The fragility of CCSS, Part 3

Oh Massachusetts!

Sorry to keep picking on you. I’m sure other states that made tweaks to the CCSS fell into similar traps. It’s just that I know your standards because we’re forced to consider them now that we’re using MCAS. There was so much care in crafting the CCSS that some of your edits have damaged what was a precise, almost fragile structure, and this one I’ll call attention to today is more like the first that I mentioned in that it introduces a mathematical error into your framework.

It’s June, so it must be time to talk about Statistics and Probability, and I want to focus on 6.SP.2. This is one of the standards where you added some exposition to make the standard more clear and explicit for teachers. While I think this was a good move, the changes you made here belie a misunderstanding of statistics that has become really embedded into the curriculum by decades of instruction based on a fundamentally wrong mental shortcut endemic to grammar school statistics. You added two parenthetical explanations to the CCSS text, clarifying that a distribution “can be described by its center (median, mean, and/or mode), spread (range, interquartile range), and overall shape.”

Comparing this language with what CCSS already had in 6.SP.5c – “… quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation) …” – in this case, you replaced “mean absolute deviation” with “interquartile range”, which is fine, but you left the note on measures of center untouched.

Why, then, did you include “mode” in 6.SP.2? Mode is not a measure of center. This is a common misconception that needs to be eradicated.

Let me say it again:

Mode is not a measure of center.

Consider the set of ten numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 9. The median is 5.5 (right in the center). The mean is 5.4 (again, right near the center). The mode is 9 (way out at the end).

Suppose the second 9 in that set now becomes a 1. The new median is 4.5 (right in the center, not a big change). The mean is 4.6 (again, not a big change). And the mode is now 1. How can we claim that the mode is in any way the center of the data set if it’s jumping from one end to the other based on a change in one value?

And now suppose the data set is: 1, 1, 2, 3, 4, 6, 7, 8, 9, 9. Now our mean and median are both 5, and our mode is … well we have 2 modes – 1 and 9. How can a so-called measure of center take on two values?

Granted these are contrived sets of numbers and not actual data, but the fact remains that Mode is not a measure of center. It just happens to begin with an M. If it was called something else, we probably wouldn’t have this problem.

Mode is a better tool for telling us about the overall shape of the data, which is elsewhere in 6.SP.2. Think of our first list of numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 9). The middle is around 5 and it’s big at the high end. We surely aren’t worried about Grade 6 students getting into skewness and kurtosis, so this way of talking about shape could be helpful.

Mode isn’t useless. (Though I’ll note it doesn’t appear in the CCSS at all.) It’s just not a measure of center.

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Functions and/or Algebra

Originally published 25 May 2018

Last year, I was working with other teachers from across the state to identify and then classify the different learner outcomes we would want students to attain in K-12 mathematics. One of the biggest debates we had was over how to organize Functions and Algebra. Is there one category of Functions and Algebra? Or two separate domains – one for Functions, and another for Algebra?

I argued for the latter. I see functions as they are introduced in Grade 8 as the beginnings of what will grow into mathematical analysis and the calculus, rather than as an extension of algebra.

In the Grade 8 Functions domain, there is a lot of overlap with the horribly misnamed (more on this another time) Expressions and Equations domain, but the two paths diverge in high school and beyond.

At its heart, Algebra makes statements about numbers and the structure of number systems. Functions are about an input-output relationship between quantities. Algebra is more static while Functions are more dynamic. (Functions also start to hint at the infinite when you start to think about their link to the calculus.)

The first Functions cluster – Define, evaluate and compare functions – contains an understanding that seems to be overlooked over and over again by middle school students: “The graph of a function is a set of ordered pairs consisting of an input and the corresponding output.” When I ask students “what does that graph have to do with that equation?” They struggle to make the connection that the graph is the map of the points that make the equation true. Perhaps by thinking of the graph as a map of inputs (the x value) and outputs (the y value, which is to say, the mx+b value), they might better understand this relationship

This line of thinking is the connexion that unifies Functions with Algebra. In fact, in “Catalyzing Change”, NCTM’s newest publication outlining recommendations for improving high school math outcomes, the authors identify key concepts in Functions, in Algebra, and in Connecting Functions to Algebra. I think this might be the best way forward. Maybe.

The fragility of CCSS, Part 2

Originally published 11 May 2018

Oh, Massachusetts.

We need to talk again.

Let’s dig in to 6.NS.4.

You made a little change here that I think did something terrible.

Here’s what the CCSS had in this slot:

  1. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

And here’s what you did:

  1. Use prime factorization to find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two relatively prime numbers. For example, express 36 + 8 as 4 (9 + 2).

 

Let’s play spot the differences.

In fact, there’s only one difference – the first 5 words of your standard. But this change doesn’t make sense and fundamentally ruins this standard.

First of all, the standards are rarely prescriptive about how a task should be carried out. We saw the direction to use long division in 7.NS.2 earlier, and I was critical then. But in that case, there was a very good reason for using long division – it is the best, if not only, way to generate decimal equivalents by hand. Here, prime factorization is rarely the best strategy.

To use the example provided – which relies on the GCF of 36 and 8 – I could find the prime factorization of 8 (2^3, not too bad) and 36 (2^2\cdot 3^2, slightly more cumbersome) and compare exponents of respective primes to find the GCF is 4 = 2^2. Or I could list factors of the one-digit number 8 (1, 2, 4 and 8) and check to see if they are factors of 36 (yes, yes, yes and no) to see that 4 is the greatest common factor. This second method seems quicker.

Yes, with larger numbers, especially numbers with many factors (70 and 96 might be a good example), using factorization might be a better strategy, but there are other strategies that should be encouraged depending on the problem.

Also, using prime factorization to find LCM when we are restricted to numbers less than 12 is a bit excessive.

I get it. I mean there were probably a lot of teachers wondering where Prime Factorization fit into the standards, so you made explicit note of it where it fit best. I’m not arguing with that. I’m not happy about how it was linked. You’re also missing out on a lot of what Prime Factorization can do as a foundation for other work in number theory (answering questions like “how many factors does 120 have?”). I’d argue that, if you really wanted to reintroduce prime factorization into the standards, put it in Grade 7 Number System or Grade 8 along with exponents in Expressions and Equations, or even the High School Number and Quantity domain.

But not here.

Quick Follow-Up

Originally published 28 April 2018.

I’ve been away at NCSM this week and I don’t have time for a full post, but I wanted to put together a quick follow-up to my last note on ratios.

I sat in on a session on Ratios and Proportional Reasoning at NCSM on Tuesday. One of the points that Phil Daro made in the session was that the standards don’t call out Proportions, but Proportional Reasoning. This is a subtle distinction, but one that he said was purposeful.

“What is a proportion?” He asked. The answer came that it’s an equation that states two ratios are equivalent. Well, if that’s the case, we’re abusing the = sign, since A = B is defined for numbers A and B, and as we said last time, ratios are not numbers – unit rates are. Daro asked why not just call it an equation? Then instead of students learning a schema for solving proportions separately from a schema for solving equations, they would be able to transfer one understanding to the other.

Ratios and Unit Rates

Originally published 12 April 2018.

In my first year of teaching, while working towards my professional certification, I remember sitting in a seminar with other first-year teachers while a Grade 6 teacher explained, at length, the difference between a ratio and a rate. It was a distinction I had never thought about nor heard made before. (I must admit my recollection of middle school math is rather fuzzy – I remember a tan Pre-Algebra book with a shell on the cover, but that’s about it.) His distinction had something to do with units or something. The definition didn’t stick with me, except that I still recall thinking that the distinction seemed more like a case of school math technicalities rather than a true, meaningful mathematical understanding.

A quick check of a textbook’s glossary (I’ll refrain from naming the book, but it rhymes with “No Bath”) explains that a ratio is “a comparison of two quantities by division” and that a rate is “a ratio that compares two quantities measured in different units.” Puzzlingly, the examples for ratio include “12 to 25” and “12:25.” Last time I checked, “to” is not a division operator, but I digress, as I am wont to do. This seems an arbitrary and capricious distinction, but it’s indeed a good definition for a teacher looking to dock a quick point or two on a technicality.

Flash forward a few years to the CCSS era. The Grade 6 Ratios and Proportional Relationships domain also used the words ratio and rate to apparently mean different things. Here, though, the distinction is not between the inclusion or lack of units, but between two fundamentally different types of mathematical things. A ratio is a relationship between two quantities. It is not a comparison, but the relationship itself. It does not rely on division, and 6.RP.1 quite purposefully avoids any division or fraction notation, instead featuring “2:1” as an example. By contrast, 6.RP.2 defines a unit rate (note: not a rate, but a unit rate) as a number associated with a ratio. The unit rate is a fraction (such as 5/3) that models a ratio (such as 5 to 3). There is no attention to units in either 6.RP.1 or 6.RP.2. The third standard in the cluster calls for students to use “ratio and rate reasoning” to solve problems, almost using “ratio and rate” as one big idea, as it should.

The fragility of CCSS, Part 1

Originally published 30 March 2018.

This was the first of what became a series of posts examining how states have gone astray in making adjustments to the standards. The original title did not include “Part 1.”

Okay, Massachusetts. We need to talk.

You’ve become a national leader in K-12 education, so much so that your earlier curriculum framework was used as a point of comparison against which the CCSS was judged when it was first published. It was a bold move, as a state, to adopt CCSS at a time when your performance was so far exceeding other states, and your success has prompted Rhode Island to abandon PARCC and adopt MCAS as its accountability metric in elementary and middle school.

It has come to my attention that you, like a lot of states, made some editorial changes to the CCSS when you built your revised 2017 Mathematics Curriculum Framework. In doing so, you certainly strengthened some of the standards through elaborations and examples that were lacking in the original. However, there are a few standards where your changes are … shall we say … ill-advised?

Let’s start, today, with 7.G.4. The CCSS standard reads:

  1. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Your revision reads as follows:

  1. Circles and measurement:
    1. Know that a circle is a two-dimensional shape created by connecting all of the points equidistant from a fixed point called the center of the circle.
    2. Understand and describe the relationships among the radius, diameter, circumference and circumference of a circle.
    3. Understand and describe the relationship among the radius, diameter, and area of a circle.
    4. Know the formulas for the area and circumference of a circle and use them to solve problems.
    5. Give an informal derivation of the relationship between the circumference and area of a circle.

Now let’s dive into this. Your new 7.G.4a (listed as 7.G.4.1 above, still learning the formatting) is my biggest point of contention, so I’ll leave that for last.

It took me a few readings to untangle 7.G.4b7.G.4c and 7.G.4d. Reading b and alone, it looks like you’re trying to avoid using the word formula in defining an learning outcome for students, which I applaud. Except that then your sub-standard d makes a complete turnaround (in Massachusetts parlance, dare I say “bangs a U-ey”?) and doubles down on the knowledge and use of the formula. My best guess is that you’re trying to motivate an understanding beyond just the formula, but I’m not sure what that could be. I mean, what other relationship is there among the radius, diameter and area of a circle? These three sub-standards are needlessly complicated, in my opinion, showing a construction that I can’t tie to anything else in the standards at any grade.

Now 7.G.4e is a simple restatement of what’s in the CCSS standard, so I’ll come back to 7.G.4a. You’re attempting to formally define a circle within the standards. I think it’s a good idea to require students to know a formal definition of a circle at this grade level, but this definition is not correct. What does it mean to “[connect] all of the points”? If I choose one of these points, to which other points am I connecting it? Am I connecting to the “adjacent points” along the circumference of the circle or am I connecting it to every single point in the set, including connections across the circle as well?

And talking of “adjacent points” opens a new can of worms. How far apart are these adjacent points? Since there’s arbitrarily many of them along the circumference, they’re arbitrarily close together. Well then how do I connect points that are already essentially touching? (Stillwell’s infinity rears its ugly head again!)

Elsewhere in the standards, students are asked to know definitions, but there are very few definitions presented in the standards themselves. Why not reduce a to “Know the formal definition of a circle [as a specific set of points].”? If you’re attached to the language in b–d, that’s fine, even though I think it’s excessive. E is fine.

I hate Pi Day

Originally posted 14 March 2018.

There. I said it. I hate Pi Day.

No, I’m not one of those Tau Day fundamentalists (though while I do think that Tau is a better constant than Pi), I just have real problems with how Pi Day is handled in schools.

For starters, pi first appears (implicitly) in the standards in 7.G.4. (I have a lot more to say about the geometry in this standard, but I’ll come back to that next time.) I think that a lot of caution needs to be used around pi in Grade 7 because irrational numbers aren’t fully explored until 8.NS.1. Remember from last time that the idea of repeating decimals is explored in Grade 7, so there’s at least some hooks for talking about irrational numbers around 7.G.4, but any work being done to introduce pi for Pi Day prior to Grade 7 or 8 may lead to more misconceptions or misunderstandings that would need to be remediated later.

I have a bigger problem with what tends to become the focus of work on Pi Day – low-level tasks such as memorization of numbers or formulas and other simple representations of basic concepts. So often we search for higher-rigor tasks in math, and exploring pi lends itself to so many options – a Pi/Tau debate, research projects into unsolved questions about pi, unexpected connections between pi and other areas of math, etc.

A quick aside – Doug Clements makes the argument that preschool students who are verbal counters are not simply rote counting – that there is an understanding even at that early age that there is some quantitative meaning baked into the number sequence. Contrast that with rote chanting of the alphabet – the order of the alphabet is arbitrary and traditional, not based on any particular rule beyond “it’s alphabetical order,” to make a circular argument. Memorizing the digits of pi, I’d argue, falls closer to this idea of rote learning rather than learning with meaning.

Pi Day’s focus on the irrationality of pi is also troubling. As I said last time, looking at irrationality implies some treatment of the infinite. Even rejecting Stillwell’s argument that the infinite implies a more advanced math, treating infinity or infinite decimal expansions is a concept well out of reach of many students, and can’t be a one-day diversion.

Pi (or Tau) is a thing of mathematical beauty. I’d rate it either the third or fourth most important number in the universe (behind 0, 1 and possibly e). It’s dangerous to address it in such a throwaway fashion.

Infinity rears its ugly head

Originally published 2 March 2018.

Last time, I recommended a book – George and Velleman’s “Philosophies of Mathematics”. I probably should have cited the authors more explicitly. I’d like to recommend another text this time – Stillwell’s “Elements of Mathematics.” It serves as a survey of elementary mathematics, with an aim of delineating the difference between elementary and advanced mathematics.

Stillwell argues that the boundary between the two lies in the use of the infinite – that once infinity enters the discussion, the topic begins to enter the realm of advanced mathematics. I don’t necessarily fully agree with him – I’d argue that the nature of reasoning and/or the level of abstraction must be considered as well, but that’s a much deeper topic to explore. That said, if we accept Stillwell, then we may be surprised to find that the infinite first appears as early as middle school.

Now, the only form of the word infinite to appear in the standards is “infinitely”, in the context of infinitely many solutions to an inequality (6.EE.8) or a system of equations (8.EE.7). But there is a hidden infinity lurking in 7.NS.2d, where students are to “[c]onvert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.” (emphasis added)

This is one of the last computation/algorithm-heavy standards in the CCSS. I’m no big fan of a standard that demands a particular computational approach, but I understand why it is called out here for students to use long division to accomplish this task. This standard asks students to really lean into Mathematical Practice 8Look for and express regularity in repeated reasoning.

This standard cements a theme in development since Grade 4 – that decimals serve as a notation for fractions. Here, students look for patterns in how specific fractions relate to decimal representations. It is not simply an anchor for the “Fraction – Decimal – Percent” conversion table worksheets of old. While, yes, for better or worse, there is an expectation that students carry out divisions to convert fractions to decimals (though notably not in reverse), the central understanding of the standard lies in the patterns that students are asked to find.

Ironically, this is one of the last appearances of positional notation for numbers, save for Scientific Notation in Grade 8. This idea that certain rational numbers form repeating decimals, whereas others terminate, is not a property of the numbers themselves, but of the way that they are written in Base Ten. One-half, for example, terminates in Base Ten, but would be written as 0.111111… in Base Three. But we’re slipping into New Math, so I’ll stop here.

Expanding the number system

Originally published 23 February 2018.

One of my favorite math courses in college was a course on the Philosophy of Mathematics. It was co-taught by a math professor and a philosophy professor, and, in true liberal arts fashion, was much more about the Why? of math rather than the What? or How?. (The professors co-authored the textbook used in the course, “Philosophies of Mathematics,” which I highly recommend. It was probably the one time in my post-secondary experience that a course using a text authored by the instructor was not an executive-level cash grab slash disaster, but I digress.)

One of the themes of the course was the formal motivation and construction of the real numbers. While that concept goes beyond the scope of middle school math, there are some interesting implications of that construction that impact the Grade 6 Number System standards.

The Number System conceptual category is a bit of a jumble, especially in grade 6. It is capstone-ing a lot of the work with computation from the elementary grades, including both whole-number fluency and division of fractions, and it starts laying a foundation for work with signed numbers.

But, while fractions were developed gradually over many years (with foundations beginning in Grades 1 and 2, and a more formal focus in Grades 3–5), signed numbers are handled, for the most part, only in grades 6 and 7. How is this possible? There are just as many operations to understand with rational numbers as there were with fractions, and there are just as many potential pitfalls.

Well here’s where the philosophy of mathematics comes in. What purpose do fractions serve? The Grade 3 standards try to put the spotlight on the fact that they fill in gaps in the number line between whole numbers. While this is true, it is horribly misleading. What’s more, numbers less than zero serve that same purpose – they expand the number line, this time in a different direction rather than between numbers. (I keep wanting to say “complete,” but that will have to wait until we get to the irrational numbers. Wait a couple of weeks.)

The real use of fractions is to be able answer division problems that don’t have whole number answers. We can find 10 ÷ 2 or 8 ÷ 2 on our Grade 2-style number line, but to find 9 ÷ 2, we need to add fractions to our number line. The standards, however, withhold this application of fractions until Grade 5, since division itself is introduced in Grade 3 alongside fractions.

Analogously, the real use of signed numbers is to be able to answer subtraction problems that don’t have positive answers. We can find 5 – 3 on our elementary number line, but elementary students confronted with 4 – 7 will likely say “we can’t do that.”

Ironically, subtraction is the more fundamental operation than division, yet “fixing” subtraction by introducing negative numbers comes years after “fixing” division. This is backwards from a purely mathematical standpoint, but it is not inexplicable. For one thing, fractions are a lot more useful in modeling the real world than are negative numbers, so even though they may be more mathematically sophisticated (if only in that division is a more sophisticated operation than subtraction), they are more commonplace and accessible to elementary students.

I’m not sure that there’s a way around this. The dominant European approach is to postpone fractions until later and work with negative integers earlier. This forces a greater deal of formal abstraction on younger students, and, while it may work, it would be a very hard sell to make this change domestically.

But is it really gone?

Originally published 9 February 2018.

Last time, I focused on the big shift in how congruence and similarity are conceived in the CCSS. Gone, I said, was the Euclidean approach, in favor of something that more closely resembles Coxeter, and is probably better connected to other branches of mathematics (transformations form a group, congruences are equivalence classes, &c).

But then there’s 7.G.2, wherein students are to “Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on construction triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.” It would seem we’ve come right back to Euclid and the Greeks with the attention to such constructions. While I’m no big fan of this standard and how it is presented, the situation here is not as firmly in contrast with the shift in grade 8 as it may at first seem.

Notice, to begin with, we have a focus on ruler and protractor rather than compass and straightedge. We are not trying to emulate the Greeks with construction by limited postulates, but to explore relationships through inquiry. My first criticism here, though, is on the explicit reference to ruler and protractor and to technology. I know that it is necessary to state expectations specifically, but here I ask why other concrete tools are discounted from the standard. Experimentation with rods of given lengths or pre-constructed angles could give students another entry point towards the key concepts in the standard.

This is also the presentation of the triangle inequality within the standards, a result that students will likely stumble upon whilst experimenting with constructing triangles based on given side length measures. The triangle inequality itself is one of those sneaky but important hooks for later math (much later in this case), so I’m not sure why it is only part of a standard here instead of a separate standard that stands on its own. It offers some low-hanging fruit in creating opportunities for students to explain and talk through their mathematical reasoning, but it also can turn into a pretty low-level, right-or-wrong item on a standardized assessment.

As I said, I’m not a fan of this standard, at least not the way it’s presented. There’s a lot of math packed into here, and I get the sense that it suffered from an effort to find some economy of language to keep the standards from seeming too bloated. But I’d almost argue that hidden substance is worse than apparent bloat.