text: Sweet Remain by Cualcultura by Dubstep published on 2012-09-12T11:47:38Z Full Description on Dubstep.NET : http://www.dubstep.net/track/sweet-remain-by-cualcultura Please Subscribe Here: http://bit.ly/LN7mCh ------------------------------------------------------------------------------------------------------------------ Follow Cualcultura: https://www.facebook.com/pages/Cualcultura/307792449309758 https://twitter.com/cualcultura SC: http://soundcloud.com/cualcultura Follow us @ http://www.Dubstep.NET Sounds like: Skrillex, Kill The Noise, Feed Me, Excision, Datsik, Downlink, Flux Pavilion, Doctor P, Funtcase, Stenchman, Cookie Monsta, Knife Party, Adventure Club, Nero, Bassnectar, Noisia, Zeds Dead, NiT GriT, DeadMau5, Dubstep, Drumstep, Minnesota, Dance Music, Bass, Bare Noize, Skream, Play Me, Rottun, Never Say Die, Heavy Artillery, OWSLA, and Mau5trap Genre Dubstep Comment by TAKEYDO nice work man 2012/09/15 08:07:47 +0000 Comment by Poolz Big!! love it! 2012/09/13 16:23:51 +0000 Comment by Lting @beatsbygt: The point of this whole website is for others to hear/find and give opinion on ones music, whether it good or bad. all feedback is positive when utilised in the right way. 2012/09/13 09:37:17 +0000 Comment by contradictionature NICE 2012/09/13 06:52:46 +0000 Comment by weed? intresting song, i really like it 2012/09/13 06:31:31 +0000 Comment by Condens8 this is dope 2012/09/13 03:52:21 +0000 Comment by ClarksoN Its a Beauty! 2012/09/13 03:31:53 +0000 Comment by Jaymeson This is awesome^^ 2012/09/13 02:17:30 +0000 Comment by Tasty Morsel Meh 2012/09/13 00:26:26 +0000 Comment by Thalius This is pretty bad.. 2012/09/12 21:15:14 +0000 Comment by UnheardOfHipHop wtf is this 2012/09/12 20:21:05 +0000 Comment by majokkororo interesting track 2012/09/12 18:21:53 +0000 Comment by NAZA Opinions, Opinions, Opinions... 2012/09/12 16:30:32 +0000 Comment by Dubstep Chick Check this out! It's hotttttt http://soundcloud.com/skunkbutterrecords/6-million-ways-to-die-1 2012/09/12 16:03:32 +0000 Comment by Steidel i dont really like this, but its definately good 2012/09/12 15:43:23 +0000 Comment by BeatsByGT @warren-palmer: I know right, this is actually garbage but it makes me think that if they'll upload something this bad then the little ppl like me might have a chance in getting something uploaded one day... --> http://soundcloud.com/beatsbygt/movement-dubstep-edit 2012/09/12 15:28:42 +0000 Comment by zirmada HELL YEAH 2012/09/12 15:20:33 +0000 Comment by Mr-E @lting: Lol finally some1 with the balls to actually say a song is bad. Unlike Electrode,Dragoonscale, or screaming chicken that think every song is "sick!". Such bull, they just do it for the followers instead of telling the truth... 2012/09/12 15:13:48 +0000 Comment by Mista bt old skool flovour avin it each and every raver come have some>>>/>>> 2012/09/12 14:12:22 +0000 Comment by Byutifu this melody is GORGEOUS! <# 2012/09/12 13:56:26 +0000 Comment by Lting @beatsbygt: nice vibe, but awefull production 2012/09/12 13:52:50 +0000 Comment by VoidDS well, this is pretty progressive that's the good side 2012/09/12 13:42:04 +0000 Comment by Ahobo1 Never heard anything like this before! 2012/09/12 13:13:21 +0000 Comment by BeatsByGT i personally don't know what dubstep.net was thinking when they uploaded this........ its actually really bad and some parts are out of key.... 2012/09/12 13:08:45 +0000 Comment by ConMuni@ Buen trabajo man ^^ 2012/09/12 12:49:33 +0000 Comment by DJ Narkut Relaxing music =D 2012/09/12 12:48:29 +0000 Comment by Dan Schellinger This is really sweet! 2012/09/12 12:23:16 +0000 Comment by Weekly Waves || Chillout sweet ç 2012/09/12 11:58:57 +0000 Comment by Clément Bindzi Dope !! 2012/09/12 11:54:49 +0000 Comment by Dubstep Follow Cualcultura: https://www.facebook.com/pages/Cualcultura/307792449309758 https://twitter.com/cualcultura SC: http://soundcloud.com/cualcultura Follow us @ http://www.Dubstep.NET 2012/09/12 11:48:33 +0000


text:
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Date: 1842
Description: This hand-colored lithograph of the second Fort Howard, with Indians canoeing on the Fox River, shows the hospital built 1834-1835 outside the stockade on...
Date: 1853
Description: Illustration of the new museum building, also known as "Bruen's Block," on the corner of East Washington Avenue and South Pinckney Street. Named for W.D. B...
Date: 1885 ca.
Description: The home and fur-trading post of Col. Dominique DuCharme. The portion of the structure on the left was built by DuCharme in 1791. The right portion was add...
Date: 1806
Description: Prehistoric Indian fortification on Hurricane Creek.
Date: 1862-04
Description: Notes include: Sketch of Battle of Island #10. Sketch of Fort Pillow Tennessee. Gen. Pope had command of the land troops.
Date: 1873
Description: Exterior view of the W.P. Storms residence with a horse-drawn wagon in front of it.
Date: 1838 ca.
Description: Pencil sketch of a stone house with a porch the full length of the dwelling, supported by 6 columns.
Date: 1850 ca.
Description: Chippewa Indian Sugar Camp. Plate 61, preceding p. 199, vol. I, The Indian tribes of the United States: their history antiquities, customs, religion, ar...
Date: 1858
Description: View of a Presbyterian church with a horse-drawn vehicle and multiple pedestrians in the foreground.
Date: 1865
Description: Plan and elevation of the post hospital at Camp Randall.
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Description: Engraved image of a drawing by Henry Sandham showing six bicycle racers on a track with a large crowd of spectators in the stands and along the side of the...
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Date: 1876
Description: View looking southeast over river from Elmore and Kelly's Elevator.
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Description: Pencil sketch of a cross-section of the Dells of the Wisconsin River.
Date: 1863-04-04
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Description: Charcoal portrait of George Baxter Burrows. A Wisconsin State Senator (1878-1882) and real estate broker, he willed his twelve-acre Lake Mendota frontage e...
Date: 1844
Description: Cross section drawing of a Wisconsin lead mine.
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subset: commoncrawl


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|Term Name
|Hidden Surface Removal The CPU removes hidden surfaces before sending meshes to the videocard. Because the CPU can remove entire meshes at a time, this can save a lot of time when checking individual triangles.
|Heads-Up-Display. This means status information which are always visible like health or ammo information.
Indexed Color
|When color information is stored in a look up table that contains the colors red, green and blue information.
|Indot refers to non-quad and non-tri polygons. For instance a pentegon without triangulation or quads. (This is only appreant during modeling in a 3d application that supports this feature.)
|Because the electron guns that draw pictures on TV screens were initially too slow to draw the screen in one pass, the first lines at the top of the screen would be fading by the time the last lines were drawn, pictures on TVs were drawn using what is known as interlacing. First the odd lines down are drawn (line 1, 3, 5, etc.). Then the even lines are drawn (line 2, 4, 6, etc.). The image shown by one pass is known as a field, and the complete image drawn by two passes is known as a frame. Standard TV broadcasts run at 30 frames per second (fps). In an effort to boost hype for a product, sometimes ads or press releases would state that their game was "60 fields per second", instead of 30 fps. Most computer monitors are non-interlaced as well as many arcade screens.
|Using a ratio to step gradually a variable from one value to another.
|An approximation between two known values. In graphics, this is a process in which the software increases the resolution of an image by first filling the image with blank pixels and then coloring the blank ones based on the color values of its surrounding pixels.
Inverse Kinematics (IK)
|The process of creating realistic positioning of a complex object, such as an arm, based on the positioning of a lower-level node in the skeletal heirarchy, such as the twisting of a hand. This process works in the reverse of forward kinematics.
Isometric Tiles
|One way of drawing 2D tiles so that it appears that they look 3D as they are layered with depth. In computer games this is usually a misnomer as isometric means a 1-1-1 perspective. Normally in games the depth ratio is less so that the tiles do not seem as wide.
|This is an ISO standard by the Joint Photographic Experts Group. JPEGs work with color resolutions of up to millions of colors (also called 24 bit color). When you save an image as a JPEG, it gets compressed, making the file smaller. However, JPEG compression is lossy, so each time you save a JPEG, you lose some data and reduce image quality. When saving a JPEG, you can choose a compression level from low to high. Low compression gives you better quality but a larger file size. High compression gives you a smaller file - but less quality.
Lambert Shading
|A shading model where the light intensity is calculated at one point along the polygon and is used to shade the entire face. Also commonly refered to as flat shading.
Left Handed Coordinates
|A version of the Cartesian coordinate system where positive X points right of the origin, positive Y points up from the origin, and positive Z points beyond the origin.
Lens Flare
|A refraction that appears in a camera lens normally when it is directed at the sun. A favorite added effect by many computer artists.
Light Map
|A special type of image map that, when applied, effects the intensity of the texture of an object.
|Usually refers to a line segment -- i.e., a one dimensional span of space between two points.
|Level of Detail. Often pertaining to the complexity of 3D models, such as having less or more polygons when the model is closer or farther from the camera.
|A coordinate that represents a point in a light map.
|Scripted (non-interactive) 3D animation comprised of pre-set camera motion, character motion and environment changes. Mostly used for cutscenes within a game in place of pre-rendered CGI animation or film. Typically uses a game's real-time 3D engine to depict a portion of the game story in a non-interactive cinematic representation. Gamasutra Article
|A premium modelling program produced by Alias, it is used by several companies in the games industry as their primary modelling tool. A free version is available at alias.com
|A 3d modelling term that refers to a model as a series of polygons.
Milkshape 3D
|A popular 3D modelling package. It was originally made for modding games, and can export and import a wide variety of popular game formats. It supports textures and texture coordinate mapping very well, and has a nice SDK. The registered version is currently $25. http://www.swissquake.ch/chumbalum-soft/
Mip Mapping
|This process takes a texture and breaks it down into smaller pieces such as 1/2 the size, then 1/2 the size of that, and so on, so that different textures can be used at different distances from the texture. This helps the texture retain what its overall image should look like at different distances.
Mode X
|A tweaked VGA card graphics resolution that displayed graphics at 320x240 which created a square pixel resolution. The term Mode X was coined by Michael Abrash.
|Usually referring to an object that is created out of a number of polygons, splines or NURBS which to create an object in 3D.
|A 3d engine developed in the late 1990s based on DirectX. Now known as 3d State. It has generated a large following of programmers and is praised for it's ease of use and compatability with many programming languages and compilers such as Microsoft Visual C++, Borland C++, Delphi, and Visual Basics. More information can be found at thier website at www.3dState.com.
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Check in from 14h / Check out prior to 10h
Hotel Garni St Georg Seefeld: Find photos, reviews, information, amenities and compare prices for 4 stars hotels in Seefeld, Austria.
Located in Seefeld, Hotel Garni St Georg Seefeld is within a 20-minute drive of Innsbruck Kranebitten Airport and provides a swimming pool, a Jacuzzi and a Finnish sauna. This comfortable hotel also offers a bowling alley and outdoor tennis courts, and is an easy drive from Leutasch and Reith bei Seefeld.
Hotel description
Hotel Garni St. Georg caters for families with children, providing a playground and putt putt golf. Those staying at can take advantage of the hotel's recreational activities, such as billiards and a dart board.
St. Georg Hotel provides spacious rooms equipped with a telephone and a radio, plus all the necessities to ensure a comfortable stay. There are also a number of rooms geared towards families.
Hotel Garni St Georg Seefeld offers a golf course and is close to the area's popular attractions. It is within a 20-minute stroll of Seefeld in Tirol Train Station, which allows convenient access in and around Seefeld.
Hotel features
TV, Billiards, En suite, Private Bathroom
Dry Cleaning, Safe-Deposit Box, Designated Smoking Area
Fitness Room/Gym, Swimming pool, Massage / Beauty Centre, Tennis Courts, Golf Course, Sauna, Casino, Water Activities, Fishing, Playground, Bowling Alley, Jacuzzi, Massage, Spa & Wellness Centre, Pool Indoor
Karaoke, Dart Board, Golf (miniature), Horse Riding, Hiking, Private Toilet, Games Available
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Surprises for PCH Deputy Even Before the Prize Award
Let the Questions Begin…
It started even before I left for my prize award trip. With a week of preparation and anticipation, I shared the news with my nearest and dearest that I was going to be a Prize Patrol Deputy for PCH. I expected to be asked where I was going and what I needed to bring but instead I was showered with surprising questions such as:
Did I win?
Can you make me the winner?
They really exist?
Don't they hire actors or models?
…and my favorite…
Were you paid extra to say you are going on Prize Patrol just to make me think it's real?
All of their questions were very easily answered by explaining that YES, the PCH Prize Patrol really does exist and we do go out on prize award duty in the blue blazers with the prize mobile and the dozen roses and colorful balloons and, of course, the Big Check. The people on TV are not actors, but are real employees, Dave Sayer, Todd Sloane and Eve Fish, all of whom I have had the pleasure of meeting in the halls where I work every day. I then explained that my family could not win since I am a PCH employee, that the winner is picked randomly by a computer program, and that, NO, I am not paid any extra to say that I am going.
They seemed almost disappointed that there was no "smoke and mirrors" to it all … just real people, like me, giving out real prizes to real people, like you (unless you're a relative or co-worker, of course!).
Oh well, I'm sure this is only the beginning of the surprises I'll encounter on my prize award journeys for PCH.
Laurel U.
Prize Patrol Deputy
16 thoughts on "Surprises for PCH Deputy Even Before the Prize Award"
Is this person Mr. KEN MOORE, a PCH employee? Just got an email from him saying "I am a winner", is this a scam?
The Golden Opportunity winner revealed NBC Has been blessed And acted up
Bring The Dream And Mega WIN
Love to meet you pruze patrol
The Newest Deputy
This is the opportunity to let everyone know My Dream Job is to be a. PCH Deputy


text:
Benna Sayyed
Marisol Castillo was planning to meet her estranged father in person for the first time this month. After a deadly shooting Tuesday night, she is left with only memories of pleasant moments shared on the phone and online with dad, Joe Valero.
Valero and his girlfriend, Lupita Casteneda, were gunned down at a barbecue in front of their home on Prince Street.
Clovis Police Chief Steve Sanders said Wednesday that Enrique Deleon, 25, is being charged with murder and child endangerment for allegedly committing the crime.
Castillo, 19, talked with Valero for the first time over the phone three years ago after connecting with his family on MySpace. She lives in Seguin, Texas.
"It's really sad that he didn't get to live that long. I have two kids, and they will never get to meet him. I really wanted them to meet him," said Castillo.
"He was always telling me that he loved me a lot and wanted to see me. He liked to laugh a lot and joke," said Castillo.
Castillo's last contact with her father was last week on Facebook. Messages on Facebook indicate Castillo was also in contact with Casteneda online, who was evidently trying to help Castillo work on forgiving Valero for his absence in her life.
Laura Lopez, whose brother was married to Casteneda, described her as a wonderful person.
"Lupita was very loving, very caring. It's just so sad because of her kids," said Lopez, who met Casteneda six years ago and the two became close.
Lopez said she had a pleasant phone conversation with Casteneda, a mother of four, Tuesday afternoon.
Lopez said she never met Valero in person but was aware that Casteneda had a 9-month-old son with him.
Cassandra Montayo baby-sat for two of Casteneda's children and helped her move into her residence on North Prince.
The two became close after Montayo lost her mother in February. Montayo said Casteneda consoled at the time.
"I don't understand it. I never knew of her hating on people. When I talked to her she talked about her kids," Montayo said. "When I saw her she was always happy. I never saw her mad," she said.


text:
NEW YORK Starbucks Corp.'s share price climbed on Friday as Wall Street welcomed news that Nelson Peltz and his related investment funds under Trian Partners had amassed a small stake in the ailing coffee giant. The activist investor also upped his stake in The Cheesecake Factory Inc. to about 4.8 percent of the casual-dining company's 67 million total shares outstanding, according to filings with federal regulators.Peltz and his affiliates said in a securities filing that the ...
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text:
Arjun Opens About Heroine's Suicide Episode
Action king Arjun is well known for his generosity towards other celebs in snags. But there is a clout over him regarding carrying this old heroine to hospital when she attempted suicide. Listen to that episode in his own words now.
Long back, heroine Gajala has consumed sleeping pills while shooting for a Telugu flick in Hyderabad due to her personal issues. This happened at the stroke of midnight and our heroine's cousin immediately called hero Arjun, for whose film she is shooting for. As Arjun couldn't find any helpers at that time he himself has carried her to hospital in his car and lifted her all the way to ICU. The actor says that media has carried different versions of stories regarding the whole episode, but in reality there is nothing between him and Gajala, except the fact that he saved her life.
'I'm happy today that Gajala's life was saved and she too respects me a lot for the whole thing. Rest of things, I don't take them to heart', says Arjun on a concluding note.


text:
Chef Maximo Tejada sautés diced bacon, bell peppers, onion, crushed garlic and ground cumin in olive oil. He then adds diced canned tomatoes, blanched black beans and chicken stock, rice and water and cooks the mixture until the rice is tender. He adds additional browned, bacon and chopped cilantro.Adapted from "Seduced by Bacon," by Bob Lape and Joanna Pruess
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text:
Lamb's liver and bacon with Onion gravy
Makes 4 servings
Prep Time 15 mins
Cooking Time 25 mins
- 450g/1lb Lamb's liver(½"-12mm slices)
- 8 rashers of lean bacon
- 1 Medium onion
- 20ml Olive oil
- 100g Plain (all purpose) flour
- Salt and black pepper
Ingredients for the gravy
- 600ml Boiling water (use the water from the onions)
- 1 Beef stock cube, I use Oxo
- 250ml semi skimmed milk
- Bisto gravy granules
- Dash of Tabasco sauce
- Dash of Worcester sauce
- Black pepper
- Squirt of Marmite
- Pinch of garlic granules
Method for the gravy
- First, prepare the onion gravy.
- 1 medium onion, peeled and roughly chopped, add to a saucepan with 250ml water, bring to a boil then simmer for 10 to 15 minutes or until the onion is transparent.
- Take 600ml of boiling water (including the liquid from the onions) and pour into a saucepan, keep on a low heat
- Crumble a good beef stock cube into the water and stir
- Add a good half cup of semi skimmed milk (this gives a lovely creamy texture)
- Now add a splash or two of Worcestershire sauce
- Do the same with some Tabasco sauce
- Add good squirt of Marmite or Vegemite
- Add a small pinch of garlic granules
- Finally add a good helping of freshly ground black pepper
- Turn the heat up and sprinkle in some gravy granules to thicken
- Use a balloon whisk to stir, until it comes to the boil, turn down the heat to minimum and leave to simmer for a few minutes, stirring now and then. This gives it time for all the flavours to amalgamate.
- Heat a little olive oil in a deep frying pan, add the bacon.
- Season 100g of plain (all purpose) flour with salt and black pepper, cover the liver with the seasoned flour then add the liver to the frying pan with the bacon, cook on a high heat for 2 minutes, turning halfway through.
- Stir in the onions and add about 900ml gravy.
- Bring to a boil, simmer for 10 minutes
- Serve with peas, brussel sprouts and mashed potato
- Large, deep frying pan
- Medium size saucepan
- Balloon whisk
Shopping List
- 450g/1lb Lamb's liver(½"-12mm slices)
- 8 rashers of lean bacon (not smoked)
- 1 Medium onion
- 20ml Olive oil
- 100g Plain (all purpose) flour
- 1 Beef stock cube
- 250ml semi skimmed milk
- Bisto gravy granules
- Dash of Tabasco sauce
- Dash of Worcester sauce
- Black pepper
- Marmite
- Garlic granules
{ 6 comments… read them below or add one }
- January 13, 2010 at 9:31 am
lovely recipe Im going to try this later, I will let you know how ot goes.
- May 19, 2010 at 11:05 am
Great info, thanks for useful article. I am waiting for more
- January 17, 2011 at 1:26 pm
spot on me and my family have it all the time
- February 27, 2012 at 10:38 am
nice but very salty .I PUT too much gravi granuals to thicken
- February 23, 2013 at 2:52 pm
Perfect we love this recipe
Leave a Comment


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Minnesota W-4 Form
On July 14th 2010, MN Dept of Revenue introduced a new withholding form - W-4MN. Some employees may be required to fill out the W-4MN in addition to the Federal W-4. If the employee claims the same number of Minnesota allowances as federal and does not request additional Minnesota withholding nor claim exemption from Minnesota withholding, there is no need for the employee to complete Form W-4MN. Use the number of federal withholding allowances to determine the Minnesota withholding.
You must have employees complete Form W-4MN in addition to the federal Form W-4 if they:
- claim fewer Minnesota allowances than federal allowances (Minnesota allowances cannot exceed the number of federal allowances),
- claim more than 10 Minnesota withholding allowances,
- request additional Minnesota withholding be deducted each pay period, or
- claim to be exempt from federal withholding or claim to be exempt from Minnesota withholding
When to send copies of Form W-4MN to MN Revenue:
You must send copies of Form W-4MN to the department if the employee:
- claims more than 10 Minnesota withholding allowances; or
- claims to be exempt from Minnesota withholding and you reasonably expect the employee's wages to exceed $200 per week (Exception: if the employee is a resident of a reciprocity state and has completed Form MWR); or
- request additional Minnesota withholding be deducted each pay period, or
- you believe the employee is not entitled to the number of allowances claimed. Note: You are not required to verify the number of withholding allowances claimed by each employee.
Mail to:
Minnesota Revenue, Mail Station 6501, St. Paul, MN 55146-6501.
A $50 penalty may be assessed for each required Form W-4MN not filed with the department.


text:
I love quotations, don't you? Everything I might possibly want to say, someone else has already said it better than I ever could.
Now I've put together all of my blackboard quotes from the homeschool co-op classes, as well as a few longer quotations I used in past blog posts, and archived them in one convenient place. Browse and enjoy!
[Photo by luis de bethencourt.]
Browse by the Author's Last Name
A – B – C – D – E – F – G – H – I/J – K – L – M – N – O – P/Q – R – S – T – U/V – W – X/Y/Z – Reference: Websites and Quote Books
For me, it begins with pushing students into the verbal — that intersection of math and language. Tom Barrett's "Fizz and Martina" series pose cartoon-like problems that students enjoy. But I've been able to harness the power of the series by stealing one question from them: Explain how you solved the problem. Do not use numbers in your explanation.
Once students can verbalize their thinking apart from the numbers, I can ask them to describe patterns they see — and give those names. Suddenly, the "names", or equations are not abstract. They apply to something concrete that can be applied to the 100th term situation or 1000th term situation.
My school math was abstract because I first learned to manipulate equations and then learned to dread the application of equations to story problems. Start with stories. Find patterns. Give the patterns names (equations).
I must study politics and war that my sons may have liberty to study mathematics and philosophy.
Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
— Woody Allen
quoted in H. Eves Return to Mathematical Circles
Arithmetic is neither fish nor beast; therefore it must be foul.
— Anonymous
They mate.
Now there's three.
Mate again, now five.
And again, now there's eight bunnies.
[Now thirteen bunnies, and counting. Mom will be pissed off.]
— Anonymous
from the comments on the Fib
If inside a circle a line
Hits the center and goes spine to spine
And the line's length is "d,"
The circumference will be
d times 3 point 1 4 1 5 9.
— Anonymous
Mathematical Poetry site
Life without geometry is pointless.
Logic is a systematic method of coming to the wrong conclusion with confidence.
— Anonymous
(similar to a comment by Morris Kline)
Natural numbers are better for your health.
— Anonymous
Simple Simon met a pi man
Going to the fair.
Said Simple Simon to the pi man,
"You have unusual ware.
The pie's I've seen before were round
But, gosh, your pi's r2."
— Anonymous
Mathematical Poetry site
The human mind has never invented a labor-saving machine equal to algebra.
— Anonymous
From The Quote Garden
In both math and writing, the core idea that you are trying to express exists somewhere in the aether. In both math and writing, you start out staring at the blank page, trying to figure out how to summon the idea, make it yours, and commit it to the page.
In both math and writing, you make false starts (unless you are very lucky) and work hard (unless you are very lucky) to express the idea with precision and clarity. In both math and writing, your familiarity with the idea that you are trying to express and your prior practice at expressing ideas can sometimes give you a head start in knowing in which direction to start.
Math is writing. Most of math is persuasive writing; math is an exquisitely structured argument.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
There's a tendency for adults to label the math that they can do (such as identifying patterns, choosing between competing offers in a supermarket, and challenging statistics published by the government) as "common sense" and labeling everything they can't do as "math" — so that being bad at math becomes a self-fulfilling prophecy.
— Mike Askew, Rob Eastaway
Old Dogs, New Math: Homework Help for Puzzled Parents
Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.
— Richard Askey
quoted in Elementary Mathematics for Teachers
One thing to keep in mind is that mathematics is a story and that teachers are a story tellers; the teaching and curriculum sequences are there to help you with the structure of the story. If you can bring the story of mathematics to life then you will have a much better chance of reaching all your students. What is easier: memorizing the story of the three little pigs, or learning to tell the three little pigs story on your own? Which is more satisfying?
— Scott Baldridge
Department of Mathematics, Louisiana State University
comment on the ProfoundUnderstanding Yahoo group
Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.
In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R's. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.
In other words these children, by avoiding the early drill on combinations, tables, and that sort of thing, had been able, in one year, to attain the level of accomplishment which the traditionally taught children had reached after three and one-half years of arithmetical drill.
— L. P. Benezet
The Teaching of Arithmetic I: The Story of an experiment
There is no philosophy which is not founded upon knowledge of the phenomena, but to get any profit from this knowledge it is absolutely necessary to be a mathematician.
I recognize the lion by his paw.
— Jacob Bernoulli
on reading an anonymous solution to a problem,
which he realized was Isaac Newton's work
It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can would once have believed in witchcraft.
— Jacob Bernoulli
Ars Conjectandi
Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in the narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!
— Jacob Bernoulli
Ars Conjectandi
The solution "cost me study that robbed me of rest for an entire night."
— Johann Bernoulli
of a problem that stumped mathematicians (including his brother Jacob) for years
But just as much as it is easy to find the differential of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found.
How dare we speak of the laws of chance? Is not chance the antithesis of all law?
— Joseph Bertrand
Calcul des probabilités
Brain: an apparatus with which we think we think.
— Ambrose Bierce
The Devil's Dictionary
Teaching any subject has a funny way of educating the teacher at least as much as the student.
— Chris Birk
How I Became a Better Writer Thanks to Distracted, Hungover College Kids
The most effective and powerful way I've found to commit math facts to memory is to try to understand why they're true in as many ways as possible. It's a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.
Actually, a close friend of mine describes this same experience: he couldn't learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can't remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.
— Ben Blum-Smith
Comment on Math Mama's post What must be memorized?
…a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.)
The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.
— Ralph P. Boas
[You will have to scroll down a bit to find Boas' essay.]
Lion Hunting and Other Mathematical Pursuits
Logic is invincible, because in order to combat logic it is necessary to use logic.
— Pierre Boatroux
Today I said to the calculus students, "I know, you're looking at this series and you don't see what I'm warning you about. You look and it and you think, 'I trust this series. I would take candy from this series. I would get in a car with this series.' But I'm going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it."
— Alexandre Borovik
quoted by Pat Ballew in A Nice Presentation of the Harmonic Series
Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing.
— R. C. Buck
Teaching Machines and Mathematics Programs
American Math. Monthly
I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.
— James Caballero
Everybody a mathematician?, CAIP Quarterly 2 (Fall, 1989)
quoted by Donald McQuarrie, Mathematics for Physical Chemistry
In mathematics, the art of asking questions is more valuable than solving problems.
Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."
— Lewis Carroll
Alice in Wonderland
"Can you do addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?"
"I don't know," said Alice. "I lost count."
— Lewis Carroll
Through the Looking Glass
"When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean — neither more nor less."
"The question is," said Alice, "whether you can make words mean so many different things."
"The question is," said Humpty Dumpty, "which is to be master — that's all."
— Lewis Carroll
Through the Looking Glass
If a child is to keep alive his inborn sense of wonder, he needs the companionship of at least one adult who can share it, rediscovering with him the joy, excitement and mystery of the world we live in.
Education is the key to unlock the golden door of freedom.
You can only find truth with logic if you have already found truth without it.
— G. K. Chesterton
The Man Who Was Orthodox
I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.
— Agatha Christie
An Autobiography
"I think you're begging the question," said Haydock, "and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that, you would go round the bend. Let me assure you of that!"
— Agatha Christie
The Mirror Crack'd (Miss Marple Mysteries)
Free math (Available here Monday through Friday). But you must bring your own container, and you must fill it with much or little according to its capacity and the amount of work that you are willing to do. The learning assistant (sometimes euphemistically called a "teacher") will provide expertise, advice, guidance, and will set an example. But in the final analysis it is you who must do the work needed for your learning… Here it is — this wonderful stuff called math. If you want it, come and get it. If you don't want it, kindly step out of the way — as not to impede the progress of those who do. The choice is yours.
— L. M. Christolphe, Jr.
quoted in the Mathematical and Educational Quotation Server
We make a living by what we get, but we make a life by what we give.
I had my seniors write me an essay about their relationship with math. Not much instruction, just tell me what math was like growing up, good, bad, whatever. In maybe 10 cases they all pointed out that somewhere between elementary school and middle school math went from something they could see and understand to something they no longer got. Every one of them said the same thing, I loved math until middle school. What in the world changes in middle school?
I am in the habit of beginning each class by apologizing to my learners. I'll teach the class better next time because of what I learn from my interactions with them and from their feedback. I remind them that they are free to take the class next year – when it is improved. No one takes me up on that, but it sets the tone that I expect to grow as an educator.
— David Coffey
What's in a Name?
The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.
He who asks a question is a fool for five minutes;
he who does not ask remains a fool forever.
Learning without thought is labor lost;
thought without learning is perilous.
Too often, kids learn a distaste for the subject without ever having the chance to see what there is to love in mathematics. For too many, the understanding of math isn't particularly enduring, while their dislike of the subject is.
— Katherine Cook
My Mathematical Autobiography
I do my best to make my students think, but they still try to become good little algorithm followers.
— David Cox
The 2 Product Property
Intuition is great. Inductive logic is great. But it just isn't enough. Back it up. Verify it. Embrace the conflict that arises when what you thought was true turns out to be, well, not so much.
How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.
"I suppose you are two fathoms deep in mathematics, and if you are, then God help you. For so am I, only with this difference: I stick fast in the mud at the bottom, and there I shall remain."
— Charles Darwin
quoted in the Platonic Realms collection
We use only 10% of our brains… Imagine how smart we would be if we used the other 60%!
I was x years old in the year x2.
— Augustus De Morgan
(when asked about his age)
It is not enough to have a good mind. The main thing is to use it well.
Each problem that I solved became a rule which served afterwards to solve other problems.
Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.
Mathematical thinking is not the same as doing mathematics — at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box — a valuable ability in today's world.
— Keith Devlin
Introduction to Mathematical Thinking
These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music. A page of sheet music represents a piece of music: the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds. The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.
— Keith Devlin
Introduction to Mathematical Thinking
What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.
— Keith Devlin
Should Children Learn Math by Starting with Counting?
When a kid is feeling bad about being stuck with a problem, or just very anxious, I sometimes ask him to make as many mistakes as he can, and as outrageous as he can. Laughter happens (which is valuable by itself, and not only for the mood — deep breathing brings oxygen to the brain). Then the kid starts making mistakes. In the process, features of the problem become much clearer, and in many cases a way to a solution presents itself.
— Maria Droujkova
Natural Math discussion of math club activities
Math happens when we notice similarities and differences. This is math proper. You purposefully create differences, keeping similarities, and observe what happens. There are layers and layers of noticing to be had. We need to return to activities again and again to reach more layers. That's why geeks are often told, "You have too much time on your hands!" when an outsider realizes how much time is spent with a single activity. There are riches to be had ONLY if you spend the time.
One's work may be finished some day, but one's education never.
—Alexandre Dumas
—[or perhaps, Alexandre Dumas?]
Logic, like whiskey, loses its beneficial effect when taken in too large quantities.
— Lord Dunsany
In J. R. Newman (ed.), The World of Mathematics
There's a tendency for adults to label the math that they can do (such as identifying patterns, choosing between competing offers in a supermarket, and challenging statistics published by the government) as "common sense" and labeling everything they can't do as "math" — so that being bad at math becomes a self-fulfilling prophecy.
— Rob Eastaway, Mike Askew
Old Dogs, New Math: Homework Help for Puzzled Parents
I have seen something else under the sun: The race is not to the swift or the battle to the strong, nor does food come to the wise or wealth to the brilliant or favor to the learned; but time and chance happen to them all.
Opportunity is missed by most people because it is dressed in overalls and looks like work.
There is no expedient to which a man will not go to avoid the labor of thinking.
Any fool can know. The point is to understand.
Gravitation cannot be held responsible for people falling in love. How on earth can you explain in terms of chemistry and physics so important a biological phenomenon as first love? Put your hand on a stove for a minute and it seems like an hour. Sit with that special girl for an hour and it seems like a minute. That's relativity.
I have no particular talent, I am only inquisitive.
Pure mathematics is, in its way, the poetry of logical ideas.
In a cat's eye, all things belong to cats.
— English proverb
When I was a child, the Earth was said to be two billion years old. Now scientists say it's four and a half billion. So that makes me two and a half billion.
Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.
There is no Royal Road to Geometry.
— Euclid
… I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. … True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand …
Now I will have less distraction.
— Leonhard Euler (1707-1783)
[upon losing the use of his right eye]
Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – … whose sum was given by Leibniz as 1/2, although others disagree. … Understanding of this question is to be sought in the word "sum"; this idea, if thus conceived — namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken — has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.
The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
— Leonhard Euler
quoted in the Mathematical and Educational Quotation Server
A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some "dreaming" time.
— Howard Eves
An Introduction to the History of Mathematics
An expert problem solver must be endowed with two incompatible qualities: a restless imagination and a patient pertinacity.
— Howard Eves
In Mathematical Circles
There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every new discovery in mathematics results from an attempt to solve some problem.
— Howard Eves
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians
I was born not knowing and have had only a little time to change that here and there.
It's amazing that this vision of math as "getting to the right answer on your first try" even exists. I have to make, unmake, remake so many mistakes to get where I'm going. I think all mathematicians work that way. Somehow, a big part of the experience of math is trouble. Frustration is the status quo. But when you get something—the thrill!
— Dan Finkel
Good Mistakes, Constant Mistakes
A math student's best friend is BOB (the Back Of the Book), but remember that BOB doesn't come to school on test days.
— Josh Folb
(I couldn't find a link about the man himself, although many people like his quote enough to include it on their webpages.)
Imagine that you wanted your children to learn the names of all their cousins, aunts and uncles. But you never actually let them meet or play with them. You just showed them pictures of them, and told them to memorize their names. Each day you'd have them recite the names, over and over again. You'd say, "OK, this is a picture of your great-aunt Beatrice. Her husband was your great-uncle Earnie. They had three children, your uncles Harpo, Zeppo, and Gummo. Harpo married your aunt Leonie … yadda, yadda, yadda.
— Brian Foley
Times Tables – The Worst Way to Teach Multiplication
Two roads diverged in a wood, and I —
I took the one less traveled by,
And that has made all the difference.
— Robert Frost
The Road Not Taken
That vast book which stands forever open before our eyes, the universe, cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.
— Galileo Galilei
quoted by Clifford Pickover, A Passion for Mathematics
Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals — the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.
— Martin Gardner
quoted by G. Simmons, Calculus Gems
In no other branch of mathematics is it so easy for experts to blunder as in probability theory.
— Martin Gardner
quoted in Behind Monty Hall's Doors: Puzzle, Debate and Answer?
The continuing hullabaloo about the "new math" has given many a parent a false impression. What was formerly a dull way of teaching mathematics by rote, so goes the myth, has suddenly been replaced by a marvelous new technique that is achieving miraculous results throughout the nation's public schools.
I wish it were true — even if only to the extent implied by entertainer (and math teacher) Tom Lehrer in his delightfully whimsical recording on "The New Math": "In the new approach, as you know, the important thing is to understand what you're doing, rather than to get the right answer."
Indeed, there is something to be said for the old math when taught by a poorly trained teacher. He can, at least, get across the fundamental rules of calculation without too much confusion. The same teacher trying to teach new math is apt to get across nothing at all…
— Martin Gardner
Foreword to Harold Jacobs' Mathematics: A Human Endeavor
I can recall the deep satisfaction I felt on the all-too-rare occasions at school when the concepts or formulas fell into place. It seemed an entirely different discipline from writing, where something arises from a blank page through a combination of hard work and patience, with a sliver of creativity. With math, the experience is more like discovering something that's always existed and finally decided to stop playing hard-to-get.
— Ralph Gardner
Making Math Fun (Seriously)
Doing math or computer programming at a professional level is a lot like writing. Sometimes it flows naturally and fluently, and sometimes it is blocked and it is like struggling to lift a huge boulder to get anything done.
The toughest thing for a homeschooler is the same as for a school teacher – shifting from a weak tea vision of math being grinding calculations to a rich frothy mug of math as an active way of thinking.
— John Golden
Elementary Homeschool
The ultimate goal of mathematics is to eliminate all need for intelligent thought.
— Ronald L. Graham
quoted in Out of the Mouths of Mathematicians
by Rosemary Schmalz
As important as mathematics is, it is a distant second to the need for good reading comprehension. We teachers so often hear students summarize a course by saying, 'I could do everything except the word problems.' Sadly, in the textbook of life, there are only word problems.
— Herb Gross,
quoted by Jerome Dancis in
Reading Instruction for Arithmetic Word Problems
The essence of mathematics is not to make simple things complicated but to make complicated things simple.
"I hate fractions."
"They probably hate you, too. The question is, which one of you will be master."
— Jonathan Halabi
"I hate fractions" at jd2718
A child learns to count spoonfuls, learns to count people, learns to count fingers, learns to just plain count, and in the process acquires the abstract concept of, for example, "two." The child takes ownership of this concept, and can reapply it freely. As adults we may take "two" for granted, but we have never met it, never touched it, never tasted it. It is one of the first completely abstract concepts that we ever owned.
— Jonathan Halabi
Outlook on teaching math
If the host is required to open a door all the time and offer you a switch, then you should take the switch. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood. My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home.
— Monty Hall
quoted in Behind Monty Hall's Doors: Puzzle, Debate and Answer?
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
Many teachers are concerned about the amount of material they must cover in a course. One cynic suggested a formula: since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick.
Board games are a celebration of problem solving, and problem solving is at the heart of a quality mathematics education… The mathematics might be hidden, but I guarantee you that it will be there.
— Gordon Hamilton, MathPickle.com
Commercial Games 8
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician's patterns, like the painter's or the poet's, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.
— Godfrey H. Hardy
A Mathematician's Apology
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
— Godfrey H. Hardy
A Mathematician's Apology
There are many things you can do with problems besides solving them. First you must define them, pose them. But then, of course, you can also refine them, depose them, or expose them, even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn't lead to it, so you found new goal they do lead to. It's called play.
Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings.
— David Hawkins
The Spirit of Play [pdf, 1.4MB]
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians
(Inscription for a monument at the crossroads.)
Here lies, extinguished in his prime,
a victim of modernity:
but yesterday he hadn't time—
and now he has eternity.
Problems worthy
of attack
prove their worth
by hitting back.
One cat just leads to another.
In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.
— Herman Henkel
[Did he perhaps mean Hermann Hankel?]
quoted by Noah benShea, Great Quotes to Inspire Great Teachers
Mathematics is a game played according to certain simple rules with meaningless marks on paper.
— David Hilbert
quoted by Nicholas Rose, Mathematical Maxims and Minims
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.
Stu came to my office looking for a new major. Stu is bad at math and can't handle the math sequence required of business majors. So Stu was wondering what majors require the lowest level math sequence that counts towards graduation.
I listed a few.
Stu was disappointed. Stu pointed out that you don't usually think about people in those fields as making a lot of money. Stu lamented that everything that is in demand requires math.
— Rudbeckia Hirta
Learning Curves blog
To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.
Repetition progressively frees the mind from attention to details, makes facile the total act, shortens the time, and reduces the extent to which consciousness must concern itself with the processes.
— E. B. Huey
quoted in Elementary Mathematics for Teachers
When my students complain that I'm smarter than them, I counter that I'm just at a higher level of misunderstanding.
A great many people think they are thinking when they are really rearranging their prejudices.
…the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not be in indulged in by one who is to have a profession to follow for his subsistence.
— Thomas Jefferson
quoted by J. Robert Oppenheimer in the Mathematical Quotations Server
I don't love math nearly as much as I pretend I do when I'm teaching it or blogging about it or trying to enthuse my kids. I just believe — ever since an eye-opening university-level Mathematics in Perspective course — that math is taught VERY badly, bumbled and fumbled and as a result we have this societal fear of what is, essentially, a great big GAME.
— Jennifer in MamaLand
Spotted (myself!) around the Web…
Nothing produced such odd results as trying to get even.
A good student is one who will teach you something.
The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization.
The greater our knowledge increases, the greater our ignorance unfolds.
Where there is matter, there is geometry.
Two contrasting attitudes:
non-math person: "Math is so abstract." i.e. "hard to understand"
math person: We abstract *in order to understand.*
Part of our job is to teach people this latter mentality.
No knowledge of probabilities helps us to know what conclusions are true. There is no direct relation between the truth of a proposition and its probability.
Polya has become the Marx and Lenin of mathematical problem solving; a few words of obeisance need to be offered in his name before an author can get down to the topic at hand.
— Jeremy Kilpatrick [pdf 97KB]
Logic is the art of going wrong with confidence.
— Morris Kline
In N. Rose, Mathematical Maxims and Minims
If you could lead through testing, the U.S. would lead the world in all education categories. When are people going to understand you don't fatten your lambs by weighing them?
— Jonathan Kozol
at Westfield State College's 157th Commencement
The Good Lord made all the integers; the rest is man's doing.
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.
— Pierre-Simon de Laplace
quoted in H. Eves Return to Mathematical Circles
Experience is the hardest kind of teacher. It gives you the test first, and the lesson afterward.
The only teaching that a professor can give, in my opinion, is that of thinking in front of his students.
But in the new (math) approach, the important thing is to understand what you're doing, rather than to get the right answer.
Someone asked me if I was ever sorry I had chosen mathematics. I said, "I didn't choose! Mathematics is an addiction with me!"
As soon as you understand 2 x 4, you can't believe there was a time when you didn't understand it.
I told myself, "Lincoln, you can never make a lawyer if you do not understand what demonstrate means." So I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what "demonstrate" means, and went back to my law studies.
— Abraham Lincoln
quoted by William Dunham, The Mathematical Universe
The mathematical question is "Why?" It's always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.
— Paul Lockhart
author of Measurement
Math class is not terribly different from shop class. Both are all about tools and toolboxes. Math class is no more about mathematics than wood shop is about craftsmanship and design.
In American elementary mathematics education, arithmetic is viewed as negligible, sometimes even with pity and disdain—like Cinderella in her stepmother's house. Many people seem to believe that arithmetic is only composed of a multitude of "math facts" and a handful of algorithms. . . Who would expect that the intellectual demand for learning such a subject actually is challenging and exciting?
— Liping Ma
Arithmetic in American Mathematics Education: An Abandoned Arena?
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.
— David Marain, MathNotations
Odds and Evens Week of 12-1-10
Procrastination is the art of keeping up with yesterday.
It is a great nuisance that knowledge can only be acquired by hard work.
The best teacher is not the one who knows most, but the one who is most capable of reducing knowledge to that simple compound of the obvious and wonderful.
The physical five oranges goes up the ladder to the picture of the five oranges which goes up to the representation of the five oranges as a numeral. This points in the direction of a definition of abstraction: when we abstract we voluntarily ignore details of a context, so that we can accomplish a goal.
You don't understand anything until you learn it more than one way.
It seems quite unrealistic to judge a curriculum by its general outline, or to judge a course by its syllabus. We can "cover" very impressive material, if we are willing to turn the student into a spectator. But if you cast the student in a passive role, then saying that he has "studied" your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does.
Arithmetic is being able to count up to twenty without taking off your shoes.
Whatever you do, for goodness sake, don't tell them the right answer. (Like, ever. Let them come to consensus. Learn how to ask helpful questions without giving away the store.) Unless for some reason you want to completely shut down discussion. And thinking.
— Kate Nowak
Virtual Conference on Core Values: Mistakes are Made
The clearer the teacher makes it, the worse it is for you. You must work things out for yourself and make the ideas your own.
— William F. Osgood
quoted in Out of the Mouths of Mathematicians
by Rosemary Schmalz
Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive.
— Ronald L. Graham, Donald E. Knuth, Oren Patashnik
Concrete Mathematics
I love mathematics … principally because it is beautiful, because man has breathed his spirit of play into it, and because it has given him his greatest game — the encompassing of the infinite.
— Rózsa Péter
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians
Being in very nature God, He did not consider equality with God something to be grasped, but made Himself nothing…
I know too well that these arguments from probabilities are imposters, and unless great caution is observed in the use of them, they are apt to be deceptive.
The more I work and practice, the luckier I seem to get.
— Gary Player
quoted in Precalculus Mathematics in a Nutshell
The mathematician does not study pure mathematics because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful.
— Henri Poincaré
quoted by Theoni Pappas, More Joy of Mathematics
No other statistical puzzle comes so close to fooling all the people all the time.
— Massimo Piattelli-Palmarini
on the Monty Hall problem, quoted in The Power of Logical Thinking
A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem.
— George Pólya
How to Solve It
Beauty in mathematics is seeing the truth without effort.
It is better to solve one problem five different ways, than to solve five problems one way.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
— George Pólya
How to Solve It
Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.
— George Pólya
How To Solve It
The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving…The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.
— George Pólya
Mathematical Discovery, Volume II
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the class. He writes a, he says b, he means c; but it should be d. Some of his sayings are handed down from generation to generation.
- "In order to solve this differential equation you look at it till a solution occurs to you."
- "This principle is so perfectly general that no particular application of it is possible."
- "Geometry is the science of correct reasoning on incorrect figures."
- "My method to overcome a difficulty is to go round it."
- "What is the difference between method and device? A method is a device which you used twice."
— George Pólya
How To Solve It
Circles to square and cubes to double would give a man excessive trouble.
Mathematics is a world created by the mind of men, and mathematicians are people who devote their lives to what seems to me a wonderful kind of play!
You have to make them laugh. You must never underestimate the power of laughing in a maths classroom.
— Carol Roberts
quoted in Laughing lesson Adult learning
In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.
— Hugo Rossi
Mathematics Is an Edifice, Not a Toolbox
A mathematician's work is mostly a tangle of guesswork, analogy, wishful thinking, and frustration. And proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks.
The 50-50-90 rule: Anytime you have a 50-50 chance of getting something right, there's a 90% probability you'll get it wrong.
I used to think that math was some kind of inaccessible, abstract magic trick, a sort of in-joke that excluded us common folk, but now I realize that math is completely not that at all. The reality of math as most of us know it is like that story where three men are standing in a dark room touching different parts of an elephant. None of them has the full picture because they're only perceiving individual elements of the whole animal. The reality, I'm discovering, is that math is just like that elephant: a large, expansive, three-dimensional, intelligent, sensitive, expressive creature. The problem is that most of us have been standing around in that dark room since about kindergarten, grasping its tail, thinking "this is what math is and, personally, I don't think it's for me." We've been blind to the larger, incredibly beautiful picture that would emerge if only we would turn on the lights and open our eyes.
— Malke Rosenfeld
The Elephant in the Room
The study of infinity is much more than a dry academic game. The intellectual pursuit of the absolute infinity is, as Georg Cantor realized, a form of the soul's quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment.
It may be that the race is not always to the swift, nor the battle to the strong – but that is the way to bet.
At age eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.
— Bertrand Russell
The Autobiography of Bertrand Russell
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying 'No, that's wrong.' Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.
— W. W. Sawyer
Vision in Elementary Mathematics
Earlier we considered the argument, 'Twice two must be four, because we cannot imagine it otherwise.' This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.
People often make mistakes when they reason about things they have never seen. Imagination does not always give us the correct answer. We can only argue correctly about things of which we have experience or which are reasonably like the things we know well. If our reasoning leads us to an untrue conclusion, we must revise the picture in our minds, and learn to imagine things as they are.
When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination. Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.
— W. W. Sawyer
Mathematician's Delight
If you cannot see what the exact speed is, begin to ask questions. Silly ones are the best to begin with. Is the speed a million miles an hour? Or one inch a century? Somewhere between these limits. Good. We now know something about the speed. Begin to bring the limits in, and see how close together they can be brought. Study your own methods of thought. How do you know that the speed is less than a million miles an hour? What method, in fact, are you unconsciously using to estimate speed? Can this method be applied to get closer estimates?
You know what speed is. You would not believe a man who claimed to walk at 5 miles an hour, but took 3 hours to walk 6 miles. You have only to apply the same common sense to stones rolling down hillsides, and the calculus is at your command.
— W. W. Sawyer
Mathematician's Delight
The desire to explore thus marks out the mathematician. This is one of the forces making for the growth of mathematics. The mathematician enjoys what he already knows; he is eager for more knowledge.
—W. W. Sawyer
Prelude to Mathematics
And here is a bonus—an article by Sawyer about Hardy: A Mathematician's Apology Revisited
Go down deep enough into anything and you will find mathematics.
— Dean Schlicter
[Does anyone know who he is? Plenty of people quote him, but I can't find a single link about the man himself.]
If you would make a man happy, do not add to his possessions but subtract from the sum of his desires.
— Seneca
Paraphrasing is one of the most important skills we can teach junior high and high school students. Often they want to rush into interpreting and reacting to a text even before they know what it means. We teachers sometimes suffer from the delusion that since a student can read the words on the page, he or she understands what's been read. But that's not always true.
— Nick Senger
Quotes about Reading for Students to Paraphrase
I had the most beautiful set of theories you ever knew when I started out as a schoolma'am, but every one of them has failed me at some pinch or another.
— Anne Shirley (fictional)
Anne of Avonlea by Lucy Maude Montgomery
We've all heard that a million monkeys banging on a million typewriters will eventually reproduce the entire works of Shakespeare. Now, thanks to the Internet, we know this is not true.
— Robert Silensky
If we are to teach mathematics at all, real success is not possible unless we know that the subject is beautiful as well as useful. Mere utility of the moment without any feeling of beauty becomes a hopeless bit of drudgery, a condition which leads to stagnation.
What would mathematics have amounted to without the imagination of its devotees—its giants and their followers? There never was a discovery made without the urge of imagination—of imagination which broke the roadway through the forest in order that cold logic might follow.
What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?
I don't want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that — by some mysterious agency — capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.
— Ian Stewart
The Magical Maze
A second reason why few students ever realize that there is mathematics outside the textbook is that no one ever tells them that. I don't blame the teachers. If your students are having problems remembering how to solve quadratic equations, the wise teacher will stay well clear of cubic equations, which are even more difficult. A process of self-censorship sets in. In order to avoid damaging the students' confidence, the texts do not ask questions that the methods being taught cannot answer. So insidiously, we absorb the lesson that every mathematical question has an answer.
It's not true.
Our teaching of mathematics revolves around a fundamental conflict. Rightly or wrongly, students are required to master a series of mathematical concepts and techniques, and anything that might divert them from doing so is deemed unnecessary. Putting mathematics into its cultural context, explaining what is has done for humanity, telling the story of its historical development, or pointing out the wealth of unsolved problems or even the existence of topics that do not make it into school textbooks leaves less time to prepare for the exam. So most of these things aren't discussed.
— Ian Stewart
Letters to a Young Mathematician
Mathematics is a vast adventure; its history reflects some of the noblest thoughts of countless generations.
— Dirk J. Struik
A Concise History of Mathematics
I have studied many philosophers and many cats. The wisdom of cats is infinitely superior.
Math is the beautiful, rich, joyful, playful, surprising, frustrating, humbling and creative art that speaks to something transcendental. It is worthy of much exploration and examination because it is intrinsically beautiful, nothing more to say. Why play the violin? Because it is beautiful! Why engage in math? Because it too is beautiful!
— James Tanton
Thinking Mathematics
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks… Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
— Silvanus P. Thompson
Calculus Made Easy
How often might a man, after he had jumbled a set of letters in a bag, fling them out upon the ground before they would fall into an exact poem, yea, or so much as make a good discourse in prose? And may not a little book be as easily made by chance as this great volume of the world?
"[M]athematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary schools learning to detest mathematics… They return to the elementary school to teach a new generation to detest it."
— Time magazine, June 18, 1956
quoted by George Polya in How to Solve It
It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.
A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction.
For example, coins, nuts and buttons are clearly distinct and countable and for this reason are convenient to represent relations between whole numbers. The youngest children need some real, tangible tokens, while older ones can imagine them, which is a further step of intellectual development. That is why coin problems are so appropriate in elementary school. Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be (and in Russia are) common already in middle school. Trains, cars and ships are so widely used in textbooks not because all students are expected to go into the transportation business, but for another, much more sound reason: these objects are easy to imagine moving at constant speeds and because of this are appropriate as reifications of the idea of uniform movement, which, in its turn, can serve as a reification of linear function. Thus, we can move children further and further on the way of de-reification, that is development of abstract thinking.
— Andrei Toom
Word problems: Applications vs. Mental Manipulatives
I finally get it — you don't have to worry about memorizing a bunch of formulas if you just understand where they come from. You can always figure them out again.
— Unidentified student in Doug's class
from the comments on Kate's post Formulas? What Formulas?
Cats aren't clean, they're just covered with cat spit.
— Unknown
Dogs believe they are human.
Cats believe they are God.
— Unknown
I got rid of my husband. The cat was allergic.
— Unknown
People have this notion that math is about getting a right answer, and the testing really emphasizes that notion. And that's such a bad way to approach math because it makes it scary. When you look at little kids, they pose their own questions. They say, "Ooooh, what's bigger than a million?" And they think about things their own way. At school, the teacher poses the questions, and the students answer their questions. Schooling is not a natural environment for learning.
— Sue VanHattum, Math Mama Writes
Richmond Math Salon: A Sweet Sampling
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.
The length of your education is less important than its breadth, and the length of your life is less important than its depth.
The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.
— George Washington
quoted by William Dunham, The Mathematical Universe
Follow the path of the unsafe, independent thinker. Expose your ideas to the dangers of controversy. Speak your mind and fear less the label of 'crackpot' than the stigma of conformity. And on issues that seem important to you, stand up and be counted at any cost.
Teaching is the royal road to learning.
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
— Hermann Weyl
The American Mathematical Monthly, November, 1992.
I will not go so far as to say that constructing a history of thought without profound study of mathematical ideas is like omitting Hamlet from the play named after him. But it is certainly analogous to cutting out the part of Ophelia. For Ophelia is quite essential to the play, she is very charming. . . and a little mad.
— Alfred North Whitehead
quoted in The Viking Book of Aphorisms
The study of mathematics is apt to commence in disappointment… We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this greatest science eludes the efforts of our mental weapons to grasp it.
— Alfred North Whitehead
An Introduction to Mathematics
From the very beginning of his education, the child should experience the joy of discovery.
Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark, unexplored mansion. You enter the first room of the mansion, and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch. You turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark.
— Andrew Wiles
Who finally solved Fermat's Enigma
Mathematics: a wonderful science, but it hasn't yet come up with a way to divide one tricycle among three little boys.
Probable-Possible, my black hen,
She lays eggs in the Relative When.
She doesn't lay eggs in the Positive Now
Because she's unable to postulate how.
Black holes are where God divided by zero.
Five out of four people have trouble with fractions.
The excitement of learning separates youth from old age. As long as you're learning, you're not old.
— Rosalyn S. Yallow
quoted in Rosalie Maggio (ed.), The Beacon Book of Quotations by Women
Reference: Websites and Quote Books
Several of the quotes below have been pulled from these wonderful math and education websites:
- Furman University Mathematical Quotation Server
- MAA Digital Library Mathematical Quotations
- Mathematical and Educational Quotation Server at Westfield State College
- Mathematical Quotations
- Platonic Realms Interactive Database of Math Quotes
And if you enjoy these quotes, you may want to check your library for…
- A Dictionary of Quotations in Mathematics
- Dictionary of Mathematical Quotations
- Great Quotes to Inspire Great Teachers
- In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes
- Mathematically Speaking: A Dictionary of Quotations
- Memorabilia Mathematica: The Philomath's Quotation Book
- Statistically Speaking: A Dictionary of Quotations
- Teacher's Treasury of Quotations
- The Music of Reason: Experience the Beauty of Mathematics Through Quotations
- The Teacher's Quotation Book: Little Lessons on Learning
- Out of the Mouths of Mathematicians: A Quotation Book for Philomaths


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The Allure of Search Funds – Greg Geronemus
January 17, 2013 12:17 pm
Greg Geronemus, principal at Footbridge Partners, talks with Bloomberg's Deirdre Bolton about the allure of search funds. They speak on Bloomberg Television's "Money Moves."
An interesting concept, very directed and focused. It will be interesting to track their success rate.


text: I was excited to enter the studio today for day number 35 of 90 straight days of hot yoga at Moksha Yoga LA, today's class had live music by Dalton and he rocked it. It was a 75 minute class with teacher Joe and Dalton, it was wonderful. Such a vibrant community of people practicing [Read More…]


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Do you think you have a small home? Wait till you see the smallest house in the world: The Keret House. It just finished and is located in Warsaw, Poland. The micro-house comes with a kitchen, a bedroom and a bathroom. It measures 3.02 feet at its narrowest point and 4.99 feet at its widest point, for a total floor area of 44 square feet.
"It's something that is very, very compact" said Keret, "but it has in it all the stuff that a house needs."
More info kerethouse.com
Photos by: Andreas Meichsner, Centrala


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Other Names: Chem, Kemu, Khem, Khnum Patron of: the creation of people and animals.
Appearance: a man with the head of a ram.
Description: Khenmu formed a triad with Anuket and Satis, and was possibly a Nubian god originally. The worship of Khenmu dates to the earliest of times in Egypt, the Unas Pyramid Text indicates that his cult was already old when that ancient document was written. Called "the Great Potter," Khenmu was the creator of people. He sculpted them out of clay from the Nile, held them up so that Ra could shine his life-giving rays upon them, and then placed them in the womb. His wife was the lioness-goddess Menhit, and their son was Hike. Originally a primal force deity of creation like Ptah, his role was later modified to fit him into the pantheon of the state religion.
Worship: Worshipped throughout Nubia and Egypt, his cult centers were Elephantine, Sunnu, Abu, and Semnut.
See also:
Khnum, Potter God of the Inundation Silt and Creation by Caroline Seawright
Who are we?
Tour Egypt aims to offer the ultimate Egyptian adventure and intimate knowledge about the country. We offer this unique experience in two ways, the first one is by organizing a tour and coming to Egypt for a visit, whether alone or in a group, and living it firsthand. The second way to experience Egypt is from the comfort of your own home: online.


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From: Executive chef Oscar del Rivero, Talavera Cocina Mexicana, Coral Gables, FL. Yield: 1 serving. 2 oz. Tequila 100% Agave Azul Tequilana Weber ½ oz. fresh lime juice ½ oz. aloe vera juice ½ oz. simple syrup 1 oz. hibiscus tea as desired for garnish, lime wedge Make a homemade simple syrup by combining equal parts sugar and boiling water. For a lower calorie version, substitute with Splenda or sugar alternative of choice. Mix all ingredients in a ...
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