a b s t r a c tWe used a teaching experiment to evaluate the preparation of preservice teachers to teach early algebra concepts in the elementary school with the goal of improving their ability to generalize and justify algebraic rules when using pattern-finding tasks. Nearly all of the elementary preservice teachers generalized explicit rules using symbolic notation but had trouble with justifications early in the experiment. The use of isomorphic tasks promoted their ability to justify their generalizations and to understand the relationship of the coefficient and y-intercept to the models constructed with pattern blocks. Based on critical events in the teaching experiment, we developed a scale to map changes in preservice teachers' understanding. Features of the tasks emerged that contributed to this understanding.
In this chapter, the author highlights the tools used in an online environment geared toward practicing teachers earning their master's degree. The focus will be on web-based platforms and the importance of both synchronous and asynchronous online learning. The author offers existing and new online instructors' ideas on how to structure their own synchronous learning environments. She will support her suggestions with relevant research references. Samples of teachers' representations used during class sessions will be included so the reader has a visual understanding of how the tools work. Asynchronous ideas will also be discussed for those who integrate both styles in their teaching.
This article investigates two children's intuitive thinking in solving multiplication problems from different educational backgrounds. One of the children is in a southern elementary school in the US. He was given the same problems both in first and second grades. The other child was a first grader in a southwest region of China, and she was given the same problems. The findings reveal a variety of intuitive thinking in solving the multiplication problems through addition beyond direct modeling and counting strategies. The authors also discussed how different educational backgrounds in early elementary mathematics education may affect children's intuitive ideas and reasoning in solving multiplication problems. The study implies the importance of understanding children's intuitive ideas of multiplication and highlights potential opportunities for developing children's understanding of multiplicative thinking and algebraic thinking in earlier stages of arithmetic learning.
Do you find that your students sometimes struggle when communicating their mathematical ideas? Do you believe that developing good spatial sense is important in the mathematics classroom? To jump-start a mathematical conversation among you and your students, complete this exercise: Glance at figure 1a for no longer than three seconds, then draw what you saw. When you are done, look at the image and compare it with your drawing. What did you see? How did you draw it? Ask a colleague to do the same. Did he or she draw the diagram differently? How did you describe the image, and how did your colleague?
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