I finished Jo Boaler’s MOOC last week. I thought it was very good. It is worth checking out yourself (tip: watch the videos at double-speed), but here is a summary of what I learned:

- It is important to foster a “growth mindset,” rather than a “fixed mindset.” I was familiar with Dweck’s work before, but I feel I understand it better now.
- Writing “I am giving you this feedback because I believe in you and want you to improve” once on a student’s paper at the beginning of the semester seems to have a huge positive impact.
- If you want to encourage a growth-mindset, it is very important that you ask open questions. Boaler gave the following example: instead of giving the students a particular rectangle and asking to find the perimeter, ask students to draw a rectangle with perimeter equal to 26. These tasks both “test” the same concept, but the former has only one right answer, while the latter has many.
- One of the problems that struggling students have is that they do not know that they can decompose numbers to help them. For instance, they do not know that 12*7 can be solved by doing (10+2)*7=10*7+2*7; they tend to believe that this is not allowed. “Number talks” are useful in demonstrating that this sort of manipulation is
*encouraged*; a number talk is basically asking students to do a mental calculation (like 12*7), and then hearing all of the different ways that students calculated it. Then students see that there are many ways of doing the problem, and all are acceptable.

I am teaching a “math for liberal arts” course this semester, and I basically have it designed. However, after going through this MOOC, I am planning on tweaking all of my prompts to make them more open. I don’t think that I will be very good at doing this, but it will be good practice for me. Additionally, I think that I will incorporate weekly “number talks” with them (I will definitely do this in my spring course for elementary education majors).

There was a lot more in this course, but these were the highlights for me. I found that this course was definitely worth the amount of time required; Boaler did a nice job of giving just enough “homework” to be useful, but not so much that it was overwhelming (in fact, the homework was pretty minimal, time-wise).

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Tags: Growth Mindset, Number Talks, Open Questions

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August 30, 2013 at 4:37 pm |

I don’t have much experience with number talks, but I strongly agree with the first three points. Every time I use a question of the type you mention in point 3, I am delighted with how much insight I gain from the student answers. That sort of flip does take the difficulty level of the question up a notch or two as well which is something one has to be mindful of.

August 30, 2013 at 6:00 pm |

Yep. I find this also makes it difficult to write such questions (as in point 3): I find that I either make it too closed so that there is only one answer (as is usually done), or so open that it is barely asking anything.

After writing this, I realize that my problem might just be that I need to become comfortable with a problem that is “so open that it is barely asking anything.” That is probably actually the ideal.

September 4, 2013 at 4:22 pm |

What sort of questions do you have in mind with “so open that it is barely asking anything”? My experience has only really been with ones similar to your Joe Boaler example where there is a set of correct answers that you could easily whip up a quick program to check if any given answer is correct.

September 4, 2013 at 6:19 pm

Leave it up to Joss to force me to really think more deeply about off-hand comments I throw out. . .

I suppose I mean this: in calculus, I often ask questions like “What is the derivative of f(x)=3x^6+2x^3-7?” To trivially make this more open, I suppose you could ask: “What do you notice about the derivative of f(x)=3x^6+2x^3-7?”

I suppose that I would be afraid that a student might answer something like “It is denoted by ‘f'(x).'”

So the problem is that I have a particular thing in mind (“f'(x)=18x^5+6x^2”), and I am afraid that students will answer in a way that circumvents what I want to find out. But—now that you forced me to think about this more—I have come to realize that the problem is that I was too myopic: the point was not for me to see that the students could get “f'(x)=18x^5+6x^2,” but rather for me to figure out if the student could generally find derivatives of polynomials. So here is a better open question:

“Give an example of a function f(x) such that f'(x) is a fifth degree polynomial. Justify your work.”

This question is much more open, as there are many more correct answers. It also tests the same skills as my “f'(x)=18x^5+6x^2.” Better yet, it tests MORE skills, since a student would have to realize that they needed to start with a sixth degree polynomial.

On Wed, Sep 4, 2013 at 11:22 AM, Solvable by Radicals

September 6, 2013 at 4:15 pm

I would be very curious to see what fraction of students that are able to correctly answer

“What is the derivative of f(x)=3x^6+2x^3-7?”

would be unable to correctly answer

“Give an example of a function f(x) such that f’(x) is a fifth degree polynomial. Justify your work.”

The second one seems like it would be much more challenging for a student that is just learning these concepts.

September 6, 2013 at 4:42 pm

That’s what I would think, too. But apparently there is more pressure with the first (there is only one right answer!), and students can relax more with the second.

But I think that maybe I should study this! I will ask for IRB approval. Thanks for the research idea. Let me know if you want to collaborate on a simple study. Bret

On Fri, Sep 6, 2013 at 11:15 AM, Solvable by Radicals

September 16, 2013 at 2:20 am

Sure. Let’s talk about this offline (well, outside of blogs offline)

September 16, 2013 at 6:32 pm

Wilco.

On Sun, Sep 15, 2013 at 9:20 PM, Solvable by Radicals

April 19, 2014 at 3:18 pm |

I am curious if you guys ever did the research. I teach middle school and have struggled with the same issue of writing good questions.

April 21, 2014 at 7:08 pm |

Not yet, although I haven’t forgotten about it yet. But it is a pretty low priority for me, so it likely won’t happen for a couple of years (if at all).

But I will keep considering it, now that I know more people than just Joss and I are interested.

Bret