Thursday, March 10, 2016

Using Desmos to self-check

I have had many students use desmos to self-check their work.  One way in which they do this (and I stress that they do this AFTER they work out a mathematical thought) is to type each equation into desmos and check to see if the solution is the same each time.  For example:

I'm not overly fond of this method because I want students to be able to develop their own self-checking abilities and not rely on someone or something to judge their work; however, I can see some value in this method providing scaffolding as students gain confidence in their own abilities.

Another way in which students use desmos to self-check is by graphing complicated equations to see if they match.  For the same reasons, I see drawbacks, but some value in this as well.  I think there is value in helping them find connections between graphs and equations.  The example below shows how one student explored completing the square.

This led the student to wonder why the graphs didn't match.  Eventually they figured out that they need a coefficient of one on the quadratic term before they can complete the square. Link

How do you use desmos?

Monday, February 15, 2016

Desmos - Tech Tool Requirement for Every Math Teacher and Student

Throw away your graphing calculators!  Desmos offers a free graphing calculator that is very easy to use and they regularly add new features.  In terms of ease of use and features, there isn't another calculator that compares.

I discovered the graphing calculator shortly after it was launched in 2013.  I eagerly showed friends, family, and colleagues what could be done with it.  Here is the first graph I created with it (and 2 derivations):


Can you generate the graph I made?  Post in the comments if you can.

Desmos has a library of activities for teachers or you can create your own for your students here.  My personal favorite are the Marbleslides activities.  My students have explored these for hours of fun.

In subsequent posts I will highlight a few of the ways in which I use Desmos with my students.

Thursday, February 4, 2016

Tech Tool Criteria

Andrew Stadel poses the question: "What's your [tech tool] criteria?"

I am excited to see what the 21st century holds for the way in which teaching and learning will change.  It's hard to say what that might look like in 50 years from now, but I believe that replacing the thoughtful, empathetic, wise, intuitive teacher will be difficult, if not impossible to do.  What excites me is the ability of technology to:
  1. Automate non-thinking tasks/processes.
  2. Organize and inform teachers of student's thinking processes.
  3. Deliver, organize, and adapt learning opportunities in interesting ways.
  4. Utilize the teacher more effectively and efficiently.
The first 3 can basically be summarized by #4.  Utilizing the teacher more effectively and efficiently is the ultimate tech-tool guide for me.  What that looks like will vary by what is meant by "effectively" and "efficiently".  Teaching and learning philosophy form the crux of any tech-tool.

If you believe that students absorb information through rote practice, then the purpose of the technology becomes a tool to automate an endless supply of multiple choice, auto-graded problems.  This does indeed allow the teacher to be more effective and efficient--according to that theory of teaching and learning.

Contrast that with the adult learners I worked with this past week.  In particular, I worked with one woman in her 50's and one woman in her 20's on a similar topic--developing number sense through estimation.  It was fascinating to see how difficult it was for both of these women to push past old misunderstandings and create new understandings.  The brain is amazing.  Both of these women would begin to grasp the concept, but would almost immediately revert back to past misconceptions as they compared what they were now beginning to understand to what they previously thought was true.  They were perplexed, or, to put it in education lingo, they found themselves in disequilibrium.  The teacher is essential in this process of helping a student find disequilibrium.  Tech-Tool Requirement #1:  Help students find disequilibrium.

I have often thought that the best type of learning and teaching occurs in one-on-one situations.  But at the school district scale, this seems impossible.  However, the computer may make this a more likely scenario.  One of the challenges in teaching is finding the students who are in disequilibrium to help guide the delicate transition to understanding.  Tech-Tool Requirement #2: Help teachers find students in disequilibrium.

There is great value in the struggle to learn, but unproductive struggle is not desirable.

One of the most frustrating aspects of teaching is the desire to reach every student, but the lack of time and focused attention to do so.  Students enter the classroom with a broad range of skills, misconceptions, and experiences.  Tech-Tool Requirement #3: Help teachers address the individual needs of each student.

This could be done by organizing an effort to quickly display student thinking for the teacher to analyze--the most difficult thing to teach a computer to do and, therefore, the most important role of the teacher.  Are there several students with nearly the same misconception?  Imagine having access to that kind of information.  I think there is value in having such a diverse student-body, but the ability to target individual needs, thereby allowing the teacher to be more effective and efficient, is what I believe will be one of the most important roles in technology-assisted learning.

The potential ability of technology to automate non-thinking tasks/processes has enormous benefits in freeing up the teacher to focus on more important tasks.  As teaching and learning philosophies are improved and refined, perhaps some of the things teachers worry about will become irrelevant.  Paper and pencil, in my opinion, are not very intuitive and technology has the potential to provide more effective ways of communication between student and teacher.  What I mean by this can be described by the frustration I feel as I struggle to communicate ideas to another person and can't find simple, intuitive tools to get the job done.

I often find mathematics difficult to communicate through paper/pencil and especially tech-tools because they are too clunky, counter-intuitive, complicated and time consuming.  But I am excited to see what the future holds because I believe that the poor quality of many current tools is due to the foundation of philosophy.  Tech-Tool Requirement #4: Improve and simplify the process for displaying, discussing, and rethinking ideas.

I envision some kind of digital discussion/thinking/communication board that extends to infinity in all directions.  Though some tools currently have something similar to this, in my opinion what they lack is simplicity, ease of use, and intuitive design.  

What do you think?  What are your tech-tool requirements?

This is my response to Andrew Stadel's blog so be sure to check that out here: Divisible by 3 [Andrew Stadel]: Tech Tool Criteria