined than our old standards, but not exactly sure what to make of the fact that old curriculum cornerstones are apparently no longer relevant.

To be fair, number patterns do seem to be still seem to be something that needs addressing, but basic identification of pattern, translating a pattern into letters, identifying the pattern unit, and extending a pattern are apparently no longer important skills. Were they important before? I'm not sure... They were definitely tested skills.

When I noticed that estimation was missing from the common core, I decided to go ahead with activities that required students to make estimates anyway, working on the assumption that students' overall number sense and therefore other common core math skills would be developed in the context of that activity as well.  (See an example here.)  When I noticed that visualizing numbers, a la "quick images" was not part of the common core, I decided that the activities were still worthwhile and could be justified by the standards about understanding place value in base ten.  These missing elements seemed to support the common core standards that remained.  Estimation and visualization are also skills that I imagine students will use in their own lives.  I feel justified in continuing to teach these lessons. 

But what about patterns?  I'm still torn.  Are these skills that students will have to use in life? Are they skills that underlie other math lessons.  For years I feel that teachers have been told that math is "the study of patterns." --Something I believe was articulated by the famous mathematician A. N. Whitehead.  Is this truly so, or is it only a handy justification? 

I have noticed over the years that some children seem to have an innate ability to recognize and work with patterns, be they colors, letters, shapes, numbers, or what-have-you, while others truly struggle with pattern activities.  Is this ability correlated with ability in mathematics later in school? 

For now I suppose, I'll continue on with my pattern lessons, but I'd love to know what others think.  Should I drop these lessons and develop others more focused on number.  Maybe I should keep the lessons, but teach them outside of my daily math block?

Pattern Work

Math choice today with the gang busy puzzling over patterns, while I pulled small groups at the round table. The choices: making and recording snap cube patterns (also circling the pattern units and translating them into letters) and completing cube pattern puzzles (recreating a cube pattern on a mat board and then extending it). At the back table we played "What comes next?" with first graders deciding on a colored cube that would come next in my snap cube pattern and with second graders drawing figures to complete illustrated patterns.

Scoring "How Many of Each?" Problems on a Rubric



Today we worked on "How Many of Each?" Problems again.  This time, though, we got a bit more abstract, so I began by telling the class that we'd be working with made-up stories. 

I also introduced a very simple scoring rubric.  Just because a task is open-ended doesn't mean there shouldn't be some performance standards. 

Here's my deal:  For every accurate answer a student provides, I mark a star on the paper.  Some students will work very hard to get that one star.  Other students will wind up with ten or twelve stars because they have found many possibly combinations.  Additionally, for every way in which they show me their thinking, I add a smiley face.  If there is a picture, it gets a smiley face.  If there are words, it gets a smiley face.  If there are numbers, it gets a smiley face.  If there is a chart, it gets a smiley face.  Obviously, four smiley faces might indicate a bit of overkill, but I'd rather have a student over-explain than under-explain as they are beginning to develop the habits they'll take into testing situations. 

And here are some of the stories we worked on today:

• There were 10 pets.  Some are dogs and some are cats.  How many of each could I have?
• There were 12 animals.  Some are cows and some are pigs.  How many of each could I have?
• There were 13 fruits.  Some are apples and some are pears.  How many of each could I have?
• There were 14 crayons.  Some are red and some are blue.  How many of each could I have?
• There were 16 toys.  Some are cars and some are boats.  How many of each could I have?
• I have 20 cents in nickles and pennies.  How many of each could I have?
• I have 25 cents in dimes, nickles, and pennies.  How many of each could I have?

Introducing "How Many of Each?" Problems

 

"I have 6 toy ducks hidden under this box.  Some are green and some are red.  How many of each could I have?"

"How many of each?" problems were some of the first open-ended problems that I used in the classroom, prompting me to feel that it is possible to differentiate math instruction without driving myself crazy teaching multiple lessons or juggling small groups.  I believe that they are a great way to challenge students at a variety of levels within the same context, and I have posted more about my experiences with "How many of each?" Problems here.) 

"How many of each?" problems are also some of the first open-ended problems that I introduce to students each school year.  I expect, of course, that students will bring a wide range of understanding to the table.  First graders who are only a few months out of kindergarten have limited experiences recording their thinking on paper.  Second graders who were in my class last year understand a lot more about what I expect in terms of paper work and their number sense is usually much more advanced.  In order to help all of my students find success on our first day working with these problems, I begin with a very concrete example.  (More abstract problems can come over the next few days.) 

In the school basement a few years back I discovered a stash of toy rubber ducks in a variety of colors-- I find them to be the perfect concrete context for this lesson.  Are rubber ducks required?  No, of course not.  Teddy bear counters, bottle tops, crayons... if it comes in a variety of colors, it would probably work well. 

I begin by showing students the ducks and we do a little counting.  Then students hide their eyes and I put some ducks under a box.  They guess how many are under the box and we check and count.  Students hide their eyes again and this time I hide 5 ducks-- 2 red, 3 green.  Now I ask the first "How many of each?" problem-- "There are five ducks under this box.  Some are red and some are green.  How many of each could I have?"  As students suggest combinations we use the extra ducks to show possibilities, and to stay organized we keep each combination on its own piece of construction paper.  Once we have all the possible combinations covered, we have a big reveal complete with drum roll.  

And just as there is a range in the group of mathematical capability, there is a range of social/emotional maturity.  It is seemingly inevitable that one of my youngest will crow that he was "right" about what was under the box and another will tear up a bit because he suggested a combination other than the one I had hidden.    We go on to have a conversation about how there were lots of great ideas about what could have been under the box and how getting the "right" answer was really just lucky. 

After all of this opener, we finally get to that open-ended workshop question.  The kids close their eyes, I hide 7 ducks, some red and some green, and they are sent off with blank paper to show how many of each there could be. 

At first as they work, I circulate, but as students begin to finish, I ask them to check in with me and show me their work.  My policy is to say something positive and encouraging... "I like how you drew all the ducks."  "I like how you labeled your picture."  "I like how you wrote down the numbers."  ... and also push them to just a little further... "Could you label your picture?"  "Could you add those numbers?"  "Could you think of another possibility?"  Some students take another piece of paper or two or three or more and staple their pages into a booklet of possibilities. 

When the work period is over, we gather once more on the rug.  We share some different strategies and answers.  We have a final big reveal. 

Shopping!

Ten years ago when I first started teaching, I came upon some great resources left behind in the classroom by a previous teacher.  These included the Box It or Bag It Mathematics : Teachers Resource Guide series(Burk, D., Symonds, P., Snider, A., 1988).  A cornerstone activity in this series-- the shopping project--is one that I try to use regularly as well.   A shopping project is the ultimate open-ended activity and it works well for a wide range of learners.  I've used it with just first graders and I've also used it with wider multi-age spans of first- through fourth-graders. 

Here's how it works:

First I collect five or six interesting art elements.  Sometimes these are pieces of construction paper cut into different shapes-- red triangles, blue squares, yellow circles, etc.  Sometimes these are squares of paper, squares of fabric, squares of foil.  I make sure to have lots of each element.  Next, I make price tags for each element--  1 cent, 2 cents, 5 cents, 10 cents, 12 cents.  Lastly, I prepare zippered sandwich baggies full of pretend coins for each student.  I just put in random handfuls of coins so that each child has plenty of money, I don't bother counting it.  

When I introduce the project to the class, I show each material and the price and model the process of developing a piece of artwork and purchasing the different elements.  Here I think aloud as I create a shopping list, pay for the pieces, and arrange them on a base piece of construction paper.  Then I give each child a big piece of construction paper to use as a base, a bag full of money, and a piece of paper for their own shopping list.  I send students off to plan, and as they finish, I open the store for shopping. 

The tricky part is keeping the line moving at the store.  If you can have help running the store from an assistant or volunteer, all the better! 

I find that younger students rely mostly on pennies for shopping, older students are fascinated by getting change back on their purchases (and often misunderstand this process as getting free money).  The most advanced students can be challenged to look at finished pieces of artwork and calculate the total cost of each project or compare two projects and determine which one cost more to produce.  Some students might be able to handle a budgeting aspect to the project, but I prefer not to include that element because of the social issues that crop up once you introduce scarcity into the classroom.  (Again, see this great post by Lori of Camp Creek Blog.)

Money Jars

Today's read-aloud, A Chair for My Mother by Vera B. Williams, is about a family that saves up coins in a jar to buy a new soft comfy chair after losing many of their belongings in a fire.  We talked about how we might save up coins in a jar and did a little counting of my large demonstration coins in a big plastic jar.  Then we compared the total coins in two jars. 

Math choices included this worksheet where students filled the jars using coin stamps, totalled the amount of money, and indicated whether jar A or jar B had a greater total by filling in the appropriate circle.  This open-ended task allowed students to adjust the amount of money they had to count in simpler or more complex combinations.  Younger children stamped only with pennies.  Older children counted quarters, dimes, nickles, and pennies. 

Small groups at the round table met with me and we worked on counting real coins in babyfood jars, then recording and comparing the amounts in the jars.  Youngers circled the jar with the greater amount while olders began to work with greater than (>) and less than signs (<). 

Introduction to Money

Today we did a little introduction to money-- coin identification, so that over the next few days we can do additional work counting and comparing amounts of money.  I started with a whole-class overview on how to identify a penny, nickle, dime, and quarter.  I have a set of large demonstration coins, and we examined each coin in turn, discussing features that students noticed.  Size and color are obvious markings, but the class was most interested in talking about the presidents featured on each coin and the fact that statue of Abraham Lincoln can be seen sitting inside the Lincoln Memorial on the back of the penny.  I actually did not know this about the Lincoln statue until a student told me about it.  I'm not really sure how much the other kids actually understood about the idea that you could see a tiny speck that is supposed to be a giant statue in a building that according to another child "looks like a trolley car."  Then again, the point of the lesson wasn't in national monuments or presidents, but in learning to identify coins.  Attending to these details definitely helped the children differentiate amongst the coins!  After our brief discussion, I handed out baggies with one of each coin and asked the children to hold up each coin in turn.  They were very successful with this little test.  The workshop part of our day included several math choices, one of which was to make your own set of pretend coins out of brown and gray construction paper.  Students really got into this activity and many tried to include Abraham Lincoln sitting inside the Lincoln Memorial!  This being an open-ended task, there was a range of quality in the coins produced, but I told them I needed to be able to identify the coins they made without being told which was which.  Again, in most instances, they were successful.

The Mathematician's Chair

First it was an Author's Chair... Then it was a Scientist's Chair... Now it is a Mathematician's Chair. 

Early on in my teaching I adopted the practice of having a few students share their work at the end of writing workshop time.  I saw other teachers using a special "Author's Chair" to keep students' attention focused on the child who was reading.  It appeared to make such a difference, that I quickly decided that it would be a helpful tactic in my own classroom. 

A few years later, I realized that my students were full of terrific hypotheses and theories during our science lessons, but they were not really terrific about paying attention to what others were saying.  A stroke of inspiration hit and our "Author's Chair" became a "Scientist's Chair."  Soon students were incredibly excited to share their ideas for the priviledge of sitting in the special chair and the group was much more attentive to those ideas being shared. 

Then when our school district's state math test results weren't great, and we teachers sat around together pouring over the data, it became pretty obvious that our students needed to improve their ability to communicate mathematical thinking.  The multiple choice scores were fine, but the open-response scores were not so hot.  Lots of discussion ensued about how we might do this, but in the end the only real conclusion reached was that we all needed to give our students more practice with open-response questions. 

Of course, I didn't rush right back to my classroom and institute daily open-response math journaling as had been recommended.  Although I definitely believe open-response questions play an important role in our math classrooms, I don't believe that all math concepts are best taught in this way.  Plus, there are only so many hours in the school day, and besides, young first graders don't all have the requisite patience for lots more pencil and paper work. 

Then I went to a Math Solutions workshop and got chatting with a third grade teacher from another school who casually mentioned that they'd started asking their students to answer open-response questions orally rather than on paper in order to save time. 

Ah-ha.  Yes.  My students might not be ready to write about math daily, but they could definitely talk about math daily.  And so the Mathematician's Chair was born. 

Now I try to build in time at the end of each math lesson for the Mathematician's Chair.  Here students take turns explaining how they know a number is even or odd, describing the results of a survey, or demonstrating an addition strategy.  And just as the chair focused the audience of writers and scientists, it provides a focus for thinking about mathematical ideas. 

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. 
 
 
 

Even and Odd

Even and Odd... a pesky concept...

It's so abstract that I really like to ground it in a meaningful context.  For the past few days we've begun to do some serious partner work around here-- playing the math card games War and Double War , reading with partners, talking to our "circle partners" during read-alouds, etc.  Pairing up students is a natural way to consider the concepts of even and odd.  If there are an even number of students, everyone will have a partner.  If there are an odd number of students, someone will be leftover and need to join a pair, making a group of three.  Today we began to work with this concept, but we can come back to it regularly during our morning attendance routine. 

Here is the open-ended problem-solving bit of work that I asked the class to work on today:


"There are an odd number of children in Mr. Brown's class.  How many children might there be in the class?"

Most of the younger children found an answer by drawing pairs of people and then counting them up.  Those who were a little more advanced found multiple answers.  Many of the older children simply gave me a list of odd numbers. 

A few students went on to more complex questions like:  "Mr. Brown's class has an odd number of children.  Ms. Green's class has an even number of children.  When the two classes get together, will there be an odd number or an even number of children?  Show your thinking in pictures, numbers, and words." 

Comparing Capacity

Today's goal-- work on comparing volumes and in the process practice counting and writing numbers.  I filled a giant plastic bin with as many different sizes and shapes of empty containers as I could gather.  I labeled each container A through Z.  (Multiples of a particular size/shape were labeled with the same letter.)  Then I challenged the class to determine which of a pair of containers was larger.  We discussed various possibilities for determining a container's volume and settled on filling the containers with various math manipulatives and comparing total capacity.  Then students set to work.  Was this lesson actually differentiated?  I'm not certain, but if I were to try to make the case that it is, I'd label it an open-ended problem-solving activity.  Because students chose for themselves which containers to compare and chose for themselves which math manipulatives to measure with, each child adjusted the level of counting and comparing difficulty for themselves. 

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.