Exemplars & Common Pitfalls
1: Requiring Automaticity with Basic Number Facts
The four arithmetic operations for whole numbers cannot be mastered if the single-digit addition and multiplication facts (and corresponding subtraction and division facts) have not been learned to automaticity. For multiplication and division, only eleven states (plus Common Core) use key words or phrases such as: automaticity, memorize, instant, or quick recall. Another fifteen states either fail to mention these â€œmath factsâ€ or specify only that students be able to compute them. But â€œfluencyâ€ with calculating the basic facts is not the same as instant recall.
These states specifically require fluency of multiplication and division facts.
- Develop quick recall of multiplication facts and related division facts and fluency with whole-number multiplication (grade 4)
- Develop and demonstrate quick recall of basic addition facts to 20 and related subtraction facts (grades K-2)
- Extend their work with multiplication and division strategies to develop fluency and recall of multiplication and division facts (grades 3-5)
- Immediately recall and use addition and subtraction facts (grade 3)
- Immediately recall and use multiplication and corresponding division facts (products to 144) (grade 4)
In the development of arithmetic, students are expected to be able to use different methods of computing, but fluency is not required.
- Use flexible methods of computing, including student-generated strategies and standard algorithms (grade 3)
- Use flexible methods of computing including standard algorithms to multiply and divide multi-digit numbers by two-digit factors or divisors (grade 5)
- Demonstrate fluency with basic addition and subtraction facts to sums of 20 (grade 2) (This can be interpreted as either computational fluency or instant recall. This lack of specificity means that some students might not be required to actually internalize the basic facts.)
Kentucky only requires students to use “computational procedures” and fails to require instant recall.
- Students will develop and apply computational procedures to add, subtract, multiply and divide whole numbers using basic facts and technology as appropriate (grade 5)
2: Mandating Fluency with Standard Algorithms
Arithmetic forms the foundation of K-16 mathematics, and whole-number arithmetic forms the foundation of arithmetic. The proper goal for whole-number arithmetic is fluency with (and understanding of) the standard algorithms.
The Following states specifically require students to learn and use the standard algorithms:
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition (two 3-digit numbers and three 2-digit numbers) and subtraction (two 3-digit numbers) (grade 2)
- Multiply multi-digit whole numbers through four digits fluently, demonstrating understanding of the standard algorithm, and checking for reasonableness of results, including solving real-world problems (grade 4)
- Demonstrate an understanding of, and the ability to use, standard algorithms for the addition and subtraction of multi-digit numbers (grade 4)
Only seven states explicitly expect students to know the standard algorithm for whole-number multiplication as their capstone standard for multiplication of whole numbers. But twenty-four states explicitly undermine this goal by offering, even expecting, alternatives to the standard algorithm:
- Use a variety of strategies to multiply three-digit by three-digit numbers (grade 5)
This standard fails even to mention the standard algorithm, and thus leaves little confidence that students across the state will master this essential content.
Other states pay homage to the standard algorithm while still avoiding the goal:
- Solve multi-digit whole number multiplication problems using a variety of strategies, including the standard algorithm, justify methods used (grade 4)
Here, while the standard algorithm is mentioned, students can clearly move on without having mastered it.
3: Getting Fractions Right
After the foundation of whole-number arithmetic, fractions form the core of mathematics. Only fifteen states even mention common denominators, something essential in the development for adding and subtracting fractions. Likewise, standards specifying fractions as division or requiring mastery of the standard algorithms are rare.
The often-confused concept of fractions as numbers is intro-duced early and clearly.
- Understand a fraction as a number on the number line; represent fractions on a number line diagram
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line (grade 3)
The arithmetic of fractions is carefully developed using mathematical reasoning.
- Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 Ã— (1/4), recording the conclusion by the equation 5/4 = 5 Ã— (1/4) (grade 4)
- Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd) (grade 5)
Similarly, in Connecticut fractions are well developed.
Common denominators are introduced explicitly:
- Compare fractions by finding a common denominator (grade 5)
When fractions are introduced, they are not explicitly introduced as parts of a set or a whole.
- Identify and model representations of fractions (halves, thirds, fourths, fifths, sixths, and eighths) (grade 3)
In the case of adding and subtracting fractions, standard procedures and fluency are not required, nor are common denominators developed.
- Add and subtract fractions having like and unlike denominators that are limited to 2, 3, 4, 5, 6, 8, 10, and 12 (grade 4)
- Solve single-step and multistep practical problems involving addition and subtraction with fractions and with decimals (grade 4)
The standard algorithms are not required. Students are instead encouraged to develop their own algorithms. And common denominators are never mentioned.
- Develop and analyze algorithms for computing with fractions (including mixed numbers) and decimals and demonstrate, with and without technology, computational fluency in their use and justify the solution [sic] (grade 6)
4: Allowing Calculator Use in the Early Grades
Standards should require that students master basic computation in the early grades without the use of technology.
Recall and use basic multiplication and division facts orally and with paper and pencil and without a calculator (grades 4-6)
Technology is introduced early and included often in the standards, undermining studentsâ€™ mastery of arithmetic. Standards also seem to give students the choice to always use a calculator.
- Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers (grades 2-6)
- Select and use an appropriate method of computation from mental math, paper and pencil, calculator, or a combination of the three (grades 3-6)
5: Including Axiomatic Geometry in High School
The best standards directly address and require students to prove theorems, and they should mention postulates and axioms.
- Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180Â°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point (high school)
- Write simple proofs of theorems in geometric situations, such as theorems about congruent and similar figures, parallel or perpendicular lines. Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contrapositive (Geometry)
Classical theorems of geometry are not specifically included. If proof is mentioned, the foundations are not well covered, and such basic theorems as the Pythagorean Theorem are not proven.
Congruence and similarity are frequently missing, as in:
- Determine congruence and similarity among geometric objects (grades 9-10)
- Determine angle measures and side lengths of right and similar triangles using trigonometric ratios and properties of similarity, including congruence (grade 10)
Our review ofState Standards