Slow at Math ≠ Bad at Math

*Note: This is a recycled post from my personal blog.

“Speed ISN’T important in math. What is important is to deeply understand mathematical ideas and connections. Whether you are fast or slow isn’t really relevant.” – Laurent Schwartz, mathematician

If you haven’t seen the video by Jo Boaler and some of her Stanford students entitled “How to Learn Math: Four Key Messages”, you definitely need to. Besides the four powerful messages (which I will list below), it has some great stories and quotes, one of which is the one I have above.  Jo Boaler has done powerful research and written some terrific books on mathematics and learning math (one of my favorites being “What’s Math Got to Do with It?” and the video about these four key messages in math is so interesting.

Here are the four key messages about learning math (I highly recommend you watch the video to clarify and define each message a bit more):

  1. Everyone can learn math at high levels
  2. Believe in yourself (your beliefs about your abilities actually changes the way your brain learns)
  3. Struggle and mistakes are really important in learning math
  4. Speed is NOT important
All of these speak directly to the way we still, sadly, often teach and learn mathematics. One that really struck out for me was #4, speed is not important. I remember my own daughters struggling with the timed math tests – i.e. you have a minute to try and solve 100 times tables, or complete as many addition problems as possible. Very stressful, very ridiculous, and to top it off, they were penalized with poor grades if they couldn’t reach the arbitrary goal of “x amount of problems in 1 minute”. It still goes on and students memorize and stress over these timed math drills. Why? It’s ridiculous. If we continue to do this to students, then they begin to believe they are bad at math (see #2 above), which leads to them thinking they can’t learn math (see #1), and therefore leads to them giving up when problems get tough (see #3). A self-fulfilling prophecy.
So – I ask those math teachers out there who continue to put pressure on students to perform mathematical skills in a timed matter, where speed is important – stop. Just stop. Focus on what mathematics should be – understanding why those calculations matter, what they are related to, how they help us solve real-world problems. Help students make connections.
I know I keep coming back to it – but the Common Core Mathematical Practices seem to embody these four key messages. No where in there does it say students have to be able to do ___calculations in _____ minutes. Math is NOT about speed – it’s about the struggle, perseverance, conjectures, connections, and applications that help students solve relevant, real-world problems and see the beauty and need for mathematics.
Check out the video here

Multiple Representations on the Casio Graphing Calculators

One of the key things we try to help students with when studying functions is the idea of multiple representations – i.e. graphical, symbolic (equation) and table.  Ideally, we want students to be able to discern what the function represents or looks at no matter what representation they are given, and to be able to find patterns and important components about that functions from all representations.  Students should never learn about functions just through graphing, or just through symbolic manipulations or just through looking at data points in a table – they should be able to go back and forth and determine which representation is the most useful for the situation.

Unfortunately, too often, the emphasis is on one representation at a time, or at most 2. Let’s look at the graph and find the minimum, maximum, or intersection. Or, let’s find the roots of a quadratic by factoring, or symbolic manipulation. Or, here’s a table of points, where are the x-intercepts or the y-intercepts? Ideally, we want students to be able to look at all of these representations simultaneously so that they see the relationships between the representations and come to understand what the points represent in the table, in the equation, or in the graph.

Technology is one way to show all these representations at the same time, and then quickly manipulate and explore. There are obviously many technology tools out there, but as I have stated in previous posts, the most accessible technology tool for most students and teachers is the graphing calculator, not only because of it’s affordability, but because it is a tool most students have readily available.  It would be nice if all students had computers or tablets for daily classroom use, but that is still NOT the reality.

I have put together a quick video showing Casio’s three graphing calculators – the fx-9750GII, the fx-9860GII, and the CG10/20 or Casio Prizm, and how they can display the equation, graph and table representations of a function on one screen. No matter which model you have, you can achieve the same functionality, allowing students to work with multiple representations and explore relationships quickly and efficiently.

Check it out:

Understanding Math – It’s All About Perspective

I love to explore TedTalks as there are so many interesting ones that expose you to new ideas. TedTalks are great to use with students as well, because they can spark conversations, provide some real-world applications, and engage students in learning. I am always looking for any math-related TedTalks, especially when I can connect them to concepts students might be about to explore or have already explored and I want to provide an interesting connection.

There was a newer talk posted by Roger Antonsen entitled Math Is the Hidden Secret to Understanding the World where he talks about how mathematics is all about patterns and the idea of finding patterns is how we use mathematics to understand the world. Asking the questions of how does this work, and why does this work.  Representing something with patterns and then changing the perspective of that pattern can lead to really interesting things.  One of my favorite lines of Antonsen’s is “If you change your perspective, and take another point of view, you learn something new about what you are watching, or looking at, or hearing”. He does a great example of looking at a very common equation: x + x = 2x and realizing that this ‘equation’ is actually two different perspectives – one additive, one multiplicative. He goes on to give several examples, and one the whole talk really brought out to me is this idea that if students are allowed to explore and describe and explain their own understanding of patterns in the ways that make sense to them, i.e. their representations, they might have a better understanding of the mathematics themselves. Representing numbers as patterns of pictures or sound – fascinating and engaging. When he looks at fractions from the perspective of music or sound, the ‘sound’ of 4/3 is really beautiful and makes sense. The different perspectives are what allow us to understand the mathematics.

Obviously, the Common Core comes to mind immediately – those Standards of Mathematical Practice that I love! If we look at just a couple things from these practices you can see Antonsen’s idea of changing perspective to understand and make sense of mathematics (and the world):

  • Students look at problems from multiple entry points (i.e perspectives)
  • Students reason abstractly – i.e. abstract what they know and apply it to make sense
  • Students model with mathematics – i.e. use different perspectives to represent something mathematically
  • Students look for and make sense of structure
  • Students look for and express regularity – (patterns)

Common Core practices really speak to Antonsen’s idea of understanding by finding patterns and using different perspectives to make sense of the world. He does a great job of both visually explaining, using mathematics as his example, of how changing perspective helps opens you up to understanding the world and becoming a more empathetic participant in it. It’s all about perspective.

Here is Antonsen’s TedTalk – worth a watch!