- start with good questions
- use vertical non-permanent surfaces
- use visible random groups of three

The first problem I tried this with was actually one Peter Liljedahl had shared with us at the Ontario Mathematics Coordinators Association's conference.

**Using the numbers 1 to 10, and the operations +, –, ✕, ÷ plus another one of these operations, create five number sentences that have the following solutions:**

**17 2 21 3 2**

(For what it is worth, this question is a lot easier to understand when the instructions are given orally!)

The students got stuck into this problem immediately. Being free to move around the room I was able to listen to some impressive number sense and logic. I'd go so far as to say that the students were more accomplished at this than adults who I have given the task to.2 2 2 2 9

and later

10 14 1 20 16

As students quickly solved this, I then asked them to consider a 10-by-10 array with the top square missing and, from there, to consider the general case:

Many groups looked at of squares of different sizes and noticed that these were one less than a perfect square. It was a nice opportunity for me to introduce sigma notation:

One group gave me math bumps though as they saw the number of squares of different sizes in a different way which connected beautifully to the difference of squares:

*They are not sitting passively copying a note, they are actually being mathematicians.*It is something I have seen before in a Grade 9 class (this post) and I also saw recently in a Grade 12 calculus class. Students were asked to sketch a graph of any function they wanted and to then use what they know of the derivative to sketch the graph of the derivative (without resorting to deriving by first principles). Students were first sorted visibly into random groups by being given a card and then having to find the corresponding representations:

They then got stuck into the task:

and as they did so, they began to challenge themselves more:

Remember, the students had to choose their own graphs for this activity. The discussions that took place were a joy to behold. Independently, they began to hypothesize that the derivative of the graph would of a degree one less than the original graph. They carefully considered the key points of the original graph (e.g. turning points) and used these clues to plot where the corresponding values would be on the graph of the derivative. As they worked through this, it was clear to me that they were developing a solid understanding of the connections between the graph and its derivative: I am not sure this understanding would have the same clarity if they were to jump into deriving through first principles. The next day they consolidated their learning and their teacher, Leanne Oliver, sent me these photos:

The upshot of all of this is that these three components of creating a thinking classroom are having a real impact on students' learning:

- they are doing more math
- they are taking more risks
- they are developing collaborative skills
- and they are enjoying it!

Our challenge now is to make sure that all students experience the power of a thinking classroom.

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