Rationale
What exactly is a "deep understanding of mathematics" and the idea of "learning deeply"? If I was to liken this concept to reading I would say it is the difference between knowing that the letters t-r-e-e spell tree verses a deeper realization that the word tree also relates to the big green thing out in the yard, and how to use of word tree by creating a sentence or story with it. It is where students move past simply knowing words in a sight list to understanding what all the words in a story truly represent. In mathematics, this can mean, for example, that a student does not memorize the fact that 2+2 is 4 but I understands that by taking two separate groups of 2 and combining them, they will always obtain 4, or that 4 things can be broken into two equal groups of 2. A deeper learning of mathematics involves comprehending how numbers work without memorizing "facts" rooted in our base 10 system.
This deep understanding what this unit strives to accomplish. In standard curricula, regrouping is taught as two different chapters, one on regrouping in addition and another on regrouping in subtraction. With in this unit I will be combining concepts I would have previously taught separately. By this I mean to teach addition and subtraction together, allowing students to see how addition and subtraction work together within the base 10 system. While it is very natural to think of a letter as part of the alphabet, it is not so common to incorporate the understanding of numbers with place value within this base 10 system, i.e. thinking of 13 as 10+3, or 57 as 50+7.
By simultaneously teaching addition and subtractions with place value, the student can begin to learn the language that mathematics offers us. This can be seen in the ideals of teaching two digit numbers as some tens and some ones or teen numbers as one ten and some ones (as in the examples above). Place value should not be taught simply as a vocabulary lesson, but used as a tool to teach addition and subtraction. The place value system and addition and subtraction complement each other and work together. This means rather than spending four weeks on chapter one in addition, four weeks on chapter two for subtraction and four weeks on chapter five for place value for example they can be blended together to spend 12 weeks digging deeply into what it means to add and subtract within our base 10 number system. It should be noted that with the deep learning of these concepts, speed with facts may not begin to appear early on. However, with the depth of understanding students will experience, they will successfully be able to move on to more complicated mathematical concepts later. Initially speed might be sacrificed for a deep and lasting understanding.
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