# 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):

• 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.

# Start the Year with Math Tool Explorations

In my last post, I talked about creating a classroom culture that promotes classroom discourse. The time to begin this is the start of the school year, when students are getting to know each other, getting to know you, and getting use to the ‘learning’ environment. My suggestion was to start these first couple of weeks with activities that are not necessarily content related, but more ‘interest’ related, because students are much more comfortable talking to others, i.e. strangers, about summer movies or vacations than they are about math. There’s no worry about not knowing something or saying the wrong thing – remember the focus in these first two days is to help students learn to communicate with each other, work together, and understand that your classroom is about learning together and everyone has a voice. Those first few times of group work or collaboration, keep it simple – remember the focus is on creating a ‘safe’ place for discussion – listening and sharing with others.  After they’ve had some time, then you begin to introduce content to the mix.

One of the things I liked to do during those first few days of school as part of this classroom culture, was to introduce the technology and other tools we would be using during the year. This allowed students to get familiar with the tools in a non-threatening, simple math focused way, ask questions, get help, and basically become comfortable using the tools. It provided a lot of engagement and interaction as well.

For example, since I used Sketchpad weekly, I made sure that first week of school classes were in the computer lab a couple times and playing around with Sketchpad (I did not have laptops or iPads…just a lab that I signed up for 2 times a week).  It was very informal these first few times, focused on getting familiar with the program and the tools of the software so that when we did eventually use it as part of our content, they’d have had their playtime already and we could focus on using it to learn math. Same thing with protractors, compass, rulers, tangrams, calculators and whatever other tools I might be using for the subjects I was teaching (usually Geometry & Algebra). We would have “explore time”, where students would be given some simple math objects or activities – i.e. angles, sides, shapes, treasure hunts, and work together to use the tools to measure angles, sides, etc. They had to confer with each other if they got different answers, help each other if they weren’t sure how to read/use a tool, and I would also do some review of how the tools were used as well (particularly the protractor).

For the calculators, they would have a scavenger hunt – things like find the on button, graph a line, solve an equation, etc. and they would have a quick-start guide to help them. Even if students had their own calculators that were different, the scavenger hunt still worked and helped everyone learn to use their own calculator.  Just letting them explore on their own and teach each other helped foster collaboration, brought out some class leadership skills, and let provided some valuable time learning tools that saved time in the long run when those tools would be needed for more difficult learning of math concepts.

Today’s classrooms I realize might be a bit more diverse with technology tools – especially if students are bringing their own devices.  But – it’s still important to provide time for students to explore and play with tools. Providing some focused math explorations (i.e. measuring objects or creating a scavenger hunt), even if students have different devices, still allows for them to get comfortable with tools they will be using for learning. It also

allows you to understand the different devices and tools students have and get a handle on where you might need to make some adjustments.

It’s pretty easy to create simple math tool explorations – think of the tool, what will students need to use the tool for, and then create a couple of ‘to-do’ activities. Provide Cheat-sheets for the more advanced tools (i.e. calculators or Sketchpad software for example) that students can reference, but really rely on students figuring it out on their own or working together to figure out how to use a tool. That’s part of the challenge, and part of the classroom culture creation – depending on each other when you are stuck or want to verify your solution.

Examples:

Protractor:

• Create a sheet with several different angles and have students measure
• Give them some specific angle measures that they then need to create the angles
• Give them shapes (polygons) and ask them to measure the angles

• Have them do basic constructions – i.e. draw a circle, make a segment, construct a line, make two lines meet at a 90 degree angle, construct a triangle
• Measure things they constructed
• Play with all the tools and describe what each one does

Calculators (graphing for this example) (provide a reference/cheat sheet, for example: http://www.casioeducation.com/resource/pdfs/PRIZM_quick_start_guide.pdf) (Some very basic sample questions below just to give an idea)

• Where is the on button (describe location)
• How do you turn off the calculator?
• What happens when you push the Graph Menu?
• How do you enter a y= equation?
• Enter 3 and 3/4 into your calculator – change it to a decimal – how did you do it?