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.

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