“Figuring It Out” Is Where the Learning Happens

 I often get asked the question when doing webinars, workshops, or professional development training, “is there a quick way to do that” with Sketchpad, TinkerPlots or Fathom? For example, last week we had a webinar on functions, and the question came up about whether Sketchpad had a quick way to plot inequalities, since it is relatively easy to plot an equation using Sketchpad’s Graph menu. The short answer is no – there is not a quick way for inequalities. It involves using mathematical properties to construct the shaded regions of inequalities. Can it be done? Absolutely. But you have to figure it out using mathematical properties and some advanced features of Sketchpad.

Another example is constructing things such as regular polygons. There’s a quick way to construct general polygons, using the polygon tool, but to construct regular polygons, it requires knowing mathematical properties and using those to construct the regular polygons you desire. Figuring it out becomes the challenge. Using the math to create more math – that’s where learning happens.

Constructing a Square

As an example of this, every year when I was teaching geometry, my students were given the quadrilateral project with Sketchpad, instead of a test, where they were required to construct all the special quadrilaterals and make custom tools. It required knowing their properties and figuring out how to use those properties and many of the menus, such as Construct or Transform, to construct and create each custom quadrilateral tool. It required figuring it out. Every year I was blown away by their creativity and the math my students knew and demonstrated.

Similar questions come up about TinkerPlots and Fathom. Why can’t you just tell TinkerPlots to make a bar graph? How do you quickly find the equation for the line of best fit in Fathom? While these things can be done, a quick way is NOT the best way. Figuring it out is where the learning occurs. Dragging and moving data to discover relationships and maybe determine that a bar graph is, in fact, NOT the best graph, is way more powerful than clicking a button and making a bar graph.

Back to my beginning example about plotting inequalities. In the webinar evaluation, it was suggested that the next version of Sketchpad have that option – a quick way to plot inequalities.  My personal answer is no. Figure it out. Use math. The power of Sketchpad, and all our software is that they are NOT an app or program where you click a button and an answer appears – they are dynamic and require the use of mathematics to make something happen or answer a question. That’s learning. That’s building understanding.

Dynamic Graphs

Granted, for all three of our software products, there are plethora of activities out there where someone has created a “prepared” sketch that can do these advanced things with the push of a button. But someone used math to create those features.  They started with a blank screen and used the power of dynamic mathematics and mathematical properties to figure things out. Shouldn’t that be a goal for students as well?

I am not against prepared sketches and documents – they are vital when just starting out and helping teachers and students start using technology in a meaningful, content-focused way. I use them all the time in trainings as they are powerful, content-related activities to help teachers integrate dynamic mathematics. It’s a great idea to start learning with some of our prepared sketches and documents, especially as a beginner. What I want to emphasize is try to move beyond those to constructing mathematics on your own. Don’t always look for the quick way or button to make it easy. Take  learning math and using technology further than a magic box with all the answers at the click of  a button.  Don’t be afraid of the blank screen and using math to create more math. Challenge yourself and your students to ‘figure it out’ – you will be amazed at the learning that happens.

6 thoughts on ““Figuring It Out” Is Where the Learning Happens”

  1. “Tell me, and I will forget. Show me, and I may remember. Involve me, and I will understand.” – attributed variously to Confucius and Sun-Tzu

    “Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.” – Polya

    One problem with having too much prebuilt and “quick/easy” is that you reduce the level of “involve” and increase the level of “show” or “tell”. That’s the caution in the first quote.

    The second quote reminds us why struggle (especially within the context of project-based learning) is useful. Something is only a dead end when you are too focused on a predetermined goal with a predetermined time limit (e.g., checking off another tick on the CCSS standards list). In the context of messy real-world goals, some (not all) dead ends lead to real breakthroughs.

    Part of what a good teacher does is encourage students to engage potential dead ends to transform their thinking – and redirect them when there’s a risk of getting stuck too deeply in a rut. Sketchpad, Fathom, and TinkerPlots support this kind of focused exploration much better than automated tutotial multiple-choice programs – and better than if it had a plethora of prebuilt tools for complex situations.

    Thanks for a good article about learning and building understanding.

  2. I have been looking over the Common Core curricula and trying to figure out how we’re going to move as a geometry staff towards the transformational approach. On our (teacher) end of things, this requires a very different approach to thinking about how translations, reflections, rotations, and dilations are used. One nice thing I like about examining someone else’s prepared sketch is to unhide everything, see how the construction idea works, then adapt that to a new situation or tool I want to create. The movement command has really opened up a lot of avenues for how to make the learning experience more alive.

    When I first started GSP it was a great tool for me so students wouldn’t have to read my handwriting all the time, but it was very static. As I’ve constructed models to illustrate to our department about how CC fits with the technology we have available, it has really made me think about the proof process, how this conclusion is based upon a previous conclusion, etc. I’m not where I want to be, but think I’m/we are moving in the right direction.

    1. Glenn,

      The transformational content in the CCSS is definitely a concern, especially as it’s been pushed down to middle school and focuses on rigid transformations, not something typically taught. Not sure if you caught our webinar this week that was actually on this very topic, but might be a great one to check out the archive for – it has some good ideas and activities. http://www.keycurriculum.com/get-to-the-core%E2%80%94constructions-and-transformations-with-sketchpad

      I love the unhide option – I do it all the time myself to figure out how they did something. Sometimes it’s beyond me, but I have definitely learned some new skills by looking at the hidden elements in a premade sketch.

      Good luck with your journey – I think you are right – we are moving in the right direction. I just hope others who are not quite as willing to explore and learn and adapt get on board or we will end up staying where we are.

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