An integrated approach to first grade arithmetic is described. It consists of a coordinated development of the three pillars of the title, which are (i) strong conceptual grasp of the operations of addition and subtraction through word problems, (ii) computational skill that embodies place value understanding, and (iii) coordination of counting number with measurement number. The ways in which these three parts interact and reinforce each other is discussed. This approach is highly consistent with CCSSM standards recently released in the United States by the Council of Chief State School Officers.
In a second part, a sketch is given of a further development of these key ideas in later grades. Increasing understanding of the arithmetic operations leads to increasing appreciation of the sophistication and underlying structure of place value notation, eventually making links with polynomials. Linear measurement becomes the basis for developing and exploiting the number line, which later supports coordinatization. Throughout, consistent attention should be given to interpreting and solving increasingly involved word problems. Successful intertwining of these three strands supports the later learning of algebra, and its links to geometry.
- Word problems
- Place value
- Counting-measurement coordination
- Number line
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Irrational numbers, which, with a few exceptions such as some square roots, π and e are not encountered by non-mathematicians, but which can be articulated into an elaborate hierarchy.
However, the final stage of understanding, in which the base ten units are written as powers of 10 using exponential notation, linking place value notation with polynomial algebra, can not take place before 6th grade, when exponential notation is first introduced (6.EE 1).
Alternatively, if we select the 5-element set as the unit, then the first two sets represent 2/5 and 3/5 respectively, and the equation would read$$(1/2) \times (2/5) + (1/3) \times(3/5) = 2/5,$$
which is also a true equation, representing 2/5 as a weighted average (not a sum!) of 1/2 and 1/3.
The unit attached to the product is then the product of the units attached to the factors.
At this point, it might be a good idea explicitly to discuss the issue of associativity of multiplication, that it does not matter how we group the factors in these (or any) repeated multiplications, the result will not depend on the grouping. Thus, 10,000=10×1000, but just as well, 10,000=100×100. In fact, associativity of multiplication is a somewhat subtle property, and its justification using geometric models involves volumes of 3 dimensional bricks. See for example (Epp and Howe 2008) for a fuller discussion.
Unfortunately, this basic principle is not explicitly enunciated in the CCSSM. One hopes that this defect will be remedied in the next revision.
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Howe, R. (2014). Three Pillars of First Grade Mathematics, and Beyond. In: Li, Y., Lappan, G. (eds) Mathematics Curriculum in School Education. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7560-2_10
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