LEARNing: Linear Functions Investigation – #AskDon’tTell

I used to think I needed to carefully scaffold each algebra learning experience into a series of steps so that each learner would learn.  Now I think I should listen more and talk less.  What if I practiced inquiry and student-directed learning? What if I lead by following the learners questions? How might I set up opportunities for learners to explore and think PRIOR to my show-and-tell show?

What if I gave my learners the following TI-Nspire document and asked them to explore it for 10 minutes? What if I asked them to jot down observations, patterns, and questions  that come to them as they play with this document? What would they learn? What would they ask me? What should I be prepared to ask instead of answering their questions? Can I spend an entire learning episode answering questions with questions to facilitate student learning?

Want to explore the investigation? Here’s how:

  • Clicking on the screenshot should enable you to download the TI-Nspire document and open it if you have the TI-Nspire software on your computer.
  • Clicking on the Launch Player button should open a player file where you can interact with the document without having TI-Nspire software. (Be patient; it is a little slow to launch.)

I would love to hear your thoughts and feedback.  Also, what questions would you ask, and what questions do you hope your learners ask?



  1. My own experience is that “free orientation” (to use van Hiele’s terminology) tends to occupy kids for between 20 seconds and 2 minutes tops. It helps them a lot to be given some goal or curious thing or something to get them started… but I could well imagine that, in the right hands (i.e. yours) a class could be trained to be curious and capable of exploring.

    van Hiele posited the steps for effective learning (and moving forward to a higher cognitive state) to be in the order of

    1. Information (play with and be introduced or discover the elements of the domain under consideration)

    2. Guided Discovery (we give them some goal or direction to focus them as we desire – and I get your point about letting them lead us!!)

    3. Explicitation (terrible word but essentially putting what they are discerning into their own words, causing a conceptual “gelling” or even congealing as the ideas start to form some sort of cohesive “mega-concept” – greater than the individual pieces previously perceived)

    4. Free Orientation – play with the materials/tools/concepts in the sense of generally exploring (after having had the direction, and starting to make sense of it all) leading finally to

    5. Integration – where the discrete parts come together into a cohesive whole, building a new “network of relations” as he terms it.

    These ideas are explored in more detail in an old paper of mine…


    Interestingly, this theory – distinct from the better known van Hiele levels of thinking – was the result of Dina ben Hiele-Geldof’s doctoral work – she was the one interested in how teachers might assist students to move to higher cognitive levels, while Pierre was the one interested in the neo-Piagetian levels of thinking. Dina died, I think back in the 80s, but Pierre continued on for many years and pulled it all together into a cohesive whole – and tends to get the credit!


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