# The Power of Math Exploration

If I had a dollar for every time I hear “I would do more hands-on, inquiry, problem-solving, collaborative learning, in math class if I ________________________ (insert any one of the following):

• had more time
• didn’t have as many students
• didn’t have to get through the ‘curriculum’
• had students who would actually talk
• if I didn’t have to make sure they were ready for the test
• if I didn’t have to review all the things they didn’t learn from last year…..
• ….the list goes on…….

I would be a very wealthy woman. What is mind boggling to me is there is so much research out there that shows students do better when they learn for understanding and not for memorization, which means learning through context, through inquiry, through problem-solving, through struggle. Time is one of the biggest ‘road-blocks’ teachers throw out there, and granted, there definitely is a time crunch to get all the content in before those dreaded assessments. What I try so hard to get across to the teachers I work with, is that you can  save time by taking time – you actually can ‘cover’ more ground by teaching from a more contextual, experiential, problem-solving way. As students make connections and problem-solve, they are able to learn more efficiently and more than one concept at a time because they are working from a connected-math view point instead of the single-skill/concept at a time approach we traditionally provide.

An example from Geometry: (this is using Classpad.net, free math software)

Concept – identifying polygons, and then what’s the difference between congruent-sided polygons versus regular polygons (identifying what a regular polygon is).

Activity: Using the drawing tool, have students draw examples of 3-side, 4-sided, 5-sided (and more….) polygons.  At least 2 of each kind that look ‘different’. Can be convex or concave

• Have students compare their shapes noting similarities and differences and coming up with definitions – attaching specific words to their definitions like convex, concave, closed, etc.
• Now have students use the arrow tool, and select one of their triangles, and the Adjustment menu to make all sides congruent. Then, choose a second triangle and Adjustment and make the shape a ‘regular’ polygon. What do they notice? Have them measure sides and angles and compare to others.
• Do the same for two different 4-sided figures (so Adjust congruent, then adjust regular), the 5-sided, etc.  Each time compare the two on their paper, and then compare to others, and try to come up with what the difference is between congruent-sided polygons and regular-polygons.
• Come to group consensus, and by the end of class students have manipulated, explored, collaborated and defined several things: polygons, convex polygons vs. concave, triangle, quadrilateral, pentagon,….regular polygon, congruent sides, etc.

An example from Algebra: (this is using CG50 Graphing Calculator (CG10 is similar):

Concept: Parent Function and Vertex From of a Parabola

Activity: Students graph the parent function of a Parabola (y=x^2) and then graph another in standard form using variables for coefficients.

• Have students use the modify feature of the graphing calculator to animate the different coefficients (one at a time)
• Observe what changes in that coefficient does to the parabola by comparing the modified to the parent
• Make conjectures and compare with other students till consensus is reached.
• Do this with all the coefficients.
• Have students then test out their conjectures by providing them several equations of different parabolas and, based on their conjectures, determine the shape, direction and location of the parabola BEFORE they do anything, and then test their guesses by entering in the calculator.
• Time saver: Doing this activity with linear equations first will then give students a general understanding of transformations of functions which they then extend and solidify with quadratics, which then can be easily extended into other equations, like the absolute value function. Time saver!

Obviously I am using technology here, because technology allows for conjectures to be made and tested very quickly. But technology is just a tool that is appropriate in some instances, but there’s so much that can be done without technology as well. You can make math much more of an exploration just through your own questioning (i.e. why do you think? can you explain that more? Are there other ways to do this?) and by providing students a chance to puzzle things out on their own, ask questions, use tools (so objects, paper, pencil, etc).

One of my favorite things to do is to provide them with a situation that has lots of information, but no question (basically, find a rich math task, but don’t give students the question(s)). Students then write down all the things they notice, such as quantities, relationships, etc. and then come up with their own wondering’s and questions. Then you let them choose a path they want to explore (this works well with small groups or partners). Usually it ends up that there are several different questions and solutions generated and explored using the same information. When students then share their findings, you find that there is a lot of math going on, which leads to some really interesting class discussions – some you yourself might not have thought of. You can then maybe even give them the question that might have been given in the problem – by that time students may have already explored it and if not, by now they have a real sense of what information in the problem will help them and they are more willing to actually solve the problem.

The key here – students only become problem-solvers if they are given the opportunities to explore math, make their own connections, and collaborate with others to verify their thinking. The more you give them opportunities and provide tools and resources and challenging problems, the more efficient they become at using math, connecting math concepts, and viewing math as a connected whole instead of isolated skills and facts. Take the time….it’ll come back in the end.