## Best Math Blog Posts: March, 2012

I thought I’d write a different sort of post this time: a round-up of some of the best recent math teaching blog posts.

I share these posts via my Twitter and Facebook accounts already, but those posts soon move off the top of the stream, and are lost pretty much for ever. But by sharing here on the blog, these links can remain accessible for much longer.

By the way, I’d love to hear what you think; I am planning to write similar posts in future weeks, if they are useful and interesting to you, my dear reader.

Math Blog Post Roundup

Fawn Nguyen [Teaching Math in Middle School] > Always Sometimes Never

• Students sort mathematical statements into three piles: those that are always true, those sometimes true, and those that are never true.

Brilliant math activity for any grade from @fawnpnguyen: Always Sometimes Never ow.ly/9PnSB#mathchat

I linked to this article on March 24, 2012 via Twitter, and it was far and away my most clicked tweet all week. I guess other teachers agree that this is a wonderful article, explaining a simple activity that any math teacher could use with their class. Fawn’s students worked on statements which included “p + 12 = s + 12″ and “If you divide 12 by a number, the answer will be less than 12”. I love these statements, and the activity, for several reasons:

• the statements themselves are easy to understand at first, which will help develop “buy-in” by students.
• I would expect just about every student to be happy to get started.
• None of the statements can be answered immediately via some standard routine procedure.
• Each one requires a level of intuition, investigation, even lateral or creative thinking.
• For many statements, there is an obvious answer; and like many obvious answers, it isn’t always true.

David Ginsburg [Coach G’s Teaching Tips] > There Are No Stupid Questions, But…

• How teachers respond to students’ questions may have a big impact on how likely students are to ask questions in the future.

Another great post from Coach G: There Are No Stupid Questions, But… ow.ly/9PmVZ

David is another blogger whose writing I admire. He manages to get right to the heart of an issue for teachers, grab your attention, and then get the reader to honestly think about his or her own teaching. This particular post is a great one if you care about the impact your comments have on students’ feelings of well being and self esteem.

I have to say, this isn’t really “Questioning 101”, but more like “Questioning 404”, for teachers or preservice teachers who have understood the basics of eliciting students’ responses, but realize that there is a higher standard to aim for. Key statement by David: “[students will] never feel such freedom unless we as educators value their input rather than just evaluate it”. Amen!

Malke [The Map is Not the Territory] > All in Good Time

• The author’s 6-year-old daughter asks to be taught to play the penny whistle.

Music, math, reading, kids develop at different rates > All in Good Time @mathinyourfeetow.ly/9A9bb #edchat

Malke always writes interesting posts, illustrated with lovely photos of her daughter and the activities they share. This post caught my attention because of the focus on playing music, and also learning math, two of my loves. The key point: given the freedom to choose when to learn something, children will often reveal when they are ready, “all in good time”.

• Ms Cookie finds out that her students’ difficulties with similar triangle questions had less to do with the math, and more with fundamental misconceptions about shadows.

Not understanding shadows interferes with learning similar triangles… from Ms Cookie http://ow.ly/9Pwmn #mathchat #scichat

I love science as well as math, and so this post was a fascinating one to me. Students really do struggle with scientific concepts quite often, but even so I was surprised at their ignorance about shadows and how they are caused. How often do we think difficulties in math problem solving are caused by lack of math knowledge, when they might come from misunderstandings of the question itself.

Dr Mike Hartley [Math Games for Kids] > Is Math The Primum Movens?

• Could mathematics explain the existence of the universe?

Is Math The Primum Movens? | Philosophical post > Does math alone explain existence? ow.ly/9Pp33 #mathchat

If you enjoy philosophical discussions and the study of mathematics, you’ll like this post. Dr Mike poses some really tricky questions about ultimate reality and First Causes (Primum Movens, in Latin). Is God the Cosmic Mathematician? You’ll have to decide.

That’s all I have space for this week. I hope you’ll follow the links to read other bloggers’ posts, and let me know below your own thoughts. If you have a favorite blog or two, let me know and I’ll check it out.

##### Photo References:

I follow Dan Meyer’s blog quite closely, and find the discussions over there really stretch my thinking sometimes about how we teach math, and the best ways to engage students in thinking.

Dan Meyer on the Khan Academy

I first encountered Salman Khan on his TED video, perhaps like lot of others. (Incidentally, that’s also how I first heard of Dan Meyer, watching his TED talk.) I found Sal Khan’s methods surprising and challenging, and incidentally, his business practices pretty remarkable also. If you look at his site, it’s hard not to be impressed by the sheer volume of material he has there, with a huge list of videos, all free for watching.

Recently Dan has posted a couple of articles about the Khan Academy:

Dan points out several really important points about the Khan academy’s approach, including an apparent shift in emphasis from supporting the work of teachers via flipped lessons to supplying an entire curriculum for students. Crucially, Dan comments that students actually find watching the Khan videos quite boring, which surely is a critical flaw in the program.

“Flipped Classes” – a Solution to Bad Teaching?

To summarise, in case you haven’t been keeping up with this debate, the idea put forward by Khan at the TED conference which has captured the attention of many educators, is “flipped classes”. In this model, instead of the teacher teaching in class and then assigning practice work for homework, students watch the teaching at home via Khan’s videos online, then in class the teacher gets to follow up the video presentation, offer one-on-one tutoring help, and generally support and troubleshoot students’ learning, freed from having to spend hours planning and teaching didactic lessons.

What’s the philosophical idea behind Khan’s approach? Note the low-tech quality of the videos: it can’t be able visual engagement, hooking students with exciting music, animations or the like. No, what Khan is attempting, without really admitting it, is to produce a set of perfect teaching videos. If you like (and I doubt you do), a teacher-proof syllabus. How does that strike you? I find it insulting: why does Mr Khan feel that a disembodied voice track and a screen showing the teacher’s written notes for a math process is better than what real teachers do in a real, physical classroom, with students who are present in the same space?

The only way to accept KA as a replacement for what teachers in general do in classrooms is if you subscribe to the idea that most teachers suck at teaching math. If that premise is accepted, then the idea that a single source of “expert instruction”, delivered uniformly to all students, could supply all the teaching might look pretty attractive.

However, critics point out, often with some heat and passion, that there are several problems with this scenario:

• lecturing to students is not the best pedagogical approach to teaching
• video recordings lock every student into a single lesson for each topic
• there is no opportunity for students to ask questions of the video teacher, to have something explained again, other than replaying that part of the video

What do you think?

## So, You Will be Teaching my Grandchildren? Job Interview Questions

I’m going to be a grandfather, this coming August.

This momentous event is not unexpected, since our daughter has been married to her wonderful husband for 3.5 years. Nevertheless, it has made me think again about small children I have a particular interest in, and what happens to them at school. This post is just a touch indulgent, but I invite you to join me in thinking about the qualifications for teaching children.

What do children need from their math teachers? How important is it? If you really were going to be my grandkids’ teacher, these are the questions I’d be keen for you to answer:

Math Teacher Job Interview: Questions for Candidates

• How did you do at math when you were at school?

It’s not important if you were an “A” or a “F” student; you could become a brilliant teacher of math from either starting point. Now, on balance, I’d rather that you got great marks at math, since you obviously “get it” and will be able to understand math more deeply than most. But either way, I’d like an answer to the second part of this question:

• Do you understand why lots of people find math difficult, scary or downright horrible?

To teach students who really don’t understand math, and therefore most likely hate doing math, you need to be able to empathize with them. Did you ever feel that learning math was pointless, illogical, impossible or just plain frustrating? If so, great! Can you put that knowledge to work as you help students who feel that way? If not, are the ways for you to learn what it feels like to students who hate math?

• Do you love math? If not, are you at least enthusiastic about math?

Not everyone loves math. Even math geeks get that. But teachers of math have a responsibility to encourage students who do love it, who will go on to do amazing, important math later in their lives. I have decided to never discourage a student who has, or is developing, a passion for something, anything almost. If a student loves dancing, or BMX bike riding, or astronomy, or writing poetry, I will do everything in my power to support, encourage and promote success for that student in that endeavour. How can you do that in math?

• Which is most important to you: understanding, or correct answers?

We should never pretend that correct answers aren’t important. But just as giving a poor person a fish is a lousy substitute to teaching that person how to catch fish, focusing on correct answers should come after we are sure the student understands the math, never before. Great math teachers have the attitude that correct answers are the product of correct thinking, and in some cases aren’t even that important, providing the student understood the process.

• Do you teach “tricks and shortcuts” to math processes?

This is really the corollary of the previous point. A shortcut (eg, to convert a decimal to a percentage, move the decimal point two places to the right) will lead a non-understanding student to the correct answer, providing (and this is the deal-breaker right here) the student can remember the correct trick to use. But the shortcut or trick will let the student down in situations in which the math is not so obvious, and thinking has to be applied to solve a problem. Good mathematicians do use shortcuts, but that happens way, way, way after they understood why the shortcuts worked.

• How important is memorization of times tables and number facts to you?

Since calculators first became a consumer item, teachers have had a challenge with this question. Do we really need to spend the time that some of us remember from our own school days on learning times tables, or can we now let calculators handle that process? Answer: unless students have fast, accurate, confident recall of basic number facts and times tables, they will be hampered and slowed down by the need to reach for a calculator every time they need one of those facts. If you have to know “13 x 24″, by all means use a calculator – it will be quicker. But if the question is “6 x 7″, recalling the answer quickly will save huge amounts of time and cognitive load.

By the way, to see a series of classroom-tested, systematic workbooks for teaching all the basic facts and times tables in 10 minutes a day, visit our store.

• What math teaching blogs, forums, wikis or Facebook pages do you read?

In most places where teaching is a profession, a certain amount of professional development is required every year. Where once that PD came in the form of visiting academics to run workshops and sessions after school, nowadays it should also include use of online communities and other sites, where other teachers and educators meet to exchange ideas and encourage each other. Some examples from hundreds of great sites:

I already wrote about the necessity of math students knowing their number facts. The way some non-math people talk, this is an “either-or” situation: either you learn times tables by rote, or you use technology. My view: the true answer belongs in the middle somewhere. Yes, students need to learn number facts. No, they don’ t need to use unthinking rote methods (see here for more on this). Yes, they should use technology. A great math classroom will have lots of appropriate technology: calculators, iPads, iPods, laptops, data projector and interactive whiteboard, internet connection (duh!), etc., etc. The technology opens up unlimited possibilities for connecting classroom math to real life math, for using microchip tech to make more complicated math possible, for practising skills using online interactive widgets, and so on and so on.

• Do you assign math homework? What sorts?

Homework is a huge topic which I don’t have space to address adequately here. But for the purposes of this post, let’s agree that homework is a great way to connect your math lessons with parents and other family members. By sending home suitable activities to apply and practise the math being learned in class, you make space for child and parent/grandparent/carer to have conversations about math. My advice: make the homework something every child can succeed on, so that it becomes an enjoyable part of family life, not one of the biggest frustrations. Build in activities for children to talk about math at home, for the benefit of child and adult.

• Do you integrate math into other subjects’ lessons? What about integrating other subjects into math lessons?

Math can seem a pretty lonely subject in some classrooms. It can be the only one which lots of students don’t like, the only one done solely using textbook or worksheets, the subject most likely to lead to the question “When are we ever going to use this?”. You can help to change these perceptions by showing students how useful math really is, by allowing math to serve other subjects and help solve their particular problems. For example: in social studies or geography classes, use math to examine statistics and graph the data on how people behave, how the environment is being affected by pollution, etc. The reverse is also worth doing: include real-life examples from other subjects when preparing problems and investigations for math class.

How did you go? Do you agree with my views? Would you be happy to teach my grandchildren? Leave a comment below and let me know…

Photo Credits:

• Grandfather and boy:  © iStockphoto.com/Pavel Losevsky

[related-posts]

## Are Wind Farms The Solution? Do the Math!

It’s Agreed: Teachers Should Teach Environmental Responsibility

Teachers around the world are expected to teacher their students to be responsible citizens, to reduce their impact on the environment and to support sustainability. For one example, see the new Australian Curriculum’s Cross-Curriculum Priority: Sustainability.

Great, good stuff. In fact, it would be difficult to imagine a teacher anywhere who did not believe in protecting the environment for future generations, and teaching today’s students to be responsible adults in the future.

But does mean that we should agree to promote every green cause or environmental suggestion, “just because”. Some ideas are really dumb, including those with a green label applied. How can we tell them apart? Do we just let others decide for us, or do we think for ourselves? I reckon we should use math to check out the facts – and teach our students to do the same.

Will Wind Farms Fix the Energy Crisis?

A lot of environmental activists and politicians are currently promoting wind energy as the solution to global warming and the world’s reliance on dirty fossil-fuel-generated energy. Are they correct?

The answers aren’t all in yet, but it is important to do some basic research, rather than just believing the promoters. There are some great sites out there with lots of information on outputs, days of output, daily averages, hourly averages, etc.,  which give a much better idea of what is actually happening.

Will wind energy on its own replace other forms of electricity generation?

Other considerations for developing an informed basis for developing an opinion could be factors that are unique to wind farming. Such as:

This is the final podcast episode that was recorded on our trip in Europe. Future episodes will be videoed in the classroom.

Podcast Location:

South Lanarkshire wind farm. You’ll notice that the Google maps photo doesn’t show the wind turbines; the photo must have been taken before they were constructed. If you check the Street View, you will see them.
View Lochhead (Wind) Farm, South Lanarkshire in a larger map

So Where’s the Math?

Teaching about environmental issues is a great opportunity to promote socially responsible outcomes in your students’ lives. It is also a wonderful opportunity to put those mathematics skills to really practical use. Environmental projects always produce controversy, because they always involve costs to someone or other, along with the benefits. In order to reach sensible, informed conclusions, we should return to the data, and ask “What are the facts?”. Your students are ready to answer questions of concern to them, that are matched to their level of maturity. Use math to help them decide!

With your class you could do some activities such as actually planning a family’s daily needs based on the output from turbines on different days. How is industry affected? What is the daily consumption, or even the hourly consumption of power of a city? Are its needs steady or fluctuating? How does this affect the power industry? With these ideas it would be possible to follow through on an idea and make a great math project where the application of math is almost limitless.

The Wind Farm Performance site has lots of real-time stats and graphs on Wind Farms:

What do you teach in environmental studies that could use some reality in the form of data? I’d love to hear your ideas – please leave a comment below.

Related Videos

Explanation of the components of a wind turbine:

Turning the Weather Into Power (article + Flash animation) – click image for link:

Danish wind turbine fails in storm:

Wind energy in west Texas, Wind Turbines:

Wind Farms in the Media This Week:

Photo Credit

## 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].

## 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