Sara Van Der Werf is a teacher in Minnesota, who has been teaching middle school and high school students for the past 24 years. She writes about teaching on her highly-recommended blog. For the past three years, Sara has collected photographs of math mistakes taken from everyday life, both from the world around her and from the internet. The mistakes come from stores, signs, newspapers, TV, advertisements, etc.). She shares each year’s collection on her blog:
Adult numeracy teachers can use the mistakes in several ways:
- As a warm-up in the beginning of class, or as students get settled coming back from break
- As a way to introduce a topic in math.
- Ask students, What do you notice? What do you wonder?
- Post a bunch of them on the walls of your classroom (or in the halls of your program) under the question, Can You Find the Mistake?, and watch as curiosity and discussion take hold.
Benefits/Messages students get from looking at math mistakes:
- Precision matters
- There is a lot that can be learned from mistakes
- It’s important to do/check the math yourself
- Students are not the only ones who make mistakes in math
- We can learn from our own mistakes as well as the mistakes of others
- Reflections on the math around us in our everyday lives
- Presenting mistakes are often helpful in talking about why things are important. For example, in the examples below, you’ll find graphs with problems with the increments of their vertical axes, which can help clarify the importance of increments for students, both as readers and creators of graphs.
The only thing I don’t love is calling them “Math Fails” or “the Wall of Shame”. I want to help students feel comfortable making mistakes and too many of them arrive in our classes with feelings of failure and shame when it comes to math. But calling them “Math Mistakes” works for me, especially with the frame of what do we notice and what can we learn from them.
Categories of Math Mistakes
Sara has a few different kinds of mistakes that tend to come up often. Below you can see a description and a few examples of each.
$0.99 vs. 99 cents vs. .99 cents
Sara has a lot of pictures showing mistakes using decimals and the ¢ cent symbol. For example, according to the sign in the photo below, a pound of bananas costs less than half of a penny.
Proportional Reasoning and Unit Rates
There are also a lot of photos of mistakes, often in advertisements, dealing with proportional reasoning.
We live in a data-rich world, but there is no shortage of incorrect graphs out there. Whether the mistakes are made by accident or by design, they offer opportunities to help students learn to be more discerning consumers of data. For more ideas on how to use graphs in this way, check out this short video: How to Spot a Misleading Graph
What impression does this graph give about the number of people on welfare and number of people with full time jobs?
Putting aside the fact that there are some of the same people in both columns (who work low paying full-time jobs and still require public assistance to survive), how does the more mathematically correct graph below tell a different story? How are the graphs different? How does the Fox News graphs mislead and distort the data?
Percentages are connected to proportional reasoning, and are challenging for many adult numeracy students. The percentage mistakes in Sara’s collections, like the ones below, offer us a chance to explore common student mistakes and misunderstandings.
The collection has several mistakes on receipts, which are great for making the case for students learning to do the math themselves, but also for mental math strategies of using benchmark percents (10% and 1% in the example below).
Sometimes harder to spot, I find the geometry mistakes are good at exploring conceptual understanding. For example, the diameter of Mars might be about half the diameter of the Earth, but since they are both spherical, saying that Mars is half the size of the Earth is not true. But then how could we compare the sizes of Mars and Earth and what information can we figure out form what’s given in the diagram.
This one involves a conceptual understanding of the relationship between the diameter and circumference of a circle. One of those numbers is not correct – can we correct it? But which one is more likely to be the mistake? And what’s the volume of this tree anyway?
Please use the comment section below to tell how you might use photos of math mistakes with your students.
Browse Sara’s collection and then once you share them with your students, tell us which photos you chose, how you used them and how your students responded.