| |
|
|
 |
|
|
| Didax "Class Ideas" Newsletter Archive |
|
|
|
Welcome to another month of Class Ideas. Fall has fallen and school is in full swing! This issue will help you with your Math Assessment plans. You'll find two great articles, one from Kathy Richardson about Critical Learning Phases and how they are important in assessing math skills, and one about using Number Talks for Math Assessment. As always, there's a special offer for Class Ideas subscribers, great Internet resources to help you find more information, and downloadable activity pages.
Don't forget: October celebrates measurement with National Metric Week being October 10-16! Almost all countries base their measurements on this system. This month offers you a chance to celebrate and compare and contrast the metric and imperial measurement systems. Check out the NCTM at www.nctm.org/meetings/metric-week.htm for ideas.
I love to hear from you, so feel free to send me an email if you have any comments on this newsletter or suggestions for future ones!
Anna Mullen, Editor |
| |
| Dead End Skills or Foundational Understandings? |
|
 by Kathy Richardson
Those of us working to improve the mathematics education of elementary students have seen many positive changes in the teaching and learning of mathematics. More and more teachers are excited about teaching mathematics and eager to give their students meaningful experiences. Teachers no longer view being "good at math" as high scores on timed tests or accuracy when following procedures. Teachers recognize that students who are good at math have number sense and can use what they know to solve problems. These students are confident and able to work flexibly with numbers. They search to make sense of whatever mathematics they are asked to do and can transfer what they know to new situations. They are persistent when working with mathematics that challenges them. They look for connections and can clearly communicate their thinking.
Yet there is much evidence that too many children are not making sense of mathematics. For them, mathematics is a maze of confusing rules and procedures, an obstacle course that must be completed in order to move forward in school. Teachers estimate that only about 1/5th of their students are good at math; 2/5th can do the math they are assigned but without much insight or understanding and at least 2/5th of their students are overwhelmed and struggling. Middle school teachers paint a bleak picture as they describe the lack of preparation their students have when they enter their classrooms. Throughout the country, test scores begin to slump in 4th grade and take a deep dive in middle school.
What is going wrong? Some have suggested that children have not been given enough practice with basic skills. Some say the standards at each grade level have not been high enough to prepare young students for what they will be required to do as they move through school. My own research has led me to a different conclusion. I have found that in spite of many improvements in the teaching of mathematics, children are still evaluated on their ability to get right answers rather than on the level of mathematical thinking they have achieved. There is an assumption that children know what they need to know if they can get right answers. However, learning to get right answers is not enough if the procedures children learn are, for them, empty of mathematics.
When teachers look only for right or wrong answers to determine the level of a student's achievement, that which is actually learned (or not learned) remains invisible. Teachers then base their instructional decisions on "illusions of learning" rather than on meaningful information about what children really know and understand. Thus, many children do not learn the underlying mathematics and are left with dead end skills rather than the foundational mathematics they need for future success.
What are dead end skills? Dead end skills are those skills that allow a child to get right answers on particular problems, but which do not help them transfer what they know to other related situations. For example, many young students can tell the answer to 10 + 6 instantly but do not know how many leftovers there would be if they made a ten train from 16 Unifix cubes. Some students know 3 x 4 is 12 but see no relationship to 12 divided by 4 or 30 x 40. Children have dead end skills when they solve the problem 14 x 3 correctly and then, when presented with the problem 114 x 3, solve it as though it were a totally new problem.
My extensive research over the past 30 years with students from preschool through 5th grade from many diverse communities has allowed me to identify the foundational ideas that children need to understand if they are to know more than dead end skills. These are those ideas that children must understand if they are to be able to think and reason with numbers and build on this knowledge when confronting more complex concepts. I call these important stages of thinking Critical Learning Phases and have developed a series of nine assessments, Assessing Math Concepts, that identify which Critical Learning Phases are in place and which are not. Since the assessments identify Critical Learning Phases that all children move through as they learn to understand number concepts, they provide important information to all teachers, no matter what math program they are using.
When teachers use these assessments with children, there is a consistent response from them regardless of what part of the country they teach in or what the level of their student's academic achievement. Teachers react by saying, "I thought my children already knew this." The evidence that challenges their assumptions and uncovers the illusions of learning is there for them to see for themselves.
Educators from around the country who attended the first two Assessing Math Concepts Institutes held this year in Bellingham, Washington learned to identify the levels of understanding children must have if they are to gain a strong foundation in mathematics. They practiced using the assessments first through the use of video and then had opportunities to assess children themselves. Working with the children convinced them that "These children have the same responses as seen on the videos." They saw that the Critical Learning Phases are predictable and they learned that children don't know what we too often assumed they knew. These educators returned home to find the same is true for their own students in their own schools.
If we are going to raise achievement in mathematics in ways that allow children to build on what they know and thus maintain high levels of achievement throughout their schooling, teachers must focus on the mathematics they want children to learn and not just on whether they are able to get right answers. Once teachers have identified what children really know and what they need to learn, they will be able to provide appropriate instruction that will give children a solid foundation on which to build, ensuring success for all students. |
| |
| Number Talks: Thinking with Numbers |
|
 What young children know and understand can never be fully determined through paper and pencil tasks. Teachers can get much more complete and useful information if they watch and interact with the children while they are doing mathematical tasks. Number Talks are one such way to interact with the children. How the children respond reveals their level of understanding.
Number Talks provide opportunities for children to work with computation in meaningful ways. During Number Talks, the teacher presents various problems to groups of children and asks them to share the processes they used to figure out "how many." Number Talks can be held either with the whole class or with small groups. When children are working with the whole class, they will have opportunities to experience a wide range of problems and many different ways to solve them. When working with a small group, the teacher can make sure all the children have the opportunity to share their processes if they wish, and can more easily tailor the problems to meet the needs of a particular group.
Helpful Hints for Implementing Number Talks
1. Do number talks every day but for only 10 minutes. A few minutes more often is better than a lot of minutes infrequently.
2. Ask questions such as...
? How did you think about that?
? How did you figure it out?
? What did you do next?
? Why did you do that? Tell me more.
? Who would like to share their thinking?
? Did someone solve it a different way?
? Who else started the problem this way?
? Who else used this strategy to solve the problem?
? What strategies do you see being used?
? Which strategies seem to be efficient, quick, simple?
3. Experiment with using the overhead, the whiteboard, chart paper, etc.
4. Consider having students "circle up" in chairs or on the floor.
5. Give yourself time to learn how to...
? record student solutions
? listen to and observe students
? collect notes about student strategies and understanding
6. To help determine what numbers or problems you select use what you learn from previous number talks as well as the focus of your daily classroom instruction.
7. Do number talks with yourself and others to try new strategies and increase your own confidence.
8. Name/label the strategies that emerge from your students:
? Use doubles
? Break apart numbers
? Make it simpler
? Use landmark numbers (25, 50, 75, 200, etc.)
? Use a model to help
? Use what you already know
? Make a "10"
? Start with the 10's
? Think about multiples
? Think about money
? Traditional algorithm
? Counting on
9. Use related problems: 3 x 14, 3 x 114, 3 x 1014 or 7 + 8, 27 + 8, 107 + 8 or 3 x 7, 6 x 7
10. Do number talks in small groups
11. Ask students to "Do as much of the problem as you can.?
12. Give students lots of practice with the same kinds of problems.
13. Use numbers for subtraction and addition that require students to work past a ten or hundred.
Example: 56 + 7 = 87 - 9 = 25 + 6 = 94 + 8 = 106 - 8 =
14. Give students opportunities to add and subtract 9, then 8 etc., using 10 as a friendly number to work with.
Example: 68 + 10 = 78 so 68 + 9 = 77
15. Expect students to break apart numbers, not count on their fingers. Show them how.
6 + 8 (think of 6 as 4 + 2; add the 2 to 8 to get 10 and just add the remaining 4 to get 14)
16. Show the strategy you used. Make sure they know it's not "the" way, just another strategy.
17. Give students larger numbers so they can give "estimates.?
18. If you use chart paper, write down the student's name next to their solution. Keep track of who is participating and their strategies. Use the following as a "sorting" or assessment guide:
? Can figure it out (by counting on, using an involved strategy etc.)
? Beginning to use efficient strategies (can complete some of the problem efficiently)
? Just knows or is using efficient strategies
19. Create a safe environment. When children feel safe, they are comfortable sharing an answer even when it's different from everyone else's.
20. Provide concrete models (snap cube "trains", base 10 blocks, money etc.)
21. Give opportunities for children to "think first" and then check with the models.
22. Have students occasionally record their thinking and the steps they use to solve a problem.
23. Encourage self-correction; it's okay to change your mind, analyze your mistake, and try again.
24. Provide number stories.
25. Be curious; avoid making assumptions.
26. Give number talks time to become part of your classroom culture. Expect them to follow the usual learning curve stages. "Keep on keeping on" and you will get positive results! |
| |
| Downloadable Math Activity Pages |
|
 This month's downloadable activities come from two popular World Teachers Press series, Math Speed Tests and A Blast of Math. Both of these series offer fun, yet useful, quiz-type activities. Math Speed Tests assesses your students' knowledge of basic operation facts. A Blast of Math assesses math listening comprehension skills. They both come in a range of grades, so find activities appropriate for your students and you're off! |
| |
|
|
| |
| Math Assessment Internet Links |
|
 Great assessment resources are available online. Here are a few links to get you started! From a long list of articles available at pbs.org to DiscoverySchool.com and their rubrics and assessment links, you're sure to find some useful information. |
| |
|
|
| |
| Humorous Happenings |
|
 Thanks so much to all of you who have sent in your stories! It's been great getting a laugh out of the emails I've been receiving. Now it's time to share them with all of the Class Ideas readers!
R.M. Cameron writes:
One morning in class, we were celebrating the birthday of one of my first grade students. The conversation moved on to students' zodiac star signs. One little girl said, "I'm a Leo," another said "I'm a Gemini," and another little boy said, "I'm not sure, but I think I'm a Vegetarius."
And this from J.N. Darwin
My preschool class was enthusiastically helping to plant seedlings in their own special garden. Little fingers carefully prepared the soil. Willing helpers pushed the wheelbarrow transporting seedlings from the trunk of my car.
Back for a second load - no keys! Must be in the trunk?? Disappointed faces. Frustrated teacher. Activity reluctantly postponed. Dash home at lunchtime with fellow teacher to locate spare set of keys. Open trunk - seedlings aplenty...no keys.
Puzzle solved a few weeks later when a beaming little face presents me with a set of dirty keys. While pulling out weeds, he'd found them buried in the plot! |
| |
| December Newsletter Theme: December Holidays |
|
| Don't miss next month's issue of Class Ideas. It will be packed with information on December's chilly holidays. From Christmas to Kwanzaa, we'll have great activity pages, Internet resources, and an informative article! |
| |
|
|
|
|
|
|
|
