Here is the new version.
]]>And here are two documents outlining the major changes from previous versions.
Note that the University of Arizona has deleted the website that used to host the progressions. I am working to have that url redirect here.
]]>8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on a straight line.
The comma indicates that the clause “whose graph is a straight line” is nonessential for identifying the noun phrase “linear function.” It turns the clause into an extra piece of information: “and by the way, did you know that the graph of a linear function is a straight line?” This fact is often presented as obvious; after all, if you draw the graph or produce it using a graphing utility, it certainly looks like a straight line.
When I’ve asked prospective teachers why this is so, I’ve gotten answers that look something like this:
We know that a linear function has a constant rate of change, $m$. If you go across by 1 on the graph you always go up by $m$, like this:
So the graph is like a staircase. It always goes up in steps of the same size, so it’s a straight line.
This is fine as far as it goes. It identifies the defining property of a linear function—that it has a constant rate of change—and relates that property to a geometric feature of the graph. But it’s a “Here, Look!” proof. In the end it is showing that something is true rather than showing why it is true. Which is to say that it’s not a proof.
Still, the move to a geometric property of linear functions is a move in the right direction, because it focuses our minds on the essential concept. We all know that any two points lie on a line, but three points might not. What is it about three points on the graph of a linear function that implies they must lie on a straight line?
Because a linear function has a constant rate of change, the slope between any two of the three points $A$, $B$, and $C$ is the same. So $|BP|/|AP| = |CQ|/|AQ|$, which means there is a scale factor $k =|AQ|/|AP| = |CQ|/|BP|$ so that a dilation with center $A$ and scale factor $k$ takes $P$ to $Q$, and take the vertical line segment $BP$ to a vertical line segment based at $Q$ with the same length as $CQ$. Which means it must take $B$ to $C$.
But (drumroll) this means that there is a dilation with center $A$ that takes $B$ to $C$. Dilations always take points on a ray from the center to other points on the same ray. So $A$, $B$, and $C$ lie on the same line.
I don’t really expect students to get all of this, at least not right away. I’d be happy if they understood that there is a geometric fact at play here; that seeing is not always believing.
]]>
8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on a straight line.
I could write a whole blog post about that comma in the first sentence, but for now I want to focus on the question of what exactly are the student expectations entailed by this standard. Certainly students should be able to recognize that $y = mx + b$ defines a linear function; and they should be able to show a function is not linear by finding points on the graph with different slopes between them. (By definition, a linear function is one with a constant rate of change, that is, a function where the slope between any two points on its graph is always the same.) However, the following PARCC released item suggests the possible expectation that students be able to tell if a function is linear or not purely from looking at its defining equation.
My guess is that the writer of this item was thinking that students would detect non-linear functions by noticing the functions with squared and cubed terms (they could also use the point method but that doesn’t seem likely). That’s not a good path to lead them down. Sure, $y = 5-x^2$ is not linear. What about $y = 5 – x^2 + (x+1)^2$? By the look of it it is even more not linear! But of course it is in fact linear, because the expression on the right is equivalent to $2x + 6$. The grade 8 standards don’t expect students to be making such simplifications as expanding squares of binomials.
This is another example of the confusion between expressions and functions. The expression $(x+1)^2 – x^2$ has non-linear terms in it, but the function it defines is linear because it is equivalent to $2x + 1$. Equivalent expressions define the same function.
There’s another confusion revealed in this item, the confusion between equations and functions. Look at option C. Is it intended to be a distractor? Will a student who chooses it be marked wrong? Such a student would have a case for protest on the grounds that $-3x + 2y = 4$ is not a linear function because it is not a function, it is an equation. Certainly if you choose to think of $x$ as the input and solve for $y$ to get the output you can think of it as a function, which would indeed be linear. You could also go the other way around and choose $y$ as the input and get a different linear function. It is conventional when $x$s and $y$s are floating around to think of $x$ as the input and $y$ as the output, but you can flout convention without being mathematically incorrect.
The moral of this story is, I suppose, that it is easier to tell when a function is not linear than to tell when it is linear. Testing for non-linearity involves just picking a few points on the graph; testing for linearity involves picking every possible pair of points on the graph and verifying that the slope between them is always the same. It’s instructive to do this with $y = mx +b$: if $(x_1,y_1)$ and $(x_2,y_2)$ are two different ordered pairs satisfying this equation, then $$\frac{y_2-y_1}{x_2-x_1} = \frac{{mx_2 + b} – (mx_1 +b)}{x_2-x_1} = \frac{m(x_2-x_1)}{x_2-x_1} = m.$$
OK, next week I’ll write a blog post about the comma.
]]>In the course of working on an article with the same title as this blog post for a publication about Felix Klein, I did a Google Image search on the word “function,” with the following results.
I find this fascinating for a number of reason. First, notice the proliferation of representations: graphs, tables, formulas, input-output machines, and arrow diagrams (those blobs in the second row). A corresponding search in a French, German, or Japanese gives a very different result. Here is the Japanese one (where I searched google.co.jp for “関数”).
This has a much greater focus on graphs, and no arrow diagrams. The French and German ones are even more focused on graphs; a search in Spanish, however, gives results similar to English.
What does all this mean? Well, I’ll be discussing this more in the article I’m writing, but for now I have just a couple of observations.
First, the English language search reveals a preoccupation with examples of relations that are not functions, with examples of graphs, arrow diagrams, and tables. I have always found this preoccupation perverse: the examples are artificial, and there is very little to be gained from them except an ability to answer questions about them on tests. This preoccupation is not evident in the searches in other languages.
Second, take a look at these two representations, from the English and Japanese language searches:
.
Together they represent a case study in designing representations. Can you see what’s wrong with the one on the top? A function is supposed to have only one output for every input. Does an apple slicing machine have that property?* The representation on the bottom, on the other hand, clearly represents how to think of addition as a function with two inputs.
I’m interested to hear readers’ thoughts on the representations that come up in these and other searches. Maybe someone who can navigate Baidu can tell us what the Chinese results are.
*To be fair, the authors of the web page with the apple slicing machine are clearly aware of the problem. But their contortions to get around it only emphasize the fundamental flaws in the representation.
]]>A number of years ago there was a popular piece by Alison Blank titled Math is not linear, which gave a number of ideas about the order in which we teach mathematics. A curriculum writer has to grapple with the fact that, although math is not linear, time is. Hermione Granger’s time turner does not actually exist. Tuesday comes after Monday, and Tuesday’s lesson comes after Monday’s lesson and, in the end, a teacher has to decide what to teach on each day; that is, they have to decide on a linear order in which to teach mathematics. The gist of “Math is not linear” is that that order need not be a dry march through a logical hierarchy of topics. You can, as Blank says, go on tangents, foreshadow topics to come, connect back to previous topics, and give students problems that create a need for a new topic. These are all great ideas.
Our question is: what other ideas do people have to make sure that the sequence of lessons in a course makes sense to students and makes sense mathematically? Do you recommend any books or articles that might help answer these questions? We have some ideas and will be writing some posts about them, but want to hear from the community as well. Please feel free to share your thoughts in the comments, or on Twitter with @IllustrateMath, #timeislinear.
]]>
In our earlier posts, we argued that diagrams can help students see the structure of a problem and understand why it can be represented by division. However, diagrams are rarely efficient for carrying out the resulting fraction division. For example:
Mateo filled a 1 pint measuring cup with water until it was $\frac{7}{16}$ full. If a recipe calls for $\frac23$ pints of water, what fraction of the recipe can Mateo make with the water in the measuring cup?
Drawing a diagram for this problem is not the most efficient method (try it!). A student who has learned to see this as $\frac{7}{16}\div \frac23$ (through working with diagrams) would most efficiently calculate that value using the invert-and-multiply rule without worrying about a diagram for that particular calculation.
Last time we argued that the “How much in one group” interpretation with the right kind of diagram can help us see why dividing by a fraction is the same as multiplying by its reciprocal.
For example, a diagram that represents a situation where $\frac25$ of a number is $1\frac34$ can show that we can multiply $1\frac34$ by $\frac52$ to find that number.
What if we were to think about this from a completely algebraic perspective? By the definition of division,$$1\frac34 \div \frac25 = x$$ means that: $$\frac25 x = 1 \frac34.$$
To solve an equation like this, we simply multiply both sides of the equation by the multiplicative inverse of $\frac25$:
$$\frac52 \cdot \frac25 x = \frac52 \cdot 1 \frac34.$$
In other words: $$ x =1 \frac34 \cdot \frac52.$$
We are not claiming that students need to be able to make a formal argument like this in order to justify the general rule for dividing fractions! But they do, eventually, need to be able to solve specific equations of the form $$\frac25 x = 1 \frac34.$$
Students who can solve equations flexibly might find the solution by rewriting an unknown factor problem as a division problem: $$x = 1\frac34 \div \frac25,$$ or by multiplying both sides of the equation by the reciprocal of $\frac25$: $$\frac52 \cdot \frac25 x = \frac52 \cdot 1 \frac34.$$
Both methods were implicit in many of the fraction division problems students have been conceptualizing with the help of diagrams, although there may not have been an explicit equation in those problems. Using equations formalizes, makes explicit, and encapsulates the implicit understandings. So students who investigate fraction division with diagrams should have the opportunity to make connections to algebraic approaches as well.
Fraction division is a topic that students encounter at a key time in their transition from their work in elementary school arithmetic to their study of algebra as generalized arithmetic in middle school and beyond. Appropriate use of diagrams can help them understand how fraction division relates to their earlier study of division of whole numbers and when a problem can be represented by fraction division. Diagrams can also mediate students’ transition to a more structural, abstract understanding of fraction division that is represented using numeric and algebraic expressions and equations. In general, diagrams can play a key role in helping students make the transition from arithmetic to algebra, as we have illustrated in the particular case of fraction division.
]]>The first of these diagrams is more familiar to students because it reflects their past work, but the second is more productive for understanding “dividing by a unit fraction is the same as multiplying by its reciprocal.”
Why is the first one more familiar? In grades 3 and 4, students study both the “how many in each (or one) group?” and “how many groups?” interpretations for division with whole numbers (see our last blog post for examples). In grade 5, they study dividing whole numbers by unit fractions and unit fractions by whole numbers. But, as we mentioned in that post, in grade 5 the “how many groups?” interpretation is easier when dividing whole numbers by unit fractions because students do not have to worry about fractions of a group. Going from $3 \div \frac12$ to $1\frac34 \div \frac12$ using this interpretation feels fairly natural:
The main intellectual work here is seeing that $\frac14$ cup is $\frac12$ of a container, but because the structure of the problem is the same and that structure can be easily seen in the diagrams, students can focus on that one new twist. The transition also helps students see that “how many groups” questions can be asked and answered when the numbers in the division are arbitrary fractions.
So the “how many groups” interpretation is useful for understanding important aspects of fraction division and has an important role in students’ learning trajectory. It enables students to see that dividing by $\frac12$ gives a result that is 2 times as great. But it doesn’t give much insight into why this should be the case when the dividend is not a whole number.
The “how much in each group” interpretation shows why. Here are diagrams using that interpretation showing $3 \div \frac12 = 2 \cdot 3$ and $1 \frac34 \div \frac12 = 2 \cdot 1 \frac34$.In fact, the structure of this context is so powerful, we can see why dividing any number by $\frac12$ would double that number: $$x \div \frac12 = 2 \cdot x = x \cdot \frac21$$
This is true for dividing by any unit fraction, for example $\frac15$:In the diagram above, we can see that $1\frac34$ is $\frac15$ of a container, so a full container is $1\frac34 \div \frac15$. Looking at the diagram, we can see why it must be that the full container is $5 \cdot 1 \frac34 = 1 \frac34 \cdot \frac51$.
With a little more work to make sense of it, we can use this interpretation to see why we multiply by the reciprocal when we divide by any fraction, for example $\frac25$:In the diagram above, we can see that $1\frac34$ is $\frac25$ of a container, so a full container is $1\frac34 \div \frac25$. We can see in the diagram that $\frac12$ of $1\frac34$ is $\frac15$ of the container, so our first step is to multiply by $\frac12$: $$1\frac34 \cdot \frac12$$
Now, just as before, to find the full container, we multiply by 5:
$\left (1\frac34 \cdot \frac12 \right) \cdot 5 = 1\frac34 \cdot \frac52$
This shows that dividing by $\frac25$ is the same as multiplying by $\frac52$!
There is nothing special about these numbers, and a similar argument can be made for dividing any number by any fraction. Now students, instead of saying “ours is not to reason why, just invert and multiply,” can say “now I know the reason why, I’ll just invert and multiply.”
Next time: Beyond diagrams.
]]>If we say that $a \times b$ means $a$ groups of $b$, then
[Pause here and see if you can come up with a “how much in one group?” story problem for $1 \frac34 \div \frac12$.]
How do these two interpretations of division come into play as students learn about fraction division? In grade 5, students solve problems like $6\div \frac12$ and $\frac12 \div 6$. What’s nice about problems involving a whole number divided by a unit fraction or a unit fraction divided by a whole number is that we can think of them using the same structure that we thought about division of whole numbers.
Notice that these are both a “how many groups?” division problem, and because there is always a whole number of unit fractions in 1, the solution will be a whole number of groups (so students do not have to worry about fractions of a group). If we write equations to represent these problems, that can also help us see the structure:
$$? \times 2 = 6$$
$$? \times \frac12 = 6$$
Notice that these are both a “how many in each (or how much in one) group?” division problem, and students don’t have to worry about fractional groups because the whole number in the problem is the number of groups.
Again, with equations:
$$3 \times ? = 6$$
$$3 \times ? = \frac12$$
So in grade 5, students can build on their understanding of whole number division without having to grapple with fractional groups, so long as they understand both of these interpretations of division.
In grade 5, students also learn about fraction multiplication, so they do encounter fractions of a group, but they are not required to put these two understandings together until grade 6 when they extend their understanding of division to all fractions. This provides some scaffolding for students on their way to understanding division of fractions in general.
Let’s look at these two interpretations of division for $1\frac34 \div \frac12$.
Here are two possible diagrams to represent these two interpretations of division:
Next time: how the different interpretations of division and diagrams can be used to understand the “invert and multiply rule” and other approaches to understanding this procedure.
]]>Conceived initially as a project at the University of Arizona to illustrate the standards with carefully vetted tasks, IM has grown into a not-for-profit company with 25 brilliant and creative employees and a registered user base some 40,000+ strong. Our partnership with Open Up Resources (OUR) to develop curriculum started a little over 2 years ago when we submitted a pilot grade 7 unit on proportional relationships to the K–12 OER Collaborative, as OUR was then known. In the fall of 2015, not understanding that it couldn’t be done, we agreed to write complete grades 6–8 curriculum ready for pilot in the 2016–17 school year.
One of the things I love about the curriculum is the careful attention to coherent sequencing of tasks, lesson plans, and units. The unit on dividing fractions is an example, appropriate to mention in the middle of this series of blog posts with Kristin Umland on the same topic. It moves carefully through the meanings of division, to the diagrams that help understand that meaning, to the formula that ultimately enables students to dispense with the diagrams. It illustrative perfectly our balanced approach to concepts and fluency. Kristin and I will be talking about that more in the next few blog posts.
]]>
Write a story problem for $1 ¾ \div ½$.
[Pause here and think about the answer yourself.]
Many people find it hard to come up with a story problem that represents fraction division (including many math teachers, engineers, and mathematicians). Why is this hard to do? For many people, their schema for dividing fractions consists almost entirely of the “invert and multiply” rule. But there is much more to thinking about fraction division than that. So much in fact, that we can’t say it all in a single blog post. This is the first of several musings about fraction division.
Consider this problem:
If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?
You probably know instantly that this is a division problem and that the answer is 6, because you know your times tables, and specifically you know that $2 \times 6 = 12$. If we say that $a \times b$ means $a$ equal groups of $b$ things in group, then a division problem where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” problem. Here are some other questions that ask “how many groups?”
Some people think that the last one feels like a trick question because you can’t even fill one completely. Because we know the answer is less than one, we could also ask it this way:
So a division problem that asks “how many groups?” is structurally the same as a division problem that asks about “how much of a group?”, but because of the way we speak about quantities greater than 1 and quantities less than 1, the language makes the structure harder to see.
What other ways might we see the parallel structure?
Diagrams:
Equations: $$? \times2 = 12, \quad ? \times \frac14 = 1\frac12, \quad ? \times \frac34 = 1\frac14, \quad ? \times 1\frac14 = \frac34.$$ The diagrams don’t have the language problem. In all cases the upper and lower braces show the relation between the size of a container and the amount you have. Whether a whole number of containers can be filled (diagrams 1 and 2), a container plus a part of a container can be filled (diagram 3), or only a part of a container can be filled (diagram 4), the underlying story is the same.
Many people think of diagrams primarily as tools to solve problems. But sometimes diagrams can help students see structure or reveal other important aspects of the mathematics. This is an example of looking for and making use of structure (MP7).
The equations have an even clearer structure, but more abstract. They all have the structure $$\mbox{(quantity of containers)}\times\mbox{(size of a container)} = \mbox{(how much you have)}.$$
The intertwining of the abstraction of the equations and the concreteness of the diagrams is a good example of MP2 (reason abstractly and quantitatively).
Coming up next week: what else are diagrams good for?
]]>Parentheses, and order of operations, tell us how to read the meaning of an expression, how to parse it, not what to do with it. In the expression above, every matched pair of parentheses contains something of the form $$
(\mbox{blob}) * (\mbox{another blob}), \qquad (\mbox{where * stands for $+$, $-$, or $\times$}),
$$ unless the blobs are just numbers or letters, in which case we don’t surround them with parentheses. We always know exactly what things we are adding, subtracting, or multiplying. Starting with the outermost parentheses, we see it contains the sum of 2 and a blob. Looking inside that blob we see that it contains a blob minus another blob. And so on. The structure of the expression can be represented in a diagram:
So what is order of operations about, and why do we need it? Well, that’s a lot of parentheses up there, so it is useful to have some conventions about when things are understood to be a blob, without actually putting in the grouping symbols (blobbing symbols?). First, any sequence of multiplications and divisions is understood to be a blob (that’s the precedence of multiplication and division over addition and subtraction). Second, in a sequence of additions and subtractions, or of multiplications and divisions, you read from left to right. (Actually, there is disagreement about this last one in the case of multiplication and division, but never mind.) The first rule allows us to write the expression above as
$$
((3\times x\times x- 7\times x) + 2).
$$The second rule allows us to leave out all the remaining parentheses. And, of course, we have other conventions about representing multiplication by juxtaposition, and about exponent notation, which allow us to write
$$
3x^2 – 7x + 2.
$$
Calling it order of operations is problematic because it can be misconstrued as suggesting that there is a specific order in which you must perform operations. There isn’t, except insofar as you sometimes have to wait to perform an operation until you have calculated all the blobs in it. But, for example, there is no law that says you have to do the multiplications first in $101\times56-99\times56$ and, in fact, it is more efficient to factor out the $56$ and do a subtraction first. Order of operations tells us how to read this expression: it’s a difference of two products, not a product of three factors the middle one of which is a subtraction. But it doesn’t tell us how to compute it. The word “order” in “order of operations” is best understood as referring to order in the sense of hierarchy, as in the diagram above.
Outside of textbook school mathematics the order of operations is a matter of common law, not constitutional law, and it’s a bad idea to make a federal case out of it on assessments. For example, dinging a student for interpreting $x/2y$ as $x/(2y)$ rather than $(x/2)y$ would be unreasonable; many scientists would do the same thing. If there is any danger of ambiguity we should put the clarifying parentheses in.
A few final thoughts:
And, by the way, the url mathematicalmusings.org also points to this blog.
]]>But what happens with an equation like $3x + 2 = 3x + 5$? In this case, the hanger diagram is a physical impossibility: the right hand side will always be heavier than the left hand side. I can imagine that students who have an idea of an equation as “the left hand side is equal to the right hand side” might be confused by this situation, and think this is not a proper equation. Especially when they reduce it to $2 = 5$. Students learn to say that this means there are no solutions, but it’s hard to make sense of that response rule without understanding what’s really going on with equations.
The fact is, an equation with a variable in it is neither true nor false, because it is merely a phrase in a longer sentence, such as “If $3x + 2 = x + 5$ then $x = \frac32$.” This sentence is true, but the phrases within it are not sentences and have no inherent truth or falsity. When we perform the same operation on each side of an equation, we are not only preserving the truth of the equal sign but also preserving the consequences of the equal sign. If we use if-then language when talking about equations, then we can make sense of equations with no solutions. A sentence like “If $x$ is a number satisfying $3x + 2 = 3x + 5$ then $2 = 5$” makes perfect sense. It’s the mathematical equivalent of “If the moon is green cheese, then I’m a monkey’s uncle.” It’s a way of saying the moon is not green cheese . . . or that there is no solution to the equation.
The middle schooler’s version of if-then language might not always use the words “if” and “then.” You might say “Imagine there is a number $x$ such that $3x + 2 = x + 5$. What can you say about $x$?” Just as you say “Imagine this hanger is balanced and the green triangles weigh one gram. How much do the blue squares weigh?” I think it’s a useful approach with students to remember that equations are a matter not just of truth, but of truth and consequences.
]]>Trouble is, all this is really hard to explain to middle schoolers, so people invent contexts. One context I’ve seen has something to do with sending out bills. If you receive 5 bills for 3 dollars then you have $5 \cdot (-3) = -15$ dollars. Sending out is the opposite of receiving, so if you send out 5 bills for 3 dollars, you have $(-5)(-3)$ dollars. But once you receive payment, you have \$15. So $(-5)(-3) = 15$.
One problem with this is that you have to buy more conventions to believe it: the convention about negative amounts of money representing debt, the convention about negative receiving being kinda sorta like sending out. That’s a lot of conventions to prove something that is, as I said, a convention itself. Another problem is that all this context really shows is that $-(-3) = 3$, five times. The multiplication in this context is really just repeated addition; it doesn’t work for numbers that are not integers. You can’t send out 5.6 bills.
There is one context that I think does a better job here, and that is $\mbox{distance} = \mbox{speed} \times \mbox{time}$. This does work with non-integers, and you can make sense of all of the quantities involved as negative numbers. Let’s assume that an object is moving along the number line, and that you measure its position at different times, setting your stop watch to 0 when it passes through the origin. Negative distance is distance to the left; negative speed is speed from right to left; and negative time is time before you started measuring. (Later we use the terms displacement and velocity, but there’s no need to introduce them right away.)
So if the object is moving at $-5$ m/sec, where is it at time $-3$ seconds? Well, it’s moving from right to left and it has 3 seconds before it hits the origin, so it is 15 m to the right of the origin. So $(-5)(-3) = 15$.
Was I cheating there? Is this context subject to the same objections I made about the money context? Didn’t I just make up a whole bunch of conventions about negative distance, time, and speed? I think these conventions pass the cognitive sniff test better. They don’t seem as artificial to me. You can really make quantitative sense of negative distance, speed, and time. It feels more like the real world and less like an accountant’s convention. (No offense to accountants intended.) In a way, we have replaced the mathematician’s desire to have the properties of operations continue to hold with the physicist’s desire to have the laws of physics continue to hold.
So where is the distributive property in all of this? I think it is built into our physical intuition about this context. If I travel for 3 hours, and then for another 2 hours, I can figure out how far I have gone by just adding the times and multiplying by my speed, or I can add the distances traveled in each time period. That’s the distributive property. If you dig into the reasoning I gave for the object moving at $-5$ m/sec in the light of this common sense, questioning each claim, you end up with something not too far from the mathematical reasoning I gave earlier.
By the way, this is the approach we take in the Illustrative Mathematics middle school curriculum. Finding contexts for mathematical ideas that are faithful to the mathematics is difficult and requires real sensitivity to both the mathematics and the way students think. Our brilliant curriculum writing team is up to that challenge.
]]>The Common Core emphasizes this unity by treating decimals as just a different way of writing fractions, e.g. in 4.NF.C: “Understand decimal notation for fractions, and compare decimal fractions.” In this view, 0.3 is not a new sort of number, just a different way of writing the number 3/10.
This leads to some difficulties in the use of language, because at some points in the curriculum you do want to distinguish between decimals and fractions, for example when you ask a student to write 4/5 as a decimal or to write 0.125 as a fraction. (“You told me it’s already a fraction!” the smart student might reply.)
The IM curriculum writing team was talking about these difficulties the other day and Cathy Kessel had a useful comment:
There’s a developmental issue. When fractions are introduced, the distinction between number represented and representation is blurred, and similarly for decimals (finite, then repeating). But, when the two types of representations are seen as representing the same thing, then the thing and its representations start to separate more.
Because we want students to develop a conception of the number behind the representation, we start out saying decimals are also fractions. Later we build a negative addition to the number line and add the opposites of fractions. Once we have a robust conception of the number line, inhabited by rational numbers, we want to talk about different ways of expressing those numbers: fractions, decimals, infinite decimals, expressions involving square root symbols and exponents. So we start to distinguish between fractions and decimals, not as numbers, but as forms for expressing numbers. We initially suppress their role as forms in order to gain a robust conception of number; once they are firmly attached to that conception we can distinguish between them.
They only way to do this without giving multiple meanings to the same words would be to invent new words and be consistent in their use. This harks back to the distinction between “numeral” and “number” in the New Math, which didn’t take hold.
]]>3.OA.A.1. Interpret products of whole numbers, e.g., interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as $5 \times 7$.
The parent had every right to be upset: a correct answer is a correct answer. Comments on the post correctly pointed out that, since multiplication is commutative, it shouldn’t matter in what order the calculation interpreted the product. But hang on, I hear you ask, doesn’t that contradict 3.OA.A.1, which clearly states that $6 \times 3$ should be interpreted as 6 groups of 3?
The fundamental problem here is a confusion between ways of thinking and ways of doing. 3.OA.A.1 proposes a way of thinking about $a \times b$, as $a$ groups of $b$. In other words, it proposes a definition of multiplication. It could have proposed the other definition: $a \times b$ is $b$ groups of $a$. The choice is arbitrary, so why make it? Well, there’s an interesting discovery to me made here: the two definitions are equivalent. That’s how you prove that multiplication is indeed commutative. It’s not obvious that $a$ groups of $b$ things each amounts to the same number of things as $b$ groups of $a$ things each. At least, not until you prove it, for example by arranging the things into an array:
You can see this as 3 groups of 6 by looking at the rows,
and as 6 groups of 3 things each by looking at the columns,
Since it’s the same number things no matter how you look at it, and using our definition of multiplication, we see that $3 \times 6 = 6 \times 3$. (We leave it as an exercise to the reader to generalize this proof.)
None of this dictates the way of doing $6 \times 3$, that is, the method of computing it. In fact, it expands the possibilities, including deciding to work with the more efficient $3 \times 6$, as this child did. The way of thinking does not constrain the way of doing. If you want to test whether a child understands 3.OA.A.1, you will have to come up with a different task than computation of a product. There are some good ideas from Student Achievement Partners here.
]]>Now to the topic of this post. There has been a lot of talk since the standards came out about what they say about multiple methods for arithmetic operations, and I’d like to clear up a couple of points.
First, the standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply and divide. But this does not mean that you have to solve every problem in multiple ways. Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn’t mean you have to go to school by car, then by bus, then walk, then bike—every single day! The point of having multiple methods available is to encourage students to think strategically about what might be the best method for a given problem, not force them to solve every problem four times.
Second, the different methods are not unrelated; they form a progression, with the ultimate goal being the standard algorithm. For example, when students are first learning to multiply two digit numbers, they might use a rectangle to represent a product such as $42 \times 71$.
This shows the fundamental role of the distributive property in multiplying multi-digit numbers. You have to multiply each base ten component into each other one. Indeed, the same rectangle representation provides a visual proof of the distributive property itself.
At some later point students might just start writing down all the partial products, without using the rectangle to derive them.
Note the correspondence between the rectangle method and the partial product method, indicated by the colors. The first row of the rectangle and shows all the products by the 2 in 42 (in red); the second row shows all the products by the 40 (in blue). The products in the partial product method are grouped in the same way. There are many ways you can order the partial products, but if you group them as I have here, going from right to left in each two-digit number, as in the standard algorithm, you make an amazing discovery: you can add up all the partial products in each group (blue group or red group) in your head as you go along. That’s because, in each case, adding the 2 to the 140 or the 40 to the 2800, there are enough zeroes in the second addend to accommodate the first, so it is easy to write down the sum right away, without writing the addends separately.
OK, so it’s not always quite this easy, because every now and then you will have to keep in mind a bundled unit from the previous step (aka carrying), but you will never have to remember that for more than one step at a time, because each bundled unit gets used up at the next step. So if you invent a notation for remembering the bundled unit (what we used to call “little 1 in the corner” when I was growing up) then you can still avoid writing down all the partial products, and just compute the sum within each group as you go along. You have just created the standard algorithm.
The different methods are not isolated different ways of doing the same thing; they are steps towards fluency with the standard algorithm, fluency that is not fragile because it is supported by understanding.
]]>We value coherence of content because we believe that a coherently arranged curriculum makes it possible for a student to see the subject as a whole, to understand the logical connections and deep structures, and to use that understanding for more efficient problem-solving and better retention of knowledge and procedures. But making it possible does not make it probable. The way students do mathematics, their mathematical practice, may have an effect on their ability to take advantage of a coherent curriculum. The CCSSM describes eight aspects of the complex construct of mathematical practice. Here we focus on two aspects, using structure (MP7) and abstraction (MP8).
Structure in arithmetic and algebraic expressions reveals what might be called “hidden meaning.” For example, writing $x^2-6x-7$ as $(x-3)^2-16$ reveals that, for real values of $x$, the expression assumes values greater than or equal to $-16$ (and it assumes that value only when $x=3$). Writing it as $(x-7)(x+1)$ highlights the values of $x$ that make the expression 0.
Treating pieces of expressions as a single “chunk” can simplify calculations; seeing that $4x^2-8x+3$ can be written as $(2x)^2-4(2x)+3$ helps one obtain the factorization from the (easier) factorization of $z^2-4z+3.$ This example can be generalized to encompass all polynomial expressions, providing students with a general purpose tool that can be used to transform a general polynomial into one with leading coefficient~1. It amounts to a change of variable in order to hide complexity, a practice that is useful all over mathematics.
Hidden meaning in geometric figures often involves the creation of auxiliary lines not originally part of a given figure. Two classic examples are the construction of a line through a vertex of a triangle parallel to the opposite side as a way to see that the angle measures of a triangle add to $180^\circ$ and the introduction of a symmetry line in an isosceles triangle to see that the base angles are congruent. Another kind of hidden structure makes use of the invariance of area when it is calculated in more than one way—finding the length of the altitude to the hypotenuse of a right triangle, given the lengths of its legs, for example.
A final example of using structure is in the view that students form of the base ten notational system. The compactness and regularity of this system make it useful for efficient computation and estimation. But in that compactness there is also the danger of superficial, and therefore fragile, grasp of procedures. The Number and Operations in Base Ten domain in CCSSM lays out a progression designed to help students learn to see the decimal expansion of a rational number as, in advanced language, a linear combination of powers of 10 with coefficients taken from integers between 0 and 9 helps. Similarly, viewing a polynomial in $x$ as a linear combination of powers of $x$ can lead to an understanding of polynomial algebra as a system in its own right. Writing $3x^2-7x + 5$ “in base $(x-2)$” as
$$
3(x-2)^2+5(x-2) + 3
$$
reveals another kind of hidden meaning in the expression.
Another theme that runs throughout a coherent curriculum is a cross-grade emphasis that helps students develop and use the many faces of abstraction. One of the most important uses of abstraction is captured in the CCSSM Standard for Mathematical Practice no.~8 (MP8), “Look for and express regularity in repeated reasoning.” It asks students to abstract a process from several instances of that process in a way that doesn’t refer to the inputs to any particular instance. Describing that process in precise algebraic language allows one to create general algorithms, equations, expressions, and functions. This practice can bring coherence to many seemingly different areas of the curriculum that often cause students difficulty.
The description of MP8 in CCSSM gives the following example:
By paying attention to the calculation of slope as they repeatedly check whether points are on the line through $(1, 2)$ with slope 3, middle school students might abstract the equation $\frac{y – 2}{x – 1} = 3$.
Helping students develop the habit of testing several numerical points to see if they are on the line and then looking for and expressing the “rhythm” in their calculations gives them a way to find the equation of a line between two points without leaning on formulas (“point slope form,” for example), and, more importantly, it gives them a general purpose tool for finding Cartesian equations of geometric objects, given some defining geometric conditions.
As another example, consider the task of building an equation. Teachers know that building is much harder for students than checking. The same practice of abstracting from numerical examples is useful here, too. For example, consider the stylized story problem:
Emilio drives from Tucson to Phoenix at an an average speed of 60MPH and returns at an average speed of 50MPH. If the total time on the road is 4.4 hours, how far is Tucson from Phoenix?
The practice of abstracting regularity from repeated actions can be used to build an equation whose solution is the answer to the problem: One takes several guesses (for the distance) and checks them, focusing on the steps that are common to each of the checks. The goal isn’t to stumble on (or approximate) an answer by “guess and check;” the goal is to come up with a general “guess checker” expressed as an algebraic equation:
$$
\frac {\text{guess}}{60} + \frac {\text{guess}}{50}= 4.4
$$
These two examples seem quite different, but coherence comes from the fact that exactly the same mathematical practice is used to find an algebraic equation whose solution solves the problem.
]]>A difficult question in designing a curriculum is to decide which topics go together. The logical and evolutionary considerations described in my previous two posts help, in that they provide guidance on the ordering of topics. But that still leaves many decisions to be made. My goal this post is to show some examples of how deep structures can guide these decisions. (See my previous post for what we mean by a deep structure.)
CCSSM in 6th grade has the following standard about percents in the Ratio and Proportional Reasoning domain:
6.RP.A.3c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
One approach to implementing this standard in a curriculum would be to have a section on percents that covers everything in the standard. But there is another possibility which attends to the difference between the parts of this sentence before and after the semicolon. The first part introduces the concept of percent. The second half involves solving problems that are tantamount to solving the equation $px = q$, where $p$ and $q$ are constants. This is related to a standard in the Expressions and Equations domain:
6.EE.B.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 non-negative rational numbers.
Thus another possibility might be to split the treatment of the percent standard into two places in the curriculum, with the introduction to percents occurring as a type of rate, in the section where ratios and rates are studied, and percent problems occurring in the section where solving equations is studied. If percents are regarded as a deep structure, one might choose the first arrangement; if rates and equations are regarded as deep structures, then one might choose the second. The second approach is the one we have taken in our soon-to-be-released middle school curriculum.
Another example of a deep structure is the profound connection between geometry and algebra. Imagine a 12 by 16 rectangle. Experiments with geometry software suggest that a square of side 14 maximizes area for the perimeter of this rectangle. If this is so, it should be possible to dissect the rectangle and fit the pieces into the square with something left over.
Trying several other rectangles of perimeter 56, a regularity emerges. Expressing this regularity in precise language leads to an algebraic identity that captures the dissection. Using an $a\times b$ rectangle, one has
$$
\label{eqagm} \left(\frac{a+b} 2\right)^2
-\left(\frac{a-b} 2\right)^2=ab
$$
This identity, inspired by geometric reasoning, can, of course, be verified in an algebra course. But its roots in geometry give it some extra meaning. And, it can be used to show how far off the rectangle is from the square.
Rather than separating the parts of this connection into two chapters or lessons, a coherent curriculum could use one story to develop both the necessary algebra and geometry, making it explicit that the main point is the connectivity of the ideas.
]]>I’ll talk about two ways in which the such evolution occurs: extension and encapsulation. Extension is a process by which a particular principle is repeatedly applied to ever broader systems, thus revealing its nature as a deep structure. Encapsulation is a process by which a related array of concepts and skills becomes encapsulated into a single compound concept or skill.
Extension is exemplified in the way that arithmetic with whole numbers is extended to fractions, integers, and rational numbers through a program of preserving the properties of operations. The fact that $(-3)\times (-5)$ is 15 is a definition, rather than a theorem—it has to be that way if we want arithmetic with integers to obey the distributive property. The properties of operations start from observation of particular instances, and evolve into powerful deeper structures under-girding the number system.
A good example of encapsulation is the development of fractions. The standard 3.NF.A.1 expects students to “Understand a fraction $1/b$ as the quantity formed by 1 part when a whole is partitioned into $b$ equal parts; understand a fraction $a/b$ as the quantity formed by $a$ parts of size $1/b$” and 3.NF.A.2 expects students to use this understanding to represent fractions on a number line. These two standards encapsulate many prior ideas and activities: dividing a physical object into halves or thirds; recognizing a geometric figure as a fraction of a larger figure representing the whole; moving from area representations to linear measurement representations; understanding the number line as marked off in unit lengths; subdividing those lengths into $n$ equal parts and thinking of those parts as a new sort of unit, an $n$th, and measuring out distances in those new units; correlating all these activities to the numerator and the denominator of the fraction.
Encapsulation builds coherence by tying what were previously disparate ideas and actions into a tightly connected structured bundle which becomes viewed as an object in its own right.
An important type of encapsulation is the evolution of representations. Mature representations are a form of encapsulation, and should be developed through a sequence of intermediate representations whose structural features preserve information about the object being represented. In early grades students might start with pictorial representations; but even then the picture should be more than a picture: it should carry information about the situation. Over time, such pictures evolve into more abstract diagrammatic representations, and eventually these diagrams are replaced by even more abstract representations such as tables and equations. The figure shows such an evolution for representations of proportional relationships in middle school. That final equation $y=kx$ is very dense with meaning, or at least it should be so for students. By the way, this is the sequence of representations for proportional relationships that we use in our new middle school curriculum coming out in July.
]]>Remember the distinction between standards and curriculum. While standards might remain fixed—a mountain we aim to help our students climb—different curricula designed to achieve those standards might make different choices about how to get there. Whatever the choices, a coherent curriculum, focused on how to get students up the mountain, would make sense of the journey and single out key landmarks and stretches of trail—a long path through the woods, or a steep climb up a ridge.
By the same token, mathematics has its landscape. CCSSM pays attention to this landscape by laying out pathways, or progressions, that span across grade levels and between topics, so that a third grade teacher understands why she is teaching a particular topic, because it will help students with some other topic in the next grade and build on what they already know.
This leads us to the first property of a coherent curriculum: it makes clear a logical sequence of mathematical concepts.
Consider, for example, the concepts of similarity and congruence. It is quite common in school curricula for similarity to be introduced before congruence. This comes out of an informal notion of similarity as meaning “same shape” and congruence as meaning “same shape and same size.” However, the fact that the informal phrase for similarity is a part of the informal phrase for congruence is deceptive about the mathematical precedence of the concepts. For what does it mean for two shapes to be the same shape (that is, to be similar)? It means that you can scale one of them so that the resulting shape is both the same size and the same shape as the other (that is, congruent). Thus the concept of similarity depends on the concept of congruence, not the other way around. This suggests that the latter should be introduced first.
This is not to say you can never teach topics out of order; after all, it is a common narrative device to start a story at the end and then go back to the beginning, and it is reasonable to suppose that a corresponding pedagogical device might be useful in certain situations. But the curriculum should be designed so that the learner is made aware of the prolepsis. (Really, I just wrote this blog post so I could use that word.)
Although the progressions help identify the logical sequencing of topics, there is more work to do on that when you are writing curriculum. For example, the standards separate the domain of Number and Operations in Base Ten and the domain of Operations and Algebraic Thinking, in order to clearly identify these two important threads leading to algebra. But these two threads are logically interwoven, and it would not make sense to teach all the NBT standards in a grade level separately from all the OA standards.
In the next few blog posts, I will talk about three other aspects of coherent curriculum: the evolution from particulars to deeper structures, using deep structures to make connections between topics, and coherence of mathematical practice.
]]>I took the opportunity to catch up on comments in the forums; I was way behind! Thanks to all those who responded to readers’ questions. I will try to stay more on top of it. One of the things that has been keeping me busy is our work on grades 6–8 curriculum for Open Up Resources. It is being piloted this year, so that link is still password protected, but stay tuned!
Also, I am close to finishing up the Quantity Progression, the last one not yet done.
]]>The mathematics curriculum in the US has been shaped by myriad forces over the years, including the competition for market share among publishing companies, economic realities of school districts’ purchasing power, the ease with which teachers can deliver the material, traditional expectations of what mathematics classroom work should look like, and so on. Surprisingly absent from these forces is the nature of the discipline of mathematics itself. The focus of this special session was on identifying and describing the essential mathematical structures of the K-12 curriculum, as well as the key mathematical practices in the work of mathematicians that should be mirrored in the work of students in K-12 classrooms.
]]>
There are many resources available to both pre-service and in-service teachers to help them increase their content knowledge in a way directly related to how they teach. Still, there are areas of mathematical content, relevant to the core of the mathematics curriculum, that somehow fail to receive systematic treatment in any of these resources. For many teachers, this means that there are always nagging questions that go unanswered. These teachers experience mathematics as a subject that somehow never fully makes sense.
The purpose of the Noyce-Dana project was to address this issue by building a community of scholars and practitioners focused on clarifying the mathematical underpinnings of middle and high school mathematics. It was hoped is that the work of this community would be a valuable resource for all those concerned with the teaching and learning of mathematics. The project was not continued after the first year, but I hope that publishing the essays that came out of that year will serve to continue the original vision of the project.
The members of the group were Dan Chazan, Scott Farrand, Sol Friedberg, Emiliano Gomez, Roger Howe, Eric Hsu, Jim Madden, myself, Mike Oehrtman, and Harris Shultz. The essays are linked below
Extraneous and Lost Roots (Chazan-Gomez-Farrand)
Letters & Types of Representations in Algebra (Oehrtman-Shultz)
What Is a Variable (Epp)
Minus Times Minus (Gomez-Hsu)
Rule of Signs (Howe-Friedberg)
AXplusB (Howe)
Equations & Functions (Friedberg)
Equations & Functions-3 Views (Stanley)
Equations from Functions (Stanley)
Expressions-Functions-Equations (McCallum)
Functions (Howe)
Graphs-Equations-Functions (Stanley)
Order of Operations (Hsu-Madden)
Proportion (Madden)
Proportion-Euclid (Madden)
Proportionality (Madden-Oehrtman)
Two Meanings of Proportional (Stanley)
Appendix A was provided as a proof of concept, showing one possible way of arranging the high school standards into courses. Indeed, on page 2 of the appendix it says:
The pathways and courses are models, not mandates. They illustrate possible approaches to organizing the content of the CCSS into coherent and rigorous courses that lead to college and career readiness. States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own.
States will of course be constrained by their assessments. But Smarter Balanced consortium does not have end of course assessments in high school, leaving states and districts free to arrange high school as they choose. And although PARCC does have end of course assessments, they do not follow Appendix A exactly. See the footnote on page 39 of the PARCC Model Content Framework , which says
Note that the courses outlined in the Model Content Frameworks were informed by, but are not identical to, previous drafts of this document and Appendix A of the Common Core State Standards.
Furthermore, there are plenty of states not using either the PARCC of SMARTER Balanced assessments.
I hope this helps clear things up.
]]>In addition to numerous small edits and corrections, and some redrawn figures, here are some of the more significant changes:
As usual, please comment in NBT thread in the Forums.
]]>Of course, standards do have to have meaningful implications for curriculum, or else they aren’t standards at all. The Instructional Materials Evaluation Tool (IMET) is a rubric that helps educators judge high-level alignment of comprehensive instructional materials to the standards. Some states and districts have used the IMET to inform their curriculum evaluations, and it would help if more states and districts did the same.
The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:
If the only computation algorithm we teach is the standard algorithm, then can we still say we are following the standards?
Provided the standards as a whole are being met, I would say that the answer to this question is yes. The basic reason for this is that the standard algorithm is “based on place value [and] properties of operations.” That means it qualifies. In short, the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.
Additional mathematics, however, is required. Consistent with high performing countries, the elementary-grades standards also require algebraic thinking, including an understanding of the properties of operations, and some use of this understanding of mathematics to make sense of problems and do mental mathematics.
The section of the standards that has generated the most public discussion is probably the progression leading to fluency with the standard algorithms for addition and subtraction. So in a little more detail (but still highly simplified!), the accompanying table sketches a picture of how one might envision a progression in the early grades with the property that the only algorithm being taught is the standard algorithm.
The approach sketched in the table is something I could imagine trying if I were left to myself as an elementary teacher. There are certainly those who would do it differently! But the ability to teach differently under the standards is exactly my point today. I drew this sketch to indicate one possible picture that is consistent with the standards—not to argue against other pictures that are also consistent with the standards.
Whatever one thinks of the details in the table, I would think that if the culminating standard in grade 4 is realistically to be met, then one likely wants to introduce the standard algorithm pretty early in the addition and subtraction progression.
Writing about algorithms is very difficult. I ask for the reader’s patience, not only because passions run high on this subject, but also because the topic itself is bedeviled with subtleties and apparent contradictions. For example, consider that even the teaching of a mechanical algorithm still has to look “conceptual” at times—or else it isn’t actually teaching. Even the traditional textbook that Garelick points to as a model attends to concepts briefly, after introducing the algorithm itself:
This screenshot of a Fifties-era textbook is as old-school as it gets, yet somebody on the Internet could probably turn it into a viral Common-Core scare if they wanted to. What I would conclude from this example is that it might prove difficult for the average person even to decide how many algorithms are being presented in a given textbook.
Standards can’t settle every disagreement—nor should they. As this discussion of just a single slice of the math curriculum illustrates, teachers and curriculum authors following the standards still may, and still must, make an enormous range of decisions.
This isn’t to say that the standards are consistent with every conceivable pedagogy. It is likely that some pedagogies just don’t do the job we need them to do. The conflict of such outliers with CCSS isn’t best revealed by close-reading any individual standard; it arises instead from the more general fact that CCSS sets an expectation of a college- and career-ready level of achievement. At one extreme, this challenges pedagogies that neglect the key math concepts that are essential foundations for algebra and higher mathematics. On the other hand, routinely delaying skill development until a fully mature understanding of concepts develops is also a problem, because it slows the pace of learning below the level that the college- and career-ready endpoint imposes on even the elementary years. Sometimes these two extremes are described using the labels of political ideology, but I have declined to use these shorthand labels. That’s because I believe that achievement, not ideology, ought to decide questions of pedagogy in mathematics.
Jason Zimba was a member of the writing team for the Common Core State Standards for Mathematics and is a Founding Partner of Student Achievement Partners, a nonprofit organization.
]]>Working with Number in the Elementary Classroom
The following lectures are scheduled in the series on Thursday nights from 7-8pm Eastern on Adobe Connect. Watch them live with the ability to ask questions, or watch the recordings at any time:
September 25, 2014 Linda Gojak, Immediate Past President, National Council of Teachers of Mathematics, Director, The Center for Mathematics Education, Teaching, and Technology, John Carroll University “Using Representations to Introduce Early Number and Fraction Concepts”
October 23, 2014 Dona Apple, Mathematics Learning Community Project, Regional Science Resource Center, University of Massachusetts Medical School “Supporting students’ conceptual understanding about number through reasoning, explaining and evidence in both their oral and written work”
November 20, 2014 Brad Findell, The Ohio State University
December 11, 2014 Francis (Skip) Fennell, Professor of Education McDaniel College, Past President NCTM “Fractions Sense – It’s all about understanding fractions as numbers (and this includes those special fractions – decimals!) – use of representations, equivalence, comparing/ordering and connections”
January 22, 2015 Susan Jo Russell, TERC: Mathematics and Science Education and Deborah Schifter, Education Development Center (EDC) “Operations and Algebraic Thinking in the Elementary Grades”
This school year we will offer two series. In the fall we are featuring “Working with Number in the Elementary Classroom” and this spring we will offer “Incorporating Mathematical Practices into the Middle and High School Classroom.” The intended audience for these series is classroom teachers, district and state mathematics specialists, and mathematics coaches. The five hour long sessions will include 40 minutes of presentation from national experts on Adobe Connect, followed by 20 minutes of Q&A. The sessions will also be recorded for participants that are not able to join in person. The cost to virtually attend each series is $150.
Here is a flyer to circulate among friends that might be interested or to post in the staff room! Hope to see you there.
]]>First, here is the list, with an additional one from July 2009 that I missed last time (which has been the source of much confusion):
Notice that there seem to be duplicate announcements of the Work and Feedback Group and duplicate releases of the standards. What’s going on here is that there were two documents. First, in summer of 2009, the people listed in the July 2009 release worked on the document that was announced in September of 2009. That document, which was actually entitled College and Career Readiness Standards for Mathematics, was confusingly referred to as Common Core State Standards in the title of the September 2009 press release. If you take a look at it you will see that it is a draft description of what students should know by the end of high school.
Subsequently, as described in the November 2009 press release, a new process with new groups was started, to produce “K–12 standards.” These were to be a set of grade level recommendations that described a pathway to college and career readiness. For the K–12 process, there were about 50 people on the Work Team and about 20 people on the Feedback Group for mathematics, representing a wide range of professions, including teachers, mathematicians, policy makers, and one representative each from College Board and ACT … none representing for-profit providers of assessments. The members of this group are listed in a linked pdf in the press release. This is the document that was released for public comment in March 2010, as described in the March 2010 press release, and released in final form as the Common Core State Standards on 6 June 2010, as described in the final press release.
As you can see from the list of members, I chaired the Work Team for the second document. Within the work team there was a smaller writing team consisting of myself, Jason Zimba, and Phil Daro (who had all been involved in the summer 2009 document, Phil Daro as chair for mathematics). We based the standards on narrative progressions of particular mathematical topics across grade levels that were solicited from the Work Team. We circulated many drafts to the Work Team, the Feedback Group, the 48 participating states, various national organizations such as AFT and NCTM, and, in March 2010, the public (see the March 2010 release). I personally made sure that we responded to and made considered decisions about all of the voluminous feedback we received.
When you hear people claim that “the standards were written by the testing industry,” they are probably referring to the first document, because of the greater involvement of College Board and ACT. Both organizations, along with Achieve, which was also represented, had conducted research into the requirements of college and career readiness. (All are non-profits, by the way.) The problem is that some people refer to the first document in a way that suggests they are talking about the second document (i.e., the actual K-12 standards adopted by states). That is an error and a misleading one.
The two documents are different in nature, of course, since one of them is just a picture of an endpoint while the other is a progression. Feel free to compare them. One influence of the first document on the second is that in the first document you can see the first draft of what became the Standards for Mathematical Practice. And the topic areas listed in the first document evolved into the high school conceptual categories in the second. All this evolution happened under the processes for the second document, with input from the various groups described above.
I think the second document is the work of the 70-odd people listed as the Work Team and Feedback Group in the November 2009 press release. But, just for fun, I put the teams for the two documents together and counted how many of them came from ACT and College Board (no other testing organizations were represented). It comes to a total of 81 people with 7 from ACT and College Board, about 9%. So even with this interpretation the claim that the process was dominated by the testing industry is false.
]]>
First I want to draw attention to a footnote on K.OA.A.1, the first standard in which drawings are mentioned, in Kindergarten. The footnote says
Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)
Not all drawings “show the mathematics in the problem.” Here’s an example of one that does, courtesy of Barry Garelick, who has written a very interesting article about how the Common Core can support a traditional approach to teaching (you have to scroll to the second section to get to the math). It is from a 1955 textbook by Brownell et al.
The arrangement of dots and the enclosing curve have a clear purpose here in helping students learn to add by making a 10; the drawing has a mathematical purpose, it is not just a random collection of dots.
In Grade 2 we have:
2.OA.A.1. 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, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. [Bold face added.]
Some form of the boldfaced wording in this standard occurs repeatedly in the K–5 standards. The intended interpretation of this wording (e.g the “e.g.”) is to indicate that drawings are one possible way of learning a strategy and of understanding why it works. It is not intended that every single problem be accompanied by a drawing and an equation. Moreover, there is a progression in the standards, culminating in requirements that students be fluent with the standard algorithms. So there should be a progression in the use of drawings, with students eventually being able to do in their heads what they first had to do on paper, and with them eventually acquiring a robust fluency that does not require them to spell out their understanding explicitly every time.
This progression is clearly indicated by the following progression of standards about addition and subtraction from Grade 2 to Grade 4:
2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
3.NBT.A.2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
4.NBT.B.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
If you read the standards carefully for the progressions in them you can see the clear intention about drawings. They have their place on the pathway to mathematical proficiency, but are not an end in themselves … and should not become an obsessive requirement.
By the way, the standard algorithms for addition and subtraction are “strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.” The standards do not forbid them being introduced in Grade 2—nor do they require that.
I’ve gone on enough for now. Maybe I’ll do another post on drawings and fractions when I get a chance.
[corrected version added 5/6/2014]
]]>As usual, please comment by starting a new thread in the forums. I’ve created a new forum for the practice standards there.
]]>Our first Virtual Lecture Series will meet around the theme: Preparing and Facilitating Engaging Professional Development for Teachers around the Common Core, on the last Wednesday of the month at 7pm Eastern/4pm Pacific from January through May. The intended audience for this series is district and state mathematics specialists as well as teacher leaders. The five hour long sessions will include 40 minutes of presentation from national experts on Adobe Connect, followed by 20 minutes of Q&A. The cost to virtually attend the entire series is $150 which includes access to the following presentations:
January 29th: Diane Briars, President Elect of NCTM, Topic: Effective Instructional Practices to ensure all your students are “Common Core Ready”
February 26th: Bill McCallum, Lead writer of the CCSSM, Topic: Preparing K-12 Teachers for the Pathway to Algebra
March 26th: Mary Knuck, Arizona Department of Education Retired, Topic: Math Talks
April 30th: Ashli Black, NBCT and Cal Armstrong, Math Teacher Leader, Topic: Involving Teacher Leaders in Preparing and Facilitating Professional Development
May 28th: James Tanton, Mathematician and Author of Thinking Mathematically! Topic: Instilling a Love of Mathematics
Also the blog is back from a rough time over the new year. Sorry if you had trouble with any of the posts or forums, we were not as quick as we could be in renewing the domain. Let us know if you continue to have trouble accessing anything.
]]>This week we have chosen a fourth grade task about subtraction. If you think you might join, please let us know here.
Hope to see you there!
]]>I think (2) works better, but (1) is slightly more convenient. (3) is a last resort when you get frustrated.
[Update, 1/28/14: (1) stopped working, but I have found a new widget that implements a google site search, on the right. I’ll make this post unsticky now in the hope that we have finally solved the problem.]
]]>But I’ve decided it’s time take a stand against the swirling tide of insanity that threatens our work, so I’m starting a new blog called I Support the Common Core. It will provide resources, links to articles, rebuttals, and discussion to help those who are fighting the good fight. If you sign up you will be getting emails and calls for action from me and others. Tools for the Common Core will remain available for those of you who prefer a quieter life and just want to get on with your jobs.
The success of this effort depends on you. If only 10 brave souls sign up I will thank them and close down the effort. If 1,000 of you join then we can get something done (and I promise there will be jokes).
]]>A clear step after developing high-quality mathematical tasks is to develop accompanying lesson plans. I wrote seven lesson plans to accompany published tasks, all of which I tested in my classroom. My goal was to write lesson plans that guided students to the level of thinking required by both the standards and the practices.
One example is the lesson plan for the task What is a Trapezoid?, aligned to standard 5.G.B.4. A student who is able to successfully complete the task not only knows the relevant content, but can also skillfully construct viable mathematical arguments (Practice 3). The obvious question to teachers is, “How do we get students there?” The lesson plan Plane Figure Court is one possible way. In it, students serve as “lawyers,” charged with proving or disproving a particular mathematical statement. For example, the statement, “A square is a rhombus” has a lawyer arguing that this is true, and a lawyer arguing that this is false. I required that students create justifications, even if they knew their justification was wrong. The other students (the jury) decided the case based on the mathematical arguments made, not on what they thought was correct. My end goal here was to help students to recognize valid (e.g., a square is a rhombus because it has four congruent sides) and invalid mathematical arguments (e.g., a square is a rhombus, because if you turn it a little it looks like one).
The format for the lesson plans is consistent through each one. The first section includes the objective(s), an overview, and the standards to which the lesson are aligned. The second section includes a detailed lesson plan, as well as suggestions for assessment and differentiation. The third section includes commentary and relevant attachments, such as worksheets or diagrams. Some lesson plans, like Cooking Time 1, include student work.
The initial seven lesson plans are listed below, and others will be added in the future. Tasks with lesson plans will be tagged “Lesson Plan Included”, and are accessible under the “Resources” heading.
5.NF How Much Pie? / Cooking Time 1
5.NF How many servings of oatmeal? / Cooking Time 2
5.NF Making Cookies / Cooking Time 3
5.NF Salad Dressing / Cooking Time 4
5.OA Video Game Scores / The Order of Operations
5.MD Cari’s Aquarium / What is Volume?
5.G What is a Trapezoid? / Plane Figure Court
We’ve also been working on developing review criteria for lesson plans that develop Illustrative Mathematics tasks into full-blown lessons. The criteria are available here.
If you’ve taken a look at the lesson plans and the criteria, we’d love to hear your feedback by September 1.
]]>Second, thanks to work of Al Cuoco, we have updated versions of the Algebra and Functions Progressions.
]]>[Edited 4 July 2013. Please go here for the latest draft.]
]]>MP8. Look for and express regularity in repeated reasoning.
There are too many words in there: regularity, repeated, reasoning. I’ve seen a lot of people latching onto one or two of these. If it’s regular, it’s MP8! If it’s repeated, it’s MP8! If it’s both regular and repeated, it must really be MP8!! One thing that is fairly regular and repeated is generating coordinate pairs from an equation in two variables. So there are lots of fake MP8 lessons out there about generating points from a linear equation in two variables to draw the graph of the equation, a straight line. The more points, the better—it’s more repeated that way. And regular.
But that word reasoning is also important. There’s precious little reasoning involved in generating coordinate pairs from an equation. But if we turn the question around, there’s lots of reasoning. Instead of going from an equation to a line, let’s go from a line to an equation. Consider a line through two points in the coordinate plane, say (2,1) and (5,3). How do I tell if some randomly chosen third point, say (20,15), is on this line or not? Given any two points on a line in the coordinate plane, I can construct a right triangle with vertical and horizontal legs, using the line to form the hypotenuse, as shown here.
It is a wonderful geometric fact that all of these triangles are similar. (Exercise: prove this!) So, if (20,15) is on my line, then the triangle formed by (20,15) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:
$$
\frac{15-1}{20-2} = \frac{3-1}{5-2}?
$$
Oops. Not true. So (20,15) is not on the line. Let’s try (20,13) instead. If (20,13) is on the line, then the triangle formed by (20,13) and (2,1) should be similar to the triangle formed by (5,3) and (2,1). If these two triangles are similar, the ratio of their vertical to horizontal legs should be equivalent:
$$
\frac{13-1}{20-2} = \frac{3-1}{5-2}?
$$
Yes! Both sides are equal to $\frac23$. And in fact, to confirm, the reasoning works the other way: if the ratios are equivalent, then the triangles are similar, then the base angles are the same, so the hypotenuses of these two triangles are on the same line. (Exercise: prove all this, too!)
So we have a way of testing whether points lie on the same line. (This is Al Cuoco’s point tester; google it.)
After testing a lot of points, we look for some regularity in our repeated reasoning. Every one of our calculations looks the same. We can express the regularity by a general statement: to test whether a point $(x,y)$ is on the line, we check whether
$$
\frac{y-1}{x-2} = \frac{3-1}{5-2}.
$$
By our reasoning, every point on the line satisfies this equation, and no point off the line satisfies it. We have discovered the equation for the line by expressing regularity in our repeated reasoning.
All the words in MP8 are important: reasoning, repeated, regularity, and also express and look for. See this post by Dev Sinha for more discussion.
]]>There’s a lot of misinformation going around these days about how the Common Core State Standards were written. It occurred to me that a simple way of learning about the process is through the press releases from the National Governors Association during 2009–2010. If you type Common Core into the search box you will find releases detailing the initial agreement of the Governors, the composition of the work teams, feedback groups, and validation committee, the state and public reviews, and various other pieces of information. It’s not a detailed history by any means, but I would encourage readers to check information they receive against this source.
[19 June] I noticed the search feature at NGA isn’t working today, so here are the main releases for 2009–2010:
1. A report from Institute for Mathematics & Education suggesting places that might need some extra PD work.
2. Consider requesting trained teacher facilitators to deliver the Common Core Toolkit, a one-day add on to existing professional development, focused on the Common Core. This is available for K-5th grade teachers, 6th-8th grade teachers, or high school teachers and is a project of an ad-hoc committee of the CBMS.
]]>]]>Friends,
For the past two years, we’ve been working with support from the MA department of education to create a course for high school teachers that helps them implement the Standards for Mathematical Practice. The approach of the design is to take examples suggested by the high school content standards—everyday, non-exotic content that is hard to teach and that causes students difficulty—and to develop that content in ways that are consistent with the practice of mathematics as it exists outside of high school, making the topics easier to teach, easier to learn, and more satisfying for everyone.
We field tested the course with over 100 teachers in two sessions over the past two summers at EDC. The a team of 10 colleagues (teachers who work with us) taught it in pairs in 5 sessions around the state at the end of last summer. All of this led to revisions, and we’re now publishing the course and offering it nationally. A sampler is at http://mpi.edc.org/dmp-hs-sampler
This problem of turning everything into “microstandards” is a problem of long standing in education. One might even say it is the original sin in curriculum design. Take a complex whole, divide into the simplest and most reductionist bits, string them together and call it a curriculum. Though well-intentioned, it leads to fractured, boring, and useless learning of superficial bits.
Read also his spirited defense of the standards a couple of days earlier.
]]>Think about a 4-by-5 rectangle. The rectangle contains infinitely many points$—$you could never count them. But once you decide that a 1-by-1 square is going to be “one unit of area,” you are able to say that a 4-by-5 rectangle amounts to twenty of these units. A choice of unit makes the uncountable countable.
Or think about two intervals of time. One is the period of Earth’s rotation about its axis; the other, the period of Earth’s revolution around the sun. Both intervals are infinitely divisible$—$a continuum of moments. But once you decide that the first period is going to be a “unit of time,” you are able to say that the second period of time amounts to 365 of these units. A choice of unit makes the uncountable countable.
More abstractly, think of a number line. The line is an infinitely divisible continuum of points. Zero is one of them. Now make a mark to show 1. The mark shows 1 as the indicated point; it also defines 1 as a quantity whose size is the interval from 0 to 1. This interval is the unit that makes the uncountable countable. We mark off these units along the line as 2, 3, 4, 5 and so on. (Later, we go the other way from zero marking off units: $–$1, −2, −3, −4, −5 and so on.) It is not for nothing that mathematicians call 1 “the unit.”
To a physicist, measurement is an active idea about using one empirical quantity (such as Earth’s rotation period) to “measure” (divide into) another empirical quantity of the same general kind (such as Earth’s orbital period). Mathematically, one can see that measurement is linked to division: how many units “go into” the quantity of interest.
Another way to say it is that measurement is linked to multiplication, in particular to a mature picture of multiplication called “scaling,” (5.OA) in which we reason that one quantity is so many times as much as another quantity. The concept of “times as much” enters the Standards in Grade 4 (4.OA.1, 4.OA.2, 4.NF.4), limited to whole-number scale factors.
The number line picture of this is that we are progressing from concatenating lengths to stretching them too. Thus, 2 is simultaneously “one more than one-more-than-0” and “twice as much as 1.” These two perspectives on 2 are linked by the distributive property, which defines the relationship between addition and multiplication. $1 + 1 = 1\times 1 + 1\times 1 = (1 + 1)\times 1 = 2\times 1$.
By Grade 5, scale factors and the quantities they scale may both be fractional; the flow of ideas extends into Grade 6, when students finally divide fractions in general. At that point, we may consider a deep measurement problem such as, “$\frac{2}{3}$ of a cup of flour is how many quarter-cups of flour?”
The roots of all this in K$–$2 are 1.MD.2 and 2.MD:1–7, and 2.G:2,3.
When we reflect back on the geometry in K$–$2 from this perspective, we see that some of what is going on is learning to “structure space” by, for example, seeing a rectangle as decomposable into squares and composable from squares. Researchers show interesting pictures of the warped grids that students make until they get sufficient practice.
Already by Grade 3 we are dissatisfied with measurement. We want to know what happens when the unit “doesn’t go evenly into” the quantity of interest. So we create finer units called thirds, fourths, fifths, and so on. This is the intuitive concept of a unit fraction, $\frac{1}{b}$: a quantity whose magnitude is equal to one part of a partition of a unit quantity into $b$ equal parts. (3.NF) We reason in applications by thinking of the unit quantity as a bucket of paint, or an hour of time. We reason about fractions as numbers by thinking of the unit quantity as that portion of the number line lying between 0 and 1. Then $\frac{1}{b}$ is the number located at the end of the rightmost point of the first partition.
Because you can count with unit fractions, you can also do arithmetic with them (4.NF:3,4). You can reason naturally that if Alice has $\frac{2}{3}$ cup of flour (two “thirds”) and Bob has $\frac{5}{3}$ cup of flour (five “thirds”), then together they have $\frac{7}{3}$ cup (seven “thirds,” because two things plus five more of those things is seven of those things). The meanings you have built up about addition and subtraction in K$–$2 morph easily to give you the “algorithm” for adding fractions with the same denominator: just add the numerators. (And don’t change the denominator $\dots$ after all, you would hardly change the unit when adding 3 pounds to 8 pounds.)
Likewise, multiplying a unit fraction by a whole number is a baby step from Grade 3 multiplication concepts. If there are seven Alices who each have $\frac{2}{3}$ cup of flour, it is a bit like when we reasoned out the product $7\times 20$ in third grade: seven times two tens is fourteen tens; likewise seven times two thirds is fourteen thirds. Again the meanings you have built up about multiplication in Grade 3 morph easily to give you the “algorithm” for multiplying a fraction by a whole number: $n\times \frac{a}{b} = \frac{n\times a}{b}$.
The associative property of multiplication $x\times (y\times z) = (x\times y)\times z$ is implicit in the reasoning for both $7\times 20$ and $7\times \frac{2}{3}$. So is unit thinking. In $7\times \frac{2}{3}$, the unit of thought is the unit fraction $\frac{1}{3}$. In $7\times 20$ and other problems in NBT, the units of thought are the growing sequence of tens, hundreds, thousands and ever larger units, as well as the shrinking sequence of tenths, hundredths, thousandths, and ever smaller unit fractions.
The conceptual shift involved in progressing from multiplying with whole numbers in Grade 3 to multiplying a fraction by a whole number in Grade 4 might be aided by the multiplication work in Grade 4 that extends the whole number multiplication concept a nudge beyond “equal groups” to a notion of “times as many” or “times as much” (4.OA:1,2). The reason this meshes with the problem of the seven Alices is that those seven Alices don’t exactly have among them seven “groups of things,” yet they do among them have seven “times as much” as one Alice.
The step in Grade 4 from “equal groups” to “times as much,” along with the coordinated step of multiplying a fraction by a whole number, represents the first major step toward viewing multiplication as a scaling operation that magnifies or shrinks. Multiplying by 7 has the effect of “magnifying” the amount of flour that a single Alice has. In Grade 5, we will “magnify” by non-whole numbers, for example by asking how many tons $4\,\frac{1}{2}$ pallets weigh, if one pallet weighs $\frac{3}{4}$ ton. We will find, during the course of that study, that a product can sometimes be smaller than either factor.
This kind of thinking about scaling is connected to proportionality, as when we use a “scaling factor” to get answers to compound multiplicative problems quickly. For example, 1500 screws in 6 identical boxes $\dots$ how many in 2 of the boxes? What would be a multi-step multiplication and division problem to a fifth grader becomes, for a more mature student, a proportional relationships problem: a third as many boxes, so scale the number screws of by a third. We quickly have the answer 500.
A year isn’t exactly 365 days$—$nor is it exactly 365.25 days. Could any rational number express the number of days in a year? Students of mathematics run up against a similar problem when they ask how many times the side of a square “goes into” its diagonal. By Grade 8, we learn without proof that the diagonal cannot be written as any rational multiple of the side. In this way, irrational numbers such as $\sqrt{2}$ enter the discussion, and likewise $\pi$ for the quotient of the circumference of a circle by its diameter. The Greeks called the diagonal and the side, or the circumference and the diameter, incommensurable quantities. This ancient idiom, meaning not measurable by a common unit, underscores the importance of measurement thinking to arithmetic.
This is an excerpt from a larger document (almost two years old now). It’s revised here with input from Phil Daro and William McCallum. Also see the Progression document on measurement in grades K$–$5. For those interested in the scholarly literature about these questions, I’m sure it is vast$\dots$but I’ll pass along one article I came across just the other day (Thompson and Saldhana, 2003). $–$J.
]]>Towards Greater Focus and Coherence May 26-28, 2013 at the University of Arizona This is a great way to start the summer, while you are still in the classroom flow!
We are looking forward to meeting people who care about math education and collaborating with math coaches, classroom teachers, mathematicians, district math specialists, and mathematics educators.
Highlights of the conference include:
1. Perspective from Bill McCallum, lead writer of the Common Core
2. Activities that can be immediately used in your classroom, and a plan for creating similar Common Core aligned activities for students in the future.
3. Breakout sessions from classroom teachers modeling the focus of the Common Core by digging into a particular standard or cluster.
4. Highlights of the focus and coherence of different grade bands and the mathematics behind the standards.
5. Online resources to support the Common Core.
You don’t want to miss this opportunity. Reserve your spot by March 31st for the best rates by registering online.
]]>Mathematical objects are key components of content standards. Practice standards on the other hand describe student actions. Thus while we usually pay attention to nouns in content standards, for practice standards we must pay attention to verbs.
For MP7 Look for and make use of structure, “Look for” is a key phrase. Consider the task 2.NBT Making 124, which in brief asks students to decompose 124 into hundreds, tens and ones in all ways they can find (e.g. 6 tens and 64 ones). In order to be efficient or complete, students will need to use exchanges—of a ten for ones or of a hundred for tens—systematically. That is, there is a structure of systematic exchanges which students must look for and make use of in order to be highly successful. We can say that this task implicitly invites students to engage in MP7, through both its “look for” and “make use of” halves. If the systematic exchanges are suggested by the task or by the teacher before students have had a chance to search themselves, then the practice would not be fully be engaged.
For MP8 Look for and express regularity in repeated reasoning, “express” is an important verb. Consider this instructional sequence from the progression on Progression on Ratios and Proportional Relationships in which students are to consider equivalent-tasting mixtures of juice. While students may immediately notice some regularity it is the process of expression, going say from observations about a table to statements like “if we increase the grape juice by 1 cup we must increase the peach juice by 2/5 of a cup to taste the same” and ultimately to writing the equation $y = 2/5 x$, which constitutes the bulk of the mathematical work of the task. MP8 provides language to discuss this kind of expressive mathematical work.
Development of tasks and lessons involves consideration of the mathematical work students are invited to do. Content standards provide nouns to be employed in describing this work, while practice standards provide verbs.
]]>6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas $V=lwh$ and $V=bh$ to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
See if you can guess what people think the problem is before reading on.
The most recent message said quite sternly that $V=bh$ was NOT correct, and that it MUST BE $V = Bh$. This point of view is one of the starkest examples I know of the obstacles we must overcome in restoring the culture of mathematics in schools. The notion that certain letters MUST always stand for the same thing across different formulas is itself a mathematical error, a profound misunderstanding of how symbols are used. It’s like thinking the word blue must always be written in blue.
And the misconception is not harmless. Students who come to college with it do not fare well amid the profusion of symbols in their science classes, unable to see that the function $f(x) = \sin(ax)$ in their calculus class is the same as the function $A(t) = \sin(\omega t)$ in their physics class. Part of the power of algebra is that you can choose any letter you like to represent a quantity, as long as you specify what the letter stands for, and you can re-use that letter with a different meaning in a different problem.
That said, the standard does not dictate the use of any specific letters; indeed, the core meaning of the standard is not about formulas at all, but rather about finding the volume of a rectangular prism by multiplying its length, width, and height, or by multiplying the area of its base by its height. So teachers and curriculum materials can use whatever letters they like, including $V = Bh$, or no letters at all. Indeed, formulas should always have words associated with them. Naked formulas like $V = bh$ mean nothing by themselves without surrounding words, such as in the sentence
The volume $V$ in cubic inches is given by $V = bh$, where $b$ is the area of the base in square inches and $h$ is the height in inches.
Changing the two occurrences of $b$ to $B$ changes the meaning of this sentence not one whit.
[Thanks to Jason Zimba for suggesting the title of this post.]
]]>Comments as always are welcome in the relevant forums: Algebra or Functions.
]]>IM&E/Illustrative Mathematics’ next Common Core Math Conference is coming to the east coast; Syracuse, New York!
Mathematics Common Core in the Classroom March 1-3, 2013 at the Doubletree Hotel, Syracuse New York
We are looking forward to meeting people who care about math education from across the country and collaborating with math coaches, classroom teachers, mathematicians, district math specialists, and mathematics educators. Highlights of the weekend include:
1. Perspective from Bill McCallum, lead writer of the Common Core, on “What’s different about these standards?”
2. Activities that can be immediately used in your classroom, and a plan for creating similar Common Core aligned activities for students in the future.
3. Breakout sessions from classroom teachers modeling the focus of the Common Core by digging into a particular standard or cluster.
4. Mathematicians discussing the focus of different grade levels and the mathematics behind the standards.
5. Online resources to support the Common Core.
You don’t want to miss this opportunity. Save your spot today by registering online. OCM-BOCES teachers register with your district office. All others register on the IM&E website.
]]>How can mathematicians aid teachers in learning this mathematics, in collaboration with others responsible for teacher education?
Current research and experience are synthesized to answer these questions in the new report The Mathematical Education of Teachers II (MET II) from the Conference Board of the Mathematical Sciences. This report updates The Mathematical Education of Teachers (published in 2001) and extends its scope from preparation to professional development in the context of the Common Core State Standards for Mathematics.
The audience for the report includes all who teach mathematics to teachers—mathematicians, statisticians, and mathematics educators—and all who are responsible for the mathematical education of teachers—department chairs, educational administrators, and policy-makers at the national, state, school-district, and collegiate levels.
The report may be downloaded free at the Conference Board of Mathematical Sciences web site. Printed copies may be ordered from the American Mathematical Society.
The Conference Board of the Mathematical Sciences (CBMS) is an umbrella organization consisting of sixteen professional societies all of which have as one of their primary objectives the increase or diffusion of knowledge in one or more of the mathematical sciences. Its purpose is to promote understanding and cooperation among these national organizations so that they work together and support each other in their efforts to promote research, improve education, and expand the uses of mathematics.
For further information, contact CBMS director Ronald Rosier, 410-730-1426; 202-293-1170.
]]>October 2012 Berkeley Conference page Participants made classroom activities based on a particular task or set of 2-3 tasks
May 2012 New Orleans Conference page Participants made short PD units or classroom activities around a particular standard, group of standards, or cluster
February 2012 Tucson Conference page Participants made PD modules around a particular standard, cluster, or domain
]]>Please register quickly as our previous workshops have sold out!
]]>I will keep the current email notification feature for those who prefer to keep using that.
]]>[29 July 2012] This thread is closed. You can ask questions here.
]]>ccss_progression_gk6_2014_12_27
[Updated 2014/12/27]
]]>UPDATE 10/29/2015 by Jason Zimba – The wiring diagram now exists as a more fully fledged digital tool called the “Coherence Map,” found at www.achievethecore.org/coherence-map. When you click the link, you will be able to navigate the content standards via their connections, and you will also see resources keyed to individual standards, such as relevant excerpts from the Progressions documents, tasks from Illustrative Mathematics, and other open resources (this feature is meant to grow over time).
]]>[30 July 2012] This thread is now closed. Please go here to discuss this progression.
]]>Here is the information page and registration.
]]>I know somebody is going to ask about printing tasks. We don’t have that yet, but are working on it!
]]>Here is the information page and registration.
]]>9_11 Scope and Sequence_traditional1
9_11 Scope and Sequence_traditional2
9_11 Scope and Sequence_integrated1
]]>1. Registered users can rate tasks by voting them up or down.
2. Registered users can comment on tasks.
You can edit and delete your comments, although it will leave a placeholder note that you deleted your comment in case someone replies to your comment (so their comment won’t be deleted also).
We have more improvements in store, and illustrations are being added every day, so keep checking in.
]]>
In the Common Core State Standards, individual statements of what students are expected to understand and be able to do are embedded within domain headings and cluster headings designed to convey the structure of the subject. “The Standards” refers to all elements of the design—the wording of domain headings, cluster headings, and individual statements; the text of the grade level introductions and high school category descriptions; the placement of the standards for mathematical practice at each grade level.
The pieces are designed to fit together, and the standards document fits them together, presenting a coherent whole where the connections within grades and the flows of ideas across grades are as visible as the story depicted on the urn.The analogy with the urn only goes so far; the Standards are a policy document, after all, not a work of art. In common with the urn, however, the Standards were crafted to reward study on multiple levels: from close inspection of details, to a coherent grasp of the whole. Specific phrases in specific standards are worth study and can carry important meaning; yet this meaning is also importantly shaped by the cluster heading in which the standard is found. At higher levels, domain headings give structure to the subject matter of the discipline, and the practices’ yearly refrain communicates the varieties of expertise which study of the discipline develops in an educated person.
Fragmenting the Standards into individual standards, or individual bits of standards, erases all these relationships and produces a sum of parts that is decidedly less than the whole. Arranging the Standards into new categories also breaks their structure. It constitutes a remixing of the Standards. There is meaning in the cluster headings and domain names that is not contained in the numbered statements beneath them. Remove or reword those headings and you have changed the meaning of the Standards; you now have different Standards; you have not adopted the Common Core.
Sometimes a remix is as good as or better than the original. Maybe there are 50 remixes, adapted to the preferences of each individual state (although we doubt there are 50 good ones). Be that as it may, a remix of a work is not the same as the original work, and with 50 remixes we would not have common standards; we would have the same situation we had before the Common Core.
Why is paying attention to the structure important? Here is why: The single most important flaw in United States mathematics instruction is that the curriculum is “a mile wide and an inch deep.” This finding comes from research comparing the U.S. curriculum to high performing countries, surveys of college faculty and teachers, the National Math Panel, the Early Childhood Learning Report, and all the testimony the CCSS writers heard. The standards are meant to be a blueprint for math instruction that is more focussed and coherent. The focus and coherence in this blueprint is largely in the way the standards progress from each other, coordinate with each other and most importantly cluster together into coherent bodies of knowledge. Crosswalks and alignments and pacing plans and such cannot be allowed to throw away the focus and coherence and regress to the mile-wide curriculum.
Another consequence of fragmenting the Standards is that it obscures the progressions in the standards. The standards were not so much assembled out of topics as woven out of progressions. Maintaining these progressions in the implementation of the standards will be important for helping all students learn mathematics at a higher level. Standards are a bit like the growth chart in a doctors office: they provide a reference point, but no child follows the chart exactly. By the same token, standards provide a chart against which to measure growth in childrens’ knowledge. Just as the growth chart moves ever upward, so standards are written as though students learned 100% of prior standards. In fact, all classrooms exhibit a wide variety of prior learning each day. For example, the properties of operations, learned first for simple whole numbers, then in later grades extended to fractions, play a central role in understanding operations with negative numbers, expressions with letters and later still the study of polynomials. As the application of the properties is extended over the grades, an understanding of how the properties of operations work together should deepen and develop into one of the most fundamental insights into algebra. The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.
This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events.
]]>
The intent of the Illustrative Mathematics website is to present sets of tasks that, when taken together, illustrate a particular standard. Eventually, we want to have sets of tasks for each standard that:
• Illuminate the central meaning of the standard and show connections with other standards,
• Clarify what is familiar about the standard and what is new with the Common Core,
• Include both teaching and assessment tasks, and
• Reflect the full range of difficulty expected.
At the moment we have quite a few standards that only have one or two tasks. For the new contest, we are asking people to help us fill in some of the sets that are not yet complete. People can send a task for any standard that already has at least one task, and in addition to sending a task, we are asking people to explain what “gaps” it helps fill. For example, there is one task that illustrates some of the expectations in A-CED.2 “Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.” However, a task that asks students to graph equations that represent relationships between quantities would help round out the set needed to fully illustrate this standard.
Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, March 12th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Feb 13 – Mar 12, 2012.”
We have also created a permanent page with general information about the Illustrative Mathematics Contest that all who are thinking about submitting a task should check out.
Lastly, we would like to encourage any groups doing PD on the Common Core Standards to consider using this contest as a way to approach thinking about the standards and the depth of understanding that they represent. Tasks are reviewed by other professionals in mathematics education and we work with contributors to refine their submission. Listed below are tasks that have been through the review process and are now available on the Illustrative Mathematics website.
Contest Winners
We are pleased to announce a few of the winners of our earlier contests whose tasks are available at the Illustrative Mathematics website!
Dan Meyer — Doctoral student, Stanford University, California. His task is titled “8.F High School Graduation”.
Travis Lemon — Teacher, American Fork Junior High School, Utah. He has two tasks in development and two posted titled “7.RP Cooking with the Whole Cup” and “8.EE Find the Change.”
James E. Bialasik and Breean Martin–Teachers, Sweet Home CSD, New York. Their task is titled “8.EE Cell Phone Plans.”
We have more winning entries making their way through the review process now and will post them here once they are published.
]]>
Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, February 13th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Jan 30 – Feb 13, 2012.”
We have also created a permanent page with general information about the Illustrative Mathematics Contest that all who are thinking about submitting a task should check out.
Contest Winners
We are pleased to announce three of the winners of our first Contest!
Dan Meyer — Doctoral student, Stanford University, California. His task is titled “8.F High School Graduation” and is available on the Illustrative Mathematics website.
Travis Lemon — Teacher, American Fork Junior High School, Utah. He has four tasks in development. We’ll announce it here when they are ready.
James E. Bialasik and Breean Martin–Teachers, Sweet Home CSD, New York. Their task is titled “8.EE Cell Phone Plans”.
We have other people whose names will appear here as their tasks get closer to completion.
]]>
Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, January 30th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Jan 17 – Jan 30, 2012.”
We have also created a permanent page with general information about the Illustrative Mathematics Contest that all who are thinking about submitting a task should check out.
Contest Winners
We are pleased to announce three of the winners of our first Contest!
Dan Meyer — Doctoral student, Stanford University, California. His task is titled “8.F High School Graduation” and is available on the Illustrative Mathematics website.
Travis Lemon — Teacher, American Fork Junior High School, Utah. He has four tasks in development. We’ll announce it here when they are ready.
James E. Bialasik and Breean Martin–Teachers, Sweet Home CSD, New York. Their task will appear soon!
We have other people whose names will appear here as their tasks get closer to completion.
]]>
The new theme for this week is focused on Expressions and Equations and we recommend reading the progressions document for 6 – 8 as you think about the task you would like to submit. People are invited to submit tasks for these specific standards:
Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task, which must be emailed by Monday, January 9th, 2012 midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Jan 2 – Jan 9, 2012.” If your task is accepted, we will notify you the week following the deadline. We may ask you to work with us to revise the task before we accept it. People may submit multiple tasks. Any questions about the contest should be sent to the same email address with subject line “Question about Illustrative Mathematics Task Writing Contest Jan 2 – Jan 9, 2012.”
How it Should Look
The Illustrative Mathematics team would like to encourage participants to review some of the tasks already published on the website (once you follow this link, click on e.g. K-8 Content Standards with Illustrations) for standards 6.RP (Shirt Sale), 7.RP (Tax and Tip), and F.BF (Kimi and Jordan) to get an idea for how tasks should look. Because the tasks are entered into the database in a particular way, the format is important. Also keep in mind that all task submissions must include at least one complete solution; here is a word_template that shows the fields that need to be included. We will give extra consideration to tasks written by pairs or teams of people, tasks that have natural connections to other tasks related to this stream, and tasks with insightful commentary. Please submit tasks in word format or LaTeX, along with a pdf if possible. To learn more about what makes a good mathematical task, read this article by Kristin Umland.
We look forward to another great set of tasks!
]]>ccss_progression_sp_68_2011_12_26_bis
As usual, comments and suggestions are welcome. [New file with corrections uploaded 12/26/11, 11:38 am MST.]
[5 August 2012] This thread is now closed for comment. Please ask questions about Grades 6–8 Statistics and Probability here.
]]>Second, here is a report from a meeting organized by Paola Sztajn, Karen Marrongelle, and Peg Smith: http://www.nctm.org/uploadedFiles/Math_Standards/Summary_PD_CCSSMath.pdf
]]>Authors of tasks selected for inclusion in the Illustrative Mathematics task bank will receive $200 per task (not per author, sorry!) and must be emailed by Monday, December 19^{th} midnight in your local time zone to illustrativemathematics@gmail.com with subject line “Submission for Illustrative Mathematics Task Writing Contest Dec 12 – Dec 19, 2011.” If your task is accepted, we will notify you the week following the deadline. We may ask you to work with us to revise the task before we accept it. People may submit multiple tasks. Any questions about the contest should be sent to the same email address with subject line “Question about Illustrative Mathematics Task Writing Contest Dec 12 – Dec 19, 2011.”
How it Should Look
All task submissions must include at least one complete solution. We will give extra consideration to tasks written by pairs or teams of people, tasks that have natural connections to other tasks related to this stream, and tasks with insightful commentary. Please submit tasks in word format or LaTeX, along with a pdf if possible. Here is a word_template.
Things You Should Know Before Submitting
Writing a great task is an art, and tasks often benefit from multiple revisions. It would be helpful to read some of the tasks that have already been accepted at http://illustrativemathematics.org. To learn more about what makes a good mathematical task, read this article by Kristin Umland.
We look forward to reading your tasks!
]]>New Version of the CCSSM Clickable Map is now up on the Tools page!
]]>http://ime.math.arizona.edu/commoncore/
IM&E is working as a part of a collaboration between NCTM, NCSM, ASSM, AMTE, CBMS, Achieve, AFT, IAS/PCMI, MfA, NAGB, and NEA to produce a day long professional development on the Common Core. The effort is authored by teachers across the country, will be field tested by teachers, and ultimately facilitated by teachers with teachers as the targeted audience. The toolkit will have activities hitting four main goals; to see structure in the standards, to understand the Standards of Mathematical Practice, to align tasks to the standards, and to understand the language used in the standards. Activities will be developed for Elementary, Middle School, and High School teachers. As we develop tools and activities for this project we will be posting them here in the Tools section of the blog so check back soon for PD tools to further explain the Common Core in Mathematics. The toolkit should be ready for initial pilot testing this summer so stay tuned if your school or district might be interested in participating as a beta testing site for activities designed for the Toolkit.
]]>Eventually the sets of tasks will include elaborated teaching tasks with detailed information about using them for instructional purposes, rubrics, and student work. Such a fully developed set of tasks will be what we call a Complete Illustration of the standard. Right now we are trying to build up our collection of Initial Illustrations of standards, which will have the following characteristics:
The new site also allows users to register. This is not necessary to see the tasks, but if you register you will be eligible for news bulletins and various opportunities for involvement in the project that will arise over the next few months.
Go to illustrativemathematics.org to see the new goodies. (This is still a beta site, and you may encounter slowness or other problems from time to time.)
]]>If you are reading it with Adobe Acrobat, you will need the latest version (10.1). If you are using a Mac, you can also use the native Mac pdf reader Preview, or the open source pdf reader Skim.
We hope to get drafts of the remaining K–8 progressions out soon; thank you for your patience.
[File updated 2/5/12 to fix printing problems.]
[File updated 9/19/13 with corrections.]
[31 July 2012] This thread is now closed. Please ask questions here.
]]>[Edited 2011/7/09 to add pdf file.
Documents Edited 2011/12/01]
We will be releasing a draft of the Fractions progression before the end of June, and the Ratios and Proportional Relationships progression in early July.
[29 July 2012] This thread is now closed. You can ask questions here.
]]>]]>
[29 July 2012] This thread is now closed. You can ask questions here.
]]>[29 July 2012] This thread is now closed. You can ask questions here.
]]>[2012/08/31. This thread is now closed. Please ask questions here.]
]]>[29 July 2012] This thread is now closed. You can now ask questions on this progression here.
]]>More information is available online at:
http://www2.edc.org/CME/mpi/workshop.html
“Articulating Research Ideas that Support the Implementation of the Professional Development Needed for Making the Common Core State Standards in Mathematics Reality for K-12 Teachers is a newly funded NSF project that will coordinate knowledge from different fields to develop recommendations for the design, implementation, and assessment of large scale professional development systems consistent with the mathematics of the CCSS. Research results from diverse perspectives (e.g, mathematics education, organizational theory, professional development) will be articulated into a coherent framework and a set of recommendations for successful large-scale, system-level implementation of mathematics professional development initiatives. The recommendations will be disseminated through the National Council of Teachers of Mathematics. Additionally, the Association of Mathematics Teacher Educators, the Mathematical Association of America, the National Council of Teachers of Mathematics, the National Council of Supervisors of Mathematics, and the Association of State Supervisors of Mathematics are partners in this effort.”
]]>1. Read the standards, noting domains, clusters that are particularly high priority for college and career readiness (don’t include individual standards unless you absolutely must).
2. Select the most important domain or cluster at each of the grades 6-8 and themes in high school (or just some of the grades and themes if you don’t have time for all).
3. [Optional] For each one pick one or two practices that are particularly salient, and explain how it is exemplified.
4. Rule: Give higher priority to things that are harder to fix [if students come to college not having them], not things that you hate to have to fix but that are easier.
5. Write a one page common agreement on priorities that both higher education and high school (1) accept as important (2) clearly understand.
[Post edited for clarification, 2/19/11]
]]>Big question: what is going to be possible to do this time that it wasn’t possible to do before?
There are echoes of the new math in some of what is going on now, and there are lessons to be learned from more recent reform efforts.
Project at Michigan that is focusing on coherence across grade levels. The New Math had its take on coherence; NCTM Curriculum and Evaluation Standards, PSSM, and Focal Points all had their take. When I was here in California in the 1980s, California had a progressive framework, followed in the 90s by a framework which many saw as a U-turn. Teachers saw a radical shift. In the 10 years I’ve been at Michigan, teachers have had three different sets of standards. Some teachers are numb to it.
On the content side, the big thing that Common Core brings is understanding to the expectations. Banned in Michigan and many other states, because of assessment. I’m not sure that 50 years after the New Math we are any closer to figuring out how to assess understanding. We also have the standards for mathematical practice, but still not in the content. (Good to call them standards.)
To the extent that the assessment can be seen as driving attention to the practice standards, the conversations with teachers will be a lot easier. Not sure that the outline of content is any better than any other we have had. New Math and PSSM had unintended consequences (back to basics, math wars respectively).
The important issue is scalability. Teacher education is a state driven enterprise. Now that everybody is adopting the same set of standards. That allows for collaboration across institutions and across state lines. That’s a very exciting project. AMTE can be at the forefront of doing that work.
]]>Big question on number 3: what do we need to do now?
Long term efforts:
]]>“More testing isn’t necessarily better,” said Dr. Linn, who said her work with California school districts had found that asking students to explain what they did in a science experiment rather than having them simply conduct the hands-on experiment — a version of retrieval practice testing — was beneficial. “Some tests are just not learning opportunities. We need a different kind of testing than we currently have.”