From Al Cuoco, commentary on Common Core

Over the last few months, the Common Core State Standards have hit the headlines in a big way, with commentaries from educators, elected officials, columnists, and even talk-show personalities. Some argue that the Core is rotten, a liberal tactic to usurp local control. Others express faith in the Core’s ability to transform K-12 education, providing a lever needed to budge the stubborn inertia that stalls improvement. Sides have formed and the simmering rhetoric has come to a full, rolling boil.

As a former high school mathematics teacher — 25 years in the Massachusetts Public Schools — and a current curriculum developer and teacher educator, I’ve read the mounting criticism of and praise for the Common Core State Standards in Mathematics with great interest. Much of the criticism that I read is oversimplification; some of it exhibits a genuine ignorance of the daily work of teaching and learning. Very little shows familiarity with the actual content of the standards, and much of it is removed from a focus on the two groups that matter most: teachers and students.

When I first heard of the Common Core — I supplied some background research to the writing team — I saw a promise that it might become one of many tools to help us tackle some of the pervasive problems that have plagued our schools for generations. The authors delivered on that promise. And there’s one problem in particular for which these mathematics standards hold great potential: closing the vast gap between school mathematics and mathematics as a scientific discipline.

Many adults have suffered through school mathematics. At the high school level, it’s a disconnected zoo of special-purpose methods and meaningless topics that most mathematicians wouldn’t recognize. There’s the “box method” for setting up equations. There’s the “y = mx+b” method for finding equations of a line through two points. And there are many more examples of paraphernalia that has no existence outside of school. There’s also incredible flatness: everything is as important as everything else. I’ve seen a half-period wasted discussing whether or not the positive y-axis is in the first quadrant.

Common Core calls for something different. Its content standards are devoid of schoolish nonsense and free of the standards-speak that has made so many state documents unreadable. Anyone who uses mathematics professionally can read the Common Core and recognize what’s there, a direct path from the mathematics students learn in school to the mathematics used in so many fields today. There is a practice of mathematics, just as there is a practice of medicine and a practice of teaching. Before the advent of the Common Core, that practice was largely invisible in state standards documents.

The Common Core reflects the reality that the discipline of mathematics is as much a web of habits of mind — key ways of thinking employed by proficient users of mathematics — as it is a vast edifice of results. And, unlike current state standards, the Common Core specifies Standards for Mathematical Practice that give us a lens through which we can look at the content standards. The Standards for Mathematical Practice can help bring some desperately needed coherence and parsimony to school mathematics. The Standards for Mathematical Practice are not, as some claim, the “most important part” of the Common Core. Rather, as Jason Zimba, one of the lead writers for the Common Core, puts it: “Practices are an engine for focusing, and a reward for doing so.”

Because the centuries-old practice of mathematics has not appeared in an explicit way until now, the Standards for Mathematical Practice are a source of confusion among educators and school administrators. In Massachusetts, the Department of Elementary and Secondary Education has addressed this challenge head on: it partnered with our group at Education Development Center to create a 45-hour course designed to provide experience with and examples for how the practice of mathematics can work in concert with specific content.

The course revisits everyday content that has traditionally been difficult to teach and learn, and it develops that content with a focus on practices such as using precision, making viable arguments, and abstracting from examples. These habits of mind are essential to mathematics outside of school, and they provide teachers and students with tools to make school mathematics a coherent and sensible body of knowledge. Teams of teachers who have participated in the course are excited about mathematics and are now teaching it around and outside Massachusetts. Most importantly, we’ve seen that debates about the value of the Common Core disappear when teachers are immersed in concrete methods for bringing the standards to life.

So, there is promise. By raising the practice of mathematics to the same level of importance as results, I see hope that students will no longer be subjected to the numbing stranglehold of school mathematics. But Common Core is not, in itself, able to transform education. The problems go deeper than standards—they go down to the kind of mathematics teachers learn in their preparation and professional development. They go down to the oppressive working conditions that encourage quick fixes, shallow treatments of mathematics, and a profession that feels under siege. They go down to tyrannical assessments of low-level skills that cripple any chance of a coherent approach to school mathematics. And they go down to the outdated foundations on which schooling was built that advantage certain groups over others. Common Core is our first step, a blueprint for what school mathematics can be. Everyone involved in education and policy should get past this first step. Stop quibbling about the standards themselves — they are a refreshing and positive first step — and mobilize to focus resources on the rest of the picture.

BRIEF BIO INFO

Al Cuoco, Distinguished Scholar at Education Development Center, Inc. in Waltham, Mass., taught high school mathematics in the Woburn, Mass. public schools for 24 years. He received his doctorate in mathematics from Brandeis University. He is a widely published author and presenter on issues such as effective mathematics curriculum and mathematical habits of mind. He is the lead author on the 9-12 CME Project mathematics curriculum, an NSF-funded program founded on mathematical habits of mind, deeply interconnected with the Standards for Mathematical Practice.

The author thanks Eden Badertscher for our many discussions and her deep insights into the ideas in this essay.

For more information about the Massachusetts DESE course, see http://www.doe.mass.edu/candi/institutes/2013/math.html?section=3 and http://mpi.edc.org.

Links:

http://profkeithdevlin.org/2013/06/19/faulty-logic-in-the-new-math-wars-skirmish

http://mathsugaroff.wordpress.com/2013/06/12/practice-standard-3-construct-viable-arguments-and

http://www.edexcellence.net/commentary/education-gadfly-daily/common-core-watch/2013/critics-math-doesnt-add-up.html

From Al Cuoco, a commentary on Common Core

Due to an error, a commentary by Al Cuoco on the Common Core was mistakenly posted ahead of schedule. The commentary will be reposted in its entirety in January. Our apologies. The following is a short blurb from the commentary.

Over the last few months, the Common Core State Standards have hit the headlines in a big way, with commentaries from educators, elected officials, columnists, and even talk-show personalities. As a former high school mathematics teacher — 25 years in the Massachusetts Public Schools — and a current curriculum developer and teacher educator, I’ve read the mounting criticism of and praise for the Common Core State Standards in Mathematics with great interest. Much of the criticism that I read is oversimplification; some of it exhibits a genuine ignorance of the daily work of teaching and learning. Very little shows familiarity with the actual content of the standards, and much of it is removed from a focus on the two groups that matter most: teachers and students.

MPs in High School: Algebra as Bookkeeping

Al Cuoco, the director of the Center for Mathematics Education at EDC, has written some essays that show how mathematical practices can be used in high school. The essays explore examples of Common Core’s Standards for Mathematical Practice. We’ll post all the essays here in the coming weeks.

For more information about EDC programs around the Mathematical Practices, visit http://mathpractices.edc.org and http://mpi.edc.org/dmp-hs-sampler.

The second essay is on the use of formal algebra for bookkeeping, and you can use the methods to solve at least one of this site’s “We Got A Problem” problems. (Click on the link to view PDF.)

MPs in High School: Perseverance

We have two different programs at EDC devoted to the Standards for Mathematical Practice., Implementing the Standards for Mathematical Practice (IMPS) and Developing Mathematical Practice in High School (DMP). For more information, see each program’s website:

http://mathpractices.edc.org/

http://mpi.edc.org/dmp-hs-sampler

These programs involve workshops for teachers. Al Cuoco, the director of the Center for Mathematics Education at EDC, has written some essays that show how mathematical practices can be used in high school. The essays were written for these projects, but we think the essays might be useful to others working to implement the Common Core. We’ll post the essays here in the coming weeks.

The first essay is on perseverance. (Click on the link to view PDF.)

We Got A Problem #13: The Birthday Problem

At my son’s recent 5th birthday party, he got 15 birthday cards.  Sure enough, he got more than one of the same card.

Suppose everyone buys their cards at Foyerjohn, and randomly picks one of the cards.  How many different cards would need to be on sale for there to be a 50% chance that all 15 people pick a distinct card?

Harder: how many different cards would need to be on sale for there to be a 50% chance that my son doesn’t get three of the same card?

Fostering a New Generation of Confident Mathematical Thinkers

EDC’s Matt McLeod has posted some thoughts on teaching and coaching with a focus on mathematical habits of mind.  He describes some parts of an ongoing research study about how teachers’ impressions of mathematics change when teaching from a curriculum that emphasizes these habits.

http://ltd.edc.org/fostering-new-generation-confident-mathematical-thinkers

We Got A Problem #13: Deal or No Deal

This short game is played with ten playing cards: an ace, two, three, …, nine, and a joker.  I shuffle the cards and lay them face down.

Your decision: grab as many cards as you want, still face down. When you’re ready, flip them all over at once and win: $1 multiplied by the sum of the card values.

Except the joker. If you flip over the joker, you win nothing. Good day.

So, how many cards should you take to maximize your expected return? What would change if all nine non-joker cards were aces? Eight aces and a ten?

[Testing Testing] SBAC Grade 11 Practice Test #4

Each “Testing Testing” post analyzes a released test item, focusing on both the mathematics and interface involved in the new breed of exams. Our goal is to help improve the quality of these exams, especially if they may be used to inform student graduation or teacher merit pay. The mathematical analysis here is from Al Cuoco, director of the Center for Mathematics Education at EDC, from the Trevi Fountain in Rome. I provide the interface analysis.

Here’s Problem 4 from the SBAC Grade 11 Math Practice Test.

q4

There are several issues with this item. By far the greatest issue is that it doesn’t assess a Common Core standard. The closest standard is HSA-APR.B.3, and read it carefully:

  • HSA-APR.B.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

This problem doesn’t ask students to do that. It asks students to use a rough sketch of a graph to build a factorization, identifying the linear factors of the underlying polynomial.

Besides this, there are some serious mathematical issues with this item as presented. The correct answer to this type of problem should start out in one’s mind as a multiple of (x+3)(x+1)(x-2)^2(x-4) , applying the Remainder Theorem (HSA-APR.B.2). Then, you check a sixth point to find the multiple, making sure the polynomial agrees with the nonzero points on the graph (a polynomial function of degree n is determined by n+1 function values, not just its n roots). But no coordinates of a sixth point are given. A student could simply not think of this and get the problem right, or estimate the value of the function at 0 and see if the graph is approximately correct (and get it right), or assume that this is a test and read the minds of the test writers (and get it right). It’s a good thing the graph seems to pass through (0, -48), or there would be no way to give a correct answer!

Even if you could get the exact coordinates of f(0), there are other polynomials functions whose graphs contain these same points and that have an extra x-intercept off the picture. The phrase “the function for the graph” is not accurate: there is more than one. Better wording: “Find a polynomial function that could have this graph.” The phrase “Create the function” should also be avoided, since it’s not clear what it means; Common Core has students “define” a function or “build” a function for modeling or transformation.

Still, the core issue is that this problem does not directly address a standard. The problem would be much better if it assessed HSA-APR.B.3 directly: Give a function defined by a polynomial that has been partially factored, revealing some zeros, allowing for complete factorization. For example:

g(x) = 3(x^2-1)(x-2)^2

Students who can sketch the graph of g (or answer questions about the graph) have met several HSA-APR standards.

Separate from the mathematics, there are a few major interface issues with this problem. Students are only allowed to drag linear factors to the right, so it is not possible for a student to enter (x-2)^2. They must instead enter the clumsy (x-2)(x-2). Some students will wonder why they have to do it this way, others will think the interface is broken. It’s an unnecessary hurdle that will prevent some students from answering correctly even though they are capable of completing the task on paper.

Like #11, the interface does a poor job of overwriting when new objects are placed. Dragged objects “snap” into one of five positions, and if an object is already there, conflict:

q4a

We can picture a student trying to do this on purpose, attempting to get the (x-2)^2 term. It easily happens by accident and the interface for “deleting” objects is not obvious. Would the answer above be marked correct or incorrect? It’s not clear, and that is a big problem.

The SBAC Practice Tests are available for public viewing, and we are grateful to have these problems available for public comment.

[Testing Testing] SBAC Grade 11 Practice Test #11

Each “Testing Testing” post will analyze a released test item, focusing on both the mathematics and interface involved in the new breed of SBAC and PARCC exams. Interfaces can have huge effects on how problems are posed, read, answered, and scored. Posts will also provide suggestions on how to improve these exams, in terms of the mathematics presented, and also in terms of the interface students will use when taking exams.

Here’s Problem 11 from the SBAC Grade 11 Math Practice Test.  (A colleague suggests I have chosen Problems 7 and 11 as the first examples primarily because I love Slurpees.)

q11

This problem targets HSA-REI-D.10 and HSA-REI.D.11, knowing that the points on a graph are the solutions to the corresponding equation, and that the x-coordinates of the points where the graphs of y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).

There is one mathematical error in the problem. When asking for “a solution for f(x) = g(x)”, the answer is an x-coordinate, not a point — something HSA-REI.D.11 states explicitly. This placement of points should be correct:

q11a

The problem interface will not allow the third point to be placed here. A placed point “snaps” to the nearest marked point on the grid. These interface behaviors need to be described in the problem, or the results can be surprising or frustrating. For example, a student might want to place their solution to y = g(x) at (0,0). The problem doesn’t exclude them from doing this, but the interface does.

The interface also causes trouble when points are placed atop one another. It looks like this:

q11b

Note that the newly placed point is not in the same position as the original. I can’t tell whether or not this solution would be accepted on the exam, and fixing the problem requires moving away the new point, then the old point, then replacing the new point. It’s a mess. If the exam requires all three points to be different from one another, this needs to be part of the problem statement, and the interface should “push” old points back to the gray area at the bottom if a new point is placed in the same location.

These interface issues can be fixed. Eliminate the dragging, label each target point in the grid, and ask the questions about the specific points. This also allows more explicit asking about HSA-REI.D.11 rather than the previous “solution for f(x) = g(x)”.

q11_edited

This new version keeps alive what makes the original question better than multiple-choice: there are several correct answers to each part. One might consider changing this version (or the original) to require the student to provide points that are on the graph of one function and not the other: “Name a point that is a solution to y = f(x) but not y = g(x)”.  Otherwise, a student can answer all three questions correctly with “C” without assurance that they have mastered the standard.

The SBAC Practice Tests are available for public viewing, and we are grateful to have these problems available for public comment.

[Testing Testing] SBAC Grade 11 Practice Test #7

We’re starting a new category of post here called “Testing Testing”. Each post will analyze a released test item. While this would normally focus just on the mathematics of the problem, the SBAC and PARCC exams are also promising new formats and interfaces for testing, which can potentially have effects on how problems are posed, read, answered, and scored. Writing assessment items for these new interfaces will be especially challenging, and it may take more revisions than usual to build a high-quality test item.

Recently, SBAC (short for Smarter Balanced Assessment Consortium) released a set of Practice Tests, giving everyone the chance to study the testing environment and suggest improvements. I took the 11th grade mathematics exam, with 23 problems, trying to get a sense of the quality of the mathematics and the quality of the interface. Several posts will talk about items from the SBAC Practice Test, along with suggestions to SBAC on how to improve both the mathematics of the exam and the interface students will use when taking it.

Here’s Problem 7 from the SBAC Grade 11 Math Practice Test.

q7

At first glance, this problem looks pretty innocent, and has a useful testing target, specifically HSA-REI.D.11, finding approximate solutions to f(x) = g(x) when f and g are given by tables. The test designers were even careful to use function types prescribed by the standard: there’s a quadratic rule matching f and an absolute value rule matching g. This could have been a good item, expecting students to interpret the table in a meaningful way.

However, there are two serious errors in the problem. First, the problem is never stated. The problem asks to “show the interval for x in which the solution to this system of equations lies”. But there is no system of equations. There’s not even an equation at all, just a table. To get this question right, the student must figure out the intended question, the interval for x that contains the solution to f(x) = g(x). Note that this still isn’t a system of equations, just a single equation with a single variable.

Second, and even more serious, there is more than one correct answer to the problem. The only thing we are told about functions f and g is that they are continuous. We are not told that they are quadratic, absolute value, or otherwise. We cannot assume that the functions do not intersect in other intervals: in fact, they can easily intersect. These two functions match the given tables and intersect many more times:

f(x) = x^2 - 5 + 100 \sin (2 \pi x) \text{ and } g(x) = |120-17x|

Because f and g are continuous, we know they must intersect somewhere between x=5 and x=6, but we do not know for certain that they will not intersect elsewhere.

These issues are very serious and require correction. There are other minor issues: “f(x)” isn’t a function, it’s an output of the function f (see HSF-IF.A.1). Functions don’t have points, they have ordered pairs; graphs of functions have points. Neither of these minor issues spells doom for the problem, but these details can be corrected.

On the plus side, the interface for the problem is clean and the task for the student is clear (“Click the number line”). Clicking marks a unit interval in red, and clicking anywhere else changes the interval to reflect the new click.

q7a

This problem can be repaired simply by changing the text that appears on the left: “The table shows several inputs and outputs for continuous functions f and g. [table] The graphs of f and g intersect exactly once. Click the number line to show the unit interval for x in which the solution to the equation f(x) = g(x) must lie.”

I encourage you to take the SBAC Practice Test. I’ll have more to say about other problems. My hope is that through the work of the community we can improve these exams while they are still in their pilot stage.

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