### Math and International Travel

If you’ve followed previous episodes of this blog, you will know that earlier this year in the northern spring my wife and I were blessed to travel in Europe.

Traveling between countries got me thinking about the types of mathematics you do when traveling. In addition to the usual math of budgeting, cooking, scheduling events, and so on, when you travel into a foreign country you have the extra challenges of conversions.

There are two main types of conversion an international traveler will likely have to do:

• Converting currency from one country’s currency to another’s
• Depending on the two countries, converting units of measurement between metric (SI) units and British/Imperial/American Conventional units

The video was shot in a studio in front of a ‘green screen’, to allow me to overlay videos. The background shows shots I took on the ferry between the UK and France on our trip, to set the scene for traveling between countries. I thought of doing the video live on the ferry, but it was too windy and noisy.

Converting Currency

Unless money was truly unlimited (certainly not the case for me), when buying food, gasoline and souvenirs a traveler will want to work out just how much he or she is paying for an item, in a familiar unit of currency. This is done using a currency conversion factor,  a number which changes continually. As explained in the video, there will be a pair of currency conversion factors for any two currencies, which are numerical inverses of each other. For example, the pair of conversion factors for Australian dollars (AUD) and British Pounds (GBP) at the time of writing is:

• 1.52875
• 0.65413

[source: XE.com on October 16, 2011]

This means that £1 British, the more valuable of the two currencies, is equivalent to A\$1.52875; conversely, A\$1 is equal to £0.65413.

The neat thing is that one can use either of these factors to derive the other by inverting it, and one factor can be used to convert the two currencies in either direction. Thus:

• 1 ÷ 1.52875 = 0.65413
• 1 ÷ 0.65413 = 1.52875

Using these data in everyday transactions, when traveling in the UK I could convert prices to Aussie dollars pretty easily and accurately by adding a half. So a burger for £4.99 would be worth around \$7.50 in our money. Going the other way, I could take around two thirds of an Australian price to get the rough equivalent in British currency.

Of course, a calculator will do this more easily, and a travel calculator is designed to simply convert in either direction by the touch of a couple of buttons.

Converting Units of Measurement

Depending on where you travel, you may be faced with different units of measurement from those you are familiar with. Europe, Australia and much of the rest of the world use Metric units, whereas the US and the UK are largely still using British Imperial units. Traveling on UK roads I changed the settings on my GPS device to use miles rather than kilometers, but in France I changed back to kilometers. That way road signs indicating how far a town was would match what the GPS indicated.
Manual or calculator conversions of measurement units are done in much the same way as converting between currencies, and again each type of conversion will have a pair of inverted factors. One example:

• 1 inch = 25.44 mm (millimeters)
• 1 mm = 0.3931 inch
• 1 ÷ 25.44 = 0.3931
• 1 ÷ 0.3931 = 25.44

International travel is not the only context for conversions like these. Other uses for these processes are international trade, which might be the subject of investigation in a geography or economics class; and international commerce, such as purchasing goods over the internet. Given the growth in trade and the opening up of secure, simple ways to buy goods on the internet, these are relevant topics for students and a straight-forward context for multiplication and division, including discussion of the best method for these calculations.

### Where is Zero on the Earth?

This is another in the series of podcasts from our trip in Europe.

Knowing exactly where you are on the earth’s surface is pretty important for most of us, and absolutely vital for airline pilots, surveyors, engineers and cartographers. Early study of location was difficult and inaccurate, hampered by lack of technology we now take for granted, and also by faulty understandings of the earth’s shape and location and movement in space.

I have long wanted to visit Greenwich, in London, to see the place which was designated as one of the ‘starting place’ for measurements on the earth’s surface, and also the reference point for time zones.

This map shows the location of the video, and the Prime Meridian:

View Royal Observatory, Greenwich, London, UK in a larger map

Royal Observatory, Greenwich

In 1884, Greenwich was chosen as the place for the ‘Prime Meridian’, the official dividing line between the eastern and western hemispheres, the line of 0° longitude. Of course, the Equator is the equivalent line of 0° latitude, dividing the northern and southern hemispheres.

The Royal Observatory at Greenwich website includes this interesting snippet about the history of the Prime Meridian:

The Greenwich Meridian was chosen as the Prime Meridian of the World in 1884. Forty-one delegates from 25 nations met in Washington DC for the International Meridian Conference. By the end of the conference, Greenwich had won the prize of Longitude 0º by a vote of 22 to 1 against (San Domingo), with 2 abstentions (France and Brazil).

The day we visited we had to drive to Scotland and didn’t have time to go into the observatory. If you have time when you visit London, I recommend a visit to this iconic location on our planet.

### Ignorance of Number Facts “No Barrier to Success”?

This week, a flurry of news and blog articles appeared, proclaiming that children don’t actually need to learn their number facts off by heart, as not knowing number facts doesn’t stop them from being good at maths. Is this really true?

The articles in question include these:

• BBC News: “Sums tables ‘not needed for maths success’”
• The Guardian’s Teacher Network Blog: “Children don’t need to know all their number facts to succeed at maths”
•  Daily Mail’s Mail Online: “Scandal of the primary pupils  who can get full marks in maths without even knowing their times tables”
• Independent Education Today: “Primary school children succeed at Maths without knowing their tables”
• Yahoo UK: “Children don’t need to know number facts to be good at maths”

My attention was caught by a tweet by @wanstad73 curated on my daily paper over at paper.li on September 13, linking to the Guardian article. Having taught number facts myself as a classroom teacher, and now teaching preservice teachers the importance of laying a good foundation in number facts in all 4 operations, I have a strong interest in the topic.

Let’s just say I was alarmed by a claim that memorization of number facts is unnecessary for success in primary or elementary maths. Surely, my thinking goes, without knowing number facts by heart, children will be unable to tackle later maths, not just in computation, but also in geometry, measurement, probability, algebra; pretty much all mathematics topics.

“The Development and Importance of Proficiency in Basic Calculation”

I had a look at the original article by Professor Richard Cowan at the Department of Psychology and Human Development, Institute of Education in the University of London. Surely the study’s author himself hadn’t said number facts were not necessary, as these secondary reports were saying?

It shouldn’t be a surprise that media outlets have picked on the idea that number facts are not really important. The English National Curriculum requires all addition facts to 20 to be memorized by the end of Year 3. So the idea that a research study has proven not only that Year 3 students aren’t learning all their facts, but furthermore those facts aren’t really that important could be expected to catch the interest of journalists whose bosses want to sell more advertising. But is that really what the research showed?

To summarize, Prof Cowan and his fellow authors say the following:

• proficiency in basic addition and subtraction to 20 is a key indicator of general mathematical ability, which later leads to adult proficiency
• students in Years 3 and 4 in the study showed above-average mathematical achievement, yet none out of 259 knew all their number facts
• only 10% of the children themselves reported that they were recalling number facts to answer most of the questions

The report’s authors describe the differences between a traditional view of learning number facts and a progressive view. According to them, the traditional view, in vogue in the 1920s and 1930s, favours rote memorization of number facts, whereas the progressive view focuses on children ‘learning numerical principles and patterns and knowing how to use them efficiently and accurately’ (Cowan 2011, p. 4).

Rote Learning vs Developing Understanding of Numbers

A comparison is thus set up between rote learning of facts and developing understanding of mathematical principles. What can we learn from this comparison?

 Traditional View Progressive View Learning by rote (repetition) Learning through understanding Facts learned in isolation Facts learned as connected to other facts & topics Facts believed to be essential for proficiency Facts believed to be less important than understanding Forgotten facts difficult to retrieve Facts not known may be derived by thinking Memorization of number facts regarded as essential for all students The ability to work out facts from understood principles regarded as essential

Which view is better? And which one is favoured by the report authors?

Contrary to the tabloid headlines, Prof Cowan and his co-authors believe that being able to carry out basic addition and subtraction quickly (the standard used in the study was 3 seconds for a correct response) is vital for developing a wider mathematical proficiency to lay the foundations for adult mathematical skills. The authors certainly did not say that number fact shouldn’t be rapidly accessible to all students. In fact, what Prof Cowan did state (according to the Mail Online) was ‘We are not saying that fact knowledge is irrelevant’, and ‘Facts help children grasp principles, and applying principles helps children learn facts’.

Conclusions

1. Children do need to know their number facts, either via memorization or via developing conceptual knowledge.
2. While children are learning the number facts, it is quite acceptable for students to use a strategy based on conceptual knowledge to quickly work out the answer.
3. Big media is wrong to imply that number facts aren’t important after all. Children need understanding of numbers first, and then need to memorize number facts. A more accurate headline than those chosen by editors would be “Children need to understand basic number concepts to succeed at mathematics”.
4. Most primary age students will use a combination of strategies based on understanding and memorized facts, as they develop greater speed and proficiency. Not having the complete set memorized is not a significant flaw, provided the child has a set of tools to derive those facts that have not yet been committed to memory.

Cowan, R 2011, The Development and Importance of Proficiency in Basic Calculation, Institute of Education, London, http://www.ioe.ac.uk/Study_Departments/PHD_dev_basic_calculation.pdf [accessed 13th September 2011].

If you are interested in resources to teach number fact recall via an organised system of worksheets based around classroom-proven strategies, check out our eBook “10 Minutes a Day: Times Tables Worksheets“. If you would like to trial part I of the system, sign up to receive it; all we ask is for your email address so we can keep you updated on new resources and content as they become available. If you are ready to buy the complete eBook, visit our store where you can purchase it.

### Math in the Cemetery

How can you use a field trip to a cemetery to teach mathematics?

I visited Richmond Park in London with my brother, and while there visited the East Sheen Cemetery to film a podcast.

What can  you learn in a cemetery? At first glance, this may sound like a strange or even morbid suggestion. However, provided you don’t have an issue with this (and neither do the parents of your students), there is a lot to be learned from the information a cemetery offers. In fact, the headstones or other locations where details of those who have passed are recorded form a statistical database of the community, potentially a very rich and fascinating record of the history of people who have lived in the area, and the events that have affected their lives.

This map shows the location of the video. Zoom out to see its location in relation to the London city centre:

View East Sheen Cemetery in a larger map

The cemetery I visited is in London, which has had a number of critical events in its history that might be reflected in the records at a cemetery, such as:

• The Great Plague (1665 to 1666; killed 60,000 people)
• The Great Fire of London (1666; killed 16)
• World War I (1914-1918)
• World War II & the Blitz (1939-1945; 30,000 killed)
• Great Smog of London (1952; 4,000 died)

[Wikipedia: History of London]

Your local cemetery will, of course, reflect the history of your local area. This opens up lots of opportunities for studies in social studies, history, civic studies, geography, and math. In fact, mathematics can be put to good use to serve studies in other disciplines, by providing tools and methods to collate and analyse the data that is collected.

As a starting point, you could ask students to record the following data from grave records for later study in the classroom:

• date of birth
• date of death
• gender
• occupation
• cause of death, if stated
• relationship to others buried nearby
• other interesting information

Footnote

By the way, this week I have made a few changes to the site, including removing a lot of fiddly looking links and graphics from the side menu and changing the colour scheme.
The biggest change, however, is that I have canned the audio podcast. The videos will continue, but the number of downloads of the audio was much lower, and so I’ve decided to simplify my life a bit and just produce one version of the podcast. The audio track is available from this page, if you’d like it, but it’s not part of the podcast feed for subscribers. Please let me know what you think!

### Teach Roman Math

I visited Chester in North England, where my brother lives with his family (he appears briefly in the video with his wife, and my wife and I). Chester is a fascinating town, which stands on top of Roman ruins, many of which no doubt have not yet been found. Basically, whenever a new building project gets underway, archaeologists have to be called in if (or more likely when) ruins are found on the site.

The video includes two on location shoots in Chester, the first at the town’s impressive Roman Amphitheatre, the biggest in Britain; and the second on the City Wall, built by the Romans, which is still largely complete and is a lovely walk around the city.

Math and the Romans

The Roman civilization was incredibly advanced for its time, in just about any field you can name (except perhaps moral behavior): architecture, engineering, military technology and leadership, government, art and fashion, economics, and so on. In many of these fields, mathematics would have been an essential part, just as they are today.

I suggest two straightforward “Roman Math” topics you can use in the primary or middle school classroom:

• Numeration – study Roman numerals, compare and convert with our base ten system
• Geometry – study tessellations and mosaics

With older classes and classes in Europe, other topics will be possible in the curriculum, and so if you are alert to the possibilities, you can link them to mathematics also.

The map below shows the locations of the video shoot:

View Chester, England, UK in a larger map

How do you include math in your teaching of history, and ancient civilizations in particular? What other connections do you make with your students with the Romans, Egyptians, Mayans, and so on?

### Teaching Slope in the Mountains of Switzerland

Switzerland is known for its beautiful mountains and chocolate-box scenery, summer or winter. My wife and I were blessed to visit there this last spring, so I took the opportunity to video another podcast episode. We took a cable car up a smallish mountain near Lucerne; actually probably just a hill by Swiss standards, then walked down. We’d done this before on a higher mountains when we were younger and fitter, and ended up unable to walk the next day. So this time we were a bit wary of taking on too much.

So, what about the math in this setting? The cable car and the incredible mountains, and the road tunnels that go through them all got me thinking. The swiss have developed an impressive network of roads that enable a driver to travel all over the country, in spite of the mountains that threaten to prevent travel due to their sheer size and their steep slopes.

To cater for this steep topology, Swiss engineers have put in place cable cars, modified railways, tunnels and myriad other installations to respond to the terrain. Sloped paths, steps, zig-zag roads and a thousand other examples allow life to happen in among the mountains.

Here is the video. It includes a montage of varied shots of the cable car we rode on, and at the end there is an overlay of the angle of the slope itself (my apologies that the overlay doesn’t fit the slope very well – it’s an artifact of the video editor I use, due to changing from 16:9 to 4:3 aspect ratio, if you are familiar with video editing you’ll understand).

If you are interested, here is an interactive Google Earth view of the location where I shot the video:

So, back to the math. In geometry or space lessons, we teach about slope and angle, which can often be rather a dry topic without a real-life application. The slope of a ramp, a steep road, the cable for a cable car, are all such applications. Using a few simple props, your students can measure slopes and apply mathematics to analyse them and measure their stats. This can then be linked to:

• slope expressed as a ratio (eg, 1:8 – I remember these from when I was a child)
• slope as a percentage (eg, 12% – the more modern style)
• the angle of the slope
• the tangent of the angle, or the sine or cosine

• mechanical advantage in a certain slope (how much easier is it to move up one slope compared to one with a different angle?)
• technology and engineering of building ramps, tunnels, cable systems, and so on
• environmental aspects of sloped land, such as erosion
• ‘optimal slope’ for certain purposes, such as driving a vehicle, mowing a sloped paddock, walking, etc.
• aesthetic aspects of hilly or mountainous country, when compared to flat land (How does the scenery make you feel? Why do people like to take vacations in the mountains?)

Till next week…

### Teaching a Great Math Lesson Part 1: Capture Students’ Attention!

#### Great Math Lesson Series:

 Phase I Phase II Phase III Phase IV Phase V Introduce Stimulus Whole-class Activity Problem Solving Synthesis & Reinforcement Revision & Recap

This is the first of a five-part series on how to teach a great mathematics lesson, using a simple, purposeful template that can be adapted for any math topic and any age level.

## First Phase: Introduce a Stimulus

Lots of math lessons fall down in the first ten seconds: “Who can tell me what ‘ratios’ are?” Seriously, which kid or teenager is going to want to answer such a question? Later in the lesson, there will be time for lots of questions. But ask such a question in the first few seconds? Never.

You know what they say about first impressions? You don’t get a second chance to make one. Well, it’s the same with teaching. I remember starting a lesson when I was a student teacher, saying “I’m now going to teach you about ‘protecting the environment’”, or some such thing. The children were polite enough not to groan out loud, but I could see the reactions immediately on their faces: Who wants to learn about THAT?

So, what should a teacher do?

Start with something interesting, exciting, unusual, unexpected, surprising, creative or enticing – which is connected with today’s math topic. Such as:

• Fractions – dress as a chef, bring in a chocolate cake, cut it into halves, then quarters, then eighths, and so on
• Subtraction – sing “Ten Green Bottles” while animating green bottles on a PowerPoint slide
• Percents – bring out a 25% off sale flyer for a department store, tell the children you’re going to buy a new outfit, but you’re not sure if you have enough money.
• Linear equations – dress as a plumber, carry a plunger or wrench. Tell students you have a tank to fill with water. It already holds 50 liters (/litres), and water is being added from a tap at 3.6 L per minute. How can we tell how much water there will be in the tank after an hour? How long will it take to reach 250 L? Could we graph the amount of water in the tank over time?

The actual idea isn’t that important; the main thing is to grab students’ interest, connect it with the math topic, and then while they’re paying attention, start teaching. It will require some time and effort put into preparation, but the payoff should be students who look forward to their next math lesson!

Photo by author.

### Welcome to Professor Pete’s Classroom

Welcome to Professor Pete’s Classroom! I hope you find resources that are useful for your classroom, to make you the Professor in your classroom!

Watch the video:

What math teaching topics are you interested in? I’d love to hear from you – leave a comment below.

Update June 2016: This was the first post on our old website, at Classroom Professor. We’ve come a long way since then, though our core values around developing children’s understanding of mathematics haven’t changed but rather become more established than ever in our business.