Flowing under the pre-K through high school curriculum, like the ever-widening Mississippi, is a steady expansion of the number system and its corresponding basic operations.
Important milepost concepts and mathematical problems which are posed, deconstructed and solved throughout those years are like the flatboats and steamboats of Mark Twain's era which floated upon the Mississippi's waters.
One doesn't seek to control the river, because the number system and its operations exist in nature, but we can select what floats on it and where to travel. Choosing when and how to introduce concepts, what problems to pose, and where they fit is the foremost responsibility of a standards and curriculum developer.
CCSSI describes the evolution of the number system (albeit awkwardly placed at the beginning of the High School Standards, p. 58), as follows:
During the years from kindergarten to eighth grade, students must repeatedly extend their conception of number. At first, “number” means “counting number”: 1, 2, 3... Soon after that, 0 is used to represent “none” and the whole numbers are formed by the counting numbers together with zero. The next extension is fractions. At first, fractions are barely numbers and tied strongly to pictorial representations. Yet by the time students understand division of fractions, they have a strong concept of fractions as numbers and have connected them, via their decimal representations, with the base-ten system used to represent the whole numbers. During middle school, fractions are augmented by negative fractions to form the rational numbers. In Grade 8, students extend this system once more, augmenting the rational numbers with the irrational numbers to form the real numbers. In high school, students will be exposed to yet another extension of number, when the real numbers are augmented by the imaginary numbers to form the complex numbers.
With each extension of number, the meanings of addition, subtraction, multiplication, and division are extended. In each new number system—integers, rational numbers, real numbers, and complex numbers—the four operations stay the same in two important ways: They have the commutative, associative, and distributive properties and their new meanings are consistent with their previous meanings.
We acknowledge CCSSI's understanding of the river, even though those who may need this ``big picture'' synopsis the most, i.e., elementary teachers, may not read as far as the high school standards.
In this blog, we previously discussed two ``boats'': first, the concept of zero, and how CCSSI gets this milepost concept wrong, even though the above summary has the correct idea. We also previously discussed the extension of the whole numbers to really big numbers, and how CCSSI falls short in this regard as well.
Introducing concepts aligned with the number scheme and posing problems of a level of difficulty commensurate with the stage of development of the student is a central responsibility of the curriculum. It is the correct matching up of (1) number system, (2) type of problem, i.e., concept, and (3) level of difficulty which is almost completely lacking in CCSSI. We don't think this necessary synchronization was on the CCSSI writers’ radar at all. In addition, we have not seen a single recognition of this shortcoming from the pundits: the mathematicians or ``curriculum experts'' who claim to have analyzed CCSSI.
What we will discuss in this blog post is a specific example of the placing of one boat in an entirely wrong part of the river, and not one, but two benefits to be gained by better placement. We are going to present an example of a standard in the CCSSI which is completely misplaced, by years, because it mismatches two of these elements in particular: the concept and the number system. The level of difficulty may be a bit trickier to agree upon, and may become emotionally and politically charged, so we will present our opinion and some examples and let the reader decide (no, we are not affiliated with Fox News).
If the concept were off by a year, or perhaps two at the most, we might not so loudly cry foul, but when it's off by three years or more, that's really an injustice. As in most cases, it's an egregious instance of dumbing down the standards, and done, we believe, unnecessarily.
Let us begin with the triangle angle sum theorem, which in CCSSI, is presented in the eighth grade (8.G.5). The theorem states: the three angles of a triangle add to 180°.
This theorem is not trivial, and can be proven many ways, one method using the parallel postulate (more on that later). Following the proof of the angle sum theorem, students can then prove (themselves) the exterior angle theorem, which states that an exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.
One obvious consequence of the angle sum theorem is that each angle in an equilateral triangle is 60°. You can ask yourself (we certainly did) whether a curriculum should keep this fact a secret from students until the year before they enter high school. But let's leave that question aside for the moment, as well.
These two theorems make possible a whole world of highly interesting and challenging yet solvable problems. Let us see first how the National Assessment of Educational Progress, otherwise known as ``The Nation's Report Card'', presented these theorems in three separate problems and how American students fared.
2003 NAEP, 8th grade
Application of the angle sum theorem cannot get any more basic, yet this question was answered correctly by just 51% of 8th graders.
What happens when the theorem is disguised somewhat?
2005 NAEP, 8th grade
With some added complications (e.g., students have to remember what a right triangle is), the percentage answering correctly drops to 48%. (Students also could have used complementary angles to solve the question.)
If we now look at a simple application of the external angle theorem, watch what happens:
2003 NAEP, 8th grade
Here, the percent answering correctly drops to 33%, and, for that reason, the question is deemed ``hard’’ by NAEP.
Let's review what is required to understand the two angle theorems. Students need to know what a triangle is, understand what straight lines and angles are, understand that angle measure is additive, and be able to add and subtract to 180. If you're thinking what we were thinking, then you realize that all of these skills should be learned by the fourth grade or so.
Well, wait, you say. The angle sum theorem needs the parallel postulate to prove and that is an eighth grade concept. We don't agree it needs to wait. The theorem can be proven with an age-appropriate activity: Any fifth grader can tear up a paper triangle into three pieces, each piece containing one vertex, and put the three vertices together to see they form a straight angle. This is a perfectly acceptable age-appropriate proof of the theorem and completely understandable by fifth graders. In addition, students can, with manipulatives, tessellate a bunch of congruent triangles and see the same result. And, finally, there is a nifty paper folding exercise where the three vertices meet to form a straight angle. These are all nice exercises...for a fifth grade class. (Obviously, angles can be non-whole rational numbers, but it does not require extending beyond the set of whole numbers to understand these theorems.)
If we stipulate that students' concept of whole numbers and fluency of the basic operations of whole numbers should be completed by the end of fourth grade, why are theorems that only require fluency with whole numbers and their operations and knowledge of basic geometry relegated all the way back to the eighth grade?
We here at ccssimath.blogspot.com think that curriculum and standards writers just don’t understand the big picture behind what we should be doing in K-12 math education.
We think that not only are we able to, but it is our obligation to present to fifth graders problems based on these these two theorems using their already learned skills with whole numbers, rather than wait. This is because, as we have said in previous posts, non-operations applications of concepts should follow closely behind the skills to lead to abstract understanding of those concepts.
CCSSI, predictably, prefers to present fourth and fifth graders with numerous dreary operations questions ``Fluently add and subtract multi-digit whole numbers'' (4.NBT.4) and ``Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers'' (4.NBT.5), ``Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors'' (5.NBT.6); the curriculum-that-is-obsessed-with-verbal-explanations ``interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5'' (4.OA.1) and ``express the calculation `add 8 and 7, then multiply by 2’ as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product'' (5.OA.2); and the dead-end-as-posed concept ``[r]ecognize angle measure as additive'' (4.MD.7).
We are not anti-skills, but we think skills ought to be used, not explained.
To its credit, CCSSI does envision some ``word problems’’ thrown into the mix. ``Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.’’ (4.OA.3) But its feeble attempt to write a word problem? The shockingly boring examples of Table 2 (``You need 3 lengths of string, each 6 inches long. How much string will you need altogether?'')
We’ve seen word problems in fourth and fifth grade textbooks and we think they are vile, too. And we also note that CCSSI can’t present a single example of what it thinks is an appropriate ``multistep word problem.’’ In fact, in the fifth grade, CCSSI completely abandons applications of whole numbers, i.e., using whole numbers to solve non-operations problems. It has moved on to operations with fractions.
All of these CCSSI standards and the problems that they will engender are extremely non-intellectually challenging, dull, dumbed down mathematics and will (we believe) squelch, if not mortally wound, any mathematical curiosity and interest that remains in students' minds. What CCSSI wants our fifth graders to do are river problems, not boat problems.
In May 2012, the U.S. Department of Education's Institute of Education Sciences issued a press release about a ``report, Improving Mathematical Problem Solving in Grades 4 Through 8... developed by a panel of eight researchers and educators for the What Works Clearinghouse, a project of the DOE that produces classroom guides based on scientific evidence.'' The ``What Works'' guide makes ``five recommendations for improving students’ mathematical problem solving in grades 4 through 8''.
We read the report and the guidelines and examined the ``routine'' and ``non-routine'' problems that the guidelines suggest. We saw no indication of which grade it deemed each problem appropriate for (a major omission—and we think some examples belong in the third grade), but more importantly, as in CCSSI, we were unable to find an underlying rationale for any example it chose. The report focused mainly on teaching practices, i.e., how a teacher might guide students to strategize in problem solving, but we didn't see an indication that the particular examples matched up, or even considered any of the elements we are propounding.
How to write standards and/or a curriculum
Not that there is nothing intellectually stimulating to be done in the fourth and fifth grades. We think there is plenty. 4.MD.7 (angle measure is additive) is an obvious standard that allows us, if we take advantage of the opportunity, to prove the angle sum theorem and go way past what the CCSSI would have American fifth graders do.
We think fifth graders will be able to do what eighth graders formerly could not do (and if CCSSI is allowed to proliferate, continue to be unable to do), that is, apply whole numbers and their operations usefully.
[In another clueless misplacement of priorities, 4.G.2 would have students learn what a right triangle is, but it is scalene, isosceles and equilateral triangles (i.e., triangle categories based on side length) that is the more important concept in elementary school. We are clueless where CCSSI would have students learn this highly important categorization of triangles; it’s absent from the standards. Right triangles (and acute and obtuse triangles, i.e., classifying triangles based on angle measurement) are more important for eighth grade, when the Pythagorean Theorem is introduced, and high school, when trigonometry is introduced.
Otherwise, categorization of triangles based on angle measure is simply rote memorization at that grade level, without any real poseable problems. If the CCSSI doesn't want to introduce the angle sum theorem until eighth grade, why would elementary students care that a right triangle exists?]
As we stated before, by postponing the angle sum theorem until eighth grade, students won't know (i.e., CCSSI doesn't want them to know) until they are almost in high school that an equilateral triangle has three 60° angles. This aspect of CCSSI alone has caused your bloggers figurative loss of sleep. But another more useful fact related to the angle sum theorem is that the base angles of an isosceles triangle are the same (one of Euclid's theorems). We don't think this has to be formally proven in fifth grade for it to be a usable fact; fifth graders will see it is so by looking at the symmetry of an isosceles triangle. We give fifth graders more credit for common sense than does CCSSI.
While scalene triangles don't have much to offer in the way of problems (yet), and equilateral triangles are all kind of the same (important, but with limited utility in problem posing), it is the humble isosceles triangle that opens a whole world of possibilities for fifth graders to learn way above and beyond what CCSSI is currently offering, and even what NAEP has posed to eighth graders.
We will now propose to show by example, in two stages, how a curriculum should be designed: it should look backwards at what has been taught, it should know where it is planning to go, and it should bridge the divide between past and future by posing problems that require skills already learned to be used in non-obvious, creative ways. In our close look at one ``marina’’, we will show how the mild-mannered, unassuming isosceles triangle can begin to put the American curriculum back on track to be the world's best.
This is not an idle boast. We are convinced it can be done.
C1. Consider the following problem:
Now, of course to do this problem, we've assumed that several things have been learned: one must know that a right angle has 90°, and the ubiquitous equilateral triangle has 60° angles, and that one can draw a circle about a given point, say, with a compass, but we think these should be learned by the fifth grade anyway. However, it is the isosceles triangle that gives this problem heft.
[Ok, a solution for those that need it. Draw a line (!) from the point where the curves intersect to B as well. Then you have an equilateral triangle. So at C, you now have a 60° angle, and in the smaller triangle, an isosceles triangle, the vertex angle is 30°. (90° – 60° = 30°). 180° – 30° = 150°. Each base angle is 150° ÷ 2 or 75°. So angle 1 is 75°.]
We've already matched the number system and the concept (Grade Five), but we also think the level of difficulty of this problem correctly invokes three aspects of mathematics that are generally missing in American education:
(a) length, which involves a high level of concentration for significant periods of time; (b) interconnectivity, which requires recollection and utilization of different math tools that a student has acquired along the way (Think: why do you want to acquire various gizmos while playing video games? Answer: you’ll need them later); and (c) dimensionality, requiring working on a problem where the end is neither visible nor apparent at the beginning.
In addition to the question itself, there are other, less obvious aspects to this problem: it looks backward to what students have learned so far and steers students toward a direction we want them to go. A good problem should have a basis for existing and a notion where it fits into the overall educational plan.
CCSSI refers to multistep problems, but we know CCSSI does not demonstrate this level of acumen and fails to grasp what elements should be in those multiple steps.
Incidentally, don't fret if you didn't see the answer to our example right away. Notwithstanding the difficulty this problem will present to fifth grade teachers, this problem and the next one have been known to make mathematicians sweat because it took them a minute or more to solve.
C2. Consider the next problem:
[No solution given, you're on your own to solve it. It is a true ``multistep'' problem.]
Part I Summary
We know that this last problem would be considered ``difficult''. We also know the potential backlash that posing such a problem to fifth graders will cause, which we will discuss in another blog post. But we think that presenting this caliber of problem, considering the number system, the age-appropriate length of the problem and level of difficulty are the elements that American mathematics needs at all grades to make the curriculum really fulfill its promise.
We think learning these theorems and applying them belong in the fifth grade, not the eighth grade.
Let us now present a problem that brings us back down a notch in difficulty and we will discuss the second part to our plan for putting American curriculum back on course.
We submit that another type of problem, using the same two theorems in the fifth grade and the isosceles triangle, will evolve mathematical skills in an additional way, and completely beyond the limited comprehension and vision of CCSSI.
C3. Consider the following:
At first, this problem looks strikingly similar to (if not simpler than) the third NAEP problem we presented earlier, but it has some underlying features absent from the NAEP problem.
Why do we like this problem for fifth graders? Because to solve this problem uses what we call ``proto-algebra'', not simple arithmetic, skills. Specifically, students should do this problem without algebra, even though it could be solved with algebra, because two techniques, the so-called additive and multiplicative properties of equations, underlie parts of this problem. Students are solving the quintessential algebra problem, 2x + 54 = 180, as part of, but not comprising the entire problem, and without using any algebra. In addition, while so-called algebra books would have students solve, as a rote exercise, 2x + 54 = 180, and its infinite variants, as the entire question unto itself, our problem uses this underlying equation as a means to addressing a lengthier, more complex challenge. Manually solving such a problem teaches a critical concept, i.e., the underlying mechanics of algebra and algebra’s reason for being, so that when algebra (the abstraction) finally arrives, it is simply a methodization of skills previously learned, and no longer a great mystery.
[Ok, a non-algebra solution. 180 – 54 is 126. Then divide 126 by 2 because there are two equal base angles. 126 ÷ 2 is 63. Add 63 and 54, that equals 117. (Or 180 – 63 = 117)]
The reader may realize that we are presenting an idea for introducing algebra in a completely different way than the existing curricula would have it. We want to introduce the concept before the abstraction (a recurring theme in this blog), but CCSSI (as has American education for years) simply jumps into the abstraction of algebra before having students learn the concept.
Let’s look at how CCSSI deals with algebra. Let’s not forget that, along with fractions, algebra has a highly problematic history in our educational system. As we have seen before, among other things, CCSSI is attempting to ``fix’’ mathematics education (in particular, fractions and algebra) by shifting concepts earlier and making them even more abstract, sometimes to the point of being absurdly age-inappropriate.
In K.OA.1, CCSSI states, ``For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.’’
Why is it important for kindergartners to know what number makes 10 when added to a given number? Mental math? Algebra in kindergarten? We have no idea.
1.OA.1 states, ``Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.’’
In CCSSI Table 1, CCSSI takes generic addition and subtraction problems and removes one number at a time from each, to make a total of six ``algebra’’ problems. This reeks of OCD and is ridiculous for a first grader. (see also 1.OA.8 which essentially repeats 1.OA.1.) In addition, each of the six possibilities is not of comparable difficulty or utility.
Let’s look at some examples of what makes sense and what does not:
``You had 7 candies. During the night you hear a mouse and in the morning you had only 5. How many candies did the mouse steal?’’, i.e., 7 – BOX = 5, is a sensible question to ask a first grader; but
``You ate five candies, now you have three, how many did you start with?’’ makes no sense as a real problem because you always know what you start with, thus obviating the need to figure out BOX – 5 = 3 — this scenario would never have a useful application, and thus makes no sense as an exercise). CCSSI presents both of these as comparable and reasonable problems, but we disagree. First graders should solve the first problem, but not the second. And we also think that the second problem is far more abstract than the first, and not fitting for first grade. Not every possibility should be presented, just because it can. There needs to be a rationale for what we ask elementary students to understand. CCSSI presents problems without an understanding of what concept underlying those problems it is trying to convey and what it seeks to accomplish.
2.OA.1 again tries to put the unknown into every position: ``Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions...’’, again unnecessary and with no underlying plan for teaching algebra concepts.
3.OA.8 states, ``Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity.’’
This sounds like full-fledged algebra to us. Algebra in the third grade? We don’t think so.
4.OA.3 states, ``Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity.’’
Fourth grade advances from the third grade in using ``multistep’’ rather than ``two-step’’ problems. We’d like to see an example of what CCSSI considers multistep. We think CCSSI is giving us the ol’ two-step instead.
In Grade Five standards, we don’t see any algebra concepts at all. Indeed, the word ``unknown’’ doesn’t appear. Not unless 5.OA.1’s ``Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols’’ is taken to mean evaluate 2(x+3) when x=2. But we don’t think that is CCSSI’s intent.
Suddenly, in Grade Six, CCSSI returns to algebra and algebra notation with a vengeance.
6.EE.2 ``Write, read, and evaluate expressions in which letters stand for numbers.''
6.EE.2a ``Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation `Subtract y from 5’ as 5 – y.''
6.EE.2c ``Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2.’’
Isn’t this a bit much? Students have supposedly just learned about exponents in 5.NBT.2 ``Use whole-number exponents to denote powers of 10’’ and 6.EE.1 ``Write and evaluate numerical expressions involving whole-number exponents.’’ Exponents are a difficult concept; how about ramping up at a reasonable pace instead of shoving in the evaluation of s3 when s = 1/2 (a fractional base?!?) right after learning what an exponent is?
Actually, we don’t think exponents belong in fifth or sixth grade at all. They should be introduced around the seventh grade, so that in the following year exponents can be used (in a useful application) in the Pythagorean theorem.
6.EE.3 ``Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.’’
6.EE.4 ``Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.’’
6.EE.6 ``Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.’’
6.EE.7 ``Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.’’
6.EE.9 ``Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.’’
It’s enough to make us go running for the hills. Along with the other new strands in Grade Six, CCSSI promises to make sixth grade the year from hell, for teachers as well as students.
Part II Summary
By thoughtfully presenting a series of well-conceived, age-appropriate and useful proto-algebra questions throughout the elementary grades, we think that ``proto-algebra’’ concepts can be successfully taught, so that when algebra is formally introduced, it will no longer present the cliff-scaling, terrifying leap from arithmetic that has scared generations of students.
Algebra is not a behemoth to suddenly explode onto the scene; it is a concept to be learned in a general form, before the letter x ever enters the mathematical lexicon.
The American curriculum for years has had a class called ``Algebra''. Algebra is not a threshold, it is the product of evolution of thought.
To resummarize the summaries:
We presented one simple example of a misplaced concept (the angle sum theorems, both of which are applications of whole numbers and basic geometry), why it is misplaced and what more can be done with it if it's correctly placed.
We showed the following elements should be matched at all grade levels:
(1) number system
(2) type of problem, i.e., concept
(3) level of difficulty = (a) length + (b) interconnectivity + (c) dimensionality
Finally, by showing that a curriculum which not only seeks to match the elements but also repeatedly asks itself where it belongs, where it came from and where it is headed, we can fix long standing problems, like the perennial difficulty with algebra, by introducing age-appropriate proto-algebra concepts before algebra is formally introduced. A curriculum always needs to both understand and explain the rationale for what it contains and where things are.
We believe these examples illustrate our vision: By creating a comprehensive curriculum with an integrated plan, well-conceived pathways and age-appropriate challenges, the United States can produce truly mathematically literate graduates, not one-dimensional test-taking automatons.
CCSSI is missing the boat.