Pi Day Activity Part Two

It may not display well but I'm posting an image of the second part of the Pi activity I posted the other day. This kept the group engaged for over a period. It may not help them to understand the deep meaning of pi or its relation to a circle but they definitely got a sense of the decimal representation of pi and struggled with some of the birthdates, since a simple Google search will not suffice. Do they now appreciate pi more? How will I assess the learning from this? I hope you will see that this was more than just meaningless busywork to keep them quiet! See if you can find all of the mathematicians in less than 10 minutes!

Anatomy and Evolution of a Difficult Math Problem

Note: The problem below was chosen for the March 23rd Carnival of Mathematics. If you'd like to see a different kind of mathematical challenge, I recommend you also visit the posting for 3-30-07 (Challenging Geometry: Circles Inscribed in Quadrilaterals, Right Triangles).

Ever wonder how much thought goes into developing a challenging math question for a standardized test like SATs? The goal on most standardized tests of reasoning is to have a small percentage of questions (usually at the end of a section) that are more discriminating. A question accomplishes this goal if less than, say, one-third of the test-takers answer the item correctly (testing and measurement experts may have a different percent in mind here). The question is first field-tested and if the p-value (% who get it right) is too low, the question may be rejected.

Let's look at how such a challenge problem may evolve.

First Version:
322 has two identical digits. What is the next larger 3-digit number of this type?

Comments: Way too easy since the answer is the next integer, 323 and even students who are not thinking will probably not choose 333 which has 3 identical digits (333 could be 'of this type' since the wording was vague). Some may choose 344 however. Does this question have promise? Can we improve the wording and ratchet it up?

Second version:
6,333 has exactly 3 identical digits. What is the next larger such integer?

Comments: Better? We added 'exactly' to make it clear that it couldn't have more than 3 identical digits, an important distinction. I dropped the 'number of digits' info since it wasn't needed. Will many students choose 6,444 rather than 6,366? Probably, so should we stop here and field test it?

Third version:
81,111 has exactly four identical digits. What is the next larger such integer?

Comments: This should be slightly more discriminating than the previous because the string of ones is more seductive and will probably lead many to be lured into 82,222, rather than the correct answer of 81,888. The stronger student will be suspicious here, particularly if this question is placed near the end of a section. Further, this question cannot appear on the SAT since one cannot grid in a 5-digit answer (4 is the maximum). So we need one more version...

Fourth and final version:
96,666 has exactly four identical digits. If N represents the next larger such integer, what is the value of N - 96,666?

Comments: Ok, now the answer 'fits' in the grid. Experienced item writers and test constructors know that a majority of random test takers will grid in 1111 as the answer, particularly if this is near the end of a section and time is a factor. The student who has stronger reasoning AND can think under pressure will probably realize that N could be 96,999 so the correct answer is 333. Sorry to give it away, but today's post is about constructing a more difficult problem rather than challenging you! BTW, a question very similar to this has appeared on the SATs and in some SAT prep books.

So what's the point of all this and how do I or any item writer know what makes a question harder? EXPERIENCE! (I almost yelled 'TRADITION' since I watched my 13 year old perform in 'Fiddler' last week!). I've given this question or a similar version to SAT prep groups for some time now. In fact, I did it this past Saturday. The results? First group, out of 23 students, 3 answered it correctly. Second group, out of 11, ONE answered it correctly. Since the percentages were considerably less than 20%, would this question ever make it out of field testing? Well, I believe it did! From my own experience, less than 15% of students answer it correctly and my sample space consists of above-average students, in fact, fairly strong students.

How many of you are thinking that this question is unnecessarily 'tricky' and really has no place on a math test? What important math skill or concept is it assessing? Well, some questions are assessing reasoning ability and this is one of them. Does this question fairly discriminate? If the highest-scoring students got it wrong, then it DOES NOT discriminate! However, that's not what happened. 'K' was the only student who answered it correctly in my second group and I predicted she would. She has exceptional reasoning aiblity. She gave me a look of "This wasn't that hard, Mr. Marain, why are you complimenting me so much!" For her it wasn't that hard and that's the point. Is there a place for these kinds of questions on assessments when most students have little exposure to these. Well, the test is not intended to be just a reflection of homework problems from a textbook. This is precisely why many educators and leaders have challenged the SATs over the past few years and why the test was changed a couple of years ago. But there will always be a couple of these...

I'm sure many of you have strong opinions about the educational validity of this question, particularly for a standardized test. If nothing else, try it out in your classes (or give it to your spouse, child, colleague or co-worker) and report the results. If over 50% of the group answer it correctly (give them 30 seconds), I would guess you have one special group there! So, folks, does this much thinking really go into writing one little old test question!

Update on National Math Panel

As promised, I am posting, with the permission of the National Math Panel, the reply to my latest email to the Panel. Jennifer Graban sent me this last Friday and you can read it for yourself and decide if you believe that some of my recommendations and those of others are now being considered seriously. Below, I will also briefly discuss the conversation I had with Ms. Graban when I called her:

Dear Mr. Marain,

We apologize for the delay in responding to your e-mail of February 11 and acknowledge your ongoing concerns about the composition of the Panel. Understanding the experiences of teachers, particularly algebra teachers currently teaching in the classroom, is important to the Panel and its final report. I have shared your concern with the Panel Chair, Larry Faulkner, as well as other Panel members. We are currently considering additional opportunities that will involve teachers to obtain their vital input in the work of the Panel.

Please know that Panel is most concerned with and plans to carefully consider the perspectives of today's math teachers.

Once again, thank you for your interest in the National Math Panel.

Sincerely,

Jennifer



Jennifer Graban
National Math Panel Staff
U.S. Department of Education
400 Maryland Avenue, SW
Washington, DC 20202-1200


I decided to call Ms. Graban last Friday afternoon around 4 PM EST. She picked up the phone immediately and we spoke for about 10-15 minutes. To the best of my recollection here's how it went:
I began by saying that I felt it important to put a voice to all the emails and blogs and let her know that I am a real person and not the enemy, just someone who feels that not having at least one secondary teacher on this panel was a gross oversight. Continuing, I indicated that, although it would be better to appoint 2-3 dozen classroom math teachers immediately, the reality is that the panel has less than one year to complete its task and publish its final report. Therefore I suggested the importance of identifying a cross-section of a few current high school math teachers with broad experience in both urban and suburban/rural school settings. I also volunteered to provide whatever input I could to the Panel, but certainly this group of teachers should be allowed to serve in an advisory capacity, i.e., freely provide their ideas and be be able to review and make suggestions now and to the final draft of the report before its release. I expressed that this would significantly enhance the credibility of the report. She replied that this was already being considered and she seemed to concur with some of my statements. I also mentioned my call for a national math curriculum but she indicated that the panel has not yet endorsed this. I also asked her if she had noted that the textbook publishers, who were allowed to give extensive presentations to the Panel, had indicated they now had editions for EACH state or nearly so. I expressed how absurd I felt this was and she did not disagree! We left the conversation cordially.

Problems 3-5 thru 3-6-07: Geometry and Reasoning

Note: The problem below was chosen for the March 23rd Carnival of Mathematics. If you'd like to see a different kind of mathematical challenge, I recommend you also visit the posting for 3-30-07 (Challenging Geometry: Circles Inscribed in Quadrilaterals, Right Triangles).



Today's questions involve well-known ideas from geometry. Similar questions have appeared on SATs and math contests.
Some suggestions: Use it for in-class enrichment or assign it for extra credit outside of class. The first part lends itelf to a fairly simply visual approach (cutting up the square and matching the pieces), but the second is more sophisticated. Encourage the visualization but require the analytical approach as well!
Reviews: 30-60-90, equilateral, areas, symmetry, etc.

National Math Curriculum UK

Just to see how England is handling a national curriculum (for all subjects), here's a link to the Mathematics section. You will need to navigate this site for awhile to get the feel for it. You will also want to download the 90+ page pdf math curriculum document. I've skimmed it and it is fascinating. Of course much of it is not new or surprising or that different from NCTM's many recommendations or reports, but it's the overall structure and expectations for the four key stages of learning that make it unique. Why can't we benefit from something like this so that we don't have to reinvent the wheel? I'm certain that members of the National Math Panel are familiar with this but I'd be interested in their views. Is it possible for our society? Better yet, perhaps we need to begin to realize the vision of Robert F. Kennedy:

There are those that look at things the way they are, and ask why? I dream of things that never were, and ask why not.

Things that never were? But they are already existing elsewhere!

Better yet, download
A Coherent Curriculum: The Case for Mathematics by William Schmidt, Richard Houang, and Leland Cogan is available online in American Educator, Summer 2002, pp. 1-17.

This was a call to action five years ago and we're still arguing about it today...

Some updates...

1. I received a reply from Jennifer Graban of the National Math Panel. Her comments indicated recognition by the panel that there is a need for greater participation of classroom teachers, algebra teachers in particular. I decided to call Jennifer personally and we had a very pleasant conversation. Electronic communication and blogs can never replace human interaction! I'll share the email and our dialogue later...
2. I've posted a solution in the comment section to the last part of the nested radical problem -- finding an expression for N that would produce an integer value for the infinite nested radical.
3. You may want to read the comment I posted regarding Math Anxiety. It describes a method that I have found useful for conscientious students who tend to underperform on teacher-made tests. There is also a wonderful set of tips for students to prepare for standardized tests like SATs. It's entitled Ten Ways to Survive the Math Blues from Murray Bourne at squarecirclez and I will share this with the students in my school.

An Online Pi Day Challenge

Even though pi day is about 2 weeks away, here's an online challenge I gave to all the students in my school back in 2005. Not very 'mathematical' but it engaged them. I posted the problem at 6 PM on March 14th and one student found a web site and submitted her solution in under 7 minutes! Get the feeling that this generation is able to learn a slightly different way!! For those offended by the fact that pi day is celebrated in other countries on July 22nd (22/7), I apologize! I am well aware that many teachers, math departments and schools have created wonderful pi day activities and there are abundant examples of these on the web. I would like others to share their favorites as well. Learning more about pi from MathWorld or Wikipedia is well worth our time, although it does get very technical. The nested radical problem from the other day (with the square roots of 2) relates to pi! Look it up!!


Background: The first 5 places (decimal digits) in pi, after the decimal point, are 14159. Find a web site that will allow you to search millions of places in pi.

The Challenge:
(1) Write down the seven decimal places in pi starting with the digit in the 5,191,306th place.

(2) These digits are a clue to the birthday of a famous mathematician (sorry, it's not Einstein!).
Give the full name of this mathematician and how he is connected to the number pi! Your submission should include the full birth date, the name and the connection.

EXTRA: Now challenge us - find another famous mathematician who is associated with pi and ask your own 'pi' challenge question!

Is Math Anxiety Caused by More than Lack of Practice and Skill?

You may have to register first, but you might want to link to the article Understanding Math Anxiety in a recent edition of Teacher Magazine. I would be interested in the perspectives of both math classroom teachers and those outside of the profession. Many teachers I've spoken to feel that anxiety decreases with increased student preparation but from my own personal observations, I suspect there's more to it than that. Do you believe that 'performance anxiety' on standardized math tests is real or a fabrication? If real, what are the implications for the classroom teacher? Simply make our classes more rigorous and our tests harder so that students will be accustomed to performing under more pressure? Or is something else needed...
One of the cognitive psychologists quoted in the article, Robert Siegler, is a professor of cognitive psychology at Carnegie Mellon University. Note below his current participation on the National Math Panel. Here is a quote:

Students feel more anxiety in math partly because they are dealing with so many concepts and procedures that are foreign to them, said Robert S. Siegler, a professor of cognitive psychology at Carnegie Mellon University, in Pittsburgh, who has examined children’s thinking abilities in math and science. Once students realize they do not grasp a math concept, the internal pressure grows.

“Math entails certain conceptual barriers that lead people to read the same passage over and over again and not understand it,” Mr. Siegler said. By contrast, in reading a history lesson, students are likely to recognize vocabulary, themes, and ideas, even if they do not understand all the implications of a particular passage.

“You don’t feel like you totally didn’t understand it, and you’re just floundering,” he said.

Mr. Siegler is one of 17 people serving on the National Mathematics Advisory Panel, a White House-commissioned group charged with identifying effective strategies for improving instruction in the subject. The panel includes a number of cognitive psychologists, along with education researchers, mathematicians, and others.