Clarity of a concept...number and a numeral

Clarity of a concept is essential for creating interest in the subject. Children are often confused with ideas in Mathematics.

For instance take this example of...

What is the difference between a number and a numeral?
When we say there are six flowers in a bouquet, we are mentioning number of flowers.Numeral refers to the markings we use to indicate that idea of number.

That is in above case it is Six, VI or 6.

A number is an abstract concept while a numeral is a symbol used to express that number. How do your express the concept of fiveness…
· Five
· 5
· V
A number is an idea that is used to refer to amounts of things.
A number symbol is called a numeral.
This is what Dr. Math Answered…
http://mathforum.org/library/drmath/view/58756.html
Read 10 Rules for writing numbers and numerals.
http://www.dailywritingtips.com/10-rules-for-writing-numbers-and-numerals/

So, it is desirable to make concepts clear with proper examples and instances.

Polya's Army

Problem Solving is a big deal in any math class I teach, and I, like most math teachers, use Georg Polya's problem solving phases as a framework. Though I used to teach it as a four step process, I now recognize it as four phases, which problem solvers can progress through in many different ways, back tracking and skipping. The ultimate reference on this is Polya's book How to Solve It. My handout (adapted from Dave Coffey's) is here; it focuses on Polya's questions. (Questioning being another important comprehension strategy.) The modern day successor to Polya as a researcher and teacher of problem solving is Alan Schoenfeld. The link leads to his site where he generously shares a lot of his research and work. I recommend at least skimming Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making In Mathematics (pdf link), a novella of a paper. Around pp 60-67 there's an amazing section on novice and expert problem solvers and teaching interventions.



One of my calc students was taking his final early, on his way to the month duty for army reservists, and commented on how he remembers Polya's steps by a connection with the troop leadership procedure. He sees:

• understanding the problem - receive the mission, issue the warning
• make a plan
• do the plan - start movement, recoineter, complete plan
• revise and check - issue plan and supervise

How cool is that! I just had to share. Thanks, James!

Google Analytics

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One Page Wonders

One option for my students to review, in this mini-6 week semester (odd but true that it seems equally as hard for students to remember the beginning of the course), is a One Page Wonder. I can not remember how I stumbled across them first, but this page at the bookpublisher Tor was the first one I'd seen. They use an old topologists trick to turn a single sheet of paper into an 8 page mini-book that can begin at any page, also be several different four page mini books, or fold to show any one single panel at a time.

When making a story for a one page wonder, it's important to have panels that can go in various orders to get the maximum effect. The first one of my own I made was this Batman one for my son's birthday, and my daughter has made a clever Warriors (the cats) one for herself and a neat gift wonder. Doesn't take long to try, and you'll surely enjoy it - so give it a go!

What's the value for my students? Connections and synthesis are two of the comprehension processes from Mosaic of Thought, the seminal text on teaching for comprehension. (That link goes to the publisher, where there's a sample chapter. Also, they're just now having a sale, $7.50 off. Go quick!). In a one-page wonder summary for the class, you'll have to narrow down to 8 important points, and get to see them connect to each other in different ways. If I get any neat ones turned in, I'll share them here.

Here's the Batman one: (click images for the full size picture)




I even used them in a bible class (also with a vid of how they work).

Soal Ujian Nasional 2009 IPS

Detik - detik pengumuman hasil Ujian Nasional 2009 akan segera tiba, menjadi harapan semua siswa bahwa tidak ada satupun mata pelajaran yang di UN kan menjadi ganjalan ( termasuk matematika ) buat mereka melanjutkan kembali belajarnya di tingkat yang lebih tinggi. Asa yang tinggi terhadap kelulusan akan sangat dinantikan oleh semua pelajar di Indonesia, sebab Ujian Nasional telah membuat harga

Mobile Math

For the Math in Art Festival I did with Susan Walborn (an amazing teacher who's moved on to becoming an amazing retailer - must just be amazing, eh?), one of my favorite lessons was a mobile lesson (link leads to a pdf of a verrry complete 3rd grade lesson plan) based on the art of Alexander Calder and the math of area and average.

The emphasis is on the average as a balance. In calculus today, we covered center of mass, and built the connections among the moment, the weighted average and the idea of balance. Students used the ideas to create cardboard cutouts of curves and find the balancing point. They did a great job. Mine is the unimpressive cubic.

Trig Rummy

My calc students have really been struggling with trig substitution, mostly, i think, due to lack of fluency with trigonometric functions. I'm a big believer in games for skill practice, as they are an engaging context that encourages more practice and ideally offer a chance for forging new connections.

So today we played Trig Rummy. Many of my card games are based on concentration, rummy, or go fish. And occasionally Euchre. (I still think the partially ordered quadrilateral Euchre has a future.) Links for printables are after the rules. I think it would adapt to high school trig pretty easily. I'd replace the calculus relations with graphs and triangle identifications.

Trig Rummy

Objective: the player will practice and gain facility with trigonometric identities and calculus relations.

Players: 2-4

Goal: get the most cards in play.

Set up: randomize cards and deal 7 cards to each player. (You may want to introduce the game with only 4 cards as 7 is overwhelming.)

Play: On your turn you may do one of:
• take any card from the discard pile (not just the top card).
• draw a new card.
After that you may do either or both of:
• pick up any sets you have.
• play any new sets you have.
At the end of your turn, if you did not play a set, discard a card.

Special:
• if at any point the discard pile contains a complete set, the first player to otice can cal “Rummy!” and take the set out to play for themselves.
• you can not play matching operator cards, like a pair of d/dx cards.
• There is a wild card, which you can choose to be -1× (times -1), or d/dx, or ∫ ⋅ dx.

End: After the last card is drawn from the deck, each player gets one more turn from the discard pile. Then the cards are counted up and the player with the most cards played wins. No penalty for unplayed cards.

Variation: Play until one player is out.

Example plays:
• Play sin2(x) matching 1 − cos2(x).
• Play d/dx with sin(x) matching cos(x).
• Play ∫ ⋅ dx with sec(x)tan(x) matching d/dx with ln|sec(x) +tan(x)| (since both are equal to sec(x).)

If a player lays a mistaken combination and no one catches it before the next player’s turn, they stay in play but turned face down. If someone does catch it, that player has to pick up the pair.

Feel free to use a trig cheat sheet.

Rules PDF - includes a trig cheat sheet
Trig Cards PDF - for 54 2-sided cards of trig identities and calc relations. Could use for flash cards or other games also.

Another Math Blog

Which, come to think of it, would have been a good title for this blog.

I've been cosistently loving a lot of the links given by Y of X. It records the musings of Hendree Milward, a Connecticut math teacher (at Tunxis Community College?). Recently he's posted interesting links to the cool Institute of Figuring and an offbeat math poetry blog. If you read his blog, I won't have to bother putting up those links here.

Realising the presence...

Mathematics is considered as an abstract subject. Students find it dull and boring. There are so many on this planet who always say " Mathematics is not my cup of tea". Do you know the reason for this indifferent attitude towards Mathematics ?

Have you ever realised the presence of Mathematics around. One thing which is very interesting to note is...the people who say that they are afraid of Math or they hate Math or they try to run away from Math are using/applying it in daily life.

Aren't these people managing house budgets ?

Aren't they not paying bills for their dinner in a restaurant ?

Aren't all of them not using number manipulations, mathematical operations etc?

Aren't they not seeing/observing beauty of geometry all around?

There is not even a single place where the knowledge of Mathematics is not applied. It is very important to make students realise the presence of Mathematics around and how important the subject is in our daily life?

I have created a blog Exploring Mathematics Around where you can find pictures from daily life depicting the presence of Math everywhere.

Soal Ujian Nasional 2009

Setelah hampir satu bulan ngga ngeblog ( hanya diselingi oleh postingan tentang penulis tamu ) karena kesibukan menyiapkan Ujian Nasional dan rentetan Ujian lainnya, akhirnya hari ini bisa menyempatkan kembali menyapa anda semua yang udah bolak - balik ke blog ini karena ke sasar. Yup hampir sebulan sudah blog ini tidak mampu untuk berbagi soal matematika yang biasanya terus berusaha untuk

Stigler on training teachers

Michael Goldenberg over at Rational Math Ed pointed this out on his blog and it's worth spreading out.

See this thread at the Math Forum for a transcript. Dr. Stigler is the co-author of The Teaching Gap and The Learning Gap, which distill research from The International Math and Science Study (TIMSS) into fresh and relevant information for any teacher, but especially math and science teachers.

The big points to me:
-Teaching is cultural => really hard to change
-Teaching needs to change <= student achievement data
-Teaching can only change gradually, but it can change, by focusing on teaching instead of teachers, and evaluating based on students' thinking and reasoning.


A relevant quote: "A review of research by Hiebert and Grouws identifies two features of classroom instruction that are associated with students' understanding of mathematics:
- CONNECTIONS: Making mathematical relationships-among concepts, procedures, ideas-explicit in the lesson
- STRUGGLE: Students spend at least some time struggling with important mathematics."

Check it out.

A programmer's view of the Universe, part 3: The Death of Richard Dawkins

We're getting close to the end of my blog. After today's entry, I only have three left to write. After that, I'll only blog anonymously or (more likely) not at all.

This is part three of five in my "Programmer's View of the Universe" series. I struggled for a while with how best to introduce the ideas in this installment, and ultimately opted for a short story.

This is a science fiction short story. It's different from many other sci-fi stories in that it is set in the "near future", but it has realistic schedule estimates. So unlike 1984, 2001, The Singularity is Near and all the other sci-fi stories that grossly underestimated their project durations, this one is set 1000 years in the future. I.e., right around the corner.

The story is disrespectful to pretty much everyone in the world. It will create a fantastic shit storm. This is probably a good time to point out that I don't speak for my employer. [Edit: Yay for fiction! Apparently marking something as fiction placates people. Nice to know.]

The story is 18 pages (PDF from Google Docs print preview). That's not unusual for my blog, but I went ahead and published it as a standalone document.

I'd encourage you to enjoy it, but I'm old and embittered enough to know better. You probably shouldn't even read it. Just wait for someone to summarize it for you.

Installments 4 and 5 will not be short stories; they will be regular old blog rants. In them I will further develop these ideas, and I will also attempt to clear up any gross misconceptions about the story, of which there are bound to be many.

My final blog-rant entry is the only one I care about anymore. I've been working on it so hard that my fingers have started to fail. It's been tons of fun, aside from the chronic pain. It's about a neat programming language, and Emacs, and lots of other stuff. I can't wait!

Oh yeah. Here's the link to the story. I've never done a read-only Google Docs link before, so it's probably broken. Or editable. I don't know. We'll see.

This week is going to suck. People are going to be mad. Maybe I should take a vacation and come back when the whining is finished. Can someone email me and let me know when it's all blown over?

Penulis Tamu

Ujian Nasional memang sudah berakhir lebih dari 2 minggu yang lalu, akan tetapi tugas selain Ujian Nasional terus menghadang. Setelah Ujian Nasional rampung siswa juga harus menyelesaikan Ujian Praktik untuk beberapa mata pelajaran yang kemudian harus berjibaku lagi dengan soal - soal Ujian Sekolah. Yup, semua kesibukan itu benar - benar menenggelamkan saya dalam rutinitas yang tidak ada

Ken Robinson

So I showed the Ken Robinson video to my class on the first day and it was quite interesting. The video started with medium engagement, but by the end everyone was tuned in. The story about the choreographer (Gillian Lynne) seemed to really connect.



The first connections were to elements of their own stories, and were interesting. Then one of the students brought up a connection with how Robinson describes us as being educated out of creativity. Her sister is interested in being a soloist, and it took ears of retraining or untraining to get her out of what she had been taught to do as a choral member. I'd never heard of that before, but it connected exactly to the point I wanted to make.

They have been educated - and successfully so, as college calc students - in school mathematics, which has almost nothing to do with mathematics as practiced by mathematicians. (Which really needs a descriptor. Real mathematics isn't going to interest anyone. Creative mathematics? Cool vs. school or cruel mathematics?) Instead of being able to repeat what someone shows you, it should be about solving problems that you don't know how to solve. In which situations mathematicians excel, because they are entirely willing to be wrong, and even glad to be wrong if they learn from it. They're willing to, as my friend Dave says (quoting his favorite Australian, Brian Cambourne) to "Give it a go!"

As long ago as 1982, my calc instructor, John Hocking, worked mightily to convince us to have no fear. To be wrong 100 times if it teaches you something. That math was exploratory, and creative. He shared his topological research with us... which was amazing and fun. He's the reason I added the math major in the first place. Often he put it as "blank pages are just waiting to be filled."

Now can I help my calc students to do the same?

Hani

Perbedaan antara yang mustahil dan yang tidak mustahil terkletak pada tekad seseorang. -Tommy Lasorda-
Pada hari aku bertemu Hani irmawati, ia adalah gadis berusia tujuh belas tahun yang pemalu, berdiri sendirian di tempat parkir sekolah internasional di Indonesia, tempatku mengajar bahasa inggris. Sekolah itu mahal dan tak menerima murid orang Indonesia. Ia menghampiriku dan bertanya apakah aku