Now that I am a full-time tweeter, I realize that I've lost most of my followers in the blogophere but I will post every now and then.
1) Twitter enables me to publish my "puzzles" 2-3 times daily. That's probably the best domain for brief math challenges. In the end, it's still all about content. If you're on just to promote yourself, people will see through that quickly, but they will come back if you offer something interesting and substantive.
2) Anyone miss the musings of my now 4-yr old grandson? Well,when he was 3, he was in his preschool class hanging out with his 2 buddies. One started to make loud silly boy noises and the teacher reprimanded him. A moment later my grandson did the same. The teacher approached him and asked why he would do that right after his friend was told to stop. He replied without hesitation, "That wasn't me, it was just an echo of ___." This is why I told my daughter to get an unlisted number immediately!
3) Actually his about-to-be 8-yr old brother is unique in his own right. His interests in and knowledge of science, astronomy in particular, astound me. He is already an expert on galaxies and nebulae. What do you think he wanted for his birthday other than a telescope? An authentic lab coat! Of course, we obliged.
4) It would be remiss of me not to mention my 2 granddaughters. Not only beautiful and very smart, they are filled with love and joy. What a blessing!
5)Looking for some controversy here? Miss my provocative comments about the current state of math education in the US? Sorry, I'm a very boring person. All I want is a blend of procedural mastery (practice!) with conceptual understanding. Although there are numerous ways to help ALL students understand, there are established practices and principles of effective math instruction which cannot be ignored. Using multiple representations, linking to prior learning, motivating with real-world examples and guiding with questions are some of these essential components. These will never go out of style.
6) Although I am a strong proponent of a standardized curriculum, I fully recognize that this leads inexorably to teacher eval based on test scores. This politicizing of education and imposing a strict business model on an essentially human endeavor will have far-reaching negative consequences.
"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
Monday, June 6, 2011
Update 6-6-11
Monday, January 31, 2011
Odds and Evens Week of 1-31-11
- Tom Friedman writing in the Opinion Pages of 1-30-11 NYT about Singapore politics, economy and education:
...
If Singapore has one thing to teach America, it is about taking governing seriously, relentlessly asking: What world are we living in and how do we adapt to thrive. "We're like someone living in a hut without any insulation," explained Tan Kong Yam, an economist. "We feel every change in the wind or the temperature and have to adapt. You Americans are still living in a brick house with central heating and don't have to be so responsive." And we have not been.
Singapore probably has the freest market in the world; it doesn't believe in import tariffs, minimum wages or unemployment insurance. But it believes regulators need to make sure markets work properly - because they can't on their own - and it subsidizes homeownership and education to give everyone a foundation to become self-reliant. Singapore copied the German model that strives to put everyone who graduates from high school on a track for higher education, but only about 40 percent go to universities. Others are tracked to polytechnics or vocational institutes, so the vast majority graduate with the skills to get a job, whether it be as a plumber or a scientist....
It is a sophisticated mix of radical free-market and nanny state that requires sophisticated policy makers to implement, which is why politics here is not treated as sports or entertainment. Top bureaucrats and cabinet ministers have their pay linked to top private sector wages, so most make well over $1 million a year, and their bonuses are tied to the country's annual G.D.P. growth rate. It means the government can attract high-quality professionals and corruption is low.
America never would or should copy Singapore's less-than-free politics. But Singapore has something to teach us about "attitude" - about taking governing seriously and thinking strategically. We used to do that and must again because our little brick house with central heating is not going to be resistant to the storms much longer.
MathNotations' Reaction:
- ..."teach us about 'attitude'..." I'll second that. As I have often mentioned regarding Singapore'a attitude toward education: In Singapore and other rapidly developing nations, education is seen as an investment. In the USA, it is seen as an expense. Talk about attitude...
- We can no longer afford to excuse the educational performance of our students in math and science in international comparisons with the hackneyed and questionable argument:
- Singapore appears to be attracting the best and brightest into politics - our most recent congressional election attracted some stellar individuals and then some others...
- For me it's all about our nation's values. If we don't deem the education and welfare of our youngest citizens to be our highest priority, then where exactly is our future?
Bravo, Tom, for a compelling piece. Much food for thought...
- Have you been keeping up with the many Twitter Math Problems and Challenges for the past 6 months? I generally don't post answers so you need to tweet a response, send me a Direct message or retweet it for others to see. Follow me at @dmarain.
- Remember my tribute problem to Martin Gardner? Jan writes, "Has seriously no-one suggested the 120-25 variant? :)" Ah, simplicity! Or is it Occam's Razor (when you have two competing theories that make exactly the same predictions, the simpler one is the better).
- I recently asked a version of the following on Twitter:
What special qualities separated your best math teacher from everyone else?
On Twitter, I wanted a response consisting of just ONE quality. Here, I'm looking for the top 3-5. Ok, so I open the floor once again to this oft-repeated query. To focus on math teaching, I would ask you to reference only those unique skills/talents which are inherent to mathematics. I may not get any response to this, but I do believe this list will be of some use for someone. It would be particularly useful to hear from students and parents as well as from professional educators
- This little challenge for your Algebra I students is a bit long for Twitter. Let me know the outcome if you use it in class:
Start value = googol (1 followed by 100 zeros)
Multiply Start value by 5.
Add 5 to the result.
Square the result.
Divide result by 1 more than googol.
Divide result by 5.
Divide result by 5 again.
Subtract googol from result.
Final result = _______?
This is not meant to be a challenge for your mathletes!
"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
Friday, December 3, 2010
Odds and Evens Week of 12-1-10
- Here's my most recent Twitter Problem of the Day:
How many 3-digit positive integers are there in which the absolute value of the difference of their hundreds' and units' digits equals 4?
For students: Reply on Twitter, Facebook or my email (dmarain@gmail.com) by 12-6-10.
For everyone else: Comments are always welcome but please hold off on solutions until 12-6-10. Thanks!
- I've been contacted again by the Education Editor of Parent Paper magazine, a well-known publication here in North Jersey. I was asked to write a piece on helping parents to help their children with schoolwork, particularly in math. I'm reprinting here in full since it will be most likely edited down to a few sentences. Most of the general suggestions are obvious but sometimes I feel that the obvious needs to be stated. I'm basing this on my experience with 7 children, 4 grandkids and over 30 foster children.
General Suggestions for Parents Helping Children With Assignments
- TV, radio, music, any other distractions turned off when your child comes home after school.
- Establish a consistent location where they will do their homework every day -- dining room table, coffee table -- preferably in the same room as parent until they are older
- Establish a routine where the child takes out the assignment book, folders, etc., before their snack. If you do it for them, they will come to depend on you for this. Have them hand you their parent folder with all papers you're supposed to read, sign, etc.
- It's up to you but I would allow the child to have their snack while they start their homework. Be less concerned about the mess and remember, if they're not allowed to start homework until they 've finished their snack, I guarantee you that snack time will extend for longer and longer periods of time (even if you say the have to finish in 15 minutes!).
- DO NOT OFFER TO HELP THEM WITH THEIR WORK UNLESS THEY ASK! DO NOT HOVER OVER THEM - JUST BE IN THE VICINITY! ONCE YOU'VE MADE THEM DEPENDENT ON YOU, IT'S HARD TO BREAK THE HABIT!
- If they ask for help, ask them to read the directions out loud. If you then ask them what it means or what they are supposed to do, many children will reply something like, "I don't know. I don't get it. I can't do it!" You know your child best. If you believe they are capable of the assignment, you can help them get started and then say you have to do something, but you'll be around if needed.
- If you cannot make sense of what the assignment is, then ask them to explain it. If they can't, the issue may be they are not yet ready to neatly/clearly copy the assignment form the board. Address this with the teacher the following day.
- Ask the teacher whether they prefer voicemail, email or face-to-face questions after or before school. Ask them if it's ok if they occasionally email concerns.
- Establish a "social" network of parents in the class - take the initiative! Set up a class group on Facebook so that parents can help each other with clarifying assignments. Parents can routinely check in. If electronic networking is not feasible, go back to the tried-and-true getting phone numbers from 2-3 other parents thus making a smaller network. Trust me, you will need to use this often unless your child is mature, organized and responsible/independent, in which case you will be helping others!
- Keep repeating to yourself the Golden Rule of Parenting: THE MORE YOU DO FOR YOUR CHILD, THE LESS HE/SHE WILL LEARN TO DO FOR HIM/HERSELF !!
Specific Suggestions for Math
- Most children have more difficulty with the wording of the directions or of the problem than the math itself! Try to break it down for them.
- Don't be too quick to correct their mistakes. When checking over their work, try "I'm not sure about #5. Would you tell me what you did?" Most of the time they can correct their own errors!
- It is important to become familiar with your child's math program. You will probably already have heard about it through the grapevine, but you can find out what it is even before school starts by asking the office or leaving a message for the math specialist in the district. Go to any meeting the school offers to introduce parents to the math program.
- All new math programs come with extensive parent resource materials. You should receive these regularly but don't hesitate to go online and find them for yourself!
- Be prepared to ask questions, but don't start tearing the program down b/c you've heard there are problems with it. The program will not be changed in the current year no matter how parents may feel.
- Recognize that every math program, whether more traditionally skill-based or reform-oriented (more problem-solving, projects, less drill) has its merits and its weaknesses. Whether you believe there is too much emphasis on basic facts (less likely!), or not enough, you can supplement with the myriad of resources on the web.
- Don't be shy about asking the teacher for guidance with your child or with the math program itself.
- Remember: MATH IS ALL AROUND US ALL THE TIME! Ask your children lots of questions involving numbers and shapes around them. For example, "I need to cut up this square into two equal parts. I know an easy way (like this) but I think there's more than one way. Can you help me?"OR "I have a riddle. What movie comes before Toy Story 1000?" OR Place four quarters on the table. "Can you give me a dollar?" Put coins back. "Can you give me a half dollar?" etc...
- Never assume a concept is too hard for them. If simplified, they can often find a way.
SOME OF THE BEST MATH RESOURCES ON THE WEB
- Home School Math (http://www.homeschoolmath.
net/) - Incredible resource for ANY parent of ANY child! Free worksheets, math ebooks for elementary grades, extensive link list of games, interactive tutorials & quizzes, curriculum guide, andmath teaching help articles/lessons. The resources emphasize understanding of concepts instead of mechanical memorization of rules. - Let's Play Math (http://letsplaymath.net/) - Outstanding site by an exceptional educator who knows her stuff and hasd done her homework! Check out her Free (Mostly) Math Resources
- Math.com (http://www.math.com/) - Excellent resource with numerous electronic tools for mastering basics and beyond. Start with Basic Math and Everyday Math (not the well-known math program) for K-8.
acss/AES/sip/documents/ WebSitesforProblemSolving.pdf) - PurpleMath (http://www.purplemath.com/) - The best resource for algebra!
- And now for the latest offerings from my 3-year old grandson. The last time I posted his "muffin" comments, I had more views than from any math post in 3 years!
- My daughter has been trying to get him to go to sleep without her staying in the room. She told him that his 3-yr old cousin, with whom he is very close, is getting big now. My daughter commented, "Her mommy reads her a story, gives her a goodnight hug and leaves." My grandson replied, "Do you think I could do that, mommy?" "Of course", my daughter replied, to which my grandson immediately came back with, "Ok, but not tonight!"
- He is all boy, all the time. Aggressive, loves contact sports and is becoming a rabid NY Giants football fan like his daddy. He wears his Giants shirt on game day and can throw his little football with velocity. After seeing him throw the football a couple of times like a pro the other day, she said, "Wow, you threw the football really well, twice." "No, mommy, only once", he replied. "Are you sure? I saw you throw it twice", my daughter asked. "Yes, mommy, the other time was the highlights!"
"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
Monday, October 18, 2010
Odds and Evens Week of 10-18-10
Much has been happening in the world of mathematics and mathematics education. I'm only scratching the surface here.
- The passing of Professor Mandelbrot -- There is no question that this man has left an eternal "singularity" in the profession. Who among us has not been mesmerized by the computer images generated by one of his creations. He dared to think different and was not always recognized or lauded for his uncanny knack of seeing patterns no one else could. When asked to look back on his career, Dr. Mandelbrot compared his own trajectory to the rough outlines of clouds and coastlines that drew him into the study of fractals in the 1950s.
“If you take the beginning and the end, I have had a conventional career,” he said, referring to his prestigious appointments in Paris and at Yale. “But it was not a straight line between the beginning and the end. It was a very crooked line.”
- I read so many tweets/posts on Twitter, Facebook and in blogs discussing whether or not to continue teaching particular traditional topics in an algebra, geometry or precalculus class. Curriculum supervisors, math department chairs/supervisors and therefore teachers should now be guided by two key documents, particularly for Algebra 2 content:
- The Common Core State Standards Initiative for Mathematics
- ADP/Achieve Algebra II Test Overview
- You may also want to download the document from Glencoe which does a nice comparison of required content to the their text.
"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
Monday, October 4, 2010
Odds and Evens- October 2010
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
Friday, July 30, 2010
Where in the World is MathNotations?
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
Sunday, July 4, 2010
Happy 4th! SAT Problems on Twitter, SAT Math Quiz Book, Updates?
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.
Stay tuned and enjoy the summer hiatus!
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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
Friday, May 28, 2010
MathNotations Soaring With Eagles or Just For the Birds? Updates 5-28-10
NOTE: I added a new solution (see (e) below). Also, read the comments to see even more solutions. Thanks to Jonathan for pointing out my error in (d) of my results.
I'll get to that cryptic title in a moment (may be obvious to some)...
1. Remember the challenge problem I posted in the tribute to Martin Gardner a few days ago? Well, we rec'd several excellent replies and I have an additional response from a very sharp high schooler as well. Here was the problem:
Can you form 95 using each of the digits 5-2-2-1-0 exactly once? No restrictions on the arithmetic operations, parentheses, factorials, roots, logs, etc... You may combine the digits to form numerals like 12 or 120.
Mr. Lomas: 5! - (2+2)! - 1 - 0 Perhaps the most elegant since it uses the individual digits in the given order.
Robot Guy: (21-2)*5+0
Nate (high schooler): 120-5^2 Oh, the simplicity of that one! Combining digits is not the first way I thought of...
Mine so far:
(a) 102 - (5+2) Pretty simple but I wasn't thinking much of combining digits until I saw Nate's
(b) 120 -25 (Shameless plagiarism from Nate's but I couldn't resist!)
(c) (2^5)(2+1) - 0! (I posted this one already)
(d) 10^2 - 5 x (2 - 0!) (I knew there had to be a way using 100 - 5)
NOTE: JONATHAN POINTED OUT MY ERROR HERE. SEE COMMENTS.
(e) A new one: (2 + 2)! x (5-1) - 0! I felt I needed to atone for my error in (d)!
I suspect Mr. Lomas has even more! It was definitely the spirit of Martin Gardner at work here!
Keep these coming if you can find more. I'd like to see us get to 10 ways.
(2/3) egg per (hen⋅day) x 3 hens x 3 days = 6 eggs.
Get the RSS feed for this at Twitter/dmarain if you want to see the daily problems.
If you have a question about the problems or want more details about solutions, send me a Direct Message in Twitter or email me.
"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
Wednesday, February 17, 2010
Odds and Ends -- Week of 2-15-10
Catchup time...
- So where are the answers and followup discussion to the previous two problems I posted? They're coming. I plan to make a video of the investigation in the "143" problem.
- I think it would be so useful to post videos of actual lessons that demonstrate how some teachers implement explorations in the classroom at all levels from K-12. I'm sure there are some of these already online but they seem to be few and far between and I'm really referring to "best practices" here to serve as models for others. We have NCTM Illuminations for example but I'm really looking for something else here.
- More important than investigations is to see models of daily lessons which incorporate the best of what we know about effective practice. Lessons which show HOW to blend procedural and conceptual understanding, help students develop skills mastery while engaging in rich problem-solving. Easy to do? Of course not, but if we want the US to be academically competitive, we had better move quickly in this direction and use international models to guide us.
- How are you, the mathematics teacher, dealing with all of the confusing and overwhelming issues in math education today? OR have you learned to ignore all the "noise", close your door and simply go about the business of teaching? If you're able to, that is! Unfortunately some of the decisions which are being made independently of your input will have significant impact on how well you will be able to do your job today and in the future... Issues like
- "Algebra for All?" So where has that experiment gone?
- States joining consortia to develop a common standards and assessments in math
- How many consortia should there be? Will most eventually merge into one or two?
- Will common math standards ultimately lead to more consistency in content -- i.e., that which is actually taught in the classroom?
- Are math ed departments in colleges and universities adapting rapidly enough to prepare preservice teachers for the paradigm shifts which are occurring?
- Will "methods" courses in ed schools increase focus on actual content, e.g.,
"You will all prepare a lesson on the effect of changing the parameter "b" in a quadratic function. Your lesson should utilize multiple representations and include a carefully planned series of Socratic questions which develop meaning and conceptual understanding for the algorithms. Specify what actions you took to balance procedural learning with conceptual understanding. Also, be prepared to answer the essential question: "WHAT ACTIONS DID YOU TAKE AND HOW DID YOU ASSESS THAT LEARNING TOOK PLACE?"
- How can we all use emerging technologies to enhance our teaching repertoire. Regardless of whether you "tweet", "Buzz" or Facebook, or all of the above, how can networking and sharing make us more effective teachers? This is sort of obvious, but how many of us are using these on a daily basis in our planning and in our classrooms? 25% 50% More?
- What are the most effective online learning tools for mathematics, particularly middle and secondary math? Many excellent bloggers have researched and compiled excellent lists of resources, but specifically, what are the best online video sites (interactive or not)? This is crucial for me and it's something to which I want to dedicate myself.
- I choose not to get into NCLB or Charter Schools debates at this time...
- So where is my new website I've alluded to? It's taking forever to set up, but I want to do it right. More to follow...
For this blog, the essential question is, how we can make this a teachable moment in our math classes?
"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
Sunday, January 31, 2010
Can Your Students Find At Least Three Methods? Odds and Evens Week of 2-1-10
I've been working on a new website which I will share with you when ready but I haven't forgotten my faithful readers who may have forgotten me!
There are so many issues in mathematics education that it would take forever to update you on all of them, however, I know that you are already aware of most of these.
Some Significant Current Issues in Math Ed
- Moving Inexorably Towards Common Standards in Math
- Teachers Need a Clear Curriculum Map/Content Guide rather than Standards!
- Rapid Push Toward Including Several Open-Ended Questions on State or Common Assessments is Slowing Down. Can you think of the major reasons for this?
- Joel Klein's Education Equality Project whose goal is to close the Achievement Gap
Of course, most of you have already skipped down to the Challenge Problems!
The first can be tackled by middle schoolers, although many high schoolers may find it interesting and fall into a trap if not careful. The wording is challenging but your students may benefit from working in small groups.
Challenge Problem #1
a, b, c, d and e are positive integers with a ≤ b ≤ c ≤ d < e.
If a + b + c + d + e = 143, what is the least possible value of e?
Comments:
Is this merely a guess-test-revise question or is there a strategy/method your students can come up with? How would you extend this problem? change the "143" to a larger value? Change the set of integers to 4 values (a,b,c,d)? 6? k? This is an important issue. Otherwise students may see each problem as an isolated quickly solved puzzle!
The goal of the next question is to review geometry and algebra skills and concepts and to encourage a variety of approaches. I will give the answer -- the challenge for your students is to find AT LEAST THREE METHODS! The teacher may want to submit the best team's efforts to me for acknowledgment on this site.
Challenge Problem #2
P(5,1), Q(8,2) and R(a,b) determine an isosceles right triangle with point R above line PQ and ∠ PRQ the right angle. Determine the coordinates a and b. In your group, you must devise at least THREE methods!
Answer (6,7)
Methods???
"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
Note: These lyrics provoked considerable criticism back in 1949-50 but Rodgers and Hammerstein would not take them out. Do they still have relevance today?
Monday, October 12, 2009
A Rant, An Update and Model Problems for You
And the seasons they go round and round
And the painted ponies go up and down
We're captive on the carousel of time
We can't return we can only look behind
From where we came
And go round and round and round
In the circle game...
Oh, how I love Joni Mitchell's lyrics made famous by the inimitable Buffy Sainte-marie. Oh, how The Circle Game lyrics above describe my feelings about the state of U.S. math education. I feel I've been on this carousel forever. But I do believe that all is not hopeless. I do see promise out there despite all the forces resisting the changes needed to improve our system of education.
Our math teachers already get it! They get that more emphasis should be placed on making math meaningful via applications to the real-world, stressing understanding of concepts and the logic behind procedures, reaching diverse learning styles using multiple representations and technology, preparing their students for the next high-stakes assessment, trying to ensure that no child is ... They've been hearing this in one form or another forever. BUT WHAT THEY NEED IS A CRYSTAL CLEAR DELINEATION OF ACTUAL CONTENT THAT MUST BE COVERED IN THAT GRADE OR THAT COURSE.
The vague, jargon-filled, overly general standards which have been foisted on our professional staff for the past 20 years is frustrating our teachers to the point of demoralization. THIS IS NOT ABOUT THE MATH WARS. THIS IS NOT AN IDEOLOGICAL DEBATE. JUST TELL OUR MATH TEACHERS WHAT MUST BE COVERED AND LET THEM DO THEIR JOB!
BY "WHAT MUST BE COVERED" I AM INCLUDING THE SKILLS, PROCEDURES AND ESSENTIAL CONCEPTS OF MATHEMATICS. NONE OF THIS CONSTRAINS TEACHER STYLE OR CREATIVITY. BUT WITHOUT THIS STRUCTURE THERE IS ONLY THE CHAOS THAT CURRENTLY EXISTS. AND IF YOU DON'T THINK THERE IS CHAOS OUT THERE, TALK TO THE PROFESSIONALS WHO HAVE TO DO THIS JOB EVERY DAY.
UPDATES...
Results of MathNotation's Third Online Math Contest
The Common Core State Standards Initiative
NCTM's latest response to the Core Standards Movement - the forthcoming Focus in High School Mathematics
Validation Committee selected for draft of Core Standards
The results of the latest round of ADP's Algebra 2 and Algebra 1 end of course exams
It will take several posts to cover all of this...
RESOURCES FOR YOU
MODEL PROBLEMS TO DEVELOP HIGHER-ORDER THINKING AND CONCEPTUAL UNDERSTANDING
Consider using the following as Warm-Ups to sharpen minds before the lesson and to provide frequent exposure to standardized test questions (SAT, ACT, State Assessments, etc.). I hope these problems serve as models for you to develop your own. I strongly urge you to include similar questions on tests/quizzes so that students will take these 5-minute classroom openers seriously.
I've provided answers and solutions/strategies for some of the questions below. The rest should emerge from the comments.
MODEL QUESTION #1:
For how many even integers, N, is N2 less than than 100?
Answer: 9
Solution/Strategies:
Always circle keywords or phrases. Here the keywords/phrases include
"even integers"
N2
"less than".
This question is certainly tied to the topic of solving the quadratic inequality, N2 "<" 100 either by taking square roots with absolute values or by factoring. Of course, we know from experience, when confronted with this type of question on a standardized test, even our top students will test values like N = 2, 4, 6, ... However, the test maker is determining if the student remembers that integers can be negative as well and, of course, ZERO is both even and an integer! Thus, the values of N are -8,-6,-4,-2,0,2,4,6, and 8.
MODEL QUESTION #2
If 99 is the mean of 100 consecutive even integers, what is the greatest of these 100 numbers?
ANSWER: 198
Solution/Strategies:
There are several key ideas and reasoning needed here:
(1) A sequence of consecutive even integers (or odd for that matter) is a special case of an arithmetic sequence.
(2) BIG IDEA: For an arithmetic sequence, the mean equals the median! Thus, the terms of the sequence will include 98 and 100. (Demonstrate this reasoning with a simpler list like 2,4,6,8 whose median is 5).
(3) The list of 100 even consecutive integers can be broken into two sequences each containing 50 terms. The larger of these starts with 100. Thus we are looking for the 50th consecutive even integer in a sequence whose first term is 100.
(4) The student who has learned the formula (and remembers it!) for the nth term of an arithmetic sequence may choose to use it: a(n) = a(1) + (n-1)d. Here, n = 50 (we're looking for the 50th term!), a(1) = 100, d = 2 and a(100) is the term we are looking for.
Thus, a(50) = 100 + (50-1)(2) = 198.
However, stronger students intuitively find the greatest term, in effect inventing the formula above for themselves via their number sense. Thus, if 100 is the first term, then there are 49 more terms, so add 49x2 to 100.
MODEL QUESTION #3: A SAMPLE OPEN-ENDED QUESTION FOR ALGEBRA II
If n is a positive integer, let A denote the difference between the square of the nth positive even integer and the square of the (n-1)st positive even integer. Similarly, let B denote the difference between the square of the nth positive odd integer and the square of the (n-1)st positive odd integer. Show that A-B is independent of n, i.e., show that A-B is a constant.
MODEL QUESTION #4: GEOMETRY
If two of the sides of a triangle have lengths 2 and 1000, how many integer values are possible for the length of the third side?
MODEL QUESTION #5: GEOMETRY
There are eight distinct points on a circle. Let M denote the number of distinct chords which can be drawn using these points as endpoints. Let N denote the number of distinct hexagons which can be drawn using these points as vertices. What is the ratio of M to N?
Answer: 1
Solution/Strategies: The student with a knowledge of combinations doesn't need to be creative here but a useful conceptual method is the following:
Each hexagon is determined by choosing 6 of the 8 points (and connecting them in a clockwise fashion for example). For each such selection of 6 points, there is a uniquely determined chord formed by the 2 remaining points. Similarly, for each chord formed by choosing 2 points, there is a uniquely determined hexagon. Thus the number of hexagons is in 1:1 ratio with the number of chords.
MODEL QUESTION #6: GEOMETRY AND THE ARITHMETIC OF PERCENTS
If we do not change the angle measures but increase the length of each side of a parallelogram by 60%, by what per cent is the area increased?
(A) 36% (B) 60% (C) 120% (D) 156% (E) 256%
Thursday, September 24, 2009
More Challenges/SAT Practice, Core Curriculum Standards, Reminders, Comments...
Additional SAT/Contest/Challenges
Challenge 1:
HOW MANY DIGITS OF 10001000 - 1 WILL BE EQUAL TO 9 WHEN THIS EXPRESSION IS EXPANDED?
Challenge 2:
HOW MANY 5-DIGIT POSITIVE INTEGERS HAVE A SUM OF DIGITS EQUAL TO 43?
Challenge 3:
Jorge can run a 6-minute mile while Alex can run a 5-minute mile. If they start at the same time, how much less distance, in miles, will Jorge run in 10 minutes?
(Yes, you can respond with answers and solutions to these in the comments!)
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Tired of hearing about THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST!? IF I RECEIVE 10 MORE REGISTRATIONS, I MAY JUST STOP!
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The Common Core State Standards Initiative
First look here for a quick overview and here for an index to the latest draft of the standards. Of course, this blog only discusses the mathematics part of the document.
Overview
The Common Core State Standards Initiative is a joint effort by the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) in partnership with Achieve, ACT and the College Board. Governors and state commissioners of education from across the country committed to joining a state-led process to develop a common core of state standards in English-language arts and mathematics for grades K-12.
These standards will be research and evidence-based, internationally benchmarked, aligned with college and work expectations and include rigorous content and skills. The NGA Center and CCSSO are coordinating the process to develop these standards and have created an expert validation committee to provide an independent review of the common core state standards, as well as the grade-by-grade standards.
HIGHLIGHTS
- Core Concepts and Core Skills
- 11 Core Standards including the new "Mathematical Practice":
- Mathematical Practice
- Number
- Quantity
- Expressions
- Equations
- Functions
- Modeling
- Shape
- Coordinates
- Probability
- Statistics
- Each standard is broken into Core Concepts and Skills, provides research-based evidence and many illustrative examples to clarify the language
- Alignment of these standards to those of 5 representative states: California, Florida, Georgia, Massachusetts and Minnesota
- Standards reduce the number of Core Concepts and Skills in accordance with many recommendations to pare down the number of required topics to allow for greater depth
Equations | see evidence
An equation is a statement that two expressions are equal. Solutions to an equation are the values of the variables in it that make it true. If the equation is true for all values of the variables, then we call it an identity; identities are often discovered by manipulating one expression into another.
The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs, which can be graphed in the plane. Equations can be combined into systems to be solved simultaneously.
An equation can be solved by successively transforming it into one or more simpler equations. The process is governed by deductions based on the properties of equality. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.
Some equations have no solutions in a given number system, stimulating the formation of expanded number systems (integers, rational numbers, real numbers and complex numbers).
A formula is a type of equation. The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1 + b2)/2) h, can be solved for h using the same deductive process.
Inequalities can be solved in much the same way as equations. Many, but not all, of the properties of equality extend to the solution of inequalities.
Connections to Functions, Coordinates, and Modeling. Equations in two variables may define functions. Asking when two functions have the same value leads to an equation; graphing the two functions allows for the approximate solution of the equation. Equations of lines involve coordinates, and converting verbal descriptions to equations is an essential skill in modeling.
Students understand that:
-
An equation is a statement that two expressions are equal.
see examples -
The solutions of an equation are the values of the variables that make the resulting numerical statement true.
see examples -
The steps in solving an equation are guided by understanding and justified by logical reasoning.
see examples -
Equations not solvable in one number system may have solutions in a larger number system.
see examples
Students can and do:
-
Understand a problem and formulate an equation to solve it.
see examples -
Solve equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
see examples -
Rearrange formulas to isolate a quantity of interest.
see examples -
Solve systems of equations.
see examples -
Solve linear inequalities in one variable and graph the solution set on a number line.
see examples -
Graph the solution set of a linear inequality in two variables on the coordinate plane.
see examples
Very Important!
(Click on image to see a clearer view)
INITIAL MATHNOTATIONS REACTIONS
- Exceptionally clear and definitive document
- Influenced by NCTM (Curriculum Focal Points), Achieve, College Board, ACT
- Illustrative examples are of high quality
- Will serve as a basis for states' revisions of current standards hopefully creating more consistency than currently exists
- Leaving curriculum to local districts and states was a politically necessary decision, however, in my opinion, developing a reasonably consistent curriculum by grade level and/or course across districts and states from these standards may prove to be difficult and may again lead to considerable disparity. Hopefully, this will be self-correcting when standardized assessments are created as is currently being done with the End of Course Tests from Achieve
Tuesday, August 25, 2009
Update Week of 8-24-09: Contest Info
REMINDER!
MathNotations' Third Online Math Contest is tentatively scheduled for the week of Oct 12-16, a 5-day window to administer the 45-min contest and email the results. As with the previous contest, it will be FREE, up to two teams from a school may register and the focus will be on Geometry, Algebra II and Precalculus. If any public, charter, prep, parochial or homeschool (including international school) is interested, send me an email ASAP to receive registration materials: "dmarain 'at' gmail dot com."
Read Update (4) below!
Updates:
(1) The first draft of the contest is now complete.
(2) As with the precious two contests there will be one or two questions which require demonstration, that is, the students will have to derive, explain or prove a statement. This is best done freehand and then scanned as a jpeg image which can be emailed as an attachment along with the official answer sheet. In fact, the entire answer sheet can be scanned but there is information on it that I need to have.
(3) Some of the questions are multipart with the last part requiring more generalization.
(4) Even if you have previously indicated that you wish to participate, please send me another email using the title: THIRD MATHNOTATIONS CONTEST. Please copy and paste that into the title. Also, when sending the email pls include your full name and title (advisor, teacher, supervisor, etc.), the name of your school (indicate if HS or Middle School) and the complete school address. I have accumulated a database of most of the schools which have expressed interest or previously participated but searching through thousands of emails is much easier when the title is the same! If you have already sent me an email this summer or previously participated, pls send me one more if interested in participating again.
(5) Finally, pls let your colleagues from other schools in your area know about this. Spread the word! If you have a blog, pls mention the contest. If you're connected to your local or state math teachers association, pls let them know about this and ask them to post this info on their website if possible.
Note: Sending me the email is not a commitment! It simply means you will receive a registration form.
An aside...
I've been asking my kids questions every day to sharpen their minds for school which starts next week. I asked my son how he would spell, arachnophobia, the fear of spiders. He was confident he knew the first four letters: iraq....
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