I agree, a lot of it has to do with real-world, hands on concepts and using a variety of tools in creative ways. For example, I teach my first graders about measuring through lessons on measuring the things in our room, their shoes, and each other. Also check out this video playlist with some interesting math lessons that highlight teaching conceptually.
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No matter what grade level, math concepts must be introduced with manipulatives. When concepts are introduced in this way, children are able to grasp the meaning more efficiently. After manipulatives, then a transition to paper pencil, still with the manipulatives is valuable, as children can see the connection between what they have been building connects to the algorithm. Last, the algorithm can be done without the use of manipulatives. I think we frequently rush to the algorithm too soon, skipping the use of manipulatives which is where the learning takes place. Upper grade teachers frequently want to take away the manipulatives or think they are not necessary, when it is imperative that all grade levels begin with hands on concrete manipulatives. I agree that all learning needs to be real world related.
Manipulatives and hands on activities is a great way to start. Students need to first understand that numbers have value and how to associate that value with objects before they can conceptually understand the basics of adding and subtracting. Hope that helps!
Begin with a good problem task.
I agree with the group.... Manipulatives are essential, but I have done some action research that delves into using fingers as manipulatives (I teach Kinder, so a good portion of our number is within 10) AND I have found that sometimes manipulatives in my classroom can be very distracting. That is why I have worked with fingers as much as I have. They are always with you and they are not distracting. Further more....we have begun to rely on ways to create math drawings that show movement. My kinders have become good at drawing arrows if there is a "change" in the number situation of the math story. I have also found that using real situations in our class that are incidental in nature can create interest in the math story and therefore motivation in solving the problem. Some successful stories we have solved involve using situations centered around ther birthdays, ages, or comparing children's belonging or what they can do. An example is: Today is Jose's birthday. He is 6 years old. What did his birthday cake look like 2 years ago? On my smart board I post a photo of Jose and a picture of a birthday cake. Or: Sam has 8 people living in his house. Caroline has 5 people living in her house. How many more people live in Sam's house than Caroline's? Even though this comparison problem is a first grade level acc to CCSS, I have found with the right story, they can figure out how to solve it. I have found that it depends on how personal story is. Drawing the picture to match the situation and explaining the drawing is magical.
I agree that it is essential to link math with real world, no matter what grade. For this reason, I choose to give to my students topics on applications related to a chapter we learned (e.g. for financial math topic on ancient economy, for trigonometry topic on how Eratosthenes measured Earth's radius, for statistics topic on the use and misuse of statistics, for sequences topic on a mathematical paradox etc). The students present their project with ppt and after the presentation follows a discussion. The presentation is not strictly scientific (it can also be the biography of a famous mathematician, or historical information about a mathematical concept), so that it is interesting and relaxing for all students.
Another concept I use, is student teaching. Students in the role of the teacher, prepare a whole period lesson plan to review a topic. They are free to use exercises, theory, quizzes, or to bring their class mates to the board or ask them questions.
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