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| Didax "Class Ideas" Newsletter Archive |
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Another Spring, another April, and another Math Awareness Month! To help you celebrate Math Awareness Month in your classroom, Class Ideas is filled with math-related resources.
We have an article from Kathy Richardson that explores what sizes of numbers are appropriate for children. Kathy is known for her practical and insightful resources and we're excited to offer her new Assessing Math Concepts series (look below for a great subscriber-only special). There are two downloadable lesson plans from authors Evan Maletsky and Mary Kay Varley and links to some great math resources.
Math is a broad but oh-so-important topic. I hope you read on to find something useful for your classroom in this issue.
Anna Mullen, Editor |
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| Too Easy for Kindergarten and Just Right for First Grade |
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 By Kathy Richardson
Those of us who find teaching a rewarding and challenging profession know we will never have it all figured out. We always have something to ponder as we work to create the best educational experiences we can for our students. Some recent experiences raised some new questions about how children develop number concepts and led me to some new understandings.
When planning learning experiences for children, my goal is to give all the children in a classroom activities that keep them thinking and working at the edge of their understanding. Even though my intent is to keep children challenged, I often find myself recommending smaller numbers than others might suggest. An ongoing question for me has been, "What sizes of numbers are appropriate for children?"
One of the ways I know when I am providing appropriate experiences for children is to notice their level of enthusiasm and engagement. When a task is "right" for them, children approach it in a way that has nothing to do with pleasing the teacher or getting their work done. This part of the teaching-learning process became clear to me when I was working with a class of kindergarten children. The children were working with a set of eight tasks designed to help them develop facility with counting numbers to 10 and build an understanding of these quantities. Each task was presented in the form of a question, thus encouraging the children to view the tasks as problems to be solved rather than simply tasks to be completed.
The children were finding out:
How many tiles long is the yarn?
How many Unifix Cubes does it take to fill the containers?
How many paper clips fit along the lines on the Line Puzzles?
How many Unifix Cubes fit within the outlines of the Shape Puzzles?
I had seen many different groups of kindergarten children working hard with these activities in the past, but this group seemed restless and lacked enthusiasm for the work. It was as though they were pushing me to give them something more, so, I decided to expand the tasks to include larger numbers. I replaced the short pieces of yarn with longer pieces and had the children measure them with toothpicks. I replaced the small containers with bigger containers and had the children fill them with walnuts and yellow pattern blocks. In the end, I had arranged the activities so that the children were working with larger numbers at all the stations. Their interest and enthusiasm were renewed immediately. They stayed involved with these tasks for several weeks.
After this experience, I decided to expand some of the tasks I had given first-grade teachers to do with their children in the beginning of the year. I designed new Line Puzzles and Shape Puzzles so that the children would be working with numbers to 20. I suggested to the first-grade teachers that they use these and other adapted tasks early in the year with their students. When I checked with these teachers in October, none of them had used these new materials yet. Their children still needed to work with numbers to 10. I now had something new to ponder: WHY WERE THE TASKS TOO EASY FOR KINDERGARTEN AND JUST RIGHT FOR FIRST GRADE?
This question was in the back of my mind when I visited a second-grade classroom where the children were working with place-value stations designed to help them understand numbers to 100. They were using Unifix Cubes, tiles, paper clips and wooden cubes to measure a variety of things in the room. The tasks required them to organize their materials into tens and ones and record their results.
When I walked into the room, a couple of boys came over to share what they had just accomplished. They were very excited when they said, "We measured our rug, and it was 438 Unifix Cubes long! Their enthusiasm for counting to big numbers was obvious. Because most of the other tasks limited the children to working with numbers less than 100, again I asked myself, "What sizes of numbers are appropriate? Should I encourage the teacher to provide more experiences with larger numbers?" I asked the boys how they counted that long train and they told me they had broken their train into tens. While I was pondering whether the children should be encouraged to work with larger numbers, one of the boys went back to the rug to get evidence of how they had counted. He returned with his hands full of Unifix trains of ten cubes. "When we measured our rug, we broke our long train into these tens," he explained. Much to my amazement, he went on to say, "And I can count these by fives, too." He laid these trains of ten on the table one at a time and counted, "ten, fifteen, twenty, twenty-five, thirty, thirty-five and so on".
Later, I observed another child who was measuring the seat of his chair by covering it with tiles. The tiles were all white ceramic, so it was not possible to see groups of ten. To keep track of the groups of ten, the child was marking every tenth tile by laying an extra tile on top. At least that is what I thought he was doing until I heard him saying as he pointed to the tiles one by one, "Ten, twenty, thirty?" and so on until he got to 100 and marked that tile by laying another tile on top. Both of these examples of undeveloped understandings reminded me that the children still had much to learn that went beyond the simple act of counting.
My next puzzling experience happened in a grade 1-2 multi-age classroom. I decided to broaden my experiences with mathematics literature and brought in the book 1 Hunter by Pat Hutchins to use with the children. In this story, the hunter is unaware that he is being followed by a variety of animals: 2 elephants, 3 giraffes, 4 ostriches and so on to 10 parrots. In the end, he turns around and discovers all the animals behind him. The question for the children to answer is, "How many animals were following him in all?" The children enjoyed the story and got busy right away solving the problem. Some were able to write down the numbers and add them, others drew pictures of the animals and counted them, and some used Popsicle sticks. As I watched the children working, new questions arose in my mind. Why was I not more excited about this experience? Why did it have a ho-hum quality? Why did it seem that the children had been working harder when they were creating designs using from five to eight toothpicks and labeling the parts?
I thought about all of these experiences for quite awhile until I realized that children were not necessarily working at a higher level just because they were counting to larger numbers. Tasks cannot be put into a hierarchy of difficulty based solely on the size of the numbers. What made a task hard or easy was the kind of thinking the children were doing when they were working. I began thinking about number skills in terms of the following three levels:
Level One: Count and Land
Children count and count and count and count and then land on a number. They have no way of knowing whether the number they landed on is reasonable. They just count as carefully as they know how and report the last number that they said as they counted.
Level Two: Number Sense and Relationships
Children think about the quantities with which they are working. At this point, they can decide whether their answer is reasonable. They might think something like, "I thought the jar was going to hold five walnuts, but it holds eight." or "I thought the desk would be forty Unifix Cubes long, but I have already used thirty-six cubes and I am only half way."
Level Three: Parts of Numbers
Children take numbers apart and put them back together flexibly. At this level, the children know such relationships as "Seven is three and three and one more, and it is also five and two more" and "If I have six, I need four more to make ten."
What Sizes of Numbers Are Appropriate?
So why were some tasks too easy for kindergarten and just right for first grade? Because the kindergartners were working with the tasks at Level One and the first graders were working with the same tasks at Level Two. The kindergarten children were no longer challenged by counting to ten but were not yet ready to think about quantities and relationships. Instead they were intrigued with the chance to count to what to them were really big numbers. On the other hand, the first graders were focused on quantities and were interested in estimating and finding out how close their estimates were.
Why do the second graders need to work with numbers less than 100 even when they love to count big numbers? Because they need to develop number sense and relationships for numbers to 100. When the numbers are too big, they can?t decide for themselves whether their answers makes sense.
Why was the experience counting the animals in 1 Hunter less challenging than labeling the parts of a toothpick design? Because it was a Level One task; children did not need to do much thinking, just enough to complete the task. It thus became clear that the size of the numbers with which the children should work depends on the focus of the task and what children are ready to bring to the task. It is appropriate for children to work with large numbers when counting, smaller numbers when developing a sense of quantities and relationships among numbers, and even smaller numbers when learning the parts that make up those numbers.
When deciding what experiences will challenge children, we must not be unduly impressed by their work with large numbers. Counting is just one aspect of the number concepts that children need to understand and is in many ways just the beginning. It is even more important for children to develop number sense, understand number relationships and know number combinations. Children should have experiences that help them learn to count, and should be able to count to get answers when working with large numbers. At the same time, we must give them experiences with smaller numbers so that they can develop a sense of quantities and relationships. In addition, children must be given opportunities to work with even smaller numbers until they have internalized the number combinations that make up those numbers.
Providing Experiences at All Levels
We need to be aware of these different ways of working with number and plan for experiences that help children experience numbers at all three levels. For example, a child might be practicing counting to 50; beginning to work with number relationships with numbers to 15 or 20; and working with combinations of, or taking apart, 4 and 5.
As we watch children at work, we must look at what approach they are using to do the tasks. For example, one child may measure a piece of yarn by counting and landing on a number, with no thought about the quantity. This child is working at Level One. Another child may be working at Level Two, estimating the number of paper clips needed and changing his mind after measuring part of the yarn.
Two children may use different levels of thinking with numbers to determine the number of Unifix Cubes in a particular train:
One child counts the cubes by ones and tells the number she landed on.
Another child says, "There are four red and five yellow. Five and five is ten. So if I take one off the five cubes, that would be nine."
Planning Level One Experiences:
Because this level requires the least complex thinking, children do not need as much time working at this level as they do other levels. Still, they need occasional opportunities to count to what is for them a big number. You could have a counting jar that you fill with different amounts over time. The children can count the objects and compare results. You could also present stories like 1 Hunter which give children a chance to use counting to solve a variety of problems.
Planning Level Two Experiences:
Level Two experiences are important because they help children develop number sense and focus on the actual amounts with which they are working. One way to provide this focus is to have the children estimate the number of objects in a jar. After the jar is partly filled, give them the opportunity to change their minds as they consider the new information. Also present independent tasks wherein the children are asked to estimate first and then find out.
Planning Level Three Experiences:
The highest-level thinking is more likely to occur when children are working with smaller numbers than when they are working with very large numbers. We need to recognize that the work children do with small numbers is complex and important to building a strong foundation that will serve them in valuable ways in the future. It is very important not to rush children through these experiences but rather to give them the time they need to build this foundation. Internalizing is not the same as memorizing. When something is memorized, it can be forgotten. When something is internalized, it is part of the way one sees the world. Teachers throughout the country, working in many different situations, have found that most first-grade children have internalized combinations for numbers to 6 by the end of first grade and most second graders still need to work with number combinations to 10 when they begin second grade. Therefore, we must give children continuous experiences with small numbers for several years so that they can develop the facility with numbers that will allow them to move beyond counting to thinking with numbers.
As we gain new insights and understandings of how children learn, we can see more. As a result, we can design more appropriate experiences that help them develop a strong sense of number concepts.
Reference: Hutchins, Pat. 1 Hunter: New York: Greenwillow Books, 1982. |
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| Useful Math Internet Resources |
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 Each year people in the U.S. and around the world celebrate Mathematics during Math Awareness Month held throughout the month of April. This year's theme is "The Mathematics of Networks." In order to help you prepare for Math Awareness Month, we have found two great websites which provide rich activities to help celebrate math. There are also links to more information about Math Awareness Month itself and a site with a great math encyclopedia. |
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| Free Activities from Big Ideas in Math |
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 This month's downloadable activities come from a set of two new books that we're really excited about, Building Big Ideas in Math. The hands-on problem-solving activities in these books emerge from the fundamental notion of discrete math, a topic that is becoming increasingly important in school mathematics. The books offer complete lesson plans, from introductory material to sample student reactions and reproducible worksheets. Activity One is great for grades 3-5 and Activity Two is for grades 6-8. |
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| May Newsletter Theme: Conflict Resolution |
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 Keep your eyes open for next month's issue of Class Ideas. It will be filled with resources, ideas, and tips for teaching Conflict Resolution skills, a hot topic in every classroom. |
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