My final Geometry post

Summer Geometry is finally over - and not a moment too soon. I don't think either I or the kids could have stood another week of it. But we all got through it and we're still on good terms (as far as I know!) I brought them bagels and juice to eat during the final review session right before the test, and then they got to take the test and leave when they finished - just like in college! It's also much less stressful than making the students who finish early sit there in silence until the end of the period.

The test was half multiple choice, and half other (vocabulary, angle webs, constructions, and proofs). I wasn't quite sure how they would do, but overall, it went well. The average score on the final was an 82% (which is much higher than my normal algebra averages), and only 3 students failed the test (with scores in the 60s).

I know that these students will have some weaknesses in their Geometry skills due to the rapid nature of the class (and the fact that we simply couldn't get to everything), but I think that we built a strong enough foundation for them to be able to move forward in pre-calculus.

An amusing story: I am a vegan, and the students love to pepper me with questions about why, what I can eat, etc. They think it's hilarious to invite me to a BBQ or to tease me about their big mac that they had for lunch - "Mmmm... Mr. Greene, we just had the best MEAT! Don't you want some MEAT for lunch?", etc. So, on the vocabulary section, I put a word bank of terms for them to use. I had one slot left to fill with a distractor, so I thought I would just toss in "vegan" as a joke. I was pretty surprised when two students selected this as the correct answer for vocabulary questions about angles. I've taught at DCP long enough that this shouldn't surprise me at all, but it did.

Another student actually asked me about using Sketchpad at home - I downloaded an evaluation copy for her. She told me that, now that school is over, she'll get bored, and maybe she'd like to "draw some triangles or whatever". I love these kids!

Anyway, I'll be teaching primarily (or maybe even exclusively!) algebra 2 this fall, so most of my upcoming posts will probably focus on algebra 2 concepts - but I'm sure I will post on Numeracy issues as they come up.

Survey Says...

My summer geometry students are taking their final exams right now, so I have some time to post. I gave them a survey (like a course evaluation) that included questions about the homework and note-taking policies I established. Here are the results:

How often did you do your homework?
a. Almost every night 6%
b. Most of the time 17%
c. Sometimes 44%
d. Rarely 22%
e. Never 11%

Hmm... that's not so good. At least they're being honest. Which leads in to the next question...

The daily quizzes were used instead of checking the homework. How did this affect your decision to do homework?
a. It made me more likely to do my homework 18%
b. It was about the same as if homework was being checked and graded 65%
c. It made me less likely to do my homework 18%

Ok, so there didn't seem to be much strong feeling either way. However, it did seem to me that less homework was being done with the daily quiz method.

In your next math class, what do you think would encourage you the most to do your work?
a. Homework checked every day for completeness, but not graded 44%
b. Homework checked every day, and given a grade 33%
c. Homework not checked, but a daily quiz given based on the homework 17%
d. Homework not checked, no daily quizzes – just regular quizzes and tests 6%

This didn't really make sense to me. Based on the results of the previous questions, I would have thought that most students would have picked option b. I showed them these results and asked them about it - some said they didn't understand the question. I probably should have asked something like "Under which option will you be most likely to do your homework".

Did using Geometer’s Sketchpad help you understand concepts in Geometry?
a. Yes, a lot. 61%
b. A little bit 33%
c. Not really 6%

That's encouraging. I couldn't always tell if they were actually learning from the explorations; they were (at least, according to my highly scientific survey!)

Was it helpful to you to use the 3-column note-taking template?
a. Yes, a lot 72%
b. A little bit 28%
c. Not really 0%

Did the organization system for your binder make it easier for you to find your notes?
a. Yes 89%
b. No 11%

Well, that's good news! See my previous posts about the issues around note-taking and conceptual structure of the binder if you want more info. This means that I am definitely going to use this method in my Algebra 2 class this fall. There were definitely some problems - for example, I didn't devote actual class time to working on their summary sheets, which would have been helpful. I also let them organize each binder section however they wanted, but when I looked through them to assess their work, I realized that more guidance will be really helpful - at least for the first semester.

The 3-column note-taking template is broken down as follows:
On the top, students write the lesson name and essential question, as well as what section of their binder the lesson belongs in. The three columns are "concepts", "examples", and "background information". And that's it. It's quite simple, but it really seemed to help some of my students organize their notes more clearly. It also helped me organize my lessons more clearly, because when I was developing a direct instruction piece of a lesson, I would think in terms of these columns, and write my lecture notes that way. The difference between concept, example, and background info is pretty clear to me, but I realized it's not necessarily clear to the students. When I do this for a year-long class, I hope to be able to scaffold away from putting notes on the board in the 3-column format, and start having students decide for themselves where information belongs.

How often did you look up notes and examples in your binder when you were stuck?
a. Usually 6%
b. Often 39%
c. Sometimes 56%
d. Never 0%

They are still DCP students, after all. Baby steps... At least no one said "never"! I have to do some more explicit activities (and assessments) focused around having students use their notes. I'm still not sure how to build the ethic of looking something up when you are stuck, instead of immediately asking someone for help, or giving up.

What would you like for breakfast before the final?
a. Bagels 61%
b. Donuts 39%

And now we know. The donut faction was much more vocal, but the bagelites had some strong grassroot support. Thanks go to my favorite bagel place. Note: students apparently only like plain cream cheese - don't try to get all fancy on them. I thought they would like jalapeno because they are always eating hot cheetos, but of course they didn't, and I had to hear that classic DCP line - "It's because we're Mexican!".

A Smashing Success

Yesterday evening, a colleague and I drove up to UC Berkeley to visit our four students who are participating in the Level Playing Field Institute's SMASH program. Here are their goals, according to their website:

First, we prepare students from underrepresented communities to be competitive in science, technology, engineering, and math-related studies (STEM) at the University of California or schools of similar caliber.

Second, our goal is to have all students aware of graduate school opportunities and encouraged to eventually attend in order to further pursue professional, STEM-related studies or careers.

Third, through SMASH, we hope to encourage a sense of social responsibility through promotion of critical thinking, civic awareness and leadership.

It's quite an intensive program. Student are in class all day until 5. They have dinner at the dining hall, and then have a self-run student meeting at 6:30. Then, they have mandatory study hall from 7 to 9.

Apparently, they have had too many discipline problems lately - we witnessed quite an amazing student meeting. It was run entirely by the "third years" who were elected to the student council. The meeting was held outside, with everyone standing in a big circle. It started off with apologies, as over a dozen students, one by one, entered the center of the circle and said they were sorry for breaking whatever rules they broke, or for their bad behaviors or attitudes. No one laughed - in fact, they all seemed to take it completely seriously. The longer or more heartfelt apologies got sincere applause from the crowd. There were some TAs in the group (recent high school grads), but there were no staff members and no other adults. I don't know how they were able to build this kind of culture among the students, but it was amazing to see (and a little surreal).

Our students love the program. Though they struggle academically in comparison to some of the other participants, they are holding their own. What a great experience for them - living in the dorms with roommates, eating at the dining halls, spending a summer committed to improving their math, science, and english skills. It felt like Harry Potter without the lightning bolts.

Rational Geometry

I was working in a nearby coffeeshop (an air-conditioned oasis) and the guy sitting next to me noticed my stack of Geometry books. We started talking, and he told me about a book called "Rational Geometry". I looked it up on the web, and the full title is DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry, by N J Wildberger.

This professor from Sydney has looked at the logical foundations of Euclidean Geometry and found them not only lacking, but fundamentally flawed. His claim is that the basic notions of distance and angle are problematic, and that the classical trigonometry relationships (can I get a SOHCAHTOA?) are cumbersome and unneeded.

He proceeds to recreate Geometry using concepts called Quadrance and Spread, which replace distance and angle. This allows him to calculate everything using only rational numbers (he even says that maybe the Pythagoreans were right - had they thought to consider the squares of numbers as more fundamental than the numbers themselves). This is, apparently, a more accurate way to compute, and does not require the use of extensive trig tables or modern calculators. He also says that this rational geometry is logically sound, not requiring the use of any undefined axioms, and therefore is easier for students to learn and make sense out of.

On the website above, you can view pdfs for the intro and first chapter. They are quite interesting. (There is also a posting for the last chapter, but the content is, let's just say, beyond my current scope of understanding.)

I wonder if anyone has had experience with this type of Geometry, especially with regards to trying to teach it to students? Is it the wave of the future, or is it a passing fad?

Don't Get Fooled (and Learn to Fool Others!)

That was the name of the lesson for spotting valid and invalid deductions. Students had to write their own examples of both valid and invalid (converse and inverse errors) deductions. Here are some of my favorites:

• If you rob a bank, you go to jail. Tim has never robbed a bank. Therefore, he's never been to jail.

• When you run, you have a healthier life. Manuel doesn't have a healthy life. Therefore, Manuel doesn't run.

• If a guy is a soccer player, then he is cute. Hector is not a soccer player. Therefore, Hector is not cute.

• If you talk to Vanessa during class, you won't get any work done. Daisy doesn't talk to Vanessa at all. Therefore, she gets all her work done.

• If you're a boy, you must not take showers. Adolfo is a boy; therefore, Adolfo must not shower.

• If you're a monkey, then you climb trees. Sara climbs trees, therefore she's a monkey.

• If you're Mexican, you should speak Spanish. Dan speaks Spanish. Therefore, he is Mexican.

• If you are Mr. Greene, then you like Voltron. Phil isn't Mr. Greene, therefore he doesn't like Voltron.

• If you're a brunette, you like sopes. Misty does not like sopes, therefore she is not a brunette.

• If you're a rat, then you eat cheese. Alondra eats cheese, therefore she is a rat.

What to do about homework...

There is an interesting post and series of comments about homework at The Daily Grind.

I agree that homework needs to be assigned every class period. But, like every teacher, I've struggled with how to best hold students accountable for not just completing it, but understanding it. In our freshmen math courses (Algebra 1, Numeracy), we give students full credit on an assignment if it is completed and turned in on time (we don't assess it for correctness at all). We also don't accept late work, unless students have an excused absence. The purpose of this is to build the ethic of doing homework and turning it in - as many students seem to come to high school with out having done much - if any - homework in the past. We are pretty successful at getting students to turn in their work by the end of freshman year. Getting them to really think about it, try hard on questions they don't understand, and seek help when they have difficulties is another thing altogether.

In my Algebra 2 course last year, I began the policy of assessing homework by spot grading 2 or 3 random problems daily, and giving that grade to the entire homework. I always made sure to include at least one of the more challenging problems. That way, if a student skipped a hard one (or did not put much effort into it), they would at best get half credit on the assignment. Overall, it seemed to work pretty well, and I often had students come in before class to get help on problems they didn't understand. This did, however, take a lot of time (about 15 - 20 minutes of my time per class); I would give them a self-paced Do Now at the beginning of each class and grade the assignments while they worked. I liked that this forced them to be more self-sufficient, because I refused to provide any help during that time. I was able to get immediate feedback on their progress from the previous class, and I was also able to give them immediate feedback (i.e. re-teaching a specific concept that many students had trouble with). While this was nice, it was also a stressful way to start each class. And sometimes, if the Do Now assignment was more difficult than I anticipated (or if I was feeling lazy), I wouldn't grade the homework right away - and then it would usually sit in my pile until I recycled it.

In my current summer Geometry course, I am trying a different tactic altogether. I am not grading (or even checking) their daily homework. Instead, I am giving a quiz first thing each day, where the questions are either identical to ones assigned for homework, or very closely related. It is a multiple choice quiz, and they take it on Scantrons - during their break, I grade them quickly, and this is a great way to provide us all with immediate feedback. It also will hopefully help prepare them for the high-stakes multiple choice tests that they will be plagued with (ACT, ELM, STAR, etc.). Now, I know I have some students who are not doing their homework, and yet are doing fine on the quizzes. I wonder if this is ok... if they are able to do it with out completing the homework, maybe that's fine... or maybe they are not getting important practice that will reinforce their understanding... I'm not really sure. My hope is that, if they see they do poorly on a quiz, that they will have enough ganas to go and complete the homework, with the understanding that the coming test will target those same skills, and will be worth many more points.

At the end of the summer, I plan on surveying the students and getting their feedback as to what was more effective for them (most of these students were in my Algebra 2 class last year). I hope to get some honest and useful answers from them. I'll post the results.

If anyone has other strategies they like for dealing with homework, please comment.

It's Logic Week!

In my desire to create a 5-week summer geometry course that would provide students with the concepts most needed to do well in pre-calc and on the ACTs, I think I underestimated how critical it would be to provide instruction on the principles of logic and deductive reasoning. I saved this for the 4th week of the class; looking back on it now, it should have been the first week. There are so many logical concepts that I take for granted, and I kept running up against them in the previous lessons (oh yeah, you guys don't get what "if and only if" means, or why the converse might not be true, etc.)

So, that being said, I am excited to start teaching deductive reasoning this week. I am using the first chapter from Geometry by Harold Jacobs as my guide. I'm starting to like that book more and more. Each section has only a couple pages of well written text, followed by 3 problem sets. The book doesn't fit the typical standards based text pattern (i.e. formula presented, 3 - 5 different examples of problem types, drill on those problem types). Instead, the problem sets seem to be more of an integral part of the lesson itself, with problems that are continuously moving to the next level.

Here is the sequence for teaching logic:
1) Drawing conclusions (determining if a conclusion is necessarily true, not necessarily true, or false)
2) Conditional statements (understanding the "if a then b" structure, using Euler diagrams)
3) Equivalent statements (converse, inverse, contrapositive; logical equivalence)
4) Definitions (it's got a great definition of "definition" - a conditional statement whose converse is necessarily true)
5) Syllogism (if a -> b, and b -> c, a -> c)
6) Deductive reasoning (putting it all together) and proof

I've never taught this before, so I don't know what to expect. Today I did just the first two parts. Some students seemed to really get into it, but a few looked like they were trying to transfer their thought patterns into the tables by staring really hard. I'm telling them that I consider this to be the single most important thing I am teaching them all summer, that it will help them not only in all math classes, but in all their classes, and beyond school. I would like to be able to have them write arguments that have intended logical fallacies in them to try and persuade the class of something, and then have the class try to identify the inconsistencies.

What experiences have you had teaching logic to students? What activities / techniques / scaffolding do you use?

Que Viva la Jeopardy!

What can I say, it's a classic. Students love it. Why? I'm not sure. But when you put them on teams, let them buzz in, and award points, the boring old review problems seem to take on a new vibrancy. It's like putting a sprinkling of cilantro on Safeway vegetable soup (and getting points for eating it!). Anyway, I used Geometer's Sketchpad to make an interactive version (clicking on the point value sends you to the question, with the answer hidden and ready to be revealed). You need a projector for this to be useful, of course, but it was way better than using the overhead.

If you want a copy of the file to look at or modify, send me an email at dgreene AT dcp.org.

By the way, is it possible to post files on blogger given that I don't have any other website where I store them?

Socio-Constructivism: A Case Study

This case study was written by Dr. Ferdie Rivera and Dr. Joanne Becker, two professors I learned from in my credential program. I found it on a site that has a virtual treasure trove of math research papers (I would love to find time to read quite a few of them...)

It provides a prime example of what "sociocultural constructivism" is all about. It does have a lot of technical jargon in it which can make it a little tough to understand, but overall I think it is pretty clear what is happening when you read it.

This case involves the authors teaching a math analysis class the topic of solving polynomial inequalities, using the TI-89 graphing calculator as a tool. (If you are of an anti-calculator mindset, read the study - it shows a great way to mediate use of the calculator. The students are guided to understand why knowing how to use the calculator is not sufficient, and when and why they should use algebra for exact solutions - so not only do they know how to do both, they know when it's most appropriate to use each).

Our overall concern in this investigation is to provide a sociocultural basis for the meaning objectification of mathematical concepts and processes - that is, by surfacing both the social nature of technological tools and the social transactions that take place in classroom activity which assist students as a collective to establish viable shared practices and collective representations

Um, right. So that points out one of the fundamental differences between this theory and "discovery learning" - students are not left to explore aimlessly on their own, or even just in small groups. They begin an exploration typically in pairs. This is followed up by a whole class discussion, in which students propose different solutions, ideas, and difficulties that they are having. There then exists a process of negotiation, in which the students discuss and evaluate each others' ideas, which will hopefully lead to further new ideas - the teacher's job at this point is to help move the discussion along by facilitating (if needed), clarifying students' thoughts, and posing key questions. Eventually, ideas, concepts, and notations can become "taken-as-shared", which means everyone in the class ascribes the same meaning to the given concept. The second fundamental difference is that, once this shared meaning has been established, the teacher helps the students generalize and formalize the knowledge (i.e. learn and use traditional algorithms, representations, and techniques).

I find this stuff fascinating. The drawback is that it requires more class time. I do think, however, that the gains can outweigh the losses if this is done right, as the students are more likely to understand and retain concepts in the long run.
I encourage people to read this study and make comments about it.

Below is an excerpt. Notice how the comments of the pairs of students follow the logical process that a teacher would provide directly.

Developing a Model For Solving Polynomial Inequalities.
The class needed eleven 55-minute sessions to accomplish this task. The model they developed for solving polynomial inequalities graphically could be broken down into three stages below using different types of tools and in which the TI-89 served as the primary tool for the progressive evolution of the two later tools.

I Using the TI-89 as a tool for investigating the following:
(1) graphs of even- and odd-powered polynomial functions in factored form;
(2) graphs of polynomial functions in factored form that contained odd and even multiplicities;
(3) graphs of polynomial functions in factored form that contained imaginary zeros;
(4) solving polynomial inequalities in factored form graphically.
II Using a constructed Cartesian plane on paper as a tool for solving polynomial inequalities in factored form graphically.
III Using a number line as a tool for solving polynomial inequalities in factored form graphically (and the same tool was used later in the case of polynomial inequalities expressed in the general (non-factored) form.

Initially, the students relied on the TI-89 to obtain generalizations about the graphs of polynomial functions subject to certain restrictions (see (I) above). They also used it to solve inequalities and to see the significance of knowing how the x-intercepts played out in the solution process. The TI-89 enabled them to develop their initial ability to describe and to reason perceptually about graphs of polynomial functions and their relationship to solving polynomial inequalities that were all initially expressed in factored form. In establishing a model for solving inequalities graphically, two additional shifts took place, and both shifts were unaided by the graphing tool. When the students were prompted to solve a polynomial inequality
independent of a TI-89, Pair 3 suggested for the class to draw a Cartesian plane, plot the real x-intercepts, and use what they initially learned about the graph of the corresponding polynomial function to draw a sketch of its graph, and then to write down the intervals in which the inequality made sense. It took students some time to accomplish this because they had to calculate specific points on the graph. A number of them obtained values for y by beginning with x = 0, 1, 2, 3, and so on,
which did not make sense in many cases of polynomial functions and, hence, did not gain much support from the class. One collective practice that emerged from a whole-group discussion came from Pair 4 who suggested obtaining points that lie
between x-intercepts that the class immediately accepted. A second collective practice came from Pair 5 who suggested that to solve a polynomial inequality graphically, a rough sketch of its corresponding graph together with all the x-intercepts
was all that were needed and that none of the other points mattered.

The Village

In a school in LA, Black students are participants in a "village" program that is seeming to provide them with a stronger sense of community. One of the aspects of the program is confronting the students with actual data, for the purpose of motivating them to improve:

At one of their first meetings with students, teachers projected on a big screen test-score comparisons for white, Asian, Latino and black students, and those learning English as a second language. Many of the black students were shocked to see themselves at the bottom.
...
"My kids came home talking about the statistics and how low we were, and it hit them really hard," said Zola Chrenko, Chris' mother.
...
[critical and/or negative] views, however, have been tempered by impressive gains in test scores, reductions in dropout rates and improved behavior among Cleveland's African American students. Scores on the Academic Performance Index jumped 95 points in two years, from 569 in 2003 to 664 in 2005, according to the California Department of Education. The districtwide average among all students in 2005 was 649, department statistics show.

In 2003, 36% of black students at Cleveland passed the math portion of the California High School Exit Examination. The figure rose to 81% in 2006.

So, this is what I've been trying to get at in my previous post. I think that showing students data is a good start, and it needs to be supported in a comprehensive way. It seems like Cleveland High School might be on to something, and according to the article, other schools and districts are taking note.

Thanks to Joanne Jacobs for providing the link!

Summer geometry update: so far, so good

Though I am teaching a lot of material to my students, they seem to be handling it well. After the first week of class, the grades are: 7 A's, 4 B's, 5 C's and 2 F's.

We went through a lot of the basics of geometry:
- Points, Lines, Planes
- Angle types (adjacent, linear, vertical, supplementary, complementary) and their properties
- Angles created by transversals, and all the theorems associated with parallel lines cut by a transversal
- Basic formulas for perimeter, area, and volume, integrated with algebra
- Triangle sum theorem, exterior angle theorem
- Lots of word problems and angle webs too

This is a short week due to the 4th of July. We will be working on triangle inequalities, the Pythagorean Theorem, special right triangles, and the like.

Hopefully, their energy will stay strong as we move through the curriculum. It's interesting to see how some students love working with Sketchpad and learn a lot by using it, and others really feel like they don't learn from the guided explorations. This just gives me further belief in the idea that both guided exploration and direct instruction are necessary. I have reassured the students who are not enjoying Sketchpad that all concepts we explore in Sketchpad will also be taught through direct instruction with lots of examples for them to practice. There is definitely a benefit to having a 5-hour class!

Why should I learn math?

One of the main components that my Numeracy class is still lacking is the piece that will help motivate students to want to learn math, and to understand why improving their math skills is so critical. I'm not talking about learning math so they can find discounts at the store or give the correct tip at a restaurant - or even so that they can pass Algebra. Though these things are important too, I'm talking about math in the larger social context.

The average incoming level for our freshmen is around 5th grade. Our mission is to get them to a 4-year college. This requires not just development of their academic skills, but it also requires a shift in their thinking and self-perception. Our students come in with comparable literacy levels; however, in our society, it seems much clearer to people that being able to read and write is an essential skill. There are plenty of well-educated people who happily admit that they can't do math, but none that laugh about their inability to read a book.

In our Verbal Reasoning class (i.e. learning to read), students are exposed to works by authors who come from backgrounds similar to their own, writers who describe how they were able to overcome their obstacles and succeed in education and in life. In that class, they often discuss the direct connection between their literacy levels, their success in high school and college, and the impact that they will have on their families and communities in the future. In that class, many students begin to care about their reading levels, and they want to improve - not just for a grade, but because they have begun to internalize what it really means to be unable to read in our society. They begin to understand that learning to read is critical if they want to reach their goals.

This is what I want to happen in our 9th-10th grade Numeracy sequence, and I don't really know how to get there. Let me give an example of one activity that I've developed - this is the kind of thing I want to implement regularly. We watch the movie "Stand and Deliver" - yes it's a bit cheesy, but it's a good movie and my students (95% of whom are latino/a) relate to the students in the movie. I teach them about what AP classes are, and we talk about AP Calculus specifically. Then, I give them the data that I download each year from the AP website - the scoring data, nicely broken down by ethnicity. I have them calculate percentages of students passing and failing for each ethnicity, and we see that the percentage of whites and asians passing is always about double that for latinos and blacks. We talk about why it happens this way, and we reflect on their own experiences in math. I try to help them connect these ideas back to their own actions and behaviors in math, and their goals for the future. When this starts to happen for students, I think the rate at which they develop their math skills will increase more rapidly, because they will be more intrinsically motivated.

I want to make it clear to them that learning math skills is not important just to be good consumers, not just so that they can pass Algebra 1 and move forward, not just so that they can get accepted to college, but so that they can start to change the way our society functions. Clearly, my one little Stand and Deliver activity is not going to do that. That's why I want to develop more activities that are similar to this, and expose students more regularly to them. I want them, by the end of the year, to be eager to see how much their math scores have improved just as they are eager for their reading results.

I am going to keep working on this during the summer and into next year (and beyond, I'm sure). If anyone has ideas, activities, comments, suggestions, or questions, I would love to hear them. As I come up with things, I will post them here too, in the hopes of getting your feedback.

Progress in the Math Wars?

Is this old news? I just read about the new National Mathematics Advisory Panel now, and its seems like it could potentially be a great step forward.

On April 18, 2006, President Bush issued an Executive Order creating the National Mathematics Advisory Panel. The Panel will advise the President and the Secretary of Education on the best use of scientifically based research to advance the teaching and learning of mathematics.

Modeled after the influential National Reading Panel, the National Math Panel will examine and summarize the scientific evidence related to the teaching and learning of mathematics, with a specific focus on preparation for and success in learning algebra.

The National Math Panel will issue two reports containing policy recommendations on how to improve mathematics achievement for all students.

I'm skeptical, but there are some good people on the panel, and both sides of the "math wars" seem well represented.

Linda Moran blogs on this here. Here is an excerpt (the "three scholars" are Liping Ma, Wilfried Schmid, and Russ Whitehurst):

While some views on the panel may be polarized on the two sides of the math wars, the three aforementioned scholars have already concluded that a rigorous, aggressive curriculum which is strong in both concepts and drill, taught both by discovery and direct instruction, is best.

I couldn't agree more. The preliminary report is due out by January, 2007. If anyone has more information or insight into this panel, I'd love to hear your comments.

Update: See Math Panel Watch, a blog set up for tracking the deliberations of the panel.

Also, this link comes courtesy of Vlorbik, and has lots of good info. Thanks!

Discovery Gone Bad

Coalition for a Republican-Free America: A...B...C...what?

An excerpt from the above posting:
As my teacher explained it, this new system put math into the hands of the students, rather than the instructor. The learning itself became more self-directed, with the teacher there as guide, should any problems or questions arise. We often worked in groups (yes, let’s take a bunch of sixteen-year-olds, put them in a group, and expect them to get any work done) and it was easy to let other, more math-oriented members do the work, while the rest sat by and wrote down answers. Not just answers, but paragraphs explaining why we got the answer we did. My friends and I now call it the “infamous Core math ‘explain.’” My math teacher further aggravated this problem by taking the term self-directed to mean, “I don’t have to teach, but rather, I can sit at my computer all hour and write email.”

Just so everyone is clear, I do not believe in this kind of learning. And if you ever catch me sitting at the computer emailing while my students are working, please fire me.