Uh Oh -- The 2-yr old turns 3 -- or does he!

Hey, these are precious moments and, anyway, I can't do more than one video a week!  So to take a respite. here's another anecdote, and, by the way, his 6-yr old brother and 3-yr old cousin are now demanding equal time.

So, he turned three the other day and when he went to bed that night his mommy, aka my daughter, told him: "____, now that you're three, I'm just going to tell you a story, give you a hug and kiss and say goodnight" (rather than staying with him until he fell asleep).

Knowing how this young man's mind operates, what do you think his reply was?

"Mommy, I'm only three during the day, I'm still two at night!"


Good luck to my daughter and all of his teachers!


Ok, fair's fair...
When the 6-yr old brother who was then 5, got his new bed (full size like his parents) my daughter and son-in-law heard him pacing and sighing loudly outside their door after they had tucked him in. Finally, his mommy came out and asked him what was wrong. Here was his response:

"I don't want to sleep in that bed. I'm not ready to be married!" 

This young man definitely marches to a different drummer, perhaps an entirely different band! He does remind me so much of myself at that age. I'm not sure if that's good...

Finally, my son's beautiful 3-yr old daughter went to Disney World a month or so ago. My wife asked her what her favorite ride was and she replied, "the rollercoaster." Then my wife asked if she was afraid. This was her reply:


"I was not afraid. I just screamed the whole way."


May their innocence remain forever...

--------------------------------------------------------------------------------
"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

Challenging Geometry Assumptions: Review for SAT I/II

The video below presents a more challenging 3-dimensional geometry problem which would be at the upper end of SAT I or SAT II - Subject Tests (Math I/II). The key here is to challenge students' assumptions about a quadrilateral being a square because it has 4 congruent sides, a common error. This question will also review a considerable amount of geometry: Pythagorean Theorem, Volume of cube, spatial reasoning, 45-45-90 triangles, area of a rhombus, etc.

As always, the focus is on the art of questioning, suggested instructional strategies and pedagogy, although this problem may be interesting enough to capture the attention of some students who are preparing for upcoming standardized tests. For students who need help with spatial visualization, a model could be provided or have enough empty boxes available (they don't have to be cubes!).  

I strongly urge using learning partners or pairs for the discussion. 

Benefits include:

(1) Students feel less tentative when offering ideas to one other person or in a small group.

(2) Instead of posing conceptual questions to individuals, receiving little or no response except from the most confident or capable, you can pose a question to a learning pair: "Julie and Jason, what is needed to insure that ABCD is a square?" They should be given a few moments to think and confer before responding. The stronger student will usually explain it to the other. If neither can respond, they can say, "Pass!"

(3) The biggest advantage of student dialog is that often our explanations simply don't click with several students, but they do make sense to others. Those who "get it" can usually explain it in terms that their peers understand better, a benefit to both the "explainer" and the "explainee"!



By the way, the question posed near the end of the video is worth pursuing if time permits:

"Without calculating the areas, is the area of the non-square rhombus less than or greater than the area of the square?"

The answer is less for many reasons, but we would hope they would recall the base x height formula for a rhombus. The height is maximized when the angle between the sides is 90°. Why? Interestingly, the areas are quite close: 19.6 vs. 20. I believe strongly that this is the type of higher-order question that not only reviews important concepts but promotes deeper thinking, or should I say, thinking more than one inch deep!

What are your thoughts? Would you give students the e√3 formula before a standardized test or ever?Are these videos helpful to you? If you respond both on this blog and on my YouTube Channel, MathNotationsVids, and also rate these videos, that gives me the guidance I need to improve them.


------------------------------------------------------------------------------------------
"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 a hen and a half can lay an egg and a half in a day and a half...

The full version in one of its many many variations:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.

Putting aside the silliness of the riddle, there really is some serious mathematics going in these kinds of rate/ratio/proportion problems. Rather than solve the "hen" problem for you, I'll leave it to my readers to solve it by their own favorite methods. By the way, the answer to this riddle is in the description of the video below on my YouTube channel. Sorry 'bout that!!

Instead, the video below, which appears on my YouTube channel, MathNotationsVids, presents a developmental approach to a more complicated ratio problem for middle schoolers and beyond. I'm far more interested in your thoughts about the teaching strategies than I am about the problem itself. Please understand, further, that I am not suggesting the method shown in the video is efficient nor would it make much sense for the upper level math or science student. See comments below the video for further discussion of this.


The Problem in the Video Below:


If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days? Assume all workers build at the same rate.




More Advanced and Efficient Algorithms


(1) We assume from the "constant rate" assumption in  the problem that the number of houses (H) which can be built varies jointly as the number of workers (W) and the number of days (D).
Thus, H = kWD.

Substituting, H=3, W=10 and D=60, we obtain:
3 = k(10)(60) or k = 1/200. Note that the units of k are Houses/(Workers x Days).
We can interpret k to mean that 1/200 of a house can be built by 1 worker in 1 day. Thus, k is not only a constant but actually represents a rate. Another way of expressing this rate is
(1 House)/(200 Worker-Days) or the reciprocal version:
(200 Worker⋅Days)/(1 House)

Substituting the new set of values into the relationship H = (1/200)WD, we obtain:
5 = (1/200)(W)(40) or W = 25 workers.

(2) This can be made even more efficient using the "factor-label" (dimensional analysis, etc.) format:

(200 Worker⋅Days)/(1 House)) x (5 Houses)/(40 Days) = 25 Workers!

(3)  I could also exploit the inverse variation between W and D, but that's for my readers to bring up or for another video!

I see these efficient methods as "black box" methods for some students. Developing a deeper understanding of direct and inverse variation is far more important for the younger student.



-------------------------------------------------------------------------------------------------------------------------------

"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)

Odds and Evens Week of 4-19-10

I've decided to post a few headlines from various sources (Ed WQeek, NYTimes, other blogs, etc.) over the past few weeks. I will not post links to most of these or make comments at this time. I believe they speak for themselves...

• Stand and Deliver Teacher Dies of Cancer
• Is Teaching A Marathon or a Sprint?
• 
  
  A Letter For Anyone Concerned About K-12 Mathematics Education 
• U.S. Needs Better-Trained Math Teachers To Compete Globally
• U.S. Falls Short In Measure of Future Math Teachers -- American college students earned a C on a new test comparing their skills with their counterparts in 15 other countries.
• Teacher Training No Boon For Student Math Scores -- A major federal study concludes that intensive teacher professional development in math did not immediately improve test scores.
• The Boys Have Fallen Behind
• Merit Pay For Students Fails to Raise Score, Study Finds -- Research suggests that payments can boost achievement if they reward behavior conducive to learning, rather than test scores themselves.
• Both Value and Harm Seen in K-3 Common Standards -- While some view the proposed expectations as valuable guidance, others worry that they are inappropriate for youngsters.
• What They're Watching -- The Ten Most Popular Course Lectures Available on You Tube
• An Open Mind -- Putting free courseware online was a first step in reimagining education. What now? Wiki U’s, smart courses and, maybe, learning

A new website, LearnBoost, is launching 0n 4-19-10. It looks like the developer, Matthew Hunter, and his team put extensive thought and planning into this. It compiles some of the top news stories in education and readers can move their favorite articles up the ladder by voting on them.

So where are MathNotations math contests for 2009-10. Sorry, folks, I'm just too overwhelmed to put this together for the year. However, I have been considering having my 2-year old grandson compose some questions!

Anyone actually looking at the videos I've been posting on my You Tube channel, MathNotationsVids?  I can see a few views but, without any responses or comments, it's hard to tell if this is worth continuing. I probably will because I enjoy it!

Stay tuned!

-------------------------------------------------------------------------------
"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

An SAT Problem or Teaching/Learning Beneath the Surface

Ok, so I don't have another anecdote from my grandson today so I'll have to get back to mathematics -- problem-solving, teaching and learning.

The video below deals with an algebraic equation in 2 variables which should be straightforward for your stronger Algebra II or Precalculus student. But will it be? I invite you to predict how many in your classes will answer it correctly, then try it out. After all, it is multiple choice, so statistically some should get it right by some means or other!

Hopefully, the purpose of the problem and the video will become clear to all of you. If we want our students to demonstrate better reasoning and an understanding of important ideas in math, we need to feel comfortable in teaching for meaning and understanding. This doesn't mean we stop teaching algorithms and procedures, however. Exactly what all this means and how to do it is the reason for this blog. I certainly never claimed to know the answers or any other mystical secrets. I only know that I never gave up trying.  Sometimes my efforts failed miserably, but I hopefully learned from these attempts.

It would mean a lot to me if you share your thoughts here or on my You Tube channel, MathNotationsVids, where you will find my other videos.




Note: Another subtle point I should have made in the video---
y(x-4) = 0 → y = 0 OR x = 4
It is important for us to stress this point and distinguish it from "AND" logic. If the equation were in the form:  y² + (x-4)² = 0, we would have (y = 0) AND (x = 4), whose graph would be the single point (4,0). Another instance where an exercise on the board can lead to a rich, fruiful and profound discussion. If all of this is seen as taking too much time away from content, remember this is precisely the kind of change in curriculum and instruction that Prof. Schmidt has been trying to tell us about for over 15 years! Well, I'm preaching to the converted, aren't I...
-------------------------------------------------------------------

"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

Same 2-Year Old, Different Day!

With all the material I'm getting from my 2-year old grandson, I may as well give up my math blog and start a new blog featuring "anecdotes about grandchildren"! I betcha' some of these would go viral like no math article ever would!

Anyway, while I am preparing my next "exciting" video, I decided to share with you what my grandson said to his grandmother (better known as "Mimi" and also known as my wife) this morning.

His mother (my daughter) and grandmother (my wife) took him this morning for his first blood test (for allergies). My daughter needed "Mimi" to hold her son as she thought she would pass out!  So, my wife had him sit on her lap. When he realized that the needle was for him, the screaming began and he tried to jump out of "Mimi's" arms. Fortunately the whole procedure took only 30 seconds. My grandson jumped off my wife's lap, wiped his tears away and in a serious tone stated to Mimi, "You're fired!" My wife was stunned, never expecting to be fired from being a grandmother! There's definitely some Donald Trump in that boy.

Of course, a few minutes later all was forgiven and they all lived happily ever after.

I realize grandparents think all their grandchildren are special and anything the children say must be a gem worth repeating a thousand times. But, I do believe this young man is quite a character.  By the way, my daughter has no idea where he got the expression "You're fired!" from, unless of course he is watching the Apprentice on the sly!

---------------------------------------------------------------------------------------

"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

Two-year olds say the 'darndest' things!

A respite from Pi, sorta...

Ok, so I'm a mad crazy proud grandpa (Pop-Pop to be exact) but I could not resist telling the following story (Reader's Digest fare!)...

My daughter is a PTA rep and had to take her two-year old to the budget meeting with the superintendent yesterday. There were about 50 people in the room with my daughter up front. After the superintendent painted a gloomy picture of the needed budget cuts for next year, he asked if there were any questions. Everyone was quiet except for my grandson who raised his hand. The superintendent looked whimsically at him and asked him what his question was. He stood up and asked the superintendent if he was going to eat the blueberry muffin he had in front of him!

The superintendent graciously gave it to him which my grandson devoured in short order. After a few minutes, the superintendent kindly looked back at my grandson and asked, "Do you have any more questions?"  Can you guess the reply?  That's right, he asked if there were any more muffins...

Can you guess how to motivate students like this!




"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

6 Divided By Pi^2 and Relatively Prime Integers - A Video Derivation

Jaime didn't just teach math. Like all great teachers, he changed lives

– Edward James OlmosSOURCE: THE SAN JOSE MERCURY NEWS5 DAYS AGO

The current post presents a non-rigorous video derivation of a formula in mathematics which might create 'shock and awe' in kids of all ages. Oh, alright --  in people like me! This 3-part video is the followup to the post from March `16th -- Pi Day, More Videos on Counting, etc...



The following description comes from the YouTube Channel, MathNotationsVids:

Designed for anyone who has a passion for mathematics, this derivation of a classical result in math is suitable for advanced middle schoolers through undergraduate math. Further, teachers may want to show this to Math Clubs/Teams. This 3-part video builds on results from previous videos, is related to a post on MathNotations and is dedicated to the "Greatest Teacher in America" -- Prof. Jaime Escalante who passed away a few days ago.

Part I




Part II



Part III




Comments, Notes:

• I hope you will see in these videos a central theme beyond the content involved -- a fundamental heuristic in teaching mathematics: When introducing an abstract concept or in deriving a formula or theorem or rule, avoid heavy symbolism and work with simple concrete numerical cases before generalizing results. I believe this has validity at all levels of math instruction.
• This topic ties together so many apparently unrelated topics in mathematics in a wondrous and surprising way. Perhaps it will inspire a budding mathematician as it did me...
• Visit my YouTube channel (see above) and please comment on these if you feel you want to see more. They are fairly labor-intensive (do you really think I'm speaking extemporaneously!) but they are worth it if someone enjoys them. The quality of the videos can still improve much more but this is just a beginning...

"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)