Price Per Item Calculations

Yesterday I introduced the subject of money in the elementary math class. Talking about buying things is a fairly popular part of math class, although our conversations should lean toward thrift, not wild abandon.

We have to be careful in the curriculum not to use any brand names or illustrations that might imply we are endorsing a certain product. And lately, we have had to remove all references to sugary foods.

Here's a typical lesson on unit pricing. This requires higher order problem solving skills. [click on the image to enlarge it]

We have co-mingled prices given "by the ounce" and "by the pound" to give kids practice in unit conversions. They should understand the relationship of ounces to pounds. They must know the number of ounces in a pound. They need to do a division problem to learn the cents per ounce.

In Clarissa's birthday roller skating problem, we provided the beginning of a story and kids are asked to finish the story, then create several of their own problems. We have provided teachers with a sample problem and how to find the answer.

This is a relatively complex problem where data is given in the narrative portion of the story. One tricky part is for the student to remember that Clarissa must pay for admission too, in addition to her 14 friends.

As the roller rink is giving a discount of one dollar per person, you must remember to deduct that from the total of the entrance fees. In this example, we took the dollar off of the two prices ($10 becomes $9; $15 becomes $14) but we could just as easily have waited until the end and subtracted $15 ($1 x 15 admissions).

In most comparison shopping situations, you are in a store, faced with choices, and must make a decision relatively quickly. The ability to calculate these types of prices in your head is a useful skill.

Once in a while you have time to analyze things more carefully. This earlier blog on the cost of a cup of tea is an example of a complex cost-per-unit problem solved over the course of an evening or two.

This challenge interested me, because the tea was from the same source and of the same type. Creative study of the purchasing options indicated we could cut the price from 58 cents to 16 cents per cup, for the same tea!

The price of everything?

In Excel Math elementary math curriculum we help kids explore the concept of careful shopping. In fact, schools are encouraged to teach thrift (not greed or covetousness) as a desirable character trait and perspective.

Thrift is:

• The quality of using money and other resources carefully and not wastefully.
• Wise economy in the management of money and other resources
• Frugality
• An association or organization that helps people save money (credit union, bank, etc.)
• From Old Norse, from thrífa "to grasp, get hold of"

In the process of teaching thrift we investigate a variety of concepts:

• price comparisons
• determining unit prices (if N items cost $X.XX, how much is one item?)
• reverse unit pricing (if one item costs $X.XX, how many items do you get for $Y.YY)
• the razor and the blade pricing (printer is free but toner is expensive)
• quantity discounts and bulk buying
• reading advertising and sales materials
• common discounting techniques (percentage off the list price, 2 for 1, etc.)
• calculating sales tax
• how interest is charged on loans
• how interest is paid on deposits
• not spending more than you have in your bank account (no deficit spending)
• additional, after-purchase costs (energy consumed, maintenance, repairs, etc.)
• tipping or gratuities

We mention profit and loss too, even though 3rd grade math class is not designed to teach economics. We avoid things like subscriptions, airline tickets, rental cars, used items, negotiation strategies, etc. These limitations are not necessarily based on the youthfulness and inexperience of our students. We just don't have enough time. In reality, there are SO MANY THINGS we might buy that NO ONE can possibly understand all the options.

Here are some numbers shared in a recent TED speech by economists Tim Harford and Cesar Hidalgo.

Tim suggested that for many centuries, our brains needed to compare 300+ items, yet now we are faced with 10,000,000,000 potential choices. Whether this number is accurate or not, we generally only have time to study for a purchase just before we buy that item. We cram. And sometimes we later end up with buyer's remorse.

Once our students understand a bit about the items they want and their prices, they have to complete an exchange with the merchant. The price is usually X plus taxes Y which becomes total Z with possibly an additional tip A.

Depending on their funds, students can use one or more payment methods:

• coins
• bills
• check
• cash card
• debit card
• credit card
• gift certificate
  
• purchase order 
• traveler's check
• put it on account

We don't have the time or ability to get into all these items in detail but we provide an introduction for students. I'll give some examples tomorrow.

Who decides which goes first? Part IV

I have a few more observations to make about the mathematical implications on which is the first day in a week? (our customer's question), and how we teach kids about weeks and calendars in the Excel Math curriculum.

In kindergarten, we introduce words that describe adjacent days. Then we connect them with the days of the week.

Days of the week and "adjacent day words" today, tomorrow, yesterday

Finally we introduce the numbers for days, weeks, months and years. We show the "rotation" as the months of the year go by, and we hint at the seasons.

The calendar, and the words and numbers for days, weeks, months and years

Week: a period of 7 days; 168 hours [man-made]

Fortnight: 14 days, two weeks, or half of a moon cycle (14.7 days) [lunar]

Month: a time unit based on the motion of the moon; 29.5 days [lunar]

Year: a time unit based on the rotation of the earth around the sun; consisting of 12 months, 52 (non-ISO 8601) weeks, and 365.25 days. [solar]

Decade: 10 years [solar + man-made]

Century: 100 years [solar + man-made]

The week is the most familiar time-keeping interval to be set at an arbitrary length, by people in different religious traditions and cultures. This Wikipedia article describes 3-day through 10-day weeks used in various times and places. If you consider the options, the length of a "week" can't be too short (it must be longer than a day) or too long (shouldn't be be too close to a month).  By definition a week is shorter than a fortnight.

The fortnight is commonly used in the UK and other Commonwealth countries, and is a typical interval for paychecks and other social benefit payments. Since it's a two-week time period, its first day is the same as the first day of the week that country has chosen to observe.

Nobody worries much today about the first day of the month except when they don't get a benefits check. In previous times, the first day could be very significant. Some Jewish holy men were given relief from traveling restrictions on the Sabbath so they could go to the eastern edge of the mountains and see the sunset or moonrise, marking holy days and months.

Although we have been debating which day is the proper first day of the week, the actual first day of the week has little consequence to most of us, other than affecting how we read calendars. I will make sure the curriculum conveys that sense to kids.

 

For an easy conclusion of this series of blog posts, I'll portray a couple watches that tell you the sequence of days of the week.

The watch above indicates day of the week with an orange spot that moves alongside the hour markers. You can set the 7th day to be either Saturday or Sunday, as you choose. During the evening of the 7th day it speeds up, racing past 8, 9, 10, 11 and 12 to land next to the 1 at midnight.

 

 

This colorful watch is showing the time as 10:10:38 on a Saturday.  Alain Silberstein created the Smileday graphical image watches to convey a changing mood for each day of the week. 

Who decides which goes first? Part III

Today in the Excel Math blog we are tackling how the weeks fit into a year. It started with the question: Why do we say Sunday is the first day when everyone knows Monday is the first day of the week?

Providing an answer is turning into a week-long project. Luckily I started it on Friday, and not on Sunday or Monday!

In our normal calendar, weeks (a set of 7 days) fall across the boundaries of the months and years. Here's how we would normally display them in the USA, if we start the week with Monday, or Sunday [click the graphic if you want a larger version]:

We could also display the weeks using the ISO 8601 standard we talked about yesterday. Here is our year indicating the numbered weeks in rows, and an alternative with numbered weeks in columns [click the graphic if you want a larger version]:

One advantage to numbering the weeks is that you can choose certain dates in advance - for example, the company annual report deadline can be Monday of week 15 in every year. Or paychecks will be issued on Fridays of evenly-numbered weeks.

All companies in your country can use the same systems - for example, a food producer's system can say a packaged meal was prepared in 2011 during Week 25 on Day 3  (the actual ISO 8601 code is 2011W253) plus "use within 15 weeks" and a retailer's system could easily calculate the date that food will be stale.

Are there disadvantages? Here are some potential issues:

• This system has leap weeks - some years have 52 weeks and some have 53 weeks
• Some days in a year may be assigned to weeks from the previous or future year
• There are no set relationships between weeks and months

Users (people, companies, countries) can disregard these voluntary standards if they want to. Across the world there are still many calendars and many ways of dividing up a year. Here's one programmer's attempt to list them all.  Here's another programmer's site. Here is a site that will tell you the ISO 8601 week and day numbers for today.

So far (just in case you were worried) we do not teach the ISO 8601 system to elementary school kids. But as an adult, you wanted to know this, right?

Who decides which goes first? Part II

My last blog post introduced this question:

Which day of the week is considered the first day: is it Sunday or Monday? 

This is a math issue, as we teach calendars, time-keeping, etc. in math class and curriculum. After beating around the bush for quite a while describing the context of the question, I am now ready to provide an answer.  

It depends.  It depends on you, your country, and your employer. Not very satisfactory, is it?

The ISO 8601 standard recommends that everyone adopt these points when discussing dates:

• The first day of the week is Monday
• The first week of the year is the week containing the first Thursday in the year
• Dates should be presented as YYYY-MM-DD
• Dates and times are shown as YYYY-MM-DDThh:mm:ss

A minor major problem with this ISO 8601 scheme is that people tend not to use it. We have our own methods. Rather than using Year, Month, Day, we tend to say Month, Day, Year. People in other countries outside the US tend to format the date as Day, Month, Year.

• July 25, 2011 is today's date, written American style
• 25 July 2011 is how it would be written in the UK

Why does it matter?

Some decades ago I lived in the UK and received a new credit card from my bank in the US. The card said valid 5-8-90  which I believed was May 8, 1990. When I tried to use it in the UK, the merchants said it was not valid until 5 August 1990 and refused to accept it. I can confirm that this was a great nuisance.
 
After many years of confusion and completely scrambled up entries, the US Customs and Border Protection agency finally switched the blanks on its landing cards into the date format expected by foreign travelers (rather than expected by its US clerks).  Notice that they did not use the ISO 8601 format! It's Day Month Year instead.

Imagine the confusion with applications, orders, flight reservations, etc. when people try to communicate across borders with the assumption that the people on the other side use the same time and date system!

So if we (in the US or EU or elsewhere) don't follow the recommended ISO method for writing dates, why would we (in the US) adopt the recommendation that Monday is the first day of the week?

Convention / tradition / habit often overrules a politician / mathematician / scientist / programmer's desire for order and efficiency. 

PS - Did you notice the bullet point stating the first week is the week containing the first Thursday?

The weeks of the year system involves another recommendation that we Americans don't often use. Technically, it's called a "leap week calendar system." I will try to explain it tomorrow.

Notice this special "leap week calendar"watch is showing 10:14 on Monday, the 28th day of August (how do I know that?) in the 35th week of the year.

Who decides which goes first?

One of the most common discussions on the playground is "which one goes first." That's an occasional discussion among adults too - especially on the road when two lanes are converging into one. There general rules for this sort of thing but often we ignore the rules and do what we feel like doing.

It happens in the math world too. Someone has to decide who goes first. For example, someone decided that addition and multiplication are performed BEFORE subtraction and division.

In previous blogs I've discussed times and dates and calendars. ( Here's one on time zones. Here's another on leap years. And another on setting your clocks for daylight savings. )

Today's blog was prompted by a customer's visit this week, asking "why is Sunday the first day of the week in Excel Math, when everyone knows it's the 7th day of the week?"

My answer was, Not everyone knows or thinks that Monday is first. 

A DAY Seems Simple, but it's not! Bear with me now as I get into this. We need to know when a day starts in order to know when to start the week (a collection of 7 days).

WHEN DOES A DAY START?

Hebrew and Islamic traditions begin their days at sunset. Or to be more precise, some said, from the moment that the sun's disk stands distant from the horizon by the length of its own diameter.

WHAT ARE DAWN and SUNRISE and DUSK and SUNSET?

The US Naval Institute of Standards says:

• Sunrise and sunset are the exact times when the upper edge of the disc of the Sun is at the horizon. 
• Dawn occurs when the geometric center of the Sun is 18° below the horizon in the morning. 
• Dusk occurs when the geometric center of the Sun is 18° below the horizon in the evening.
• Twilight refers to periods between dawn and sunrise or sunset and dusk.

Twilight's hazy light is an effect caused by the scattering of sunlight in Earth's upper atmosphere. The brightness of twilight is subjective, depending upon on your location and elevation, the time of year and local weather conditions. Twilight is divided into 3 categories based on the angle of the Sun below the horizon.

• Astronomical twilight is the period when the Sun is between 18° and 12° below the horizon. 
• Nautical twilight is the period when the Sun is between 12° and 6° below the horizon.
• Civil twilight is the period when the Sun is between 6° below the horizon until sunrise. 

The same designations are used for periods of evening twilight.

Roman and Western European medieval monastic days began at 6:00 am. In the Western world today, we normally end and start the day at midnight (the middle of the night).

WHEN IS MIDNIGHT?

• Halfway between dusk and dawn? Not exactly. That depends on where you are.
• How about 12 hours or opposite from noon? Ok, but when's noon?
• Noon used to be when the sun was directly over your head, but that definition changed with Time Zones.
• Noon is when the sun crosses the meridian, or is at its highest elevation in the sky. Solar noon varies depending on your longitude and the date.
  

WHAT DO WE CALL MIDNIGHT?

The US government style manual recommends: use 12 a.m. for midnight and 12 p.m. for noon. Time up to midnight (but not including it) is PM and time after midnight (but not including it) and before noon (but not including it) is AM. 

The US Naval Institute of Standards says: 12 a.m. and 12 p.m. are ambiguous and should not be used.  The shortest measurable duration after noon should be designated as PM. For example, a digital clock changing from 11:59:59 a.m. to 12:00:00 should indicate PM as soon as it the 12:00 appears, and not delay the display of PM for a minute, or even a second.

Depending on your digital clock, midnight could be 12:00 or 00:00  

WHICH DAY GOES FIRST?

Traditionally, countries whose primary religion was based on the Bible (Judaism, Christianity and Islam) have generally considered the first day of the week to be Sunday.

Some countries do not use the naming structure of Sunday Monday Tuesday (Domingo, Lunes, Martes, etc.) but instead say First Day, Second Day, etc. These might use Monday as the first day.

In China, Sunday means "week day".  Monday is "first day of the (seven-day) week cycle", Tuesday is "second day of the (seven-day) week cycle", etc. When China adopted the Western calendar Sunday was at the beginning of the calendar week, but now Monday is preferred.

In Swahili a traditional day begins at sunrise, rather than sunset. It is twelve hours before Arabic and Hebrew calendars, so some countries may consider Saturday as the first day of the week.  

WHAT IS THE GENERAL RULE?

In the USA for the last 10 years or so Sunday been considered the last day of the week, as our more secular view now places Monday at the top of the pyramid of importance. When Janice Raymond created Excel Math (1978), Sunday was definitely the first day of the week in the US school systems.  As I began revising the curriculum in 2000, we reduced the places where kids were asked to count days of the week and solve math problems with those numbers.

It all comes down to how you want to display your weeks on a calendar. Most of the calendars in my house start with Sunday and end on Saturday. How about yours? There is no right answer.

The international standards organisation (ISO) decided describing the time of day was an important issue, and created Standard 8601 to answer all our questions. Here's a paraphrase of the introduction:

Although ISO standards have been available since 1971, various ways to represent dates and times are used in different countries. Where this happens across national borders, misinterpretation can occur, resulting in confusion, errors and economic loss. The purpose of this International Standard is to reduce the risk of misinterpretation. 

We include specifications for numerically showing the date and time. In order to achieve similar formats for calendar dates, ordinal dates, week-number dates, time intervals, recurring time intervals, combined date and time of day, and local time or Coordinated Universal Time, it is necessary to use numeric characters, single alphabetic characters and/or one or more other graphic characters.

The Standard provides a unique representation of any date and/or time. It retains commonly-used expressions for date and time of day, and provides unique representations for some new expressions. It can be used with computers and should eliminate errors and misinterpretation. It will facilitate interchange across international boundaries, improve the portability of software, and ease  communication within and between organizations.

Stay tuned for the next blog! It comes After Midnight.

Where does all this data come from?

It's about time to do a complete backup of the Excel Math editorial production machines. (You back up your computer too, don't you? No?) As a result, some of my machines are occupying themselves, and I will use the numbers they report as subject matter for the blog today.

Since elementary school kids are operating phones, downloading music, playing videos and games - they are more than capable of learning basic concepts of data management.

We address this to some extent in Excel Math curriculum. We have a series of lessons on cell phones filling up the memory, and some on the cost of sending many text messages when you don't have an unlimited plan. We plan to add more on this subject, as protecting your data means doing a few calculations now and then.

I did a survey of my computers and this is the result:

First editorial machine:

• 249.2 gigabytes of space
• 154.1 gigabytes of data
• 242,893 folders
• 1,038,000 files are present
• if we divide the number of files by folders, we have an average of 4.27 files per folder
• if we divide the data size by the number of files, we have 155k per file

On the second, older laptop:

• 55.9 gigabytes of space
• 52.75 gigabytes of data (Warning! Drive is almost full!)
• 138,334 folders
• 519,617 files
• if we divide the number of files by folders, we have an average of 3.75 files per folder 
• if we divide the data size by the number of files, we have 106k per file

On our server:

• 74.5 gigabytes of space
  
• 46.0 gigabytes of data
• 34,679 folders
• 355,797 files
• if we divide the number of files by folders, we have an average of 10.25 files per folder  
• if we divide the data size by the number of files, we have 135k per file 

On the main external drive:

• 245 gigabytes of space
  
• 236 gigabytes of data (Warning! Drive is almost full!)
• 238,621 folders
• 1,872,976 files
• if we divide the number of files by folders, we have an average of 7.9 files per folder  
• if we divide the data size by the number of files, we have 132k per file 

 What analysis can I make? Are there any things to learn here?

1. Both machines 1 and 2 have had active users, with email, iPhoto, iTunes, Office, Dreamweaver, Creative Suite, etc.
2. By design, our server runs no applications. It hands out files when requested, and stores them when we are finished. The data files on the server are vector graphics or pdfs. They are relatively small compared to the music, photo and video files users are creating by the thousands nowadays.
3. The external drive has several hundred thousand read-only clip art files that fill 1/3 of its space
4. I have two more drives to check before I will be finished.
5. Between the time I started the blog post and now,  I ran to the store and got a new backup drive for $100. Look how much it can hold:

Who is visiting my blog today?

Elementary school mathematics includes an introduction to statistics. In our Excel Math curriculum, we help elementary school kids collect, analyze and present statistical data.

Statistics: collecting, organizing, and interpreting numerical data; a mathematical-based science rather than a branch of mathematics.

Many statistical projects involve collecting data about a sample group, then analyzing and manipulating the data in order to make projections or forecasts about a whole population. Other projects look at data in different ways in order to best understand what's happening.

Today I will be provide some data about my blog visitors, using different types of display formats that we might use with our students.  From looking at several sets of data I learn that I've had around 35,000 unique visitors from 170 countries.

NOTE: You can check the counters along the left margin of the blog window to see what they indicate as you are reading this.

Visitors (blue) and Page Views (green) by month

These bars show the visitors (and page views) we've received in the past 17 months. This is not a sample or projection; it's a count of your computers which have talked to the computers hosting this blog. You can see the number of visitors is increasing over time, with dips during school breaks (and my vacations and/or boring blog postings).

New visitors by day over the past month

 A more detailed look at a month's data shows that the visitor count dips on the weekends.

Distribution of visitors by time zone

This chart shows a time zone map, with bars at the bottom indicating the percentage of visitors that come from each time zone. We have 10% visitors from UTC 0 (the UK), 10% from +1 (Europe) and 55% total from -5 to -8 (North America). Other data, not shown here, tells me that less than 1% of my visitors come from South America.

Distribution of visitors by distance in miles

With this chart we can see how far you travel to visit the blog. Notice there are very few visitors coming from say 3500-4500 miles to see us. Let's see why that is.

Here's a circle with a 4000 mile radius laid on top of a global map. We can see there aren't that many people living 3500-4500 miles from San Diego. Only a few in the Andes, Brazilian Rain Forests, and perhaps a few Alaskans.

Here's a pie chart showing the distribution of our recent visitors. Now that we have discovered where our visitors come from, do we know anything else?

We can see 60% of our visitors use MS Explorer, 32% use Firefox, and 8% use Safari. The operating system data reveals that about 25% use Macs, and 60% use Windows PCs. 

If I look even closer, I can see that yesterday we had several visitors using their Blackberries. This morning I looked at the blog with my iPhone just to see if I could see myself. And I could.

Ok, enough statistics for today.

Math Curriculum, Part II

Yesterday I talked about math curriculum in Mongolia, where math teachers operate in a very complex environment. They have struggled with several character sets and alphabets, under communist or other political systems, using traditional, Russian or Western education methods.

Mongolia's math challenges are daunting, but every country faces some degree of political upheaval, economic upset, language and population shifts due to immigration, changing expectations about their schools, etc.

What we share with Mongolia is an expectation that science points us in the right direction to improve education. Or to use the lingo:

Evidence-based practice refers to preferential use of interventions for which systematic empirical research has provided evidence of statistically-significant effectiveness in solving specific problems

I'm going to color the following paragraphs to represent two divergent points of view:

Traditional education, although based on the experience of generations of teachers, lacks "scientific evidence". Please notice that a lack of scientific evidence SHOULD NOT IMPLY traditional methods are less than ideal or contrary to an ideal, just that so far there is no research.

Many education policymakers believe it's vital to identify which approaches work best so those practices can be promoted (and others abandoned or suppressed). Lack of major progress in education may be the result of random, disconnected and non-cumulative practices employed by individual teachers in local school districts, each re-inventing the wheel and failing to learn "what works" based on scientific evidence. 

Opponents argue that scientific evidence is a misnomer. Knowing that a technique works in engineering or chemistry is not the same as knowing a teaching method works with students. Centralizing control of education in the hands of government, scientists or "experts" is not the answer.

Evidence-based practitioners reply that teachers should happily choose from scientifically-proven options; that their only goal is to eliminate unsound or ineffective practices in favor of those showing better outcomes. One simple solution is to approve only methods or materials about which scientifically-valid studies have been published. Educators will be encouraged (or compelled) to select from this subset of methodologies.

Traditionalists point out that we lack of complete, reliable and useful data. We can't adopt Technique A before evaluating research design, quality of implementation, the backgrounds of the researchers, the volume of data available to the public. etc. Not all evidence is equal. No all techniques lend themselves equally to scientific investigation. Here are some categories into which our teaching options may be sorted (by scientists):

1. evidence-based: randomized study design to compare new methods with existing methods, results are independently replicated, blind evaluation of outcomes, and written documentation of the methods.
2. evidence-supported: non-randomized study compares methods, results are independently replicated, blind evaluation of outcomes, and written documentation of the methods.
3. evidence-informed: case studies on similar but not identical populations outside the target group, without independent replication; documentation does exist, and there is no evidence of harm or potential for harm. 
4. belief-based (traditional): little or no published research, based on composite populations; uses religious or ideological principles, or claims to be based on accepted theory without providing a scientific rationale; there may or may not be documentation, and there is no evidence of actual or potential harm
5. potentially harmful: negative mental or physical effects have been recorded, or by referring to documentation an expert may conclude there is potential for harm

Do you see the the blind alley we're in?

There's never enough science to satisfy science! Ideally we could have scientifically-based research determine the ideal process to compare traditional and scientifically-valid methods. Instead, by design or default (perhaps with no consideration of economics, interest, or practical value), the scientific approach places traditional education methods just above "potentially harmful".

This situation reminds me of Woody Allen's movie, SLEEPER:

Doctor A: For breakfast he requested "wheat germ, organic honey and tiger's milk."
Doctor B: [chuckling] Oh, yes. Those charmed substances thought to contain life-preserving properties.
Doctor A: [in amazement] You mean there was no deep fat? No steak or cream pies or... hot fudge?
Doctor B: Those were thought to be unhealthy... precisely the opposite of what we now know to be true.
Doctor A: Incredible!

Steak, rare, with chocolate ice cream

Math Curriculum, Part I

Quick! Before the title of the blog today puts you to sleep, answer just one question:

Is anything more complex and unclear than the 2-word phrase Mathematics Curriculum?

Why are you complaining? you may ask, Isn't a curriculum just a syllabus or outline that shows what will be covered during the school year? Don't you guys write them? 

Yes we do and no, we don't. Curriculum by definition includes more than just the content being taught. We create materials to help students and teachers, but curriculum includes:

• goals and objectives
• content and supporting materials (our part)
• expected teaching methods
• desired student interaction
• assessment of students and teachers and schools

The curriculum could be viewed in at least another 3 perspectives:

• Intended Curriculum - what Standards [nation, states, communities, parents] expect
• Implemented Curriculum - what and how the schools and teachers actually deliver
• Attained & Retained Curriculum - what grows inside a student / is used by a student

This blog was provoked by a paper titled Analysis of Intended Mathematics Curriculum of Primary Schools in Mongolia  and an article in the Financial Times newspaper today: How science led teaching down a blind alley. Now do you see why I am bewildered? Mongolian curriculum!

Is science taking our educators down a blind alley?

The first paper was written by a Japanese educator evaluating math curriculum approaches in Mongolia relative to those used in the United States and Japan. It was interesting to read, partly because I know little about Mongolia other than stories from friends who mentor Mongolian teachers. I've learned a bit from Janice Raymond, the creator and founder of our Excel Math elementary curriculum. Janice now lives in Mongolia most of the year, speaks and writes Mongolian, and is writing her doctoral dissertation on Mongolian proverbs.

Some young Mongolian students

I found the Mongolian experience fascinating. For example, in 1941, the country officially changed alphabets and writing systems. Here's the vertical script that is still used in Inner Mongolia:

And the current official Cyrillic alphabet characters:

 
In a century they have repeatedly changed their educational processes:

1. 1900-1940 Using traditional script, deliver content to students who copy and memorize
2. 1941-1980 Using Cyrillic script and language; adopt European and Russian teaching and learning methods
3. 1980-2010 Teachers start using North American, constructivist, student-as-explorer, teacher-as-facilitator curriculum process

The conjunction of Math in Mongolia and Science, School and Blind Alleys is an odd one, I admit. But here is a slightly-paraphrased statement from the latter that summarizes the link I saw in the two articles:

A fascination with science  - the belief that scientific methods can solve any problem - has become an addictive mindset in education. We concentrate our study and teaching on those things of which we can be certain. The rest is either scientised to give it a semblance of certainty, or ignored as too messy and too difficult to deal with. But markets are messy. Students (and consumers and employees) are unpredictable and irrrational. There is more than one best way ... we need to bring art as well as science back into our thinking about teaching.

More tomorrow.

The whites of their eyes

It's Friday, and this is the last of my blogs on famous phrases and math.

The quote until you see the whites of their eyes has been attributed to a number of military commanders. The original source is unclear. It might have been:

• Lieutenant-Colonel Agnew to the Royal Scots Fusiliers at Dettingen in 1743
  
• Prince Charles of Prussia in 1745
• Frederick the Great in 1755
• General Wolfe on the battlefield of the Plains of Abraham in 1759
• Revolutionary War commanders Putnam, Stark, Prescott or Gridley at Bunker Hill in 1775

The meaning of this phrase is clear - wait until they are close enough before engaging in battle. But how close is that? Is there a mathematical way to convert see the whites of their eyes to feet and inches?

Prescott said in his battle report that they engaged at 30 yards. Others have put the distance at 10 yards. Still others have said the distance was irrelevant as long as they didn't miss a shot.

Hollywood producers know the importance of the right distance; a battle filmed without the enemy in view is pretty boring. Great whites of their eyes were provided in the spaghetti westerns of Sergio Leone!

For example, this view shows the whole scene, but is clearly too far back:

This is the right distance to see the whites of their eyes:

 

 

The same holds true in space battles (as this article makes clear). How entertaining would it be for a Klingon battle cruiser to fire when it's two galaxies away? You want to see everything on one screen:

Despite these movie portrayals, in real life we are seeing less of the whites of other people's eyes.

• More people are looking at Facebook and not real faces
• Sales of sunglasses keep rising
• More cars and homes have tinted windows for privacy and energy savings

If you feel the need for a few more numbers than I've provided today, examine these facts:

1. All window glass filters out ultraviolet B and transmits UV A and visible light
2. Some glass provides ultraviolet A protection without loss of visible light (tint)
3. In California, vehicle windows with tint must have light transmittance of 70% or more, the tint alone must have a minimum light transmittance of 88%, and UV A must be reduced
4. Many sunglasses have plastic lenses and don't automatically reduce ultraviolet ( "The degree to which sunglasses will attenuate sunlight and block UV varies with the physical, chemical and optical properties of the lenses")
5. European sunglasses are divided into 5 categories:

Filter category	Description	Range of luminous transmittance
		Above (per cent)	To (per cent)
0	Clear or very light tint	80	100
1	Light tint	43	80
2	Medium tint	18	43
3	Dark tint	8	18
4	Very dark tint	3	8

I have been unable to ascertain the percentage of luminous transmittance that allows the whites of their eyes to be seen.

A belt and braces

Today's phrase is belt and braces (for the English) or  belt and suspenders (for the Americans).

Notice our man on the left who is wearing a pair of suspenders as well as a belt on his pants. This double  approach is designed to help make sure he won't be caught with his pants down (unprepared).

How do I relate this old expression to the mathematics we unveil to elementary school kids with our Excel Math curriculum? 

Our whole approach is a belt and braces design. Why? This is math education. It's a critical component of learning to participate in society, and we don't want to fail at it.

Here are our two strategies:

1. We teach kids to check their own work. When our students do a set of related problems in Excel Math, they then go on to solve a Checkanswer. This check-sum number allows them to confirm that the original group of answers were correct.

Let me show you - it's easier than trying to describe it all in words. Below, you see a box with three problems. The answers are shown in red. Once all three problems are solved, students add those three answers and compare the resulting sum (32) with the number in the box to the right of the group label H (32). If their answer does not match, they go back and recheck their work on the problems in this group.

We use this process throughout much of Excel Math - introducing it in 1st Grade and carrying it on through 6th Grade. The set above is from 4th grade; the more complicated version below is from 6th:

2. When they are working on complex problems that require a series of steps, we always ask students to try to solve them twice, in different ways. One solution is the belt, the other solution can be the braces. If they solve it twice using two varying approaches, they will never be embarrassed by their mistakes.

Of course, some people feel that a belt and braces strategy is redundant and unnecessary. It can be costly and time-consuming. It's like making a back-up flight reservation, so in case one airline is unable to fly, you can just walk over to another ticket counter.

Those people ask, Wouldn't it be better to trust one, less expensive solution?

I think it's safe to say it all depends on your point of view, and the risks that are involved ... we give our kids both belts and braces in Excel Math, and it's up to them to decide whether to wear them both or not.

Businesses are run by grown-up kids. As large companies streamline their supply chains and then get caught with their pants down due to natural disasters (Japan), political upheavals (Libya), supplier issues (Boeing 787) and so on, they begin to look around for some suspenders ...

A wing and a prayer

One of our planes was missing, two hours overdue
One of our planes was missing, with all its gallant crew
The radio sets were humming, we waited for a word
Then a noise broke through the humming - and this is what we heard:

Comin' in on a wing and a prayer, Comin' in on a wing and a prayer
Though there's one motor gone, we can still carry on - Comin' in on a wing and a prayer 

This week we have been looking at math in popular phrases (hammer and nail, mortar and pestle, wing and prayer). Math is everywhere in our culture, not just inside the Excel Math curriculum we create here at Ansmar Publishers. Today my mind is fixed on an expression attributed to songwriters Harold Adamson and Jimmie McHugh in 1943:

A wing and a prayer has come to mean "we are barely able; we may not succeed; but we are trying our best and praying we will make it"

Otto Lilienthal gliding with a wing, circa 1895

Researching the term WING today, I found much more than I expected. Some of the data appears below. Your challenge is to select the numbered item that defines our phrase:

wing(s)

1. noun: modified forelimb of a bird, covered with large feathers and usually specialized for flight
2. noun: insect's organs of flight, including a vein-covered membrane growing out from the thorax
3. noun: organ of flight in certain other animals, such as the forelimb of a bat
4. noun: aircraft's main lift-generating surface OR an aircraft designed as one complete wing
5. noun: position in a flight formation, just to the rear and to one side of another aircraft
6. noun: organ or apparatus resembling a wing OR the wings of a sphenoid bone
7. noun: resembling a wing in form, function, or position; such as a sail of a windmill or a ship
8. architecture: part of a building that is subordinate to the main part
9. automotive: (British) the part of a car body that surrounds the wheels (US term is fender)
10. automotive: inverted airfoil on a race car that produces downforce at high speed
11. botanical: lateral petals of a sweet pea flower OR outgrowths on wind-dispersed fruit or seeds
12. business: an affiliate or subsidiary of a parent organization
13. clothing: insignia worn by a qualified aircraft pilot
14. farming: outside angle of the cutting edge on a plough
15. furniture: pieces that project forward from the sides of the back of a chair
16. military: tactical formation in the air forces, consisting of two or more squadrons
17. nautical: projection on the side of a ship's hull OR a jetty or dam for narrowing a channel of water
18. politics: faction or group within a political party; as in left wing or right wing
19. sports: either side of a soccer pitch near the touchline OR a player assigned there
20. theatre: space offstage to the right or left of the acting area in a theatre
21. buffalo wings: fried chicken wings with hot sauce, dipped in dressing; developed in Buffalo, NY
22. clip your wings: to restrict your freedom OR thwart your ambition
23. fear gave wings to his feet: cause of rapid motion to escape danger
24. in the wings: ready to step in when needed 
25. on a wing and a prayer: with only the slightest hope of succeeding
26. on Eagle's wings: in several Bible passages OR popular devotional song for memorial services
27. on the wing: flying OR travelling OR about to leave
28. spread or stretch your wings: to make full use of your abilities
29. take wing: to lift off or fly away OR to depart in haste
30. under your wing: in your care or tutelage
31. wing it: to make things up or ad lib
32. wing window: the US name for a quarterlight or vent window on a car
33. verb: to make your way swiftly as if on wings OR to cause to move as if on wings
34. verb: to shoot or wound a person superficially, as in the arm
35. verb: to provide with wings

What does this phrase have to do with math? Everything! I'm sure you know (perhaps from personal experience) that many students approach math tests with a wing and a prayer.

I won't try to define prayer today. We'll end with a flying wing developed by Dan Dougherty, at Cal State University in Long Beach. Dan proved he was good at math, and while still a student was hired as a designer for Northrup Grumman.

A mortar and a pestle

Stone age people in Mexico and California ground up kernels of corn, acorns and seeds in shallow depressions in granite boulders (called a metate). These items are very common in the San Diego foothills, as the native American tribes were using them until only one or two hundred years ago.

Eventually the big boulder got carved down into a tray for portability. With feet so your back wouldn't hurt so much from leaning over it. it Then it became a bowl so things wouldn't fly out so easily. The round stone got a handle so you wouldn't be bashing your fingertips all the time.

Now known as a metate y mano or mortar and pestle, variations of this tool are used in chemistry, medicine and cooking.

TV chef Jamie Oliver has become famous for his generous use of herbs, spices and olive oil. He uses a mortar and pestle all the time. For those who don't want the weight and trouble of a mortar, the personal electric spice grinder is now a favorite tool. You can grind spices or coffee beans or whatever you like. Just remember you need to be within reach of an electrical power outlet.

WHAT'S THE POINT?

I know you are wondering when I will get around the the math! Now.

Let's draw some parallels between this history lesson about cooking, and math. In the beginning, we started with tally marks. Then we drew numbers with our fingers in sand or mud ...

and when we needed to move the numbers we cut out a tablet of clay and handed it to someone else.

Eventually we carved our meanings into a long piece of wood (read about the Tally Stick) that we could share with other people.

It was a nuisance to edit these carved numbers, until some bright thinker put sliding markers on sticks and called it an abacus:

As the centuries went by lots of math tools and reminder devices came and went (just like grinders) until eventually we worked our way back around to a modern math memory stick.

And finally, now we are drawing numbers on tablets with our fingers again! Just remember you need to be within reach of an electrical power outlet...