# The Power of Visualization – Modifying Graphs with a Graphing Calculator

I have had some great discussions with teachers in my courses lately about the power of providing opportunities for students to see and manipulate mathematics as a way to test out their ideas, play with patterns, and develop their own rules and understandings. Visualization, manipulation, experimenting – all contribute to students developing deeper understanding and their own ‘algorithms’, and because of these contextual experiences, they are much more likely to recall how to do a math process than if they were just given the rules/algorithm to memorize.

In a recent final reflection, one teacher wrote, “As a high school teacher, I have always stayed away from using manipulates for fear they were “too elementary” for my classroom.”  This attitude – that older students don’t need those physical objects or need to see – that they just need to  memorize rules and practice – is sadly still prevalent today. Which is frightening really. I experienced these same attitudes and beliefs over 2 decades ago when I was teaching in  middle and high school, and bringing out my two-colored chips, algebra tiles, and Sketchpad. Allowing students to play with math, to use physical objects, and virtual objects, to represent the math and then be able to manipulate change and see what happens was always considered ‘babying’ them. Clearly that attitude is still going strong today, since as you read above,  I hear it in the courses I teach with current classroom math teachers. This despite even more tools being available to provide a way for students to experiment, play, discover, create and find the mathematical patterns and rules themselves. The tendency to just give them the rules and the process and the definitions and have them memorize and regurgitate is still very much a part of our mathematical education. What we really want to do is provide multiple ways to look at and explore math concepts, so that when students ‘forget’, they have that experience where they built the understanding to recall where they can rebuild it again. Much easier to recall something they saw or something they physically moved and connected to than an isolated, memorized fact.

In most typical high school classrooms I visit and work with these days, it is rare to find physical manipulatives (more often in Geometry, but much more rare in an Algebra 2 or Pre-calculus class for example). But – there is almost always a technology tool – whether that be the teachers projector attached to the internet, or students on tablets/laptops, or more often the case, graphing calculators of some sort. Which means there is no excuse NOT to be providing students the opportunity to visually see the mathematics, and to manipulate and explore to come up with those algorithms they are often asked to just memorize. Meaning: use the technology for more than checking answers!  Use it to help students find the patterns and connections and create their own algorithms and definitions, use it to delve deeper into the math, to gain insight, to test out conjectures and really get a sense of what all those numbers and variables mean and how they interact with each other to change the shape of a graph and what that might mean in a application of that math in the real world. Use the tools to manipulate and see the math; technology allows for students to test a conjecture quickly, make predictions and check if they are right, and explore very large and very small numbers, etc.

As an example of this, I am going to use the Graphing Calculator App (for mobile devices), since I haven’t previously used this before in any of my videos, to show the power of visualization and technology to make conjectures and immediately test them with modifying features/dynamic math capability. You can do this on our hand-held Prizm series graphing calculators  (handhelds and emulators).

Additional Note: Try our FREE new dynamic math software that is web-based – perfect for tablets, PC’s, mobile devices: ClassPad.net

# Learning from Webinars

Online learning takes many forms, of which webinars are one. For teachers, webinars are nice because they are usually content or instructional strategy focused and they are relatively short in length, so you can fit them into your busy schedule. Webinars are often live – meaning happening in real time, and you register and sign on at the designated date & time and can interact with the presenter (usually via a chat forum).  This is nice because if you have questions, you can ask them right away. But – the disadvantage here is you have to have the time to sit in on the webinar, which is often not the case, considering our crowded school and personal schedules. If you can participate ‘live’, I highly recommend it. When I worked at Key Curriculum and hosted our weekly webinars, I know the live interaction was a very positive aspect of the learning.

Let’s face it – the reality is it is not always easy to get to a scheduled ‘webinar’, even if it is the most interesting topic in the world. Which is where the beauty of technology helps out – because most webinars are recorded and archived for on-demand viewing (much like our on-demand television binge watching craze!). Most education companies or organizations that host webinars will have them archived somewhere because they WANT you to log on and watch – it’s good for business. There are a couple of good sites listed below that have some educational archived webinars that might spark your interest:

There are more out there, but this should be a great start. And – not to let Casio be outdone, we have many archived webinars as well that focus on integrating Casio technology and math content. These are free and accessible on our Casio YouTube site  We have short how-to videos on this site as well, so the way to determine a longer, content focused (and/or technology focused) video is to look at the time stamps – those that are 20 minutes or longer tend to be the webinars. I’ve included one below on Proportional Reasoning, since this is such a huge issue with students of all ages, and the presenter, Jennifer N. Morris is one of my favorite people and math teachers. Enjoy!!

# Free Online Casio How-to’s & Content Focused lessons – Great Personal Learning Resources

I am clearly on a ‘what should you do with your summer’ kick, if you look at my previous two posts. But – my belief is that summer, while a time for fun and relaxation, is also a time to brush up on some skills you may be lacking or things you want to learn, find new ideas for the classroom….basically, use the time to foster your own personal learning.

This learning doesn’t need to be expensive, it doesn’t need to be long – it’s all about improving skills or learning new ones. With that said, I thought I would remind all of you, whether you are a teacher, a student, or a parent of a student – if you want to brush up on your Casio calculator skills, we have a lot of free online tutorials and how-to’s that might fit the bill. In my interactions with teachers, I am often asked if we have ‘tutorials’ so that the teachers can support all those students coming to class with Casio calculators (because they are more affordable and much more intuitive to use).  The answer is yes!

In this post I am just going to share some links to our free online resources, and highlight a couple of the videos here as well.

Casio Education has a Youtube Channel where we post previous webinars (so these are longer and actual ‘lessons’), shorter how-to’s, and some quick reference videos and overviews as well. Here’s the link to the Casio YouTube Channel.

A couple highlights here:

Here is an example of a short look at the fx-9860 Stat menu:

Here is a much longer lesson with the Casio Prizm on families of functions:

There are also Prizm specific guided tours at this link.

And I have my personal YouTube channel where I do comparisons and how-to’s on the different calculators, so there might be something of interest there as well.

Here’s a quick how-to using the fx-991 Scientific Calculator to solve systems of equations (and use the QR code):

So – if you have a spare 10 minutes or a spare hour, there’s something for you and we will continually add to these so come back often!

# Math Fun in San Antonio – NCTM 2017 Annual

Next week is the NCTM Annual Math Conference in San Antonio, TX.  It’s a great time to go to Texas, as the weather hasn’t gotten too hot. I remember the last time NCTM was in San Antonio, when I was still teaching high school, and met up with all my teacher friends. We had such a great time, not only going to different workshops at the conference, but exploring the area (a trip to the Alamo was a must) and eating and shopping along the River Walk. I am going this year as part of the education team for Casio, where there will be a lot of fun to be had at our exhibit booth and sponsored events.  I have been going to NCTM Annuals for over 23 years (what?????), and as usual, I am looking forward to reconnecting with math educators and friends from all over the country, many of whom I only get to see this one time a year. So, it’s more than just a place to learn new ideas and collaborate with like-minded educators, it’s a time to renew friendships and share memories. I certainly am hoping to catch up with as many folks as I can, even if just to share a cup of coffee or a hug as we pass in the conference hall.

Naturally, the goal of attending a conference is to learn new things to bring back to your classroom or to the educators you work with. It’s one of the aspects of these conferences I love the most – the ‘renewed’ energy and excitement that occurs when you see a strategy that you want to take back to your class or you learn a new approach to a familiar concept that you know will resonate with your students, or you find that perfect resource for an upcoming unit. I always consider these conferences as a way to reaffirm why we teach math – seeing what others are doing, sharing stories and ideas, and leaving with at least one or two ideas that are going to spark your students creativity and understanding. For me personally, I always had a key focus (say Algebra, or Geometry or technology or manipulatives) to narrow down the workshops I went to, with the goal to find a few new resources, ideas and strategies to incorporate into my teaching over the summer so that next years classes would be even better. This type of focus helped to make ‘teaching’  a new adventure every year, even if I was teaching the same subjects, and it also made sure that as a teacher, I was always challenging myself to be better and find relevant strategies and multiple ways to help my students learn.

One aspect that I always look for is technology applications and resources. I am a firm believer in the idea that technology, whether it be a calculator, a tablet, a computer, a video, can be a valuable resource to help students both learn and develop mathematical understanding, but more importantly to visualize abstract concepts and explore ‘what if’s’.  I am sure there are many of you out there as well looking for some technology workshops as you attend NCTM this year, so I wanted to share some workshops from some of the amazing teachers that work with Casio, as these are always such great hands-on experiences.

Workshops:

• Thursday, April 6 – 9:30 – 10:30, Room 213AB Conv Center: Exhibitor’s Workshop What’s New At Casio: Viewing Mathematics through a New Prizm (or Two)
• Thursday, April 6, 3:15 – 4:30, Room 217C Conv. Center: Polar, Parametric, Rectangular Graphs – Really See the Connections! with DeeDee Henderson
• Friday, April 7, 11 – 12:00, Room Presido ABC (Grand Hyatt): Conceptualizing Polynomials with Jennifer N. Morris
• Friday, April 7, 1:30 – 2:45, Room 224 Conv. Center: Conics – The Ugly Duckling of Algebra 2 with Denise Young & Tracey Zak Johnson.
• Friday, April 7, 2:o0 – 3:00, Room 008AB Conv. Center: The Probabilities and Mathematics of “Wheel of Fortune” with Mike Reiners
• Saturday, April 8, 8:00 – 9:15, Room 006D Conv. Center: Hands-on Activities + Technology = Mathematical Understanding through Authentic Modeling with Tom Beatini

We will also be having a fun time at the booth, Thursday – Saturday, playing games, having give-aways, talking and doing mathematics with our hand-held technology, so be sure to stop by and say hi (Booth #631) and come play with math. I will be there most of the time and hope to meet some new math educators and give a hug to old friends!!

# Online Training for Casio Prizm – Get Free Emulator Software!

One of the most frequently asked questions I remember at the NCTM Regionals and NCTM Annual Conventions from teachers was “do you have any training to help me learn to use the the Casio calculators because I have so many students in my math classroom using them and I want to be able to support them?”  The short answer is yes, we do!

There are a couple of free options available on-demand now.  One is the Quick-Start guide that will help both teachers and students navigate and learn some of the basic functionality. We have quick-start guides for several of our calculators which you can find at this link: Quick-Start Calculator Guides. The other option, specific to the Prizm, is a free, self-paced, online course for teachers that let’s you learn about the calculator and specific features by working through modules. The great thing about this course, besides the fact that it’s free, is when you complete it you get the emulator software free for use on your computer for use with your classroom instruction. Can’t beat that deal!  There are of course several other free resources, but these two are a terrific way to get started.

Additionally, we will be offering some upcoming, regionally based workshops where you spend a few hours doing content-specific math activities while familiarizing yourself with the calculator. You leave with a free calculator and read-to-use lessons. All for only \$35, so that’s a pretty amazing deal as well, plus you get some professional development hours and collaboration with other mathematics educators.  Stay tuned for those.

Just like with students, starting the new year for teachers means learning new things and finding supports for your own learning and teaching. Take advantage of both free and inexpensive ways to develop new skills and get some great hardware & software to enhance your classroom instruction. Don’t forget we have a Youtube channel of free videos as well.

# Test-Driving Classroom Technology

I am a huge proponent of using technology in the classroom, particularly in mathematics. If technology is used appropriately, it can do several things: provide visuals for concepts that are often hard for students to grasp; allow for students to explore and test conjectures; provide opportunities to go beyond basic understandings and get into deeper meaning and more complex structures; provide multiple ways to practice and learn; and obviously, foster engagement. These are just some of the benefits.  There is of course a downside to technology – lack of training for teachers often leads to using technology just as a digital replacement of paper and pencil – an electronic worksheet for example – in which case, what’s the point?  That’s NOT a great use. Or using technology when it is NOT the best option or doesn’t really enhance/support the learning goals. Or using the wrong technology. Just because you have technology doesn’t mean it’s a good fit for your learning goals/standards.

In my own research, and in my personal travels around the country working in math classrooms and with math teachers, there is a wide variety of technology available, and more often than not, this technology is NOT being used to enhance and expand learning. Often times this is because technology has been purchased with no real effort to match it to learning goals or standards, and little or no training or support for how to use the technology in the classroom has been provided. Teachers are frustrated, students are frustrated, and the technology becomes just another ‘add on’ versus a true learning tool.

What is often missing in a technology implementation is the most crucial step – planning. With planning, technology aligned to objectives becomes a focus for purchasing technology that actual supports learning goals and needs. Seems obvious – but, having been an administrator, I know that often times ‘funds’ for technology are released and must be spent quickly (i.e. for me, I was told we must put our orders in by this week or we lose the funding), so often times technology is purchased that sounds good, or looks good, but may in fact not be a good fit.

Ideally, technology should support learning goals, which can only happen if you sit down with your subject leaders/teachers, look at your standards and learning goals, and then analyze the various technology options and determine which ones support those goals. And, if possible, test these technologies out BEFORE purchasing, to ensure they do indeed support learning.  This also allows you to plan for training needs of the teachers, infrastructure, curriculum and standards alignment, etc. These are important steps – often left out of the technology implementation process – and often the reason why much of the technology in school is misused and unused.  I bet if you looked around your school you would find a lot of ‘great technology’ gathering dust.

You don’t buy a new car without a test drive, so why buy technology without one? Especially when making large school/district purchases.

Here’s the link to CASIO’s loaner program – check it out and go for a drive! http://www.casioeducation.com/benefits/loaner

# Calculators – Back to School Choices

Hard to believe that summer is already over – for some students anyway.  Depends of course where you live – here in PA students don’t go back until the end of August, in VA, it’s after Labor Day, and in CA, my sisters kids are already on their 4th day of the new school year.

As a teacher and as a parent, I know the beginning of the school year involves a lot of money and making choices – new clothes, bookbags, pens, pencils, papers, notebooks, calculators.  Those long “school supply lists” seemed to get longer every year.  In part because school funding has decreased, so teachers have less to work with and count on the parents to help support classroom needs. Though, as most of you know, teachers spend a lot of their own personal money on their classrooms and students too.  Something to remember when you question the box of tissues, the dry-erase markers, and other seemingly ‘unnecessary for my child’ supplies.

These days, some of the needed supplies come with a hefty price-tag – laptops and tablets for example. Depends again on what the school/school district supplies and if your child’s school even uses such technology.  BYOD (bring your own device) schools expect you to provide these, and those that do provide the devices often require a hefty ‘rental’ fee – for obvious reasons such as damage, repairs, upkeep, etc.  But – these more expensive items are still not the norm in most schools – again, because of funding, lack of internet, lack of training on how to use these devices appropriately, etc. In my research, only about 25% of schools are using laptops/tablets on a daily basis, with most students only getting access to these a few times a month (shared laptop/tablet carts or a computer lab).

One technology tool that is still prevalent and requested, more so in middle and high school, is the calculator. Schools use to supply these, and some still do, but the cost to maintain and replace broken or lost calculators is difficult when school funding is so drastically reduced. Calculators are often added to the school supply list, like in my sisters case, where she was asked to buy two graphing calculators for her two high school students. Sometimes they request a specific brand – for no other reason than its what the teacher/school is familiar with, not considering price at all. But price matters, especially when functionality is the same and often times better. Scientific calculators have similar price tags, no matter the brand, but graphing calculators have a huge range of pricing and options – color, non-color…what to choose? Color graphing calculators in particular can be expensive – \$150 for the ‘familiar’ model. But you don’t always need to get the most expensive, ‘familiar’, requested model just because it’s on the list.  Get the one that’s going to support your child’s math learning.

Obviously I am going to promote Casio calculators here, since I am a Brand Ambassador for them.  But, as a math teacher for 25 years, I am also promoting them because they are truly a better calculator and more affordable, so why wouldn’t you make that choice? If your child needs to purchase a calculator, then just go into a store, like Walmart or Target or online – and compare. In the Scientific models, pricing is similar, so how do you choose? Well- you go with functionality and Casio is easier to use and, as in the case of the fx-300Es vs the TI-30xIIs, the Casio makes fractions look like actual fractions, lets you see tables when entering data – just a few of the things it does better. Same in elementary calculators – the fx-55plus is far superior than any TI calculator and fractions look like fractions! (i.e Natural display).

Graphing calculators are trickier – there are color options, non-color options. In all cases, Casio is much more affordable than TI. Do you need a color graphing calculator is probably the real question. For more advanced mathematics courses, the color graphing calculator is the better choice for several reasons, for example they tend to have more functionality and color displays allow for easy comparison when looking at several functions on one graph. The Casio Prizm is significantly more affordable than either of TI’s color options, and as we showed at NCTM, a previous blog, and in many comparison videos, the Prizm outperforms TI.

In most instances, particularly middle school, color is not needed, in which case you can get a Casio Graphing calculator like the 9750GII or the 9860GII for 1/3 to 1/2 the price of a T. Even without color, both of these calculators outperform and are easier to use than the TI models, including color (see videos again!) And, if you only want to purchase one graphing calculator for your child to last all through high school, both of these will get them through the highest levels of math. I guess my thoughts are why pay more for a name when you can get a better calculator at a significantly more affordable price? And if the teachers or schools say it must be a specific model, that’s only because it’s what they know – it’s NOT because it’s a better product. And in this day and age, where we are all watching our money and have to make choices, I say go with the more affordable and efficient functioning option.

As you go shopping for your back to school supplies, just remember you have choices. So choose what works best for your budget, your child, and don’t forget to pick up some extra tissues and dry-erase markers for the classroom – they are definitely needed!!