Look for fully developed K-14 math investigations, math challenges, and standardized test practice both for SATs and Common Core assessments. The emphasis will always be on maintaining a balance between skills proficiency and developing conceptual understanding in mathematics. There will also be dialogue on key issues in mathematics education.
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See top sidebar to subscribe to solutions of Daily Twitter Problems.
The following is excerpted from the essay, "When Pedagogic Fads Trump Priorities" in the 9-29-10 edition of Ed Week. The author is Mike Schmoker, an author, speaker and education consultant.
"First we need coherent, content-rich guaranteed curriculum - that is, a curriculum which ensures that the actual intellectual skills and subject matter of a course don't depend on which teacher a student happens to get...
Second - and just as important - we need to ensure that that students read, write and discuss, in the analytic and argumentative modes, for hundreds of hours per school year, across the curriculum...
Third, we need to honor, beyond lip service, the nearly half-century-old model for good lessons that all of us know, but so few consistently implement:
Good lessons start with a clear curriculum-based objective and assessment, followed by multiple cycles of instruction, guided practice, checks for understanding (the soul of a good lesson) and ongoing adjustments to instruction... multiple checks for understanding may be the most powerful, cost-effective action we can take to ensure learning. Solid research demonstrates that students learn as much as four times as quickly from such lessons.
For decades we have put novelty and the false god of innovation above our most obvious, proven priorities"...
I've been in touch with Mr. Schmoker to congratulate him for the courage to speak the truth. I hope to continue the dialog. He also takes on "differentiated instruction" and mindlessly incorporating technology into lessons as if "that will rescue poor instructional plans from failure."
I rarely say this, but, if you disagree with him, you are either wrong or hypocritical! Yup, dems fighting' words!
And now for something completely different...
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)
" You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught." --from South Pacific
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes both multiple choice and constructed response items. Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT! ------------------------------------------------------------------------------
I decided to post a video solution of the Twitter problem I posted on 8-25-10:
4 red, 2 blue cards; 4 are chosen at random. What is the probability that 2 of the cards will be red?
Because of the 140 character restriction on Twitter, the questions are often highly abbreviated and I actually consider it a "fun" challenge to write the question both concisely and clearly. Of course, as we all know about human interpretation of word problems, "clear" is in the eye of the beholder!
There's no doubt that the question above needs some fleshing out and might appear on the SAT and other standardized tests something like this:
A set of six cards contains four red and two blue cards. If four cards are chosen at random, what is the probability that exactly two of these cards will be red?
I'm sure my astute readers can improve on this wording but we'll leave it at this.
A few questions naturally pop up:
(1) Could this really be an SAT/Standardized Test question? Well, as I state in the video below, a question quite similar to this appeared on the College Board website the other day as the Question of the Day.
(2) For whom is the video intended? Everyone who happens upon it! I certainly wrote it to be helpful to students who will be taking the PSAT/SAT in the near future. Rather than simply presenting a single quick efficient solution, I demo'd 2-3 methods and indicated some important strategies and reviewed key pieces of knowledge to be successful on these harder probability questions. By the way, someone who is comfortable with probability will surely not find this question so formidable, but we're talking here about high school students or even undergraduates who struggle mightily with these.
(3) I'm hoping that the video will also serve as a catalyst for dialog in your math department. From the inception of this blog, I've never even intimated that a suggested way of explaining a concept, skill or a problem solution is in any way prescriptive. I encourage you to continue using whatever instructional methods have worked for you and to share these with our readers! However, for novice teachers or those who wish to see other approaches, I hope it will have some benefit. Of course, the video is not in a classroom. There are no students asking or being asked questions. There are no interruptions and I have a captive audience (except for my dogs who bark incessantly!).
SOME KEY STRATEGIES/TIPS/FACTS FOR PROBABILITY QUESTIONS
(1) It is highly recommended that students begin by listing 2-3 possible outcomes and to include at least one that is NOT one of the desired outcomes! This will help you to decide on a plan: organized list vs more advanced counting/probability methods. Further, you can ask yourself the key question in all counting/probability problems: DOES ORDER COUNT!
(2) Although it appears difficult for most test-takers to be systematic when making a list under test-taking conditions, preparation is critical here. If one practices several of these in the weeks leading up to the test, the chances of success improve dramatically. Did I just suggest preparation and practice could make a difference!
Where do you find these problems? Any SAT/ACT review book or my Twitter Problems of the Day or my upcoming SAT Challenge Quiz book to name a few sources...
(3) The basic definition of probability should always be in the forefront of your mind:
P(an event) = TOTAL NUMBER OF WAYS FOR THAT EVENT TO OCCUR DIVIDED BY TOTAL NUMBER OF OUTCOMES.
As indicated in the video, one can and should think of this ratio as TWO SEPARATE COUNTING PROBLEMS! Do the denominator first, i.e., the TOTAL number of possible outcomes. In the Twitter problem it is 15 if order is disregarded. Whether you arrive at 15 by listing/counting or by combinations methods, the denominator is 15 and is a completely separate question from "How many ways are there to get 2 red and 2 blue cards?"
(4) Finally, there are other methods for solving this probability question using Laws of Probabilities and/or permutation methods. I was going to make a 2nd video but I'm not so sure about that now.
An important point about the video below: I used 4 Blue and 2 Red cards, the opposite of the original Twitter problem but that won't change the final result!
As I develop my Facebook page further, I may start posting these questions there as well as my videos. Facebook allows up to 20 minutes videos, much less restrictive than YouTube's 10 minute limit.
If interested in purchasing my new Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest practice or Problems of the Day/Week.
Price is $9.99 and secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL FIRST SO THAT I CAN SEND THE ATTACHMENT!
"All Truth passes through Three Stages:
First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)
"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught" --from South Pacific
Sometimes levity is needed at the start of a new school year. In the past I have posted more serious "words of wisdom" but I'm in a more whimsical mood right now. Besides, I haven't posted anything for awhile, so here goes...
Here are a couple of my own Murphyisms I just posted on Twitter:
Murphy's Law for SAT Students: Running out of time on the last section of Math, you desperately guess C,C,C,C,C,C for last 6 answers. Of course, the correct answers turn out to be B,A,D,B,A,D.
Murphy's Law for Trig Students: You confidently apply the mnemonic "SAHCOHTAO" to your first major unit test!
The problem child will be a school board member's son.
Students who are doing better are credited with working harder. If children start to do poorly, the teacher will be blamed
The school board will make a better pay offer before the teacher's union negotiates.
Personal note: Been there, done that! Here's my own version when I was non-tenured:
As we were picketing, my poster read "We've lowered our demands -- now up yours!" Which one would you guess got picked up by local newspapers...
Law of Universal Intelligence:
The most ill-behaved student in all of a teacher's classes is always one of the bright ones he can't flunk.
For Students
If you study hard for that important examination, the focus of the exam will be 'thinking-based' and 'analytical'.
Corollary: If you memorized information, it will be useless.
If you don't study for that important examination, the paper will be content-based.
Corollary: If you don't study, every question will appear to be something you remember reading on your textbooks from a month ago, hence will appear (deceptively of course) easy, although you will not recall the exact phrasing of an answer.
The more studying you did for the exam, the less sure you are as to which answer they want
Eighty percent of the final exam will be based on the one lecture you missed about the one book you didn't read.
PLS PLS PLS ADD YOUR OWN TO THESE. MAKE THIS A REAL WIKI!
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)
You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific
Have you been wondering about the question in the title or just assuming that I'm on hiatus? No videos? No provocative comments about standardized curriculum? No interviews with the movers and shakers in math education on the national front? Most importantly, no anecdotes from my 3-yr old grandson? Is the world coming to an end?
Seriously, I suspect the world is still spinning on its axis, perhaps just a tad more tilted. And my faithful readers/subscribers have far more important concerns in their life than waiting for my next post! On the other hand, some of you know that I've discovered Twitterdom or should I say Tweedledum and Tweedledee!
What exactly have I been doing other than coping with the joys and trials of keeping my wife, 3 teenagers and a 21-year old happy? Not to mention 4 grandchildren and 3 other older children not "officially" living at home...
I. Twittering a brand-new SAT Twitter Math Problem of the Day virtually EVERY day since May 26th! Answers were provided up until about the middle of June. You can see these problems in the right sidebar (limited view) of this blog or follow me on Twitter here. I've had very interesting reactions from 8th and 9th graders in Indonesia who seem to find these problems intriguing although I really don't have a very good translation of their tweets. I think they keep calling me Papa and some are probably afraid of me!
II. Completing Volume I of SAT Math Quiz Problems. It's still in draft mode but when completed it will have 150-200 challenging math questions not previously published on my blog. Many of the Twitter Problems will be included, answers will be given for all questions and selected hints/solutions will be provided. The book will be available for download as a pdf with some copyright limitations. I will post more on this here, on Twitter and on my new MathNotations Facebook page (still under construction!).
III. I've been in contact with K.C. Yan from Singapore who has been enlightening me re the model method in Singapore Math. I strongly encourage you to check out his website Singapore Math and follow him on Twitter here. He is a remarkable individual and possesses a profound understanding of mathematics and pedagogy. He is a math coach, writer and editor. He currently conducts recreational and competition math courses and workshops for schools and enrichment centers, and educates the public against innumeracy and pseudoscience. I strongly encourage you to read his blog and, in particular, his demonstration of several non-algebraic "model" methods for solving the following question:
A farmer has twice as many ducks as chickens. After the farmer has sold 413 ducks and 19 chickens died, he has half as many ducks as chickens. How many ducks does he have now?
Who knows? Perhaps, we will one day collaborate!
IV. I've been in touch with Professor William Schmidt of TIMSS renown. We were scheduled to have a Skype conversation back in May but our schedules were at cross purposes. I'm hoping to contact him again and reschedule. Some of you know how much respect I have for his knowledge, dedication and tireless efforts to improve the math education of our children.
Talk to you soon!!
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific
Well, let's see...
I haven't posted in almost a month, I haven't been promoting our wonderful Math Carnivals, I haven't brought you any updates or controversial material, I haven't produced any new and exciting videos,...
So what have I been doing? Enjoying the heat wave here in the Northeast?
1. Finishing up my SAT Math Quiz Book Volume One which will hopefully be done before the world ends in 2012. I haven't decided yet how I will make these available to my readers or schools or students or whomever but that will all be worked out. One possibility is to send the book electronically upon payment.
2. Continuing to post a Twitter SAT Problem of the Day despite the fact that I said I would take a respite for the summer. Further, many of these problems will appear in the Quiz Book. Can't stop writing these -- please help me! These problems also appear in the right sidebar of this blog but they may be truncated. If you only get the feed for this blog then you may want to subscribe to the RSS feed for my Twitter posts.
3. Exciting new trends in math education? Actually other than states racing to the top and continued movement toward standardization of math curriculum, it's really the same old, same old. Technology will always evolve and influence math education -- that's a given -- however the nuts and bolts of what makes for effective math teaching, well, that's still the ten trillion dollar question and that's still the reason for this blog.
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific
As we wind down toward the summer my SAT Problems and Videos continue to pick up steam! Below is the latest video from you YouTube channel, MathNotationsVids. I want to thank those who voted in my survey of these videos. I am gratified but I really need more specific suggestions on how to improve these. Your comments on YouTube or here are welcome!
Note: Because I am explaining two problems on one video, I am omitting details and multiple solution paths. Therefore these videos may be useful for your students who want to practice over the summer or revisit in the fall.
The percent increase problem could be asked in a variety of ways and demonstrated using multiple representations, aka The Rule of Four. The visualization suggested in the description of the video has students physically demonstrating that doubling the edges of a rectangular solid, a cube in this case, will allow placing not only the original box inside of the bigger box, but SEVEN MORE! There's your percent increase, hands on!
I will be stopping the posted SAT Problem on Twitter on Tue 6-15-10. If I am able to sustain it, I will try to keep this up for the entire 2010-11 school year but who knows...
Finally, as posted on Twitter, I will be offering an individual or small group online course (using Skype) for the SAT or ACT Math this summer on a very limited basis. If you know of any student who might benefit from individualized instruction just email me at dmarain@gmail.com and I will provide details. This must be done ASAP however, as I will be closing this out very quickly.
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific
Please note correction to 2nd problem in the video. The correct answer is 4096 "real" values. The original answer, 13, applies to rational solutions only. Thanks to Nick Hobson for pointing out my careless error. Haste makes waste!!
Please vote in the poll at the right. Be candid in your opinion of these videos. It will guide me in the future to improve. Don't hesitate to share your opinions on MathNotationsVids and rate each video there as well. If you subscribe to my feed, please vote directly on the site. Only a few days left...
The title says it all so here is the video as promised:
Note: See above correction to 2nd problem! The video has not been corrected so beware!
Comments on 2nd problem:
If x is greater than or equal to 0 and less than or equal to 3, for how many values of x will 16^x be an integer?
As mentioned above, Nick pointed out my error. I should have restricted x to be of the form a/b, where a and b are integers, b ≠ 0. Normally, SAT questions avoid use of the term rational so they would spell it out. This problem however is very questionable for SATs. If real solutions were sought, this question would be more appropriate for a math contest. Here's one way of explaining why the answer is 4096 for real solutions:
16^x = k, k an integer → 2^(4x) = k 3 ≥ x ≥ 0 → 12 ≥ 4x ≥ 0 → 4096 ≥ 2^(4x) ≥ 1 since the exponential function 2^(4x) is increasing. This argument is reversible, so there are 4096 solutions for x, one of each integer value of k from 1 to 4096 inclusive. This solution could be written more concisely using log base 16 or log base 2 as Nick did, but I wanted to show a method without the log symbol.
Again, the video solution is WRONG as it shows only rational solutions! Well, at least i was thinking "rationally!"
I fully realize that the school year is over for some and about to end for others but these SAT Problems will be around for you or your students in perpetuity! Let me know if you like the questions. They are now appearing in the right sidebar of my blog so you will need to visit the page to see them.
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"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific
Well, today's SAT Tweet comes too late for many students taking SATs this morning (unless you're in a much later time zone) but I posted it anyway since it can certainly be used to review for final exams in Algebra 2 or whatever"∫ -ated" name you have for it in your district!
As you can see if you visit this site (rather than get the RSS feed), I'm now posting the Twitter Problems of the Day in the right sidebar. I'm new at this, don't have too many Twitter followers yet and I am learning that you need to get the word out there any way you can. Those who have replied to me seem to really like the level of these questions. I do feel the need to explain some methods to students who want them. If they're following me, they can simply send me a Direct Message or, if not, they can reply with @dmarain. I've also placed these questions in the #Math and #SAT categories on Twitter so more will be able to see them, but a lot of what's there is promotion, links and personal thoughts -- so who knows. I may also post a video or two here and on my YouTube channel, MathNotationsVids, to explain a couple of these problems using a variety of approaches both for teachers and students.
If I were a faithful math blogger I would have been announcing Denise's latest Math Teachers at Play and latest Carnival of Math 66 over at Sol's Wild About Math sites. They are in my blogroll, but I am deeply ashamed I haven't been promoting them here. So, please please please go over to Let's Play Math and Wild About Math to view the latest and greatest Carnivals! Also, look here for Denise's ranking of her most popular posts broken down by categories, a mammoth undertaking, but well worth it. Sorry for being so negligent...
Finally, I feel the need to say something that may be provocative but is absolutely necessary for my integrity and the raison d'etre for this blog:
While I have been advocating for a standardized math curriculum for the past 25 years, I know fully well that learning outcomes depend far more on teacher effectiveness than any other factor. Yes, algebra should cover the same topics in every district, however, there's coverage and then there's teaching. It is my observation that most professionals who've been on the job for awhile, whether in education, medicine, engineering or whatever, are open to receiving new information about the latest research and technology, but when you start making suggestions about the actual practice of their profession, you're sure to provoke strong negative reactions in many.
Bottom line, folks... There are ways of introducing and developing ratio concepts, for example, that are more effective than other methods. I have never pretended to know what the best practices are in every case, but I sure know what has worked better for me and what has failed. We need to be open to these ideas and accept the truism that when students do not perform there is a myriad of reasons, one of which was our failure to reach these particular students. We have the obligation to vary our methods and be the researchers in the classroom. We have the obligation to learn from our students, our colleagues, our supervisors and others who have been there and done that.
Ok, I'm off the soapbox.
Also, I must say that I find it fascinating that recounting my grandchildren's latest observations on life seems to bring in far more my readership than any math post I have published! Surely this is the ultimate tribute to Art Linkletter.
Have a great "end of the year" and an even better summer!
"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific
I hope you enjoy the math challenges and discussion on this site. If you have any questions, you can contact me using the new contact form in the sidebar.
Proper Attribution
When using or adapting any of the materials on this site which I have personally developed, please include proper attribution by indicating that "This investigation (activity, problem) was authored by Dave Marain." Please read and adhere to the Creative Commons License below. Thank you.
ZERO IS A 'WEIRDO'! (W)hole (E)ven (I)nteger (R)Rational/Real (DO) Cannot Divide by O! BUT Zero is NOT Positive and NOT Negative!
POSITIVE INTEGERS start from 1
PRIMES start from 2 (not 1)
INTEGERS can be NEGative (and zero!) as well as positive
MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d. Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term? Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence. Step 2: Identify the values which are given First term or a(1) = 3 Common difference or d = 4 Number of terms or position of desired term or n = 100 Step 3: Substitute into formula and solve a(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399
Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!
Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!
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Recently retired math educator and Supervisor of Mathematics; 30 years experience as an Advanced Placement Calculus (BC) teacher; Former Author of Math Teachers of New Jersey Annual HS Math Contest; Former K-5 Chair of New Jersey Math Content Standards and Curriculum Frameworks; Former member of Math Item Review Committee for New Jersey High School Proficiency Assessment; Experienced SAT Math Instructor and author of SAT materials; speaker at many regional and national math conferences
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