# 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.

# 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

# 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!

# #Edtech Professional Development – Comfort, Confidence & Relevance

I did an article several months ago about technology professional development and ready-to-use lessons being one way to support teachers implementation entitled Teachers and #Edtech – REady-to-use Lessons Can be A Support. Obviously, this is only one of many things schools and education leaders can do to help support technology implementation and ensure that the technology use is sustained over time as well as actually improves and supports student learning. I wanted to just share a couple more things that I found in my own doctoral work that education leaders need to consider PRIOR to purchase technology, as well as AFTER purchasing technology.

I have spent years researching, creating, and doing professional development, much of it specific to technology integration in mathematics, whether that be online learning, dynamic software or calculators. I have been all over the country (and now the world as well!) providing teachers and administrators with face-to-face workshops, online learning, and blended professional development to support technology integration. My doctoral work was specifically focused on technology professional development with math teachers and was a long term, embedded study over 3 years. From my own research, which supports other educational research findings on PD and technology integration, here’s is a condensed list of things education leaders can do (before, during and after professional development) that make a difference in the success and/or failure of technology integration into classroom practice:

1. Curriculum & district expectations
• Ensure technology used actually supports standards and content taught
• Make sure there is relevance of professional development content/resources to what teachers actually teach and do in the classroom
• Provide content-focused, ready-to-use activities/lessons that utilize the technology
• Set clear expectations from administration that using the technology was expected & supported
2. Teaching practices
• Professional development should emphasize using technology to teach specific content
• Professional development should provide classroom management and teaching strategies for using the technology
• Multiple teaching strategies need to be modeled in professional development (questioning, collaboration)
• Teachers are given time to collaboratively plan lessons and practice using technology with their content/classroom
3. Sustained Professional Development
• Long-term support must be provided
• There should be continued training on technology as well as content-focused implementation of technology
• Coaching, modeling, active learning should be key components of sustained professional development
• Teachers need sufficient opportunities to collaborate & time for practice and feedback
4. Internal & External Factors are accounted for and controlled
• Access to technology should actually be available(seems a no brainer?!). Technology integration won’t work if students access to the technology is limited.
• Teachers must believe students will benefit from use of technology (so PD emphasizes relevance) and be confident in their ability to use it (so sustained PD is provided and teachers are supported in many ways)
• Time is provided.  Time for teachers to learn and practice implementation, time for students to learn, and time for changes to take place BEFORE judgements/assessments are made
• Classroom structures need to support the use of the technology. So – class size, other competing technologies and/or resources are de-emphasized, support for changing classroom teaching strategies, etc. are all considered and addressed prior to and during implementation

The importance of providing teachers with resources they can use right away with students that are relevant to what they are actually teaching is so crucial, especially when a new technology is introduced. If the beginning of trying to use a new technology is filled with frustration and angst, the chance of that technology being a lasting education tool is unlikely. Comfort, confidence, and relevance make a big difference in the success of technology as a learning tool – if you provide those resources up front, and then as teachers see the benefit and get more comfortable with using the technology with students over time, you will see continued use. Follow that up with sustained support through collaborative lesson planning, coaching, online on-demand support and resources – so many possibilities, and you will see a big difference in the successful implementation of technology. ROI as they say – return on investment if you invest the resources, time, and support from the very start.

These are just a few things to keep in mind as you consider new technology for classrooms or as you re-consider how to support current technology implementation.

NOTE: A great example of this relevant, hands-on PD approach, with sustained support afterwards, is offered by Casio’s technology workshops.  At these mini-workshops you get to ‘do technology’ using content-focused, hands-on math activities that can be used immediately with students, you get the technology itself (Casio fx-9750GII) and you get on-demand, sustained support via our tutorials and ready-to-use lessons, content webinars, and guided tours. The idea here is teachers get their hands on technology and do content-specific activities that help them see the relevance of the technology to their teaching and student learning. They then have ready-to-resources to implement immediately, getting them more comfortable with the technology. And they have a place to find additional resources as they become more adept with using the technology with their students. The resources and support help the use of technology become an integral part of teaching practice.

# Teachers & #Edtech – Ready-to-Use Lessons Can Be A Support

I am a little obsessed with edtech and integrating technology into math classrooms. It’s what I have been doing for the last 16 years of my educational career, first within my own school and district, and then, throughout the country through my work with Key Curriculum, McGraw-Hill, Kendall Hunt and Casio. I read a lot about the infusion of technology in schools these days, but my reality, when I go to schools and districts throughout the country, is that the use of technology in mathematics education is actually very, very limited. There are of course countless reasons for this – a big one being funding. Most schools I work with have 1-2 computer labs that math teachers rarely get to use, or they have a laptop cart shared between 15 math teachers. They have calculators – sometime – most of which are broken, have no batteries, or they honestly don’t know how to use. There are also the instances where there is a lot of technology available, but the teachers don’t know how to use it, don’t have resources to support it or they haven’t had a chance to find a place where it would support their curriculum.

The reasons for not using technology are many. But – in my own personal research, one of the biggest deterrents to integrating technology is lack of training and support. A recent survey of teachers by Samsung shows teachers do not feel prepared to use technology in classrooms.  Not a surprise. Unfortunately, the majority of professional development is still the one-stop workshop, where new technology/apps/ are bought and teachers are trained for a few hours on the tool, with little or no emphasis on teaching with the tool, which is the most important aspect of technology integration. Technology is only a tool – and when used appropriately, can enhance and expand learning. This involves more than learning how the tool works. It involves looking at the curriculum and instructional goals, determining what tools (of which technology is only one) are going to provide the best fit, and then creating instruction that incorporates the tool as part of the learning, not as an add on, not as something extra we do after we learn.  This is what is missing most of the time – helping teachers make technology fit into their instruction as part of the learning, not as something extra.

One of the things I found in my research is that if teachers are provided with pre-made, ready-to-use lessons that can replace current lessons and use the new technology, they are more likely to start using it, especially in the beginning stages of learning. Lack of confidence is a huge reason teachers don’t use, or continue using, new technology – this is helped if they are given a push, especially in the first stages of learning, that allows them to use technology without too much stress – i.e. the lesson is ready to go, there are teacher notes/guidelines, and it FITS INTO THEIR CURRICULUM. In the Samsung survey, 80 percent of teachers said it would be helpful to have pre-existing lesson plans that help them easily integrate technology. I found this was one of the strongest indicators of continued integration of technology in my research.  It’s one of the things Key Curriculum provided for Sketchpad, it’s one of the things Casio provides for their calculators.  If teachers are given new technology and ready-to-use lessons that show them and students how to work with the technology while learning required content, they are much more likely to use the technology.  And – the more they use, they more confident they become with it, the more likely there will be continued implementation.

To go along with ready-to-use technology lessons, ones that scaffold learning for both teachers and students, is collaborative lesson planning. Teachers should have the chance to work together to plan lessons to incorporate technology. Again, in my own research, teachers expressed how the monthly collaborations with other teachers from around the district, as well as the online sharing community, really helped support their own efforts to integrate technology and gave them new ideas. Sharing ideas, planning for where technology is appropriate, learning from each other – all of this is powerful in helping teachers be more confident in using technology in instruction. There is no reason for teachers to reinvent the wheel for every lesson – if there is a premade lesson out there, or a lesson another teacher has tried, that will support others integrating technology, there should be sharing and collaboration.  Teaching is a profession, not an isolated, individual endeavor – we should be working together to improve and help students learn and help each other learn.

# Graphing Piecewise Functions – Casio Prizm vs. TI-84+ CE

In my explorations of hand-held calculators and how they can support mathematics learning, I want to continually share when I learn new things. Why calculators? Well, the obvious answer is because I am working with Casio. But the real answer is, if you actually go around the country and go into math classrooms, calculators are still the most-used and available technology to students.  I know, I know -we hear about iPads, tablets, laptops, etc. in use in classrooms, but the reality is these are NOT readily available to most students.  I think I did a post already about this (Calculators, A thing of the past?), but from my own personal experiences, teaching and working with teachers (some of these in the last couple of months), most math classrooms are still working with the following technology: one computer with projector/screen (sometimes a whiteboard, most often NOT), and then hand-held calculators.  And, unfortunately, not even enough of those for each student.

So – yes, despite the ‘edtech revolution’ we hear about in the news, in the real, every-day classroom, students are most often using calculators, and this will be the case for quite a while unless there is some funding-miracle, which, as we know, is very unlikely.  It’s a sad reality – as an edtech supporter, I would love more than anything all students to have access to technology on a regular basis that allows them to quickly research, explore, practice and visualize mathematics, whether that be via tablets or computers or calculators. But as most of us who work in/with schools know, that is NOT what’s actually happening in most math classrooms.  That said, let’s focus on the great technology that is accessible to a majority of students – and if not, should be, since it’s affordable, portable and can do much of the visualization and exploration that students should be doing in mathematics – graphing calculators.

Now another reality is that TI seems to be the go-to calculator found more often in schools, a lot of this due to brainwashing and really good marketing and the old “change is hard” mentality in education. I myself was a TI graphing calculator user the whole time I was teaching in public schools because that’s what we had. What I am now finding more and more, as I learn the Casio and compare it to the TI, is that I can remember what to do on the Casio way more so than I can on the TI.  That’s just one thing, though admittedly a pretty major thing.  And – while many of the steps for using the TI and Casio are often similar, the Casio is often quicker and more efficient than the TI, and can usually provide a visualization on one screen that helps make a connection which might otherwise be impossible to see when having to look at separate graphs (i.e. graphing  y= and r= on one graph).

My goal here is to point out places where Casio has an advantage over TI (and I am comparing the Casio Prizm and TI-84 CE, which are the graphing calculators most similar and also both are accepted on standardized tests). Obviously, my opinion is probably considered biased – though I am speaking as someone with over 26 years experience, one who has used many different technologies and only ever taught with the TI (Navigator included). I honestly find the Casio more fun and easier.  More intuitive. I just can’t remember where things are with the TI – it’s frustrating! As they say with many things – once you go Casio, you’ll never go back! But – I don’t think I would feel this way if I wasn’t constantly comparing the two side by side, something most teachers never get the chance to do.  With that said, here is another side-to-side comparison of the Casio Prizm and the TI-84 CE showing how to graph a piecewise function, something I believe Algebra II teachers are probably getting into about now, that helps illustrate my preference for the Casio over the TI.

# Finding Function Intersections – Casio fx-9860GII vs. TI-84 Plus: Casio IS Easier!

I admit to being a TI-83/84 user for all of my teaching.  Not because I had a personal preference for a TI, but because it was what all the schools where I taught in Virginia provided and recommended. It was in our books, it was what our state tests recommended, it’s what we were told to tell students to buy. Why? Was TI better than a Casio or an HP or any other brand for that matter? No. It wasn’t…and isn’t to this day. But – TI knew how to play the market and basically embedded themselves with publishers, testing companies, schools, to the point where today, schools & teachers & publishers are convinced TI is the better calculator option. Or maybe I should say they think it is the only option.

It’s not. It’s an option. But it’s not the better option in terms of cost and in terms of ease-of-use. In my 25 years of using TI, I can honestly say I still forget how to do most operations because there are so many steps involved or I can’t remember where the menu item lives. Obviously, as the Casio Brand Ambassador, I have a definite bias in my current opinion about hand-held calculators. However, it wasn’t my opinion until about 7 years ago, when Casio approached me, in my role as Director of PD/Education Outreach at Key Curriculum, to get some advice and suggestions for quality trainers who could help support Casio PD efforts. That’s when, for the first time, I actually used a Casio calculator and realized it was really easy and quick to learn – and I could remember things!!  All the Casio PD teachers,  many who were also Key trainers and who teach in classrooms all over the country to this day, were former TI users as well. But, once they tried Casio, never looked back and encourage others to make the switch as well. In my new role, I’ve been asking these teachers why, and the first answer I always get is “it’s so much easier for the kids to use and they remember how to use it”. The second answer I always get is “they are much better calculators in what they can do”.

In European countries, Casio has a much bigger share of the market than TI – they know about quality and ease of use. Here in the U.S. we are fighting the TI machine…it’s so embedded in our math culture and it’s hard to change, as are most things in education (think Common Core!) As Terry Walsh said in his NCTM session in Atlantic City, something a college professor told him, Casio calculators are much more user friendly, but TI calculators are more user familiar”. Sticking with the familiar and avoiding change is a disservice to our students, who deserve technology tools that are affordable and intuitive to use.

I thought it would be a fun idea, since Casio is always saying how much easier it is to use a Casio vs. a TI calculator, to actually show how. I picked two comparable graphing calculators: The Casio fx-9860GII SD (\$79.99) and the TI-84 Plus (\$139.00) to do a side-by-side comparison of a relatively common algebraic skill – finding the intersection of two functions. (Note: This is NOT a lesson on teaching this algebraic concept, so please, please, do not think this is how I would help students understand how to find the intersections or what those intersections represent in real-world context. There is a whole lot of discovery, hands-on learning, conversations, etc. that would occur if this were an actual lesson). What this video demonstrates is a step-by-step “how do you find the intersection of two functions using each graphing calculator” – nothing more, nothing less. The point being to actually demonstrate, talk-the-talk, walk-the-walk, and show ONE example of how a Casio graphing calculator is easier to use than a TI graphing calculator.

Watch the video: Step-by-Step Casio vs. TI Graphing Calculator