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Even if students learn a concept with manipulatives and simultaneously learn it with written symbols, the two may remain separate, with students never drawing the connection between them

A simple review of key conclusions makes a few things clear. First, we must temper our endorsement of manipulatives in classrooms with some caveats; there are instances where manipulatives will not speed children’s learning, and may even slow it down. Second, the objects themselves should draw attention to whichever feature is meant to convey information, for example, the length of a rod if it is meant as an analogy to number. Third, teachers should provide instruction in the use of the manipulative so that this feature is salient to students, but teachers should not be so controlling that students are merely executing instructions without thinking. In addition, students are more likely to understand the concept the manipulative is meant to convey if that parallel is made explicit to them

But many teachers struggle with their own math anxiety, and research shows that they then pass on this anxiety to their students. (That happens with parents too, unfortunately.)

small, incremental steps which made the math accessible to all students and allowed some of them to experience success in math for the first time. “Because they can master the increments, they are getting the checks and building the mindset that their efforts can amount to something. That experience motivates them to continue,” she said. By continuing, they practice more math, get more skills, and become the math people they thought they couldn’t be.

it starts small and progresses in very small steps to a very sophisticated level in a relatively short period of time

collective effervescence,” the joy of knowing they can do it, rather than the joy of just getting a high mark

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math has been overhyped as hard, and all that students and teachers need is to have things broken down properly. Many have dubbed these simple steps as “drill and kill”. But he says the steps can be made fun, like puzzles.