What follows is a pointbypoint commentary on the New York City Scope and Sequence for Mathematics. The comments were originally written in connection with a meeting between NYC mathematicians and Mr. Evan Rudall, Chair of the Children First numeracy working group, arranged by Mr. Rudall and Elizabeth Carson of New York City HOLD. The NYC Scope and Sequence was discussed, with reference to a handwritten version of the present commentary, in the Talking Points for that meeting.
As is seen from the extensive comments, my colleagues and I feel that the present NYC Math Scope and Sequence is deeply flawed. While one could attempt a detailed revision, some have questioned whether the considerable effort needed would be worth it. California has already gone through the agony of crafting a similar document (at least for grades K  7), and I have repeatedly compared the NYC Scope and Sequence with the CA Standards in my commentary. Although the CA Standards are not perfect, I believe they represent a better starting point for an effort to revise the NYC Scope and Sequence than the present document.
In any case, revising the NYC Scope and Sequence will not be a small task, though it is important. A very large proportion of students in K8 will ultimately seek entry to college level programs (including community colleges or technical institutes) where basic math skills are required to succeed. The current NYC Scope and Sequence reflects a curriculum that is inadequate to prepare students for the sort of high school work needed to qualify for such programs.
1. The Scope and Sequence lists (many) goals for math instruction gradebygrade. Sometimes these abstract goals are accompanied by brief illustrative examples indicating what is appropriate for the grade level being discussed; often there is no elaboration, and in a distressing number of cases the stated goal is vague and unclear.
The CA Standards present a list of goals in the manner of the NYC Scope and Sequence, but these are accompanied by a whole manual that provides multiple (and specific) illustrative examples for each goal listed in the main document. We could make the NYC Scope and Sequence much more useful for teachers (and parents) by following California's lead. This is another incentive for using the CA Standards as a starting point  they have already created the file of illustrative examples.
2. It would have been helpful if the bulleted items in the NYC Scope and Sequence were *numbered*. It would be even more useful if the items for each grade were ranked in order of importance, or if the goals posed for each grade and area were somehow organized to distinguish critical topics from those of lower priority. Teachers would find that particularly helpful.
3. As one progresses through the grades in the NYC Scope and Sequence there is an unfortunate trend for the writing to become increasingly vague and unfocused. By the time I got to grade 8 the writing is so terse and careless I felt as if the writing team had gotten tired of their task and were just ``phoning it in'' while reading from the table of contents of grade 8 CMP. The Scope and Sequence for the grades 4  8 is really quite lame.
4. Many items in the NYC Scope and Sequence fail to distinguish between the *skills/conceptual goals* a student should achieve by year's end, and curriculumspecific *processes* by which those goals are to be achieved. The Scope and Sequence document should be about *goals*, and should not favor processes specific to a particular set of curricular materials  e.g. TERC, CMP, ARISE, and IMP. The present version of the Scope and Sequence is clearly biased toward those programs.
An inordinate number of items begin with the word ``Explore ...'' . While a certain amount of time spent in ``exploration'' is useful, the process of exploration is hardly the whole story. Many items in the NYC Scope and Sequence would be more forceful if they emphasized *mastery* of skills or content, as in ``Know ...'' or ``Understand ...''.
5. In its present form the New York City Scope and Sequence promotes indiscriminate use of calculators as early as grade 1. This is characteristic of programs such as TERC, and is truly misguided. Use of calculators when children are first trying to understand numbers and arithmentic processes seriously undermines the development of algebraic intuition, which is crucial in higher level courses. USE OF CALCULATORS SHOULD BE AVOIDED IN GRADES K3, and a revised Scope and Sequence should reflect this.
6. For many items the suggested illustrations ``(e.g. ... '' mentioned in Comment 1 tend to be extremely simplistic, and appear to be written to conform with NCTMbased curricular materials. Exemplars accompanying a goal are important, and should be much more diverse.
7. The same item (or nearly the same) is often repeated from one year to the next. Even when the intent is to reinforce certain topics from grade to grade, the description in year 2 should represent some advance beyond the description in year 1.
After reading all this I'm reminded that sometimes ``Less is more.'' Many goals seem relatively unimportant, or are repeated verbatim from grade to grade; the main goals are not clearly identified (cf. Item 2 above) and get buried under this mountain of micromanagerial details. There must be a better way.
(A note about page format. In the sequel, the headings and text in bold constitute the complete Scope and Sequence document [www.nycenet.edu/dis/scopesequence/]. The text in normal font are the present author's comments.)
Grade K* 1. Identify written numerals to ten. OK * 2. Use ordinal number names from first to tenth. OK * 3. Explore numbers through 20. OK * 4. Use a number line to count forward and backward. OK * 5. Practice the skill of counting on from a particular number (e.g., starting from the number three and counting on by ones  3,4,5,6,7). OK * 6. Understand quantities represented by the numerals one to ten and that the last number counted in a set tells how many things are in a set. OK * 7. Using concrete materials, explore putting two sets of objects together to produce a new set whose sum is less than ten. This is constructivist boilerplate; instead of a concrete learning goal it gives a game to be played (and not a very instructive game at that). Try the CA Standards version: Use concrete objects to answer addition and subtraction questions for numbers 1  10. * 8. Compare two groups to determine which is more, less or the same. OK * 9. Develop the concept of first, middle and last within a set of three objects. OK * 10. Explore benchmark fractions as they relate to daily life (e.g., dividing an apple in 1/2, dividing a cake into four equal parts). OK * 11. Explore bills and coins. OK 
Grade 1* 1. Match words and symbols from zero to twenty. OK, but why limit to numbers 120 unless you want to restrict students to counting on their fingers and toes? The CA Standards (rightly) suggest ``numbers 1  100''. In view of item #3 below, perhaps what the writers meant is: students should become thoroughly familiar with integers 0  20, while exploring the larger range of numbers 1  100. The writers were not clear about this. * 2. Explore ordinal numbers from first to thirtyfirst. OK, but why the arbitrary cutoff at 31st? * 3. Explore numbers to 100. OK * 4. Count forward and backward up to 50 by ones and twos using concrete materials, number lines and number charts. Although extensive use of concrete objects is important in the early part of G.1, the goal described here seems needlessly restrictive. By the end of G.1 shouldn't students begin to be able to do these counts *without the use of artificial aids*? I suggest that, as with the CA Standards, the ultimate goal by end of G.1 should be: ``Count forward and backward up to 100 by 1's, 2's, 5's and 10's *without the use of artificial aids*. (Concrete materials, number lines, number charts, etc should be used early on in the path toward fluency). * 5. Learn about the meaning of each digit in a twodigit number (place value). OK * 6. Explore the concept of even and odd numbers using sets of concrete objects. OK * 7. Use the symbols < (less than), > (greater than), = (equal to), + (plus), and  (minus). Learn sums of ten using two numbers and three numbers (e.g., 7 + 3 = 10; 5 + 3 + 2 = 10). OK, but embodies 2 quite different topics, and should be split into two separate statements: Use the symbols (<), (>), (+), (), and (=) with confidence and understanding. Learn sums of ten using two numbers and three numbers (e.g. 7 + 3 = 10 and 5 + 3 + 2 = 10). * 8. Add and subtract twodigit numbers without regrouping. This goal is OK, but how are you going to achieve it (and in G.2 deal with 2digit sums like 36 + 47 which require borrowing), unless the following is made an explicit, highpriority G.1 goal? Know addition facts (sums of two numbers 1  10) and corresponding subtraction facts, *committed to memory*. This is stressed in the G.1 CA Standards. * 9. Explore 1/2, 1/3, 1/4, 1/5, 1/8, 1/10 as part of a whole or part of a collection of things (e.g., 1/2 is one out of two, 1/3 is one out of three). OK, but why not ALL the fractions ``1/2, 1/3, 1/4, 1/5, ..., 1/12'' * 10. Learn value of individual coins and begin to learn their equivalents (e.g., one nickel = five pennies). OK, but would be improved by adding: ... five pennies). Show combinations of coins that have equal value. * 11. Recognize dollars and cents notations. OK * 12. Explore making change for amounts of money up to $1.00. OK SERIOUS TOPIC OMISSION: A. ``Know addition facts (sums of 2 numbers 1  10) and corresponding subtraction facts, *committed to memory*.'' 
Grade 2* 1. Identify number names orally through 100. Very weak; has already been done in G.1. A more appropriate G.2 level goal would be: Count, read, and write whole numbers 1  1000, and identify the place value of each digit. * 2. Use ordinal numbers from first to thirtyfirst and beyond. OK * 3. Use concrete materials such as baseten blocks to represent numbers between ten and nine hundred ninety nine. Weak, and specifically biased toward the use of manipulatives promoted in TERC. In the Scope and Sequence it is not appropriate to mandate use of a particular proprietary product such as the ``base 10 blocks''. I suggest a less sectarian goal: Use words, pictures, and expanded forms (e.g. 45 = 4 tens + 5 ones) to represent numbers 1  1000. Work should become less dependent on manipulatives toward the end. * 4. Explore expanded notation for two and threedigit numbers (e.g., 325 = 3 hundreds + 2 tens + 5 ones = 300 + 20 + 5). OK, except that ``Understand ...'' is more to the point than ``Explore ...'' * 5. Explore the role of zero in two and threedigit numbers. OK, but definitely replace ``Explore'' by ``Understand'' here. * 6. Count forward up to 100 by twos, threes, fours, fives, and tens and backward by twos, fives, and tens, using concrete materials, number lines and number charts. OK if you drop the NCTMstyle implication that it should *only* be done using concrete aids. Students should be able to do this *unassisted*  i.e. orally, using no concrete aids at all  by the end of G.2. That would distinguish this G.2 item from G.1 #4. * 7. Explore the relationship between addition and subtraction. THIS TOPIC BELONGS IN G.1 because addition and subtraction are introduced there. Especially, one should not delay *committing to memory* number facts about onedigit addition! * 8. Know singledigit addition and subtraction facts. THIS TOPIC BELONGS IN G.1 because addition and subtraction are introduced there. Especially, one should not delay *committing to memory number facts about onedigit addition! * 9. Learn about the associative [e.g.,(3+4)+6=13 and 3+(4+6)=13] and commutative (e.g., 5+3=8 and 3+5=8) properties of addition. OK * 10. Add and subtract twodigit numbers with regrouping using concrete materials. OK, but phrase ``... using concrete materials'' seriously weakens this item. Delete this NCTMinspired implied restriction in acceptable methods toward goal. * 11. Explore multiplication as repeated addition and division as repeated subtraction. OK * 12. Recognize 1/2, 1/3, 1/4, 1/5, 1/8, 1/10 as part of a whole or part of a collection of things (e.g., 1/5 is one out of five objects, or 1/5 is one out of five parts). OK, but this limited list of exemplars is misleading. It would be better to say: Recognize fractions from 1/2, 1/3, 1/4, ... 1/12 as part ... * 13. Recognize dollars and cents notation to ten dollars. OK * 14. Make change for amounts of money up to one dollar. OK 
Grade 3* 1. Use knowledge of place value to read and write numbers up to hundred thousands. OK * 2. Explore expanded notation for large numbers. OK * 3. Count forward and backward to 100 by twos, threes, fives and tens. Pretty dumb goal for G.3; has already been covered in G.2 and G.1. Delete this item. * 4. Add and subtract whole numbers, with regrouping, with and without using calculators. OK, but delete the reference to calculators in G.3. * 5. Identify the sum/difference of two whole numbers as even or odd. So vague as to be pointless. A better and more specific goal would be: Find sum and difference of whole numbers between 1 and 1000. * 6. Estimate numbers by rounding using number lines, thermometers, and/or yardsticks. OK * 7. Explore multiplication and begin to learn multiplication facts. Extremely vague, and manages to avoid the main goal that should be achieved by end of G.3: Memorize multiplication facts for pairs of whole numbers 1  10. * 8. Explore the role of zero and one in multiplication. ``KNOW the role ... '', not ``Explore ...'', which is vague. * 9. Explore division and division procedures without remainders. OK * 10. Explore the relationship between multiplication and division. OK * 11. Learn about positive and negative numbers as they relate to the number line and measurement of temperature. OK * 12. Explore comparing fractions (e.g., 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12) using <, > and = symbols. OK, but exemplars are far too special, hence misleading. Try: Write mathematical statements comparing simple fractions (e.g. 1/2, 1/3, 1/4, ...) and compound fractions (e.g. 3/5, 4/7, etc) using the symbols (>), (=), (<). Interpret these comparisons using the number line. * 13. Use the terms numerator and denominator. OK * 14. Add and subtract fractions with like denominators. OK * 15. Understand the relationship between fractions and decimals (e.g., 1/4 = .25, 1/10 = 0.1). OK * 16. Relate fractions and decimals to the monetary system and metric measure. OK * 17. Add and subtract decimals with one place (tenths). OK 
Grade K* 1. Explore basic shapes in the environment such as a circular clock, a rectangular door and a square window. OK * 2. Create geometric pictures and designs. OK * 3. Compare objects based on size and capacity. OK * 4. Use comparison language, such as bigger than, lighter than, less than, and equal to. OK * 5.Explore nonstandard units of measure (e.g., use string to measure the distance around objects). OK * 6. Begin estimating (guessing sizes). OK * 7. Respond to directions about location (e.g., above, below, between). OK * 8.Describe positions while building with blocks (e.g., top, middle, bottom, inside). OK SUGGESTED ADDITIONAL TOPICS: A. Telling time of everyday events, to nearest hour. B. Demonstrate understanding of time concepts: hour, day, week, year; yesterday, today, tomorrow; morning afternoon, evening. 
Grade 1* 1. Explore two and threedimensional shapes in everyday life: square, rectangle, triangle, circle, cube, prism, pyramid, and sphere. OK * 2. Compare attributes of objects (e.g., size, shape, weight, texture). OK * 3. Explore the need for standard units of measure. OK, but would be much improved by adding the following rationale for this topic: ... units of measure *as a means of comparing two objects.* * 4. Develop familiarity with length, weight, and capacity using standard and nonstandard units (e.g., inches, centimeters, grams, handfuls, body length) through concrete experiences. OK * 5. Compare the capacity of containers using materials such as sand and water. OK * 6. Use clocks and calendars to study time to the hour, days of the week, and months of the year. I believe this activity should begin in grade level K! Do not delay it. A more appropriate G.1 goal would be: Tell time to the nearest 1/2 hour. Relate time to events (e.g. before/after, longer/shorter). * 7. Explore temperature using Fahrenheit and Celsius thermometers. OK ADDITIONAL SUGGESTED TOPICS: A. Give and follow directions regarding space and location. 
Grade 2* 1. Explore the properties of two and threedimensional shapes noting their similarities and differences. Somewhat vague. Try rewrite: Describe and classify plane and solid geometric objects (e.g. by number of faces, edges, vertices, etc), noting their similarities and differences. * 2. Explore symmetry and congruence. OK * 3. Understand the need for standard units of measure. OK, but would be more specific and stronger if you add: Understand ... measure, choosing appropriate measuring tools. In fact, there are several topics about measurement listed under G.3 that really belong in G.2. * 4. Develop familiarity with standard units of measure through concrete experiences (e.g., weigh objects using pounds, grams, and kilograms; measure liquids using cups, quarts, and liters; and measure length using inches, feet, yards, meters, and centimeters). OK * 5. Compare sets of objects using the following terms: more than, bigger than, greater than, less than, the same size, equal to, before, after, and between. OK, but stronger if you add at the end: Compare ... between. Understand the connection between these concepts and the symbols (>), (=), (<). * 6. Use clocks and calendars to measure time in days of the week, half hours, quarter hours, and minutes, using clocks and calendars. OK * 7. Measure temperature using Fahrenheit and Celsius thermometers. OK, but stronger if you add at end: Measure temperature ... thermometers, estimating values to the nearest degree. ADDITIONAL SUGGESTED TOPICS: A. Determine *duration of time intervals*, in hours (e.g. 11:00AM  4:00PM = 5 hours). B. *Measure lengths of physical objects* to nearest inch or centimeter using rulers. 
Grade 3* 1. Investigate and classify properties of two and threedimensional shapes such as squares, rectangles, triangles, circles, cubes, prisms, cylinders, and pyramids noting similarities and differences (e.g., similar and congruent figures). This is a vague catchall statement. Would be much more useful if replaced by several distinct topics, as it is in the CA Standards: Identify and classify plane figures such as squares, rectangles, triangles, polygons, circles. Identify attributes of triangles (e.g. 2 equal sides = isosceles, etc). Identify and classify various 3dimensional objects such as cubes, prisms (including rectangular), pyramids, cylinders, and spheres. Discuss similarity and congruence of figures. * 2. Investigate symmetry (reflections). OK * 3. Explore threedimensional shapes to begin to understand volume (filling space within an object). OK * 4. Through concrete experiences, estimate and measure length, width, perimeter, and area of objects using metric and customary (U.S. Standard) tools. OK * 5. Choose appropriate measuring tools for standard or nonstandard measurements. This topic should be in G.2! (I have already done this in my comments for G.2.) * 6. Identify equivalent measures within a system (e.g., 12 inches = 1 foot and 100 centimeters = 1 meter). This topic MUST be shifted to G.2! * 7. Relate the clock to fractions of a circle (1/2 is equivalent to 30 minutes, 1/4 is equivalent to 15 minutes). OK * 8. Use clocks and calendars to study time to five and oneminute intervals. OK * 9. Estimate benchmark temperatures in Celsius and Fahrenheit (e.g., room temperature, body temperature, freezing point, boiling point). OK * 10. Locate points on a coordinate grid or map. OK 
Grade K* 1. Look for patterns in number charts, designs, nature, and literature. OK * 2. Observe, describe, and explore patterns using a variety of manipulative materials (e.g., circle, square, triangle, circle, square, triangle). OK * 3. Sort and classify objects according to a rule or generalization. OK 
Grade 1* 1. Recognize, describe, create and extend geometric and number patterns. OK * 2. Sort and classify objects according to a rule or generalization. OK, but this is a verbatim repeat of K, #3. It would help to specify some examples to indicate what is appropriate for G.1. * 3. Explore more than one object belonging to one set (e.g., five fingers to one hand, two eyes to one face). This item is really dumb, as well as vague. Rewrite to have some point. NOTE REGARDING G.1: Something important is missing. Throughout G.2 reference is made to (+), () operations, but nothing at all is said about these symbols in G.1! Teaching elementary notions of addition and subtraction, realworld interpretations, and the symbols used to indicate these operations, should be a key objective in G.1. SERIOUS TOPIC OMISSIONS: A. Understand the meaning of addition, subtraction, and the symbols (+), (). B. Write and solve number sentences (based on concrete situations) that involve the symbols (+), (=), (). 
Grade 2* 1. Recognize, describe, and extend numeric and geometric patterns (e.g., counting by twos, fours, fives and tens). OK, but the exemplar is bad. Instead try: Recognize ... patterns (e.g. arithmetic progressions such as 2,5,8,11, ... with various starting points and step sizes.) * 2. Explore patterns using number lines and number charts. OK * 3. Sort, classify, and order sets of objects according to a rule or generalization. OK * 4. Find the missing numbers in open sentences such as 17 + ____= 20. OK * 5. Investigate many to one correspondence such as ten pennies = one dime. OK NOTES TO G.2: Item #4 above is related to topics that MUST be introduced in G.1 but are not mentioned there. Perhaps it was meant to be implied by G.1 #7 (Functions and Algebra)? 
Grade 3* 1. Recognize, describe, and extend patterns (e.g., numeric, symbolic). OK, but pretty vague. A better version might read: ``Recognize ... (e.g. extend arithmetic sequences, geometric sequences). Discuss realworld examples of such patterns. Construct bar graphs illustrating the behavior of such sequences.'' * 2. Explore patterns in odd and even numbers. This seems a really dumb and pointless topic for G.3. If item #1 is rewritten as suggested, the present item becomes redundant and should be deleted. * 3. Use manipulative materials to model skip counting patterns related to multiplication. Really idiotic, completely slanted toward TERC, and avoids mentioning the real skill G.3 students should acquire. Delete this item entirely, and replace it with the following important goal, which has not been explicitly mentioned. Shows mastery, from memory, of multiplication tables for whole numbers 1  10. * 4. Represent and analyze patterns and functions using tables (e.g., input/output boxes). Very weak, in that it severely downplays symbolic methods. It would be a little better if you add the change indicated below. Represent ... boxes and *mathematical formulas*. * 5. Find missing numbers in open sentences (e.g., 2 x ____= 6). OK, but vague and unfocused. Try rewriting as: Solve problems involving numerical equations and inequalities (e.g. Solve 2 x ___ = 14. If 4 ___ 3 = 12, what operation symbol works in the blank space?) * Use manipulative materials to model the associative [e.g., (3+4)+6=13 and 3+(4+6)=13] and commutative (e.g., 5+3=8 and 3+5=8) properties of addition and multiplication. OK SERIOUS TOPIC OMISSION: A. Know multiplication tables for whole numbers 1  10 *from memory*. ADDITIONAL SUGGESTED TOPICS: A. Learn to represent simple relationships in mathematical form (e.g. find the total cost of multiple items given cost per unit). 
Grade K* 1. Talk about data using words such as most, least, and the same. OK * 2. Participate in collecting, discussing and displaying data in a variety of ways (e.g., bar graphs, pictographs). OK * 3. Anticipate outcomes by guessing and estimating. In the absence of any illustrative examples, it is not at all clear what this means in connection with Probability and Statistics. Rewrite or delete. 
Grade 1* 1. Collect and record data in a variety of ways (e.g., surveys, tables, pictures). OK * 2. Discuss data using appropriate terms (e.g., most/least, more than/less than). OK, but a repeat of K #1. Illustrative example should be included to indicate something new expected by end of G.1. * 3. Construct bar graphs and pictographs to display real world data (e.g., What are our favorite colors?). OK * 4. Predict the likely outcome of repeated acts (e.g., coin toss). OK * 5. Explore combinations and arrangements by solving problems such as, `How many different pairs of numbers add up to ten?' OK 
Grade 2* 1. Collect and record data in a variety of ways (e.g., survey classmates about favorite foods). OK * 2. Arrange data in tables and display data using bar graphs, pictographs, and Venn diagrams. OK * 3. Make predictions, record data from experiments, and explain outcomes using spinners, coins, and color tiles. OK * 4. Discuss certainty or uncertainty of events based on data collected over a period of time. OK, but it is silly not to combine items #4, 5 as one item. * 5. Understand that some events are more likely to happen than others. OK, but it is silly not to combine items #4, 5 as one item. * 6. Show combinations and arrangements of groups of objects (e.g., How many different sets of three numbers will add up to twelve?). Somewhat vague, and the exemplar seems quite pointless as a topic in *probability*. An example with a truly probabilistic slant might read `Show ... (e.g. How many ways can a club with 10 members choose a president and a secretary?)'' 
Grade 3* 1. Organize data using tables, graphs and Venn diagrams. OK * 2. Interpret and identify the parts of a bar graph (e.g., title, vertical and horizontal axes, bars, labels). OK * 3. Read and interpret a line plot. OK * 4. Discuss graphs found in everyday publications. OK * 5. Explore range, median, mode and mean using concrete materials. OK * 6. Predict the outcome of an experiment and compare the result to the prediction. Vague; the writers give no idea *what kind of experiments* they have in mind at this grade level, or what their purpose should be. Furthermore, it seems premature to launch into this at level G.3. I think #7 below is quite enough at the present stage. * 7. Understand and use fractional notation to show the probability of the outcome of an experiment (e.g., one out of three chances of making a specific selection is the same as the fraction 1/3). OK * 8. Explain why a game is fair or unfair. OK * 9. Develop orderly ways to determine the number of possible arrangements and combinations. OK 
Grade K* 1. Participate in math explorations and explain mathematical thinking. OK * 2. Use content specific language to describe, explain, and compare. OK * 3. Draw pictures to show mathematical situations. OK * 4. Talk about mathematics and problem solving in everyday life. OK * 5. Explore the use of mathematical tools and technology (e.g., computers, fourfunction calculators, balances). OK 
Grade 1* 1. Create and solve word problems. OK * 2. Select appropriate strategies for solving word problems (e.g., using objects or drawings). OK, but should also include *counting* as an acceptable method; restriction to constructivist methods peculiar to TERC is not acceptable. Rewrite: Select ... objects, drawings, or counting. * 3. Convey mathematical thinking using content specific language to describe, explain, and compare. So vague as to be useless. Rewrite or delete. * 4. Talk about mathematics and problem solving in everyday life (e.g., attendance, time, weather). Misleading. As written, this seems an open invitation to avoid math content entirely, and spend a lot of time just talking about math. To give it some content, I would rewrite along lines of CA Standards: Formulate arithmetic problems in terms of easily understood models drawn from everyday life (e.g. attendance, time, weather), explaining what features of the model appear as numbers and operations. * 5. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic fourfunction calculators, measuring cups, scales, rulers). The use of calculators should be avoided throughout grades K3, where students are just beginning to understand addition and subtraction on their own! A more appropriate, and clearer, version of #5 might be: Use physical measuring devices such as rulers, scales, clocks, measuring cups, and diagrams to illustrate numerical results of calculations. This revised item might well be repeated in G.2 and G.3, with some indication of the increasingly sophisticated calculations that should be done. 
Grade 2* 1. Use a variety of strategies to solve problems (e.g., using estimation, objects or drawings). Incomplete owing to its narrowly focused list of suggested methods. In G.2 students should begin to see how word problems can be represented as mathematical word equations (``open sentences''), as asserted in G.2 #4 (Function and Algebra Concepts). Therefore I believe the present item should be expanded to include the additional sentence: Solve simple word problems by describing them as open sentences, which are solved using elementary arithmetic. * 2. Use appropriate operations to solve word problems. Quite vague. What kind of operations? Algebraic operations? Erasing marks on a blackboard? Counting beans? Cf also my comments to #1 above. * 3. Discuss, justify, organize, and write about solutions to problems using content specific language to describe, explain, and compare. Idiotic edspeak. What does the second half of this sentence mean?? Furthermore, by G.2 the emphasis should be on *actually solving* some problems, and explaining how those solutions were arrived at. This item should be completely rewritten. * 4. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic fourfunction calculators, measuring cups, scales, and rulers  metric and U.S. Standard). Once again: the use of calculators should be avoided throughout grades K3. Early use of calculators will interfere with understanding of algebraic processes later on. It would be better to replace this item with an appropriately modified version of my rewrite for G.1 #5 
Grade 3* 1. Understand word problems, identifying pertinent, extraneous, and missing information. OK * 2. Use a variety of strategies to solve and represent problems/solution (e.g., logical thinking, estimation, number sense, pictures, diagrams, and charts). Would be OK if it did not completely overlook the use of *mathematical equations* in favor of purely descriptive modes of inquiry. Item should be rewritten to read: Use ... diagrams, charts, and mathematical equations). * 3. Work individually and collaboratively to discuss, justify, organize, and write about solutions to problems using content specific mathematical language. Focused entirely on process and fails to state any objective; very strongly slanted toward the TERC philosophy. The tone is changed considerably, and the correct emphasis is provided, if you replace ... and write about solutions to .... mathematical language. with ... and SOLVE problems using appropriate mathematical tools. Nowhere in the present description is it suggested that children should acquire the ability to actually *solve* problems, as opposed to writing or talking about them. This is the difference between ``math'' and ``math appreciation'', similar to the difference between ``art'' and ``art appreciation''. * 4. Recognize the use of mathematics in other subject areas such as science, social studies, and music. OK * 5. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic fourfunction or fraction calculators, measuring cups, scales, and rulers  metric and U.S. Standard, thermometers, and tape measures). Omit calculators at this level; focus on use of measuring devices, but give exemplars indicating the level of sophistication appropriate to G.3. Cf also comments to G.2 #1. 
Grade 4* 1. Use knowledge of place value to read and write numbers up to hundred millions. OK * 2. Add and subtract whole numbers, with regrouping, with and without calculators. Verbatim repeat of a G.3 item. Delete. * 3. Multiply and divide whole numbers, with and without calculators. Needs to be more specific. *How large* are the numbers they are expected to deal with by end of G.4? * 4. Know multiplication and division facts through one hundred forty four. ``Know ...'' is too vague. This should read Memorize and be proficient at using multiplication and division facts through 12 x 12. * 5. Learn about the associative [e.g., 3 x (4 x 5)=60 and (3 x 4) x 5=60] and commutative (e.g., 6 x 7=42 and 7 x 6=42) properties of multiplication. OK * 6. Learn about primes, factors, multiples, and square numbers. OK * 7. Explore division and division procedures with remainders. OK * 8. Demonstrate rounding and estimating skills. OK but pretty vague. A more explicit version might read: Round off whole numbers through 1,000,000 to nearest 10, 100, 1000, etc. Explain when and why rounding off is appropriate. * 9. Compare fractions (e.g., 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12), using <, >, and = symbols. Terrible choice of exemplars, making this item sound quite dumbed down by restricting scope to particularly simpleminded fractions. This item should be rewritten along the following lines: Compare simple fractions 1/2, 1/3, 1/4, ... 1/12 and compound fractions such as 5/5, 5/8, etc, using the symbols (>), (=), (<). Interpret these relations in terms of the number line. * 10. Compare decimals (e.g., 0.10, 0.20, 0.25, 0.75) using <, >, and = symbols. Vague, and the exemplars are too simpleminded. Item should be expanded in scope and made more specific: Order and compare whole numbers and decimals (to 2 places) using the symbols (>), (=), (<). Know how to located decimals on the number line. * 11. Compare and identify equivalent fractions (e.g., 2/4 = 1/2). The exemplar here is particularly stupid. Why not: Understand and identify equivalent fractions (e.g. 4/8 = 1/2 or 3/7 = 9/21). * 12. Identify use of fractions and decimals in daily life (e.g., .75 = $ .75 = 3/4 of a dollar). Delete this verbatim repeat of a G.3 item, or indicate what new emphasis is intended for G.4. * 13. Add and subtract decimals with two places (hundredths). OK * 14. Learn about percents as part of one hundred (e.g., twentyfive out of one hundred is the same as 25%). Another really lame exemplar. I suggest: ... 35 out of 100 is the same as 35%. * 15. Compare relationships between fractions, decimals, and percents as they relate to daily life (e.g., Ten students were asked to name their favorite sport. 1/2 chose soccer = 0.5 chose soccer = 50% chose soccer). Yet another really lame exemplar. Try: ... (e.g. 20 students were asked to name their favorite sport; 3/5 chose soccer = 60%, or 12 students). SERIOUS TOPIC OMISSIONS: A. Use concept of *negative number*. Be able to interpret negative numbers in terms of the number line and in practical settings (e.g. in counting, in ``owing'', in reading temperature, etc). NOTE: The first mention I can find of negative numbers in this Scope and Sequence occurs in G.6 #9, and then again in G.7 #3. That seems late in the game to start talking about negative numbers. However, some of my colleagues disagree, and would place this important topic in G.5. The important point is that the writers of the present Scope and Sequence do not indicate clearly where this crucial topic should be taken up. G.6 #9 presumes that it has already been covered. B. Explain *different interpretations of fractions*, including the use of charts. Explain equivalence of fractions (e.g. 3/7 = 9/21). OTHER USEFUL ADDITIONS: C. Demonstrate understanding of, and ability to use, standard algorithms for addition, subtraction, and multiplication of multidigit whole unmbers. D. Know how to factor small whole numbers (e.g. 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3). 
Grade 5* 1. Use knowledge of place value to read and write numbers up to one billion. OK * 2. Use addition, subtraction, multiplication, and division facts efficiently and accurately. OK * 3. Explore powers of ten as another way of naming numbers. OK * 4. Identify differences between prime and composite numbers. OK, but I would delete the phrase ``differences between'' * 5. Estimate whole numbers by rounding to the nearest ten thousand and decimals to the nearest hundredth. OK * 6. Explore adding and subtracting integers using the number line (positive and negative numbers). Better if replaced by ``Understand addition and subtraction of (positive and negative) integers as operations on the number line.'' * 7. Understand the concept of proper and improper fractions. OK * 8. Develop skill of changing improper fractions to equivalent mixed numbers. OK * 9. Find the greatest common factor and least common multiple of a set of numbers. OK, but would be better placed immediately following item #4. * 10. Add and subtract fractions with unlike denominators. OK * 11. Multiply decimals to the hundredths place. OK * 12. Explore dividing decimals without remainders (to hundredths). OK * 13. Relate dividing decimals to the monetary system (e.g., $4.50 divided by three). OK * 14. Compare relationships among fractions, decimals, and percents (e.g., 1/4 = 25/100 = 0.25 = 25%). Almost verbatim copy of a G.4 item. Try a somewhat more advance alternative: Interpret percent as part of 100. Find decimal and percent equivalents of common fractions, and explain why they represent the same value. * 15. Compare decimals and fractions using the terms less than, greater than, between, and equivalent. OK, but at the end you should add: ... equivalent, and explain these comparisons in terms of the number line. 
Grade 6* 1. Read and write numbers through billions. Repeats a G.5 item. Delete. * 2. Use place value and expanded notation. OK * 3. Use exponents up to five. OK * 4. Round off numbers through millions. Delete. This should have been done in G.4 (cf G.4 #8, revised), when students were first introduced to numbers 1  1,000,000 and to the concept of rounding off. * 5. Add, subtract, multiply, and divide fractions with and without common denominators, and mixed numbers. OK * 6. Use equivalent forms of fractions, decimals, and percents. OK * 7. Add, subtract, multiply, and divide decimals. OK * 8. Expand their understanding and use of special numbers such as primes, composite numbers, square numbers, common divisors, and common multiples. OK * 9. Relate positive and negative numbers to reallife situations (e.g., loss and gain in bank transactions). Convoluted. Clear if rewritten: Understand basic number theory concepts: primes, ... common multiples.'' * 10. Understand order of operations. OK SERIOUS TOPIC OMISSION: A. Solve problems involving *ratios and proportions* (e.g. find N if 4/7 = N/21 using crossmultiplication as a method, as an illustration of the principle that the same thing is being done to both sides of an identity). B. Recognize and solve word problems that involve ratios or proportions. NOTE: Ratios and proportions are not the same thing as percentages, although they are related. Furthermore, many types of word problems get cast into mathematical form using the concepts of ratio and proportion. The G.6 sequence makes no mention of this extremely important topic. 
Grade 7* 1. Read and write numbers through trillions. This item is pretty useless at level G.7, and just repeats similar items from G.5 and G.6. Delete it. * 2. Round off whole numbers through billions. OK * 3. Add, subtract, multiply, and divide positive and negative numbers. Vague, and should have been finished earlier; repeats items G.6, #5 and #7. Delete this item. * 4. Understand the inverse relationships between (1) addition and subtraction and (2) multiplication and division. OK * 5. Explore the concept of perfect square numbers and their positive square roots. As presently worded this item is overly oriented toward process (``Explore ...'' as opposed to ``Understand...''), and does not state clear goals regarding skills or concepts to be mastered. For instance, having discussed powers and (small) exponents in G.6 #3 one should now discuss the general inverse operation of taking (small order) roots, *not restricting the discussion to square roots only*. The square root should be portrayed as an example of an *inverse process*, and should receive most of the stress at this grade level, but the concept of n th root should also be mentioned at this stage. It would suffice, for G.7, to provide a few simple illustrations (e.g. cube root of 8 = 2) * 6. Understand terminating and repeating decimals. Seems weaker than necessary. Try: Differentiate between rational and irrational numbers, and understand terminating/repeating decimals. * 7. Find the percent of a number. Incredibly lame; what is this doing in G.7? Maybe what they had in mind is the following item lifted from the CA Standards: Convert fractions to decimals and percents, and use these representations in estimations, computations, and applications. With this rewrite item #9 below becomes redundant and should be deleted. * 8. Apply the associative, commutative, and distributive properties; and the inverse and identity element. OK * 9. Convert among fractions, decimals, and percents. Delete (assuming #7 above is rewritten as suggested). * 10. Add, subtract, multiply, and divide whole numbers, fractions and decimals accurately. Another item that seems out of place in G.7, and redundant. (It's the same as item #3 above, as well as a repeat of G.6 #5,#7.) Delete. * 11. Discover and apply rules of divisibility. What is this supposed to mean?? If my guess is correct, this should be reworded as follows: Discover ... of divisibility by 2, 3, 5, 9, and 10 (e.g. if the last digit is even the number is divisible by 2; if the final digit is 0 or 5 the number is divisible by 5; etc). * 12. Understand complete factorization using factor trees. Vague and jargon laden. Try being more specific: Know that whole numbers have unique prime factorizations. Be able to factor a number of reasonable size into its primes. SERIOUS TOPIC OMISSIONS: A. Understand positive and negative *whole number exponents* (e.g. 5^3, 5^(3)). NOTE: The treatment of exponents and exponent notation is not well delineated in this Scope and Sequence. This is not good, as this very important topic underlies much later work. The goals regarding this topic should be spelled out clearly and consistently! B. Understand the meaning of *absolute value* and its number line interpretation. ADDITIONAL USEFUL TOPICS: C. Multiply and divide expressions involving exponents with a common base (e.g. 5^3 / 5^7 = 5^(4) = 1/625 ). 
Grade 8NOTE: The following section is so badly written that it is hopeless to criticize it item by item. It is repetitous and vague, maybe deliberately evasive, and any serious rewrite will take time and personnel. * 1. Use scientific notation to express and compare very large and very small numbers. OK, but see also Additional Topic B below. * 2. Add, subtract, multiply, and divide rational numbers, decimals, and integers, accurately. Delete. Seems out of place as a G.8 topic, and redundant in view of G.7 #10. G.7 is the level at which proficiency should be achieved in these skills! * 3. Understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, expanded, and scientific notation). OK * 4. Develop an understanding of number theory (primes, factors, multiples). Writers of this G.8 item fail to say how this is supposed to differ from G.6 #8. Rewrite to clarify what is to be done at G.8 that differs from what was done at G.6. * 5. Explore and use operations dealing with roots and powers. Weak, vague, and emphasizes the process of exploring rather than mastery. Much more appropriate for G.8 if rewritten to say: Understand the algebraic laws governing the use of exponents in dealing with roots and powers. * 6. Recognize order relationships for decimals, integers, and rational numbers. OK * 7. Apply ratios, rates, proportions, and percentages. What is this doing here?? This is a G.6 level topic. Failure to cover it earlier than G.8 will preclude discussion of MANY interesting real world applications. It should be covered thoroughly by the end of G.7, and initially introduced no later than G.6. ADDITIONAL USEFUL TOPICS: A. Use calculators to find square roots, nth roots, and fractional powers of decimals. NOTE: This capability is necessary to effectively apply Pythagoras' Theorem, a key G.8 geometry topic, in real world situations. B. Understand the concept of *logarithmic scales* and their use in real world applications (e.g. the sizes of things in the universe). NOTE: This is a natural companion to the discussion of scientific notation. 
NOTE: The whole Geometry section for grades 48 is incoherent: the subject matter is not organized to reveal which facts are related to each other. For example, area of a triangle can only be gotten *after* one has the formula for the area of a parallelogram, which is obtained after one derives the area formula for a rectangle. Furthermore, there is no clear emphasis on the axiomatic properties of ``area'' and ``volume'', for instance: if you cut a figure into pieces, the area of the figure is the sum of areas of the pieces; if figure A is contained in figure B then the area of A is less than or equal to that of B; etc.
If one uses these basic laws (backed up by intuitive explorations with real objects earlier on), many things follow directly. NCTMbased curricula, however, eschew clear definitions and logical deduction (``proofs'') based on simple collections of fundamental laws. The fact that this allimportant topic receives no mention suggests the failings of this whole section.
Grade 4* 1. Identify and classify properties of two and threedimensional shapes including vertices, line segments, edges, angles, parallel, perpendicular, congruency, and lines of symmetry. Quite vague, and nearly a repeat of an item that first appeared in G.3. Rewrite for clarity and emphasize what's new. It might help to split into several items: Identify lines that are parallel or perpendicular Identify congruent figures and discuss the meaning of congruence * 2. Explore the properties of circles, including diameter, radius and circumference. OK, but change ``Explore ...'', which is vague and focused on process, to read ``Identify ...'' * 3. Explore the use of formulas to find the area and volume. VAGUE. Replace with more specific G.4 level objectives: Develop formulas for the area and perimeter of squares and rectangles, and the volumes of rectangular solids. NOTE: The skills listed here are required in item #5 below. * 4. Select units of measure (pounds, inches, minutes, and degrees) for estimating and determining quantities such as weight, area, time, and temperature. Too vague to be meaningful, and it seems to repeat a G.3 item. Delete or rewrite to indicate G.4 level content. * 5. Estimate, measure, and represent length, width, perimeter, and area of objects in the real world. OK * 6. Read and draw simple maps using coordinates. OK 
Grade 5* 1. Represent and create models of two and threedimensional shapes including cubes and prisms. OK * 2. Use concrete and artistic activities to explore the concepts of similarity, symmetry, and congruence in plane geometric figures. Emphasizes process rather than content. Rewrite to say: Explore the concepts of similarity, symmetry, and congruence of plane geometric figures. * 3. Develop formulas for the area and perimeter of rectangles and squares. Delete. This should be done earlier, in G.4, and I have moved it into G.4 item #3. * 4. Measure area and perimeter of triangles, regular and irregular polygons by using graph paper and square tiles. Weak and pointless. In G.4 students already should have done measurements to arrive at area formulas for rectangles. It is then easy to derive formulas for triangles, parallelograms, and polygons by dissection and rearrangement. (a much more effective approach than playing around with square tiles). This item should be replaced by: Derive and apply area formulas for triangles, parallelograms, and polygons using the area formula for rectangles plus dissection and rearrangement. * 5. Explore the relationships among diameter, radius, and circumference of circles. OK * 6. Investigate threedimensional shapes to begin to develop a method for finding the volume of rectangular prisms. Hopelessly vague and convoluted. Would be better as several items, each with somewhat different objectives: Understand concept of volume, and appropriate use of units (e.g. cubic centimeters, cubic inches). Understand the mathematical properties of volume (e.g. volume of object = sum of volumes of pieces), and derive a volume formula for rectangular prisms by reasoning based on those properties. Investigate other 3dimensional shapes. * 7. Estimate and measure length, distance, mass, volume, and capacity in realworld situations using appropriate measuring tools, and be familiar with abbreviations such as cm, in., g, L. OK * 8. Identify equivalent units of measure (e.g., 3 meters equals 300 centimeters, 36 inches equals 1 yard, 60 seconds equals 1 minute, and 2 cups equals 1 pint). What has this got to do with geometry?? Should be placed among Number Concepts. * 9. Use centimeter graph paper to explore scale drawings and relate scale to ratio. OK SERIOUS TOPIC OMISSIONS: A. *Understand and use formulas* to solve problems involving perimeters and areas of squares, rectangles, triangles and parallelograms. B. Understand the axiomatic properties of area and volume (e.g. the area of a figure is the sum of the area of its pieces, etc). Know how to use them in analyzing problems. OTHER USEFUL TOPICS: C. Measure, draw and identify angles, perpendicular and parallel lines, and triangles *using straightedge and compass*. 
Grade 6* 1. Investigate similar and congruent polygons. OK * 2. Classify two and three dimensional figures according to their properties. OK * 3. Understand the concept of parallel and perpendicular lines. This seems out of place. Parallel/perpendicular already introduced in G.4. Delete. * 4. Construct plane geometric figures using rulers, protractors, and compasses. OK * 5. Find the area and perimeter of polygons such as triangles, rectangles, and squares. Already stated verbatim in G.5. Pointless to repeat it here. Delete * 6. Find the area and circumference of circles. ``Find ... '' How?? Might be better as 2 separate items: Know formulas for area and circumference of a circle, and understand how they are derived. Understand the concept of a constant such as ``pi'', and know common estimates for this constant (e.g. 3.14 or 22/7). * 7. Find the volume of rectangular prisms. OK, but this topic probably should be done by end of G.5 (cf the rewrite of G.5 #6). * 8. Construct scale drawings. Vague. Scale drawings of what? And why?  with what mathematical concept in mind? If it is to be kept, this item should be rewitten to be much more specific. SERIOUS TOPIC OMISSION: A. Knows basic *definitions* (much neglected in CMP) of different types of triangles (right, equilateral, isosceles) and quadrilaterals (square, rectangle, parallelorgam, trapezoid, rhombus). 
Grade 7* 1. Understand the difference between similarity (same shape but different size) and congruence (same shape and size). Weak. Similarity was first taken up at G.5 level (item #2), and we are now in G.7. A stronger and more G.7appropriate version might read: Demonstrate understanding of conditions under which two geometric figures are similar or congruent. Understand implications when angles, sides, perimeters, and areas of such figures are compared. * 2. Construct and classify triangles and quadrilaterals by angles and sides. This should have been done at a much lower grade level, or else it is so badly written as to invite misinterpretation. For something appropriate to G.7 try: Identify and construct basic elements of geometric figures (e.g. altitudes, midpoints, diagonals, bisectors, etc) by compass and straightedge construction. * 3. Use geometric terms (point, line, plane, segment, and ray). OK, but rewrite to say ``Correctly use geometric terms ...''. * 4. Name and define angles and angle pairs. Vague; mystifying. The rewritten version of #2 above may serve as a replacement. Delete this item, or rewrite to serve some clear purpose. * 5. Find the area of polygons and circles; volume of rectangular prisms, cubes, and cylinders; and surface area of rectangular prisms. OK * 6. Understand coordinate graphing. Much too vague. Try: Understand and use coordinate graphs to plot simple figures, determine related lengths and areas. * 7. Construct scale drawings. Delete this verbatim repeat of G.6 #8 (and subsumed also in revised item #6 above). OTHER USEFUL TOPIC: A. Use units and *dimensional analysis* to check resonableness of answers. 
Grade 8GENERAL COMMENTS on G.8 Geometry. Basically the text of this section is impossible. It sounds vaguely like what used to be part of a 10th grade course, but it got garbled in transcription. There's no use trying to fix it itembytiem, especially with all the repetitions. I can't take the authors of this section seriously. * 1. Visualize, represent, and transform two and three dimensional shapes. OK * 2. Recognize symmetry in two and threedimensional figures. OK * 3. Analyze and extend geometric patterns such as tessellations. OK * 4. Understand and compute perimeter, area, volume, and surface area. ``Understand ... area'' of what?? In what way is this G.8 item supposed to differ from G.6 #5 or G.7 #5? If it is to remain, this item needs serious clarification of its intent for level G.8. * 5. Understand and apply the geometry of right triangles (including the Pythagorean Theorem and trigonometric ratios). OK * 6. Use appropriate units of measure to the correct degree of accuracy. OK * 7. Estimate, make, and use measurements in realworld situations. Really vague. What kind of measurements do you have in mind for G.8 level? Measure Angles? Areas? Temperatures? And which connections should be emphasized at this grade level relating the measuring process to the mathematical concepts of geometry? * 8. Determine the image of a shape under a transformation in the coordinate plane. Replace ``under a transformation'' with ``under various transformations * 9. Construct scale drawings using proportional reasoning to convert measures. Confusingly vague. What is the intent here? This item also seems to miss the mathematically important point of studying scaling. For that, see item A immediately below. SERIOUS TOPIC OMISSION: A. Understand and use *scaling laws* to determine the effect of scaling a plane figure on its perimeter, area, angles. NOTE: Some mention might be made at G.8 level about what happens in 3 dimensions, though detailed discussion might be regarded as a high school level topic. 
Grade 4* 1. Recognize, describe, extend, and create numeric and geometric patterns. Hopelessly vague. Delete it, or revise to have some specific goals (accompanied by specific examples). * 2. Represent and analyze patterns and functions using tables (e.g., input/output boxes, function tables). OK * 3. Begin to develop the concept of a variable. The concept of ``variable'' is tricky and sophisticated. In G.4 I would stick with specific skills, as in #4 immediately below where the objective is to get students used to letting letters stand for numbers. That should suffice for G.4. * 4. Use letters, boxes, or other symbols to stand for any number or object. Weak and unfocused. It should have the use of symbols in *mathematical equations* as its main goal, once the notion of ``variable'' has been introduced as in item #3. Rewrite as: Use various symbols (e.g. letters, boxes, etc) to stand for any number *in simple algebraic expressions and mathematical equations*. * 5. Find missing number in an open sentence (e.g., 7 x ____ = 56). OK, but the mathematical point (distinct from that in #4) becomes clearer if this item is rewritten to say: Solve simple mathematical equations (e.g. find the missing number in an open sentence such as 7 x __ = 56). Furthermore, it is not clear how the present item differs from G.3 #5 (Functions and Algebra Concepts). Perhaps the goal at level G.4 should be more advanced than the one stated here. SERIOUS TOPIC OMISSIONS: A. Understand use of simple equations (e.g. 3x + 5 = y) as a prescription for determining one number when the other is given. B. *Use and interpret simple formulas* in both symbolic and verbal form (e.g. Area = length x width or A = lw ), to answer questions about relationships between quantities. SUGGESTED ADDITIONAL TOPICS: C. Know *use of parentheses* in algebraic expressions. 
Grade 5NOTE: The whole set of goals in this area is pretty weak for G.5, much weaker than the G.5 goals set forth in the CA Standards. The reason is a strong aversion to the use of symbolic processes (so prevalent in TERC and CMP), evident in the way many of these items are crafted. This will not do! G.5 is the place where students should start to become accustomed to abstraction and symbolic processes (read: elementary algebra). * 1. Use patterns and functions to represent and solve problems. Terribly lame, in absence of specific suggestions. Needs serious rewrite. * 2. Show how one quantity determines another in a functional relationship (e.g., how square numbers grow  1,4,9,16, etc.) OK * 3. Understand that the relationship between two quantities remains the same as long as the same change is made to both quantities. OK, but could be said better, for example: Knows how to manipulate equations. Understands the principle that equals added to equals are equal, and that equals multiplied by equals are equal. * 4. Write and solve open sentences using letters as placeholders (e.g., 4 + a = 14). Almost verbatim repeat of G.4 item. A better goal for G.5 level might be: Use letters to represent unknown numbers in equations and algebraic expressions. Write simple algebraic expressions describing real world situations, and evaluate them by substitution. SUGGESTED ADDITIONAL TOPICS: A. Shift ``Geometry'' item #8 to here, and rewrite it for clarity: *Convert* one unit of measurement to another (e.g. inches to feet, centimeters to inches). This would also be a good place to have students construct graphs showing the relation between inches and feet, centigrade and Fahrenheit, etc. 
Grade 6* 1. Find succeeding terms in a sequence of numbers (e.g., 1,3,6,10,15?) Rewrite for clarity: Recognize numerical patterns (e.g. find succeeding terms in 1,3,6,10,15, ...). Give a verbal description of the rule governing each pattern. * 2. Understand ratio and proportion. OK * 3. Use fractional notation to show ratios. OK * 4. Develop an understanding of a solution to an open sentence with variables on both sides (e.g., 5x + 4 = 8x  5). Vague. Rewrite to say ``Understand how to solve an open ...'' instead of the processoriented (and convoluted) ``Develop an understanding of a solution to an open ...'' * 5. Graph ordered number pairs on a grid. OK * 6. Use ordered pairs of numbers to construct figures on a grid. OK SERIOUS TOPIC OMISSION: A. Solve problems involving rates, average speed, distance, and time. SUGGESTED ADDITIONAL TOPICS: B. Correctly evaluate algebraic expressions given values for the variables (up to 3 variables). 
Grade 7NOTE: The goals set forth for G.7 are really pretty weak. A number of items from G.8 should be taken up in G.7, and some important topics are not mentioned at all. I have suggested changes that would strengthen the G.7 goals. * 1. Find the missing term in a sequence and write the rule. Very weak for G.7, and repeats G.6 item #1. This should be deleted from G.7 goals. * 2. Find the missing term in a proportion where terms can be fractions, decimals, or percents. OK * 3. Describe functions and generalize them by the use of rules and algebraic expressions. Pretty lame for such an important topic. Try: Identify word problems in which one must find the missing term in a proportionality relation. Know how to solve such relations when the known terms can be fractions, decimals, or percents. * 4. Use algebra to translate verbal phrases into mathematical form. Would be OK if you add at end some mention of the use of equations: Use algebra ... form as equations or inequalities (e.g. perimeter of a rectangle is P = 2A + 2B if the side lengths are A,B). However, item #4 here is pretty much redundant if one rewrites #6 below as suggested, since linear equations and inequalities are the only ones dealt with at this grade level. So, item #4 could be deleted. * 5. Evaluate algebraic expressions. Awfully vague and uninformative. Try: Simplify and evaluate algebraic expressions using the laws of algebra. * 6. Solve an equation and check the solution set by substitution. Could be a lot better. Try: Solve linear equations and inequalities in one variable. Interpret solutions in the context from which they arose, and verify that the result is reasonable. * 7. Explore the concept of rates (distance, time, and unit pricing). Weak, and processoriented. State some explicit goals, as in the CA Standards: Explore the concept of rates and solve multistep problems involving distance, time, unit pricing, etc. SERIOUS TOPIC OMISSION: A. If you begin the discussion of functions in G.7, as in item #3 above, it is educational malpractice not to discuss *graphical representations of functions at the same time*. Defering it to G.8, item #2, is absurd. Shift G.8, #2, to here! B. Represent quantitative relationships graphically. C. Know how to graph linear functions. Explore the meaning of ``slope of a line''; know how it is determined from the equation of the line. 
Grade 8NOTE: Items for this G.8 area are so vague or terse one gets the feeling that the writers got tired of their task and were just ``phoning it in'' while reading from the table of contents of the G.8 materials for CMP. The G.8 goals deserve much more serious attention than is evident here. Many important G.8 topics seem to be missing.* 1. Identify, describe, represent, extend, and create patterns. Ridiculously vague statement! This could mean anything. Besides, numerical patterns (sequences) have already been mentioned twice, in G.7, #1, and G.6, #1. Unless there is something specific and new that should be taken up in G.8  for instance, initial discussion of mathematical induction  this item should be deleted. * 2. Describe and represent functions using tables, graphs, and equations. This item is weak and seriously incomplete (for G.8 level) unless you add: Describe ... and equations. *Understand how information about a function is read from its graph*. * 3. Analyze tables, graphs, and rules to determine functional relationships. OK * 4. Find solutions for unknown quantities in linear equations, inequalities, and basic quadratic equations. Mashes together two distinct goals. What G.8 students are expected to know about quadratic equations must be specified carefully, and the statement about linear equations leaves too much to the reader. I suggest replacing this with several items, as in the CA Standards.
Know how to set up the equation of a straight line from verbal
description of a relationship between two variables.
* 5. Apply the order of operations including the use of parentheses. Vague, and misses the point. Try: Correctly apply the rules of algebra in handling operations and parentheses to simplify algebraic expressions. * 6. Model and solve problems involving rate, average speed, distance and time, or direct variation. OK, but ``direct variation'' is 19th century jargon; the word ``proportionality'' would be more appropriate. Even so, this item is fairly vague. I would recommend the following, adapted from the CA Standards, which is somewhat more specific and demanding: Apply algebraic techniques to solve verbally presented rate problems, work problems, and percent mixture problems. SERIOUS TOPIC OMISSIONS: A. Understand the use of *laws of exponents* when finding reciprocals, taking roots, and *raising to fractional power*. Understand the meaning of fractional powers of positive numbers. B. Solve a system of two linear equations in two variables *algebraically*, and interpret the solution in terms of the graphs of the lines. C. Know the quadratic formula, its proof by completing the square, and its use to solve quadratic equations with real roots. D. Understand how to graph quadratic functions, and the significance of the xaxis intercepts. E. Apply quadratic equations to physical problems, such as the motion of an object under the influence of gravity. 
NOTE: The whole ``Statistics'' section from grades K8 is extremely inflated, and could well be put into two of the grades, maybe 4 and 8. Its underlying premise that students are in a position to make meaningful predictions or assess sampling results is mistaken, except in so primitive a sense as to be meaningless. The best they can do at this level is understand that the sample resembles the total, and the future resembles the past. Not much mathematics there.
Students should learn about graphical presentations of data, how to construct them on their own, and how to interpret such presentations. Another approach appropriate to this level would be to run them through the little book ``How to Lie with Statistics'' (which is by now antiquated in its 1920's language), or it really effective modern successor (Statistics: Concepts and Controversies, by David S. Moore, W.H. Freeman, 1991). The math required to understand all the common fallacies is simple; the practical importance of understanding this topic is great; nothing at all is said about it in this Scope and Sequence.
Grade 4* 1. Collect data to answer a question or test a hypothesis. OK * 2. Construct, read, and interpret pictographs, bar graphs, and line plots. OK, but clarify terminology by writing ... bar graphs (histograms), and line plots. * 3. Read and interpret a line graph. This verbatim repeat of a G.3 item should be deleted. * 4. Find range, median, mode, and mean using a collection of data. Ok, but even better if you add at end: Find ... data. Discuss ``outliers''. * 5. Explore realworld polls such as TV ratings and opinion polls in order to understand random sampling. OK * 6. Predict results and find out why some results are more likely than others, less likely than others or equally as likely as others. Supremely vague. In what contexts should these activities be pursued? * 7. Determine probabilities of simple events in realworld situations. Vague. By what methods are students expected to do this? Try: Using trial and experiment, *and elementary combinatorial reasoning*, determine probabilities of simple events. * 8. Display orderly ways to determine the number of possible arrangements and combinations using models, pictures, lists. Repeats a G.3 item. Try something more specific, and appropriate for G.4: Represent possible outcomes for a probability situation in an organized way (e.g. using tables, pictures, tree diagrams, etc.). 
Grade 5* 1. Use different ways of collecting, organizing, and displaying data, such as tally tables, graphs, and Venn diagrams. OK, but replace ``graphs'' by more specific ``bar and circle graphs''. * 2. Construct, read, and interpret line graphs, bar graphs, pictographs, and line plots. Items #2,3 in this G.5 area are redundant, almost verbatim repeats of G.4, #2,3. Delete this, or state new concepts or skills to be mastered at level G.5. * 3. Read and interpret double bar graphs and circle graphs. See comments to #2. Delete, or seriously revise. * 4. Use circle graphs to explore the concept of percent. Silly and useless, as stated here. Percentages have already been covered and are not a probability topic. In this G.5 area it WOULD be relevant to discuss the notions of ``frequency'' and ``relative frequency'' of events in a data set, and examine how these empirical notions might be related to mathematical probabilites. Unfortunately, the authors of the Scope and Sequence do not say that. * 5. Select, compare, and use appropriate graphs to represent data. OK * 6. Understand and identify differences among mean, median, mode, and range. OK * 7. Predict, represent, and explain probability using fractions, ratio, and percents. OK * 8. Identify events that are impossible (that have a chance or probability of happening equal to zero), events that are certain (that have a chance or probability of happening equal to one), and events that occur sometimes (expressed as a proper fraction). OK * 9. Examine random and unbiased samples such as market surveys. OK SERIOUS TOPIC OMISSIONS: A.See item #4 above for a really serious omission. B. Use fractions and percents to *compare data sets of different sizes*. 
Grade 6* 1. Collect, analyze, and interpret a variety of data. OK * 2. Construct and interpret double bar graphs, line graphs, circle graphs, histograms, and stem and leaf plots. OK * 3. Use the average (mean), median, mode and range to interpret and analyze data. OK * 4. Use estimation to check the reasonableness of results. OK * 5. Determine and predict outcomes of experiments. Vague. A more specific version might read: Identify claims based on statistical data and, in simple cases, evaluate the validity of those claims. SERIOUS TOPIC OMISSIONS: A. *Compare various samples from a population with data from the entire population. Identify situations when it makes sense to use a sample. B. Identify *different ways of selecting a sample*, and understand which make a sample more representative of the population as a whole. C. Use data to estimate the probability of future events (e.g. batting averages or number of auto accidents per driven mile). NOTE: The point of C is spelled out in my comments on item #4 of G.5, where this might first be taken up. 
Grade 7* 1. Construct and interpret a variety of graphs and explore scatter plots and box and whisker plots. OK * 2. Develop and apply understanding of mean, median, and mode in order to analyze data. Delete. Seems a needless repeat of G.6, #3. * 3. Organize data using terms such as range, interval, and frequency. OK * 4. Conduct and report on a variety of probability experiments. OK * 5. Determine probabilities of independent and mutually exclusive events. OK, but vague. By what methods is this to be accomplished? * 6. Conduct a variety of simulation techniques to estimate probability of events. OK * 7. Develop and explore combinations and permutations. OK * 8. Identify sample spaces by listing all elements. OK 
Grade 8NOTE:In this G.8 area many topics are stated in a vague and perfunctory way. If I were a teacher I'd have little idea how to procede based on what is said below. I have some suggestions, but this area needs serious thought and a total rewrite. * 1. Collect, organize, and display data with tables, charts, and graphs that are appropriate for the data. OK * 2. Construct scatter plots and box and whisker plots. Weak. Stronger and more to the point if rewritten to say: Know various ways to display data sets, including histograms (bar charts), scatter plots, and boxandwhisker plots. Construct plots based on realworld data sets, and know how to use them to compare two data sets. * 3. Consider the effects of missing or incorrect information. Mushy wording. ``Consider ... '' is aimless; ``Understand ...'' has a point. Furthermore, it would help to add a related topic in this area, so the item reads: Understand ... information. Examine the effects of outliers on the mean and median. * 4. Analyze data with respect to frequency and distribution. Vague and convoluted statement. I think the following goal is more suitable for students by the end of G.8: Understand the meaning of a histogram as a plot describing the frequency distribution of some property of a population (e.g. as a description of weight distribution, income distribution, etc.). * 5. Formulate hypotheses to answer a question and use data to test hypotheses. This is so vague I'm not at all sure what is intended. One reasonable and explicit interpretation, which fits well with the proposed rewrite of #4, is: Know how to plot histograms from real data sets. Know how to interpret histograms, and make predictions based on the information they provide. * 6. Represent and determine probability. What is this supposed to mean?? How is it related to goals specified in this area for lower grade levels? This item is impossibly vague. Total rethinking and rewrite needed. * 7. Use estimation to check the reasonableness of results. Quite vague; moreover, it hardly sounds like a topic in probability and statistics. Maybe in ``Number Sense'' (where it has already appeared several times)? * 8. Estimate the probability of an event. Not at all clear what the distinction is between this item and #9 below. Both are extremely vague: in what context are students supposed to do this? Maybe the writer had something like this in mind: Use histograms describing a population to estimate the probability of events associated with that population. * 9. Make predictions based on experimental and theoretical probabilities. See comments to item #8, which this closely resembles. * 10. Understand combinations and permutations. Incomplete. At G.8 level this really should include the connection with calculation of probabilities: Understand ... permutations, *and their use in computing elementary mathematical probabilities.* SERIOUS TOPIC OMISSIONS: A. Understand ``independence of events'' in probability, and the product law for probabilities associated with independent events. B. Solve simple probability problems with finite sample spaces using the rules for addition, multiplication, and complementation of probabilities. 
Grade 4* 1. Understand word problems, identifying pertinent, extraneous, and missing information. OK * 2. Use a variety of strategies to solve and represent problems/solutions (e.g., logical thinking, estimation, number sense, pictures, diagrams, and charts). OK, except for a crucial omission in the list of suggested strategies. No mention is made of the use of *mathematical equations as a solution strategy*; this relentless deemphasis of algebraic and symbolic methods is typical of most NCTMbased curricular materials. It is completely wrongheaded, as these methods are the heart of real mathematics! This same omission occurs repeatedly throughout all grade levels in the area of Mathematical Process. The present item MUST be expanded to read: Use ... (e.g. logical thinking, number sense, pictures, *and mathematical equations*. * 3. Work individually and collaboratively to discuss, justify, organize, and write about solutions to problems using content specific mathematical language. This item is processoriented edspeak mush, typical of TERC at this grade level. Here's a more goaloriented item from the G.4 CA Standards. In fact those standards address the issues in item #3 as two separate statements: Use a variety of methods, such as words, numbers, symbols, charts, graphs, models, to explain mathematical reasoning. Express solutions clearly and logically, using clear language and appropriate mathematical symbolism. * 4. Explain how solutions to problems can be applied to other school subjects and in realworld situations. OK * 5. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic fourfunction or fraction calculators, measuring cups, scales, and rulers  metric and U.S. Standard, thermometers, tape measures, and protractors). OK 
Grade 5* 1. Understand word problems, identifying pertinent, extraneous, and missing information. OK * 2. Use a variety of strategies to solve and represent problems/solutions (e.g., logical thinking, estimation, number sense, pictures, diagrams, and charts). This is a verbatim repeat of G.4, #2. It also neglects to mention the use of mathematical equations as a strategy. At the very least this omission should be corrected by adding: Use ... diagrams, charts, and mathematical equations. However, this item should probably be rewritten to emphasize new skills and concepts appropriate to G.5. * 3. Work individually and collaboratively to discuss, justify, and write about solutions to problems using content specific mathematical language. Delete this item. It is edspeak mush, and a verbatim repeat of G.4, #3. * 4. Solve problems systematically and logically, and develop an awareness of when estimating is more appropriate than finding an exact answer. OK * 5. Recognize the use of mathematics in other subject areas such as science, social studies, and music. OK * 6. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic fourfunction or fraction calculators, measuring cups, scales, and rulers  metric and U.S. Standard, thermometers, tape measures, and protractors). OK

Grade 6* 1. Create, analyze, and solve word problems in all of the concept areas. OK * 2. Identify pertinent, extraneous, and missing information. OK * 3. Use a variety of strategies to solve and represent problems/solutions (e.g., logical thinking, estimation, number sense, pictures, diagrams, and charts). OK, except that it omits mention of algebraic methods as a solution strategy. Revise to read: Use ... diagrams, charts, and mathematical equations). * 4. Work individually and collaboratively to discuss, justify, organize, and write about solutions to problems using content specific mathematical language. Delete this item. It is processoriented edspeak mush, and a verbatim repeat of G.4, #3. What is this doing in G.6?? * 5. Apply basic math skills to realworld situations. OK * 6. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic scientific calculators, protractors, compasses, scales, and rulers  metric and U.S. Standard). OK SUGGESTED ADDITIONAL TOPIC: A. Formulate and justify *mathematical conjectures* 
Grade 7* 1. Create, analyze, and solve word problems in all of the concept areas. OK * 2. Identify pertinent, extraneous, and missing information. OK * 3. Use a variety of strategies to solve and represent problems/solutions (e.g., logical thinking, estimation, number sense, pictures, diagrams, and charts). Delete. This is same as G.5, #3, and G.6, #4; what is it doing in G.7?? Furthermore, the use of algebra and mathematical equations as strategies has once again been omitted. * 4. Work individually and collaboratively to discuss, justify, organize, and write about solutions to problems using content specific mathematical language. Delete. This item is entirely processoriented, and tailored specifically to the ideology of NCTMbased math curricula. * 5. Apply basic math skills to realworld situations. Extremely vague. It would be more meaningful if split into two separate parts: Apply basic math skills to realworld situations, expressing solutions clearly and logically using appropriate mathematical terms and notation. Support solutions with evidence in both verbal and symbolic work. Decide whether solutions are reasonable in terms of the original situation. * 6. Talk about the uses of mathematics and its importance to their present and future lives. OK * 7. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic scientific calculators, protractors, compasses, scales, and rulers  metric and U.S. Standard). Vague, and a nearly verbatim repeat of G.6 #6. By G.7 this item should indicate some new and more advanced objectives. First of all, write ``Use appropriate ...'' in place of the convoluted ``Explore the use of appropriate ... ''. Second, this item speaks of using tools  to do what?? Either add some meaningful G.7 level exemplars or delete this item. SUGGESTED ADDITIONAL TOPICS. See Topic A suggested for G.6. If that is not made part of the G.6 goals, it certainly should become one of the G.7 goals. 
Grade 8* 1. Create, analyze, and solve word problems in all of the concept areas. OK, but emphasis slightly askew. Better if rewriten: Apply basic math skills to create, ... concept areas. * 2. Identify pertinent, extraneous, and missing information. OK, but a verbatim repeat of G.7 #2. Nothing new at level G.8? * 3. Use a variety of strategies to solve and represent problems/solutions (e.g., logical thinking, estimation, number sense, pictures, diagrams, and charts). Once again, the use of algebraic methods and mathematical methods as solution strategies go unmentioned. Add: Use ... diagrams, mathematical methods and algebraic techniques for solving them). * 4. Work individually and collaboratively to discuss, justify, organize, and write about solutions to problems using content specific mathematical language. Verbatim repeat of G.4 #3, G.5 #3, G.6 #4, G.7 #4. As before, it is completely process oriented, and slanted explicitly toward the philosophy of NCTMbased curricula such as CMP. Delete this item. * 5. Apply basic math skills to realworld situations. This seems a reworded version of #1 above, and serves no additional purpose. Delete it. * 6. Verify and interpret results of a problem. OK, but extremely vague and should have been stressed at earlier grade levels than G.8 * 7. Conduct research about a group of objects by studying a few of them (sampling). Awkwardly worded, and pretty vague (the phrase ``glittering generalities'' came to mind). Try rewording as: Conduct research about a population of objects by studying samples. Then ADD an explicit example of what you have in mind. * 8. Talk about the uses of mathematics and its importance to their present and future lives. OK, but ``Talk about ...'' sounds aimless compared to ``Discuss ... . * 9. Explore the use of appropriate mathematical tools and technology (e.g., computers, basic scientific calculators, trigonometric function tables, protractors, compasses, and rulers  metric and U.S. Standard). This is a verbatim repeat of G.7 #7. Delete it or rewrite to indicate that the G.8 goals represent some advance beyond those of G.7. A more relevant G.8 goal might be the following, which appears in the CA Standards: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, prioritizing information, and observing patterns. 
Frederick Greenleaf
Professor of Mathematics
Courant Institute of Mathematical Sciences
New York University
Email: greenleaf@cims.nyu.edu
With assistance from
Ralph A. Raimi
Professor of Mathematics (emeritus)
Department of Mathematics
University of Rochester
Webpage: www.math.rochester.edu/u/rarm
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