Revolutionizing the math class

Okay so you’re working to create this growth mindset (see last post) among the students in your class (or your child at home), now what?

The next topic from Jo Boaler’s Mathematical Mindset that made an impact on me was how desperately we need a math education revolution in the US. Our students are consistently scoring below average in mathematics, according to PISA (Program for International Student Assessment), but more importantly than that, we’re creating an army of students who loathe and fear the subject.

pisa scores

This chart is from the 2012 assessment, but we dropped even lower in math on the 2015 assessment. Andreas Schleicher, director for education and skills at the OECD, which administers the PISA exam explains that “[US] Students are often good at answering the first layer of a problem in the United States, but as soon as students have to go deeper and answer the more complex parts of a problem, they have difficulties.” 

Let’s examine how we could encourage our students to “go deeper” in math.

Jo Boaler defines math as “a set of ideas, connections and relationships that we can use to make sense of the world.” At it’s core,” Boaler continues, “it’s about patterns.” Does this sound like the math you learned in school? Or if you’re a teacher, does this sound like the math that you’re teaching?

If I were to ask my middle school students, “What is math?” I bet their definition would be more like “math is a bunch of disconnected, seemingly irrelevant formulas taught to students so that they can solve problems that they will never face in the real world.” Or some of them might even just define math as “torture.” That’s how I would’ve defined it in school.  

Math, as a subject, is meant to involve collaboration, sense making, reasoning and connections. Conrad Wolfram, a British businessman who’s known for his work on information technology, explains that there are 4 stages of math:

  1. Pose a question
  2. Go from the real world question to a mathematical model
  3. Perform a calculation
  4. Go from the model back to the real world to see if it solves the question

Unfortunately, according to Wolfram, much of school math is spent in stage 3: performing calculations. Not only are calculations done individually in class, but dozens are sent home for kids to practices these “skills” for homework (we’ll get into the Homework Battle another time).

You can find Wolfram’s TED Talk: Teaching Kids Real Math with Computers here

So basically we sit kids down, teach them the formula (ie: how to divide fractions) then give them a bunch of calculations to perform individually, maybe sometimes throwing in some group work or “real world” word problems.

No wonder all of the fun, creativity and curiosity is stripped from the subject.

We know it’s wrong and bad for our kids, but what are some things we can do about it?

First, we can work on redefining math (make sure the activities in class actually fit the definition from above) and try to build that growth mindset of mistakes do not = failure.

Let’s assume you’re already working on that. Here are some ideas from Boaler’s Mathematical Mindset

  • Group work, group work, group work – showing students that math is a collaborative subject meant to be a combination of questions and ideas and trial and error, this also allows teachers to weave interpersonal and teamwork skills into class. I did a few big group projects in math last year. One was a “city building” project where the groups had to work together to plan their own city using specific sizing criteria (ie: the homes must be at least 1/4 the size of the courthouse). It was so cool to see what the kids came up with. Not only did they love the project, they worked collaboratively and some even went on to add infrastructure such as a city-wide bike lane, solar panels and , of course, Starbucks. 
  • Number Talks – Pose a question, and have a discussion about ALL of the different ways that students solve the problem.
    • Here’s an example of Jo Boaler doing a number talk with a group of students. 
    • Here’s another: 18x5 show your workSee all of the different ways these students solved for the same equation? This validates each student’s process and allows other students to recognize that there are many ways to get to the same answer. It also just gets kids talking about math – rationalizing and explaining HOW they got to an answer – which is immensely beneficial to reasoning skills. Talking about these methods allows students to form a deeper understanding of multiplication, addition, subtraction and division, as well as patterns. Number talks turn numbers from abstract to concrete.

(Unfortunately, I came across Boaler’s book too late in the school year so I didn’t have a chance to try these out with my math class, but a colleague started incorporating them into her class and she reported that otherwise very uninterested kids were leaning in and sharing their ideas. She also recommended that I look into Cathy Humphrey’s book, Making Number Talks Matter)

  • Meaningful homework (or NO homework!) – Boaler places value on math reflections as homework (or exit tickets). This adds a metacognitive aspect to math. Reflections allow students to think about HOW they think about the concepts they’re working on. Here’s an example of a sheet of math reflections. She also suggests homework ideas such as:
    • Find an example of the Fibonacci sequence at home or outside
    • Find examples of other patterns at home or in nature
    • Create a problem similar to the ones you worked on in class
  • Encourage students to share their answers in multiple representations (words, graphs, tables, symbols, diagrams, etc) – An example of this is in the Featured Image on this post. Sharing multiple representations allows students to better visualize the concepts, as well as see that if one way doesn’t work for them, there are many other methods to try. Boaler shared that she often tells students: “What I really like to see is an interesting representation of ideas or a creative method or solution.”
  • Aim for “Low floor, high ceiling” questions and activities. “Low floor, high ceiling” means that everyone in class can access the ideas, yet it can be taken to high levels for the more advanced students. If students “finish” the question or activity, ask them to create a similar problem but make it slightly more difficult. Here’s a Low Floor High Ceiling example called “Paper Folding”:
    • Instructions – For each part of the problem, start with a square sheet of paper and make folds to construct a new shape. Then, explain how you know the shape you constructed has a specific area.
      1. Construct a square with exactly ¼ the area of the original square. Convince yourself that it is a square and has ¼ of the area.
      2. Construct a triangle with exactly ¼ the area of the original square. Convince yourself that it has ¼ of the area.
      3. Construct another triangle, also with ¼ the area, that is not congruent to the first one you constructed. Convince yourself that it has ¼ of the area.
      4. Construct a square with exactly ½ the area of the original square. Convince yourself that it is a square and has ½ of the area.
      5. Construct another square, also with ½ the area, that is oriented differently from the one you constructed in 4. Convince yourself that it has ½ of the area.
  • Prepare students to be convincing. Ask students to first convince themselves that they came up with a reasonable solution (pretty easy to do – convince yourself that what you did is correct), then have them convince a friend (a bit more challenging), then finally have them convince a skeptic (this involves lots of explaining and showing of how they came to a solution and it also builds reasoning skills).

convince

These are just a few of the many steps we could take in the direction of sparking a mathematics revolution in the US. By shifting our focus and aiming towards a more dynamic, interactive, deep, creative and collaborative subject, students will hopefully start to show more interest and be more engaged in math class.

-Mrs. Sanford

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