MATHEMATICS
Mathematics in the primary grades is focused on training the mind to seek out the sense of things whether in arithmetic or in other areas such as geometry, symmetry, measurement, or fractions. Much of our math time is spent in search of mathematical relationships. We want our children here at Forest Ridge to become mathematical problem solvers, to know how to communicate in mathematical terms, to learn to reason mathematically (and logically) and to become confident in their ability to do mathematics. Yes girls too.

With these goals in mind, I have taken on the role of facilitator of learning rather than dispenser of knowledge. In our classroom the children are active participants in constructing mental processes (mental structures) and in explaining their thinking mathematically. I am there to gently direct and focus their attention on important observations and provide questions which lead to deeper understandings. If you visit our classroom during one of the times we are “doing” math you will see the children working individually and/or in small groups using models and manipulatives to explore concepts and processes. With the use of objects the students first develop their ideas and understandings. These experiences with visual models allow children to develop the mental structures upon which later symbolic representations are built. They are training their minds to look for the sense of things.

The three main components of my math program in first grade take place daily. In the morning we focus on arithmetic, which we refer to as ‘math folder’ time and in the afternoon we spend time building logical thinking structures which we call ‘Bubble Math’ -- essentially a math/science lab when each child is engaged in his or her own particular endeavor. The third part is CGI (Cognitive Guided Instruction), an ongoing small group activity designed to develop children’s ability to make sense of mathematics in context. I’d like to elaborate on each of these programs.

Bubble Math is what you will likely hear most about since this in one of the children’s favorite times of day. During Bubble Math each child selects a collection of objects to work with. The rules are, once you pick something, you keep it for the duration of the period. This prevents a superficial exploration that almost always prevents deeper understandings from emerging. Another rule is to build a bubble - the space all around you within one arm’s length. When working in the bubble, children do not bother other children by moving about or talking. As the children get down to business I join them, one at a time, to engage in a conversation about their work. Children enjoy this undivided attention and don’t mind being “bothered” by me. In fact at the end of math lab those children who haven’t had a chance to talk about their work seek me out to at least take a look before they put their work away. Also at the end of the work time, children often record what they have been doing through words and drawings.

On any given day you might see activities involving sorting and classifying -- things like rocks, things that float and don’t, buttons, plastic animals, cars, planes, boats, etc. Other children are using their collections to make patterns of all sorts, or to make designs and then reversing them, to order groups, or to work in a ‘double bubble’ with a partner to develop spatial relationship skills. These many and varied activities help to build mental structures. Among them are logical thinking, spatial understandings, cause and effect, Euclidean relationships, and number relationships.

To give you an idea of what this math lab looks like I will briefly describe one day in first grade using some examples. Jane is working with water. She is guessing whether or not a collection of various objects will sink or float and is testing her theories and recording her observations. She finds surprises. Allen is also working with water. He is pouring various amounts into clear film canisters to form a seriated collection. He is also planning on doing a little color mixing with the blue, red, and green bottles of water.

Nick is using his collection of two colors of unifix cubes to build a schematic of an army base but soon gets hooked into the challenge of developing patterns. What can you do with just one color of unifix cube to show a pattern? How many different patterns can be made? He is delighted when he has figured out which one I have removed from his pattern and is looking to trick me by removing a different one when I’m not looking. He has solved a problem to show pattern by using position of the cubes: standing up, sideways, and upside down. You can make an ABABABABAB pattern or an ABCABCABC pattern or even and ABCBABCB pattern. Maybe the choices are endless. We ran out of time before we found out about that. Later on in the year Nick will “invent” a double pattern using position and color. Wow!

Joe has finally gotten his turn with the Marbleworks, a collection of plastic interconnecting pieces that can be constructed in a seemingly endless variety of ways to build pathways for marbles to pass through. Can you make a marble jump from one course to another? What are the variables? How can you keep the towers stable? Can two marbles go at the same time on different tracks and arrive at the bottom at the same time and if so, how?

Amy is working with blocks. If I had to choose just one set of material that is essential in building logical, spatial, and Euclidean thinking I would choose the unit blocks. They are big and are almost as satisfying as having a really big dog to hug. They come in a variety of shapes and sizes and most importantly they are proportioned. There are blocks that are simply squares (or one unit). There are blocks that are rectangles, which equal two unit squares side by side. There are rectangle blocks that equal four units and a few that are eight units long. There are some that are 1/2 units both in rectangles and triangle shapes. The possibilities for great mind-stretching conversations and experiences are vast. I could stay with Amy and explore the wonder of the blocks for the whole lab. But I move on.

Brad is eager to show me what he has done with the set of Things that Go. He has formed a matrix with all the cars in one row, the boats in another row, the planes in another and the fire engines in still another. He has also noticed the attribute of color and put the group of yellow cars under the group of yellow airplanes, which are under the yellow fire engines. Likewise he has made a special column for the blues, greens, reds, purples, and orange. When I ask him what could go here (an empty space at the end of one row) he answers that there isn’t anything that goes there, but if I wanted to go to the store I would need to get more cars that are a different color -- maybe white or black or magenta. A year ago when he used this collection he was able to sort only by one attribute at a time. Building a matrix requires logical structures that develop over time with brain maturation and experience with objects to manipulate.

John is building a system of gears. How can he place the wheels and cogs so that when he turns the handle on one of the wheels all will move? Is there only one way to do it or many? Joy is having many possibilities. Finding the factors that determine success are also brain building. Over in the corner behind the desk Eric is working with a set of wooden shapes. We engage in a conversation about shape types and eventually end up sorting them by how many sides they have and then again by how many triangular sides they have.

Holly and Jake are using the pattern blocks in a double bubble. They build shapes and build them again upside down or they copy a complicated shape, taking turns adding parts. Bobby and Jim are also in a double bubble. They have selected the collection of ocean animals and are taking turns making identical oceans on their pieces of blue felt -- sitting across from each other. Right now they are placing them in a mirror relationship. Later they will do it the “Hard way.” When they have finished placing and arranging the animals, they take turns covering their eyes while the other removes an animal. How long does it take the partner to figure out which animal is missing? This activity builds topological and projective mental structures and is a favorite activity for most of the children. As you can see, each child is engaged in his or her own pursuit. And they get to choose. No wonder it is so appealing.

ARITHMETIC
In the morning the children work on arithmetic skills. This is a time when they grab their math folder and spend time building concepts associated with number. What are all the ways or combinations to make 5. The children move through a protocol of activities designed to build numberness. What is 5? At first the children use objects only to build models of the different ways. Danielle takes a baggie of lima beans painted green on one side. She counts them into piles of 5 and finds how many different ways there are to make 5. Two greens and three whites, one green and four whites, all whites, and all greens. Are there any other ways, she wonders. Together we look for opposites. Yes, all whites are the opposite of all greens. But what about one green and four whites? What is the opposite of that? She figures out what it could be and builds it. And what is the opposite of two greens and three whites? When all the ways are found, we arrange the groups of five in another way; all greens, four greens and one white, three greens and two whites, etc. She tells me the combinations. The next day she finds a baggie with 2 kinds of objects, perhaps red chips and yellow chips or small erasers shaped like lions and tigers. She recalls the process she went through in organizing the beans into different groups of five and replicates it with the “Two Kinds” baggie of objects. Again, she calls me over and tells me all the ways. I notice if she has organized them according to opposites, in numerical order, or randomly (much harder to discover missing combinations.) These two activities are at the object level of concept development. Danielle will next select a set of story boards and head for the carpet. Here she will build all the combinations for 5 using objects and context. There was an apple tree with 5 apples on it. One fell off and now there are 4. She makes similar stories for each of the combinations for five that she found when working with the beans and Two Kinds. She tells her stories to me. The sense of the combinations is forming in her right brain hemisphere (both in the context of addition and subtraction). Soon she will be moving to the transitional or connecting level activities. She will repeat many of the activities described above but this time working more independently and recording her work by drawing pictures of the beans or Two Kinds and gluing on the matching numeral combinations which describe them. Notice that she is not yet writing the numerals or equations. Numerals and other symbols are over there in the other brain hemisphere. The left hemisphere is good at providing names for things but does not make meaning to go with them. That is why it is so important to use objects to build the meaning in the right side before moving to the left to attach the written symbols. Eventually Danielle demonstrates understanding by doing the hand game - an activity designed to assess her fluency with the various combinations for 5. I always do the hand game with a child before moving them into the symbolic level worksheets and story problems -- the paper and pencil work you might find in a math textbook. I want to make sure a child has meaning well established. You will not see much of this number work until your child arrives at the symbolic level. Then as the children complete their number booklets they are sent home. As you will see, the tasks are rigorous -- more challenging than those pages found in math workbooks.

When Danielle completes her book of 5, she will move on to the sixes, building her understanding of what 6 is in similar fashion. All the numbers preceding 6 are encompassed in sixness. So her work with 5 will help her in making sense of 6.

Each child is moving through these protocols at his or her own pace to develop a firm understanding. In our class there might be children working on 5s, 6s, 7s, or 8s. This is not surprising when you consider that in first grade the ages of the children range over a year’s difference. Brains, like bodies, grow at their own rate. It takes time to develop mental structures that provide the ability to think logically (not just rote memorization of facts or just following directions). and that brings me to the next important component of mathematics -- story problems.

CGI
“If two trains are traveling from opposite directions at 65 miles per hour . . . “ -- don’t you just love the time you spent grappling with story problems in those text books. Homework. Do pages 236 through 239.
All story problems! Of course the hard part was trying to make sense of them. Well, the hard part hasn’t changed, but the way to build story sense and understanding of these diabolical problems has. Start with objects.

A lot of thoughtful research has gone into the study of how children make sense of story problems. I use a model developed at the University of Wisconsin called Cognitive Guided Instruction in Mathematics -- CGI for short. Here’s the deal. There are 11 different types of addition and subtraction story problems that primary children need to make sense of. They are sorted by difficulty. Generally children develop understanding of what the stories mean in the same sequential order, but not of course at the same pace. Each child’s brain develops uniquely, therefore logical thinking develops uniquely, too. And story problems require logical thinking unless a child is taught to look for key words and other clues not tied to meaning. Since I am interested in children learning stuff that will be useful in different contexts, I am interested in children learning with meaning. If you really understand it, you can apply it in contexts that are new without the help of tips or tricks.

The Wisconsin study also found that children develop solution strategies in a particular sequence, each stage being essential to true understanding. Modeling, counting, and deriving from a know fact. I want to tell you a bit more about this but first, let’s go back to the types of stories there are.

There are, of course, the kinds you see in textbooks. There were 3 birds in the tree and 2 more came. Now how many are there? There were 5 birds in a tree and 1 flew away. Now how many are there? These two story problems illustrate the easiest and first type of problem children will be able to understand. N + N = ? The start is there, the change is there. It is the result, which is the unknown part. Remember, I told you that a child will and should start with a modeling strategy to make meaning out of this problem. Take a bunch of beans or wooden cubes. Make a pile of 3. Make a pile of 2. Put them together and count them all and the result is 5. That’s modeling. Of course a child might model with his or her fingers, too. Counting them all is the tip-off. You see this with children completing worksheets. They are modeling both quantities.

As children become more confident with this strategy (typically it lasts for a year or more) they move on to the counting strategy. If they use their fingers, they use them as counters. Solving the above problem using a counting strategy would look more like this. The child thinks of the first number mentally and then counts up two more starting with three (which is in his/her head) and arrives at 5. The two fingers are there as number holders. For the second problem, a child would count down (backwards) 1 from 5. As children become more able to use this strategy I will make up problems that will encourage them to be inventive. What if there were 2 birds in the tree and 8 more came? I’m looking to see if a child starts with the larger number and why. Children who understand this strategy will be eager to tell me that it is a lot faster and easier to count up from the larger number, even though it wasn’t “first” in the story.

The second type of problems are the ones with the change as the unknown part. Frank had 5 rocks. He wanted 11 in his collection. How many more did he need? Adults thing about this as a subtraction problem: 11 - 5 = ? But children see it for what it is (if they can make sense of it). What can I put with 5 to make 11? 5 + ? = 11. See, it’s the change that is the unknown part. The strategies: Model -- make a pile of 5 and add more until you have 11. Count how many you have added. That’s modeling. Of course, at first when children are inventing this solution they forget to keep the ones they add on separate and end up with an answer of 11 every time. It is a light bulb moment when they realize they have to keep the add-ons separate. The counting strategy goes like this. Think 5. Then count on . . . 6,7,8,9,10,11. That’s 6 more. Fingers are good for keeping track of how many count-ups there were. A child that has moved on to the deriving strategy thinks 5 + 5 = 10 (a known fact) so 5 + 6 is just one more -- 11. This child is using a fact he or she knows to derive the answer. Of course some children might already know the combinations in the 11 fact family. Think of this problem: 15 + 16 = ? As an adult you will quickly derive from known facts: 15 + 15 = 30 + 1 = 31 perhaps or 10 + 10 = 20 + 11 = 31.

There is also a subtraction version of the change unknown type. 8 - ? = 3. Can you make up a problem to match that equation?

In the next type of problem the start is the unknown part. Of course this is difficult. After all, how can you start when the start isn’t there. If you can think logically and reverse the action, it can be done. But not until.

As we work on story problems I take 4 children to a corner of the carpet and give them a problem to think about. Objects are provided for children who are solving problems by modeling. Other children in the group might be using a counting strategy and others might be using facts they know to derive the answer. For example if the problem was “I have 2 sea shells and my friend gave me 8 more. Now how many do I have?” ( 2+8= ?), one child might build a set of 2 and another of 8 and put them together and count them all. Another child might count up from 2. After all, it is the start of the problem. Another child may start with the 8 first and another my have mastered the facts for 10 and just “knows”. When each child has found the answer and put it in his or her hand, I go around the circle asking each child to tell how they figured out their answer.

Being able to put your cognitive process into words is at first a bit daunting. “I just know,” they say. “My head told me,” they say. “What did it say?” I ask. “It told me to take a little guess!”

As the students become more experienced and listen to each other explain their thinking, they do come to identify how they know. “Yes, I made a pile of 2 here and a pile of 8 there and . . . .” “I counted on from 2.” “I counted on from 8. If you count on from 8 you only have to go two more.” This is a revolutionary idea. We all try it. Hmmm, something to think about. If a child is ‘ready’for that particularly piece of information, it is another strategy to try out. Usually the group will work with three or four problems before resuming their independent work in their math folders. And another group will have a turn.

Counting bags of objects is also an important independent activity. Your child selects a little box or a bag of objects and thinks about how many might be in it and then counts them, grouping them into tens and making tally marks, recording the amount on a piece of paper.

When the children complete their fact books through ten, they are ready to start place value. Place value is a very difficult and sophisticated concept, not one that is readily accessible to most first grade brains. Think of all the ancient grown-ups in societies of long ago that never thought of it. The hard part is trying to think of the 2 in 26 as both two groups of 10 and as 20 ones, and not just as 2 ones. Once children have begun the journey to understanding place value doesn’t mean they are ready for symbolic level worksheets like 29 + 12 or even 23 + 12.. Most children will simply turn that last problem into two baby problems 3 + 2 = __ and 2 + 1 = __. They can successfully write the answer without the least understanding of the total sum. Likewise children can memorize the algorithms for borrowing and carrying without having a clue. It just turns out right (most of the time). The problem is that if it doesn’t really make sense then the answer won’t either and children don’t spot answers that don’t make sense because they aren’t looking for sense. So here is my caution to parents, particularly ones with children who have a great aptitude for math. Let me decide when they are ready for borrowing and carrying (also known as “trading” or “renaming” depending on the decade you were in the primary grades.) Quite frankly, most first graders and a lot of second graders simple do not have the mental maturation for it. It is wrong to push them simply so that they can add and subtract 2-place problems. However, it will be right for some children and when the time comes, I will know it.

There are so many strands in mathematics that can fill up our days, that dwelling on something difficult that doesn’t make sense, doesn’t make sense. Instead, while waiting for those brains to mature we will investigate many mathematical concepts that do make sense.

Sometimes we will take time off to explore math as a whole class endeavor. Geometry is a good example. Geoboards -- using rubber bands on wooden square tiles with 25 nail posts. Geometric shapes can be made by stretching the rubber bands around the posts in various ways. Perimeter, area, characteristics of rectangles, and fractional amounts are just some of the concepts that can be developed with geoboards. Measurement and time are also often addressed in whole group activities, as well as multiplication and division, and even prime numbers. All using objects for concept development.

Let me say this one more time coming from years of experience and regrets; children will learn some thing if you insist on it, but they will not learn it in a way that will be useful information, which they can apply in different ways until they have developed the proper mental structures. Instead they will learn it and forget it. As a third grade teacher I have found that many children who were taught place value as first and second graders came to third grade not understanding it at all. If you really understand it, you don’t forget it.

So the question is, you ask, if it isn’t age appropriate, why is it in the curriculum. Well, that’s easy. What’s tested counts. Fortunately we don’t test first graders here at FRA but the second and third graders undergo standardized testing and some school districts do it two or three times a year. What ends up in the newspapers counts, even if it doesn’t tell what’s really important. And if the math program in a school is driven by a textbook then it is taught. You are welcome to examine the first grade textbooks that I have (and don’t use) to see what I mean. The combination of programs I use are based on the dap mathematics program which is a individualize mathematics/science program based on 20 years of research into how children’s thinking develops and CGI, also, as mentioned above, a research based hand-on program. Both are designed to teach children; not curriculum. Welcome to my world. It is a great place to be.