What are pseudocontexts, and should K-6 math teachers be concerned about them?
I came across the term in Dan Meyer’s excellent blog, in which he explores better ways of engaging students in learning math, and calls out “fake math” and poor teaching. You should go check out Dan’s work, I find it really challenging and encouraging.
As a publisher of educational content for K-6 math, I am deeply concerned about students’ recognition of the importance and the usefulness, the utility of the math they learn.
This means that I am always looking for ways to show students examples of math in real life. But does that mean I have to restrict my examples to ones where someone is asking a specific math question that matches the topic students are learning, or can I ask questions which someone might ask, but probably didn’t?
What do you think? Leave a comment below if you’d like to share your thoughts.
Recent examples of “math in real life contexts” are found in my Where’s the Math? video series, which you can see here. Am I guilty of introducing pseudocontexts?
In the busyness of classroom teaching, do you find math lessons becoming a bit stale? Are textbook lessons getting you and your students down a bit?
I believe that students crave interesting, relevant lessons, especially in math. How can we provide such lessons?
It’s a simple idea: find real math going on in the world around you and bring it to the children’s attention.
In the video I talk a bit about the new kitchen we’re having put in. There’s a heap of math that the builders have to get right, including measurement, simple arithmetic and three-dimensional geometry.
What can you talk about with your students? Here are some ideas:
Real-Life Contexts for K-6 Math: Some Suggestions
Take photos with your smartphone, add them to PowerPoint slides, ask students “What do you notice?” or “What questions are you thinking of?”
Talk about shopping experiences where you had to figure out a best buy, someone gave you the wrong change or something cool happened
Explain how you adapted a recipe for a different number of servings
Talk about your favourite sport and how rankings work, and how many points teams have to win to come out on top this season
Talk about designing something cool, such as a garden, a craft project, a greeting card or a decorated cake
What ideas have you used to bring classroom math to life? Leave a comment below.
The Australian Federal Government recently conducted a survey; have you completed your form yet? Or are you waiting to see whether the Australian Bureau of Statistics website gets attacked again?
The recent national census really caught the attention of the Australian populace, mostly for all the wrong reasons. The official website couldn’t cope with the traffic to the site on “census night”, which was entirely predictable, and at the same time it was subjected to several “Denial of Service” (DOS) attacks.
But apart from that, what can we teach children about censuses? Here are a few ideas:
Governments use censuses to find out how and where to spend money on schools, hospitals, rail lines and highways
School math includes taking surveys, collecting data and analysing the results, just like in a census
Jesus was born in Bethlehem partly as a result of a census conducted by the Romans
People expect survey-takers and governments conducting censuses to protect their privacy
What do you think?
Do you have favourite activities for teaching children about statistics? Share a comment below.
Do your students believe that math is irrelevant to their lives? Sadly, all too many of them do, especially as they reach high school.
The article I discuss this week lists 10 occasions in which major Australian supermarkets got the mathematics behind their special offers totally wrong. Sometimes the “offer” was worse than the standard price, sometimes it was exactly the same. And occasionally the offer was better than the staff member who put up the marketing ticket realised.
I love this article for its immediate relevance to students. Very few people don’t enjoy finding a bargain, but no-one wants to be ripped off. The examples here provide lots of examples for students to figure out what each offer is, what it’s really worth, and suggest a better, more honest offer.
Buyer beware! Even though stores are using computers to create special ticket offers, human beings are required to think up the actual numbers in each offer. And sadly, some staff members don’t recognize their own mistakes before letting customers find out about the specials.
What do you think?
Do you do “shopping math” in your classroom? Share a comment below.
Rain gauges measure rainfall by collecting a small sample and measuring how deep the water is. The trouble is, we are interested in very small units – in the metric system, rainfall is measured in millimetres/millimeters. How can you accurately measure such small amounts?
How can we use everyday examples to teach measurement?
Watch the video: I explain how a rain gauge amplifies the depth of water collected to make it easier to measure.
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.
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.
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:
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
cause of death, if stated
relationship to others buried nearby
other interesting information
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 your students about Roman civilization with a math connection!
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:
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
Cross-curricular links could then be made to topics such as:
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?)
What do you teach about mountains in your curriculum? Could you incorporate slope in applied lessons with your students? Please leave a comment below – I’d love to hear your thoughts.