Remembering the Reason for Memorial Day

With Memorial Day Weeekend upon us, I thought it appropriate to do a little research and posting about what Memorial Day actually represents, something I believe many of us forget amidst the weekend barbecues, beach outings, and other ‘holiday’ celebrations. Being the math person that I am, I wanted to focus a bit on “numbers” surrounding Memorial Day. Which, in retrospect, is very sad and alarming. But – it makes you appreciate our military even more and the sacrifice they and their families make to protect our U.S. Freedoms and allow us to celebrate this weekend.

I found this infographic by John S. Kerman at WalletHub that pretty much did all the work for me, so I am including the link and infographic here. Some of the numbers are fun facts and some are startling numbers that hopefully will make you stop, think, and remember those who have given their lives. When you eat that hot dog, or enjoy a sleep in on Monday because you have the day off, remember the 1.3 million service men/women who have given their lives for that right. And don’t forget at 3 pm to honor them as part of the National Moment of Remembrance.


Memorial Day 2016 By The Numbers

Source: WalletHub

Multiple Entry Points and Rich Math Tasks

I was reading this article the other day about how a strength-based approach to learning math (and learning in general) redefines who is “smart” and allows all students to succeed. In her article, Katrina Schwartz has some quotes and reflections from former students who learned math using the complex instruction method, and who were all successful.  They talked about how “math class made them feel safe, heard and able to express their ideas without fear”.  Wow – how often do you hear something like that?!!

Complex instruction is based on the idea that learning is collaborative, where students are learning using rich tasks with multiple entry points and pathways, and each student has a role and accountability. This isimage15 not a post about complex instruction however. What I was thinking about while reading the article was in fact about the Common Core Standards for Mathematical Practice and how they support what the students were expressing in the article – that math class “was a process and it required other people. It wasn’t just you and your work and not talking.”

If you actually read the 8 mathematical practices, you will notice over and over again words like communicate, justify, analyze, plan, make sense, look for entry points, reason, ask questions, make viable arguments, apply.  The practices are all about communicating and talking and finding multiple entry ways to solve problems. And working and talking with others to get there. Like complex instruction, these problems should be rich, where in fact, there are multiple entry points and possible solution pathways. Where each students strengths can support the process and help build the understanding of others. Learning is collaborative, NOT an isolating experience that a worksheet or a lecture so often create.

The Common Core gets a bad rap because so many publishers and testing companies have standardized it – by providing ‘common core problems/strategies’ that are in fact limiting and narrowly focused so they can be graded easily. When I see parents and students and teachers complaining about “common core problems”, I get so angry because what I am actually seeing are ‘forced entry points’ – meaning, rather than allowing all students to approach a problem from their mathematical strength and understanding, they are forced to choose between 1, 2 maybe 3, ways to solve a problem, which may NOT be understandable methods for them.  Therefore, NOT Common Core (or Complex Instruction). As it says in the practices: “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution”. It does not say mathematically proficient students are given specific entry points to solve a problem.

image16My hope is that with the changes in standardized testing on the horizon under the Every Student Succeeds Act, that math teachers and classrooms can truly begin to focus on students strengths and learning, not preparing for a test. To actually provide learning experiences focused on allowing students to work from their strengths. But it requires a willingness to have a noisy classroom with students talking and collaborating.  It requires  rich mathematical tasks, not standardized worksheets and drill and practice,  that truly provide multiple entry points. This in turn requires teachers who are willing to accept multiple solutions from students rather than the traditional one-way, algorithmic approach we tend to focus on. And it requires support for teachers – in resources, training, time, and expectations.

Read the article by Katrina Schwartz – it also has links to information about Complex Instruction and great feedback from San Francisco Unified School District who has made a concerted effort to teach mathematics this way. Read the Common Core Standards for Mathematical Practice. If you are not already doing so, try to incorporate some of these practices into your math instruction. It’s not what you teach, but HOW you teach, that has an impact on students.  Every student can learn math – it’s up to you to create a culture that helps them believe that.

Casio vs. TI – Finding Max/Min Points of Functions

DispCap1In this weeks’ comparison of the Casio vs. TI calculators, I demonstrate how to find a max/min of a given function, using the Casio Prizm and the TI-84+ CE. In my example I use a cubic function because it allowed me to show both a maximum point and a minimum point on the curve. Why might students be asked to find a max/min point of a function you ask?  Well, besides the obvious ‘on a standardized test’ question, what we really want students to be able to do and understand is what the max/min points mean in the context of the problem/situation. In a real-world application, how does that max/min point help us understand what is going on in the problem? The short answer is it provides specific points where ‘something’ has happened, and finding these points provides insight, allowing students to ask different questions or analyze the situation.

Here is a common example: A quadratic function might be used to model the path of a ball as it is thrown or hit, with x representing time and y representing its’ height.  So the max point in this case would be the maximum height of the ball at a given point in time before it begins its descent back to earth. We want students to be able to find that max/min point, in context, so they can answer questions or make conjectures about the ball. For instance, in this ball example, is it possible to change the angle the ball is thrown/hit to increase the max height, but keep the time the same? Being able to quickly find these max/min points so that interesting questions and conjectures can be made and students can apply mathematics in challenging and deeper ways is one benefit of using technology, such as graphing calculators. The max/min points can be a starting point for deeper exploration.

Below is a quick video on how to find a max/min point of a function (using a cubic as the example, since it has an example of both a maximum and minimum point).

More than Calculators – Teacher Support & Resources

I received a message the other day from a reader who commented on how much he liked the Prizm, but because 2016-05-13_12-52-22Casio didn’t have any resources to support the learning of the Prizm, he was a little reluctant to try it.  My first reaction was “What?!! We have a TON of resources!!”  My second reaction was to ask myself why might he think this? I was able to answer my own question when I searched for our resources – the issue being they are a bit hidden among all of Casio’s other products, (which, just so you know, is of course in the process of changing as we create a more user-friendly web-page).

In the meantime, I want you to see the great teacher/student resources we have! Let me share with you the resources we have that supports teachers (and students), from complete subject-specific or grade-specific resource books (i.e. complete lessons), so sample lessons and activities (free), to online course for Prizm (free) to webinars (free).  There are teacher-created resources and quick-start guides.  Casio WANTS teachers and students to use their calculators and get the help and support they need to use them appropriately.

  1. Free online activities and sample questions:
    • These include grade-level activities and specific Casio Prizm-vs-TI 84 activities
    • Scrolling down the page you will find sports activities for use with five different calculators
    • Keep scrolling to our Quick Start Guides for 6 of our calculators (including Prizm)
    • Keep scrolling to Subject-specific Teacher Resource Guides and Calculator Tips
    • Scroll further to see all our grade-level and subject-level resource books that contain complete lessons
  2. If you look at our products page, under Software & Additional Products, you will be able to scroll through all our grade-specific/subject-specific resource books:
  3. Here’s a short-link to our Casio Lesson Library (with teacher created activities):
  4. Short-link to Guided tours for the Prizm:
  5. 2016-05-13_12-56-04If you are interested in the Prizm, we have a whole webpage dedicated to Prizm activities and support, which includes lessons, videos, and also has the OS updates. 
  6. We have a free online course for the Prizm (self-paced).  If you complete the course, you get the Prizm (fx-CG) emulator software for free.
  7. Free webinars on many math topics (statistics, geometry, algebra, calculus, etc.)(you do have to register your email to view these, but they are free):
  8. Links to manuals for specific calculators:
  9. And let’s not forget the videos showing you how-to’s and comparisons!  and

As you can see, we have a ton of support for teachers and students wanting to use and learn-to-use Casio calculators to support their instruction and/or math learning. We hope those of you out there excited to start working with Casio calculators start using these supports. We are educators here at Casio and want you to love the calculators as much as we do!!

Plotting Vertical Lines – Casio Prizm vs TI-84

To play off of the #NCTMannual Calculator Face-off challenge, I am going to try to post a weekly comparison of Casio calculators compared to TI calculators. My thoughts on this, as expressed in my recent post, Casio vs. TI – Calculator Face-off NCTM and Beyond, are that too many teachers and schools are stuck in the mind-set that TI is the ONLY option for calculators out there.  This is clearly not the case, and also, clearly NOT the best choice in calculators when you take into consideration functionality and affordability, especially to address technology equity issues that are such a hugeconcern in schools.  Casio is simply more affordable, easier to use, and has better functionality.

With that in mind, I want to keep it up-front and center so that those of you out there in the market for calculators can actually see side-by-side comparisons and make educated decisions vs. the “it’s what we always do” type of decisions.

Side Note: We are getting some “feedback” from our TI-users, who are not happy about these comparisons – probably because they are new to being challenged. Granted, when comparing calculators from any vendor, there are going to be certain concepts or procedures that might be faster and and maybe more efficient when on one compared to the other – no calculator will always be the winner -it’s bound to happen. But – overall, and consistently, based on teacher and student feedback and our own personal experiences with both,  Casio is superior in it’s functionality, is definitely more affordable, and is easier for students to use because it is much more intuitive and there is less hunt-and-find-the-right-menu going on. It’s right there on the screen with a Casio, not hidden in apps or ‘math’ or ‘test’ buttons.  If we consider why we use technology in the mathematics classroom – it’s NOT to ‘get the answer’ (or shouldn’t be!), it’s about using technology appropriately to allow students to be more efficient in their process so they can explore, discover , make conjectures, test hypothesis, and make comparisons. Getting bogged down in “where do I find this?” is not conducive to productive student learning and exploration.

There are several Casio vs. TI comparisons videos already – all of them currently Prizm vs. TI-84.  I am adding today’s video, where I compare how to plot a vertical line on the Prizm vs. the TI-84. You will note the process is similar, with a significant difference being Casio creates a true plot of a vertical line, where you can find actual points on the plot, and TI is only a drawing. It is not possible to use the “drawing” to find specific coordinate points on the plot accurately or efficiently.

Here’s this weeks video comparison.  Stay tuned for future comparisons with the Prizm and other Casio Calculators.


“I Hate Math” – Let’s Change the Dialogue

I saw an article last week, entitled “Stop telling kids you are bad at math.  You are spreading math anxiety ‘like a virus “ .  It was all about how our culture basically perpetuates math anxiety in our children because we make it “okay” to be “bad” at math.  It’s not stigmatized like being bad at reading is, and therefore everyone feels okay admitting that math was a weakness or they don’t like math.I highly recommend you read the article). This is of course nothing new – this has been going on for probably as long as we have had schools. There have been other recent articles and videos specifically about this myth that some people are just not good at math – Jo Boalers’ work comes to mind.

So, I am not going to belabor the point. What I think would be more helpful is to offer some suggestions, specifically to parents, on how NOT to perpetuating this idea that’s it’s okay to be bad at math or fostering in your children math anxiety because you yourself might not feel confident about your own math abilities.

If I look back to when I was a child, my father was an architectural engineer and a mother who was a typical woman of her generation – high school graduate and then stay-at-home mom to five children. You would think that when we struggled in mathematics, dad would have been our go-to help, but believe me, if you ever ask an architectural engineer for help with anything, you get a history lesson on the origins of the problem and a diagram of the situation and two hours later there is still no real help! No, mom was our go-to support.  And what I remember more than anything, especially as we got into harder subjects, was my mom telling us, “well, you smart enough to figure it out.  Read about it, and if that doesn’t help, ask your teacher for help”.  Never did she say “I’m not good at math” or “math is hard, I never understood it”.  Even though she clearly did not understand or know how to help us, it was NOT because she wasn’t good at math (or whatever the subject), it was because, as she would say, she hadn’t learned it herself so let’s go figure it out together and if we can’t, then let’s get help from someone who does know, like your teacher.  Obviously now, students have the ability to use the internet to get help – there are a million tutorials and help sites out there.  But in my day, that was not an option, so we read the textooks, asked the teachers, and asked friends or siblings for help.

The key here: never once were the words “I was not good at math” ever uttered by my mom.  And it was definitely NOT okay to use that as an excuse if the math was “hard” – it was instilled in us that we could do anything with hard work, research and support when needed.  It was never okay to say “I am just not good at math because neither is mom”. Instead we were always given the confidence to try, to find the solutions, to ask for help, to struggle and maybe fail. As long as we had tried, researched, and put in the effort, that was what mattered, and that was what would help us learn, even when things got tough. Because we could do anything we put our mind to as long as we worked at it and asked for help and didn’t give up or give in to excuses.  And – I have to say, all five of us, while very different, all have succeeded in our lives and have strong math skills we use in our careers – a physicians-assistant, a math educator, an engineer, a carpenter, and a priest

So – for those parents out there who look at their children’s math work and say to themselves, “I have no idea how to help you” or “I’ve never seen this before – I use to do it differently” or “OH NO!!! I can’t help you”, here are some suggestions from my mother (yes, I called her up!).  She admits feeling and thinking these thoughts many times, but what she said she always wanted to instill in all of us was the idea that school was a place to improve yourself and get better at something.

1) Never ever, ever say “I’m not good at math (or whatever subject really)” or “Oh, that’s okay, I was never good at math (or any subject) either….” Or “I hate math”.

2) Do say, “I’ve haven’t seen this in a while.  Can you show me in your notes, or textbook, or ebook, what you’ve been doing so I can refresh my memory?”  Or “can you explain to me what you’ve been doing in class/school?”  If you get students to talk about what they think they have been learning, that often helps them recall how to do something, and it will perhaps remind you of some things as well.

3) Take time to actually read through the textbook, ebook, notes, etc. to see if that helps jog your memory.  If you are at a complete loss, then together, do some research on the internet.  There are tons of free help videos or games or tutorials for math (and other subjects).

4) Try to work through the problem (s) with your child, asking them to explain in words what they think they should do.  Often times students can articulate out loud what they have to do or are thinking, but have a difficult time putting it down on paper, so help them organize their thinking, one step at a time.

5) Let’s say you and they are still stuck – that’s okay.

  • Write out what you have tried, even if it doesn’t match what the textbook says they should do or what the teacher wants them to do. A lot of times students know how to solve a problem but are being forced to solve it a specific way that doesn’t coincide with their thought process. Have them write out what they do understand or their thought process.  Being able to explain how they understood a problem, how they approached their solution, is part of the learning process.
  • A good teacher is going to look at what they have done and know they put in the effort. If they showed their thought process, then the teacher can see where they might have gotten confused and support them to the next step.

6) If you and they are still frustrated, ask for help.

  • Ask for time with the teacher to help understand.
  • Ask for resources – many curriculums have supports for parents the teacher can send
  • Ask for supporting materials – a lot of textbooks/ebooks have online support and games that can help.

7)  Never give up – never give in to frustration in front of your child, because this just exacerbates math anxiety.  Instead, work as far as you can and then say “hey – we’ve reached a point where I think we might need some help.  Let’s ask your teacher. We are not giving up, but rather we are stuck and need a bit of help to get us to the next step.

The idea here is to instill the idea that math is doable, that learning math might require some effort and research, and may even require some outside help, but you can do it.  You can learn math.  It’s okay to struggle, because eventually you will get it.  The hardest part is to not fall back on the acceptable “math is hard, so it’s okay not to be good” or it’s okay to say “I hate math”.   Let’s change the dialogue.