Dynamic Graphing on the fx-9750GIII – Helping Students Discover and See Relationships

We talk a lot about how learning math should NOT be focused on memorizing steps and formulas, but instead on doing and discovering the relationships and building understanding. Ideally, students should be visually seeing multiple representations of concepts (tables, graphs, words, pictures, models, manipulatives, equations, etc), comparing and analyzing these representations, and making their own conjectures and ‘rules’. Instead of telling them what they should see or know or do, they are able to figure it out themselves by focusing on relationships and connections (obviously with good structured investigations and questions and activities). This is the ideal.

Research shows that when students discover the patterns and come up with rules/formulas/relationships themselves, they are much more likely to retain the information, or more to the point, more able to ‘recreate’ the experience they had and recall and/or rebuild the information. Not possible when memorizing isolated facts or skills. It’s why just teaching struggling students rules and steps and skills and making them repeat it over and over does not improve learning…..there are no real connections being made.

Algebra is a subject that many students struggle with because of the ‘unknowns’ or variables – it is abstract, and without exploring patterns and relationships of the variables in many forms, it is hard to help students understand and make sense of things like “equations to model real-world situations”. Sadly, there is still the tendency to just teach process – i.e. follow these steps, or “use the quadratic equation to get the solutions” without even talking about what those solutions mean, where they are located, why there are two (or one or none). If we just teach skills and if students just memorize processes/steps, when confronted with something similar, but not exactly the same, they can’t do it, and give up, because they have not had experiences that help them build understanding and see relationships and connections to prior knowledge.

This leads me to today’s focus – Dynamic Graphing.

Dynamic mathematics is when something in a given representation is changing (a measure, a value, a construct), and as it changes, you see the impact of that change. As an example, in geometry, this might mean a triangles three angles are measured, and as you move one or more of the vertices, the angle measures change as you see the triangle change. If you sum the three angle measures, you get 180, and as you change the triangle dynamically, even though the triangle is now different, with different measures that are changing as the triangle changes, the sum of those angles is still 180. Which leads to a conjecture.

Dynamic graphing is the same thing – it’s using variables (instead of static numbers), in equations and changing one or more of them, and then watching what happens to the graph as that variable changes. Dynamic graphing really emphasizes what a variable means – i.e. a quantity that varies, and as it varies, the graph/tables/equations also change. As an example, in a linear equation, defined as y=Ax+b, if we make A- a dynamic variable and change it, students will see the line change it’s ‘steepness’ and direction, and make a conjecture that A- must have something to do with a line’s steepness. If we change b- dynamically, they will see the line move up and down vertically (but not change it’s direction/steepness) – so they conjecture that b- determines where the line crosses the y-axis (because they notice the line always seems to cross at the y-axis at a point that corresponds to b-). Simple things like dynamically changing and visually seeing the impact of that change can lead students to more deeply understand what those coefficients/variables in an equation mean and do, so that when you give them y-=-3x+5, they already know the direction of the line (negative), how steep it might be, and where it’s going to cross the y- axis. You didn’t have to tell them – they figured it out by observing, looking for patterns, and making conjectures and then, most importantly, having discussion with others to confirm their findings.

I wanted to share how you can do dynamic graphing on the new fx-9750GIII graphing calculator. Not a new feature – it’s possible on all the Casio graphing calculators. However, I don’t think many people realize this functionality exists, especially on a black-and-white, inexpensive (but powerful!) graphing calculator. It’s something I think many people expect only mathematics software to be able to do, so I wanted to show you this feature. (And, you can do dynamic geometry too with the Geometry Add-in Menu!)

Video: fx-9750GIII: Dynamic Graphing and Visualizing Variable Changes


Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning

Here are quick links:

Teachers Rock! Show Your Appreciation in a More Personal Way – Tell Them

It’s National Teacher Appreciation Week, for those of you not in the know. In schools everywhere, teachers are probably getting nice little ‘treats’ from parents and students, or having special lunches or breakfasts brought in, or being treated to free ice cream or nice messages or pep rally’s – lots of things to show how much everyone appreciates the work they do. Obviously these celebrations and expressions of gratitude vary around the country, but there is usually, based on my own personal experiences in middle and high school, some recognition for teachers at some point during this week.  Which is great. Teachers deserve to be told how wonderful they are and what a difference they make in students lives, because they do. They do every day, whether they or you realize it.

It’s the little things that teachers do every day, which often go unrecognized, that really make a difference in students lives and learning. That extra time put in to make a lesson really engaging, that eating in the classroom during lunch to spend time with students who just want to talk or get some help, the personal money spent on supplies and classroom decoration so all students have what they need and to make the classroom a welcoming place, the smile at the door as students enter, the late hours grading, the phone calls to parents to share good news about students (yes, teachers do that!)….there are too many to list here, but every day teachers are providing not only learning experiences, but emotional and physical experiences that help to mold and build students confidence and understanding. This is what I don’t think people who have never been teachers understand – teaching is unlike any other job. You can’t just come in, do the same thing every day, and go home at the end of the work day and forget about it. Teaching is more than teaching content. There is a lot of emotion and dealing with students on so many levels, and navigating that, along with teaching content, makes teaching one of the most difficult jobs out there.

Unlike many other jobs, teachers often never know the impact they had on their students. Sure, we can see grades and scores on tests, but that is a moment in time in a students life, and we don’t often ever know if what we did as teachers has long-term impact (which we hope) as students grow and move on. We think it did. We hope it did. But often, we never know. Unless a student comes back and visits, (or, we are now friends on FB, years later!) – we never really know if the things we thought would make a difference did in fact make a difference. Which makes teaching different from many other professions, who can usually see immediate results or impact of their job. Teaching is a profession of faith – where we believe our efforts are the best we can provide and are something powerful that contributes to our students potential future selves. And though we often never know, we do believe.

What I think would be a really powerful way to show appreciation during this week is for students, current and past, to let a teacher know what it is they are doing or have done that has an impact on them or helped them. Reach out to that Spanish teacher who made class funny, and embraced your obnoxious sarcasm, and influenced your decision to become a teacher yourself, or write that math teacher who helped you survive Calculus and helped you become an engineer, or that teacher who smiled at you every day and gave you a hug so that you loved coming to school. Get your kids to write a note to a teacher (now or in the past) that made school exciting or turned them on to reading or helped them perfect their dunking. It’s those little recognitions’, those personal recollections that really make a teacher feel appreciated and know that what they do is making a difference to someone. Those of you who have been out of school for a while, it’s pretty easy to locate a former teacher via FB or LinkedIn. Those of you still in school, write a note, even if anonymously – it will brighten that teachers day and reaffirm their commitment to teaching.

The U.S. Department of Education has shared some really great videos of teachers sharing what makes them feel appreciated, so I am providing links to those here:

  1. https://youtu.be/dLZXKu8fxnc
  2. https://youtu.be/eqi_kE31tZU

My favorite is what students say about their teachers though, so I am sharing that video here:

 

Equity, Equality, and Access to Quality Education – Part 1

Back-to-school is already upon some, and for many will be starting up in the next few weeks. With that in mind, and especially with the very public conversation around school choice and ESSA and accountability for schools, I’ve decided to do a 3-part series on equity, equality, access and quality education. These are ‘buzz’ words that are thrown about in news stories and education settings, but I think often times these words or terms are used incorrectly, or interchangeably, with many people not really understanding what is really being said or what the meaning behind these terms actually might be. With that said, this first part in my series is going to focus on defining these three terms so that we are all on the same page and have a common understanding in which to move forward.

Quality Education

This term is loaded. Everyone wants a quality education for their child and schools and states strive to provide quality education for all their students. But what does this mean? What does this look like? I am going to define it here and in later follow-up posts we will dive more deeply into this.

There are many definitions out there for what quality education means. I actually had a hard time finding an ‘official’ definition, but found the term ‘quality education’ used frequently in vision/mission statements from many education organizations and school districts. Which is interesting – we use the term, yet we don’t define it, so how are we ensuring that students are indeed getting a quality education?

Here is a definition of Quality Education from ASCD (Association of Supervisors of Curriculum Development) and EI (Education International) which I think provides a strong common understanding that will connect to equity, equality and access.

A quality education is one that focuses on the whole child—the social, emotional, mental, physical, and cognitive development of each student regardless of gender, race, ethnicity, socioeconomic status, or geographic location. It prepares the child for life, not just for testing.

A quality education provides resources and directs policy to ensure that each child enters school healthy and learns about and practices a healthy lifestyle; learns in an environment that is physically and emotionally safe for students and adults; is actively engaged in learning and is connected to the school and broader community; has access to personalized learning and is supported by qualified, caring adults; and is challenged academically and prepared for success in college or further study and for employment and participation in a global environment.

A quality education provides the outcomes needed for individuals, communities, and societies to prosper. It allows schools to align and integrate fully with their communities and access a range of services across sectors designed to support the educational development of their students.

A quality education is supported by three key pillars: ensuring access to quality teachers; providing use of quality learning tools and professional development; and the establishment of safe and supportive quality learning environments. (retrieved from http://www.huffingtonpost.com/sean-slade/what-do-we-mean-by-a-qual_b_9284130.html)

Equity and Equality

The definition of equity in the dictionary is “the state or quality of being just or fair”. The definition of equality is “the state of being equal, especially in status, rights and opportunities”. So what does this mean in terms of education, especially as these two terms are often used interchangeably, when they are very different when it comes to education? Let’s look at each separately in terms of education.

Equality in education would mean that all students are treated the same and are exposed to the same opportunities and experiences and resources. This is deemed as fair because everyone is getting the same instruction, the same assessments, the same resources, the same access to teachers. However, if students are coming into a classroom with different capabilities and different backgrounds – which is the reality no matter where you are – (this means educational knowledge, socio-economic status, family support, etc.), then treating them equally is going to disadvantage most students. No one will get what they truly need to learn – most will not get the appropriate supports and opportunities they need to be successful and to learn to their full potential (as examples, those with special needs would not get the additional supports needed and ‘gifted’ students would not be exposed to more challenging learning experiences they might need).  Everyone gets the same and so everyone suffers to some extent.

Equity in education means that all students get what they need from education, meaning instruction, assessments, resources are distributed so that every students individual needs are met in a fair way so all students can be successful. This relates to the statement above, under quality education, that students have access to personalized learning so that their educational needs are supported, allowing them to be prepared for future success, whether that be a career, college or some other aspiration. So unlike equality in education, equity in education is not the same for everyone, rather it supports everyone with what they need. A students socio-economic status, gender, race, or ability level do not prevent their access to education resources and opportunities. Equity does NOT mean equal. Equity implies an education for each child that meets their specific needs,  both pedagogically and developmentally, so they can be successful in their future endeavors no matter where they live or what their economic status might be.

Access

Access to education is closely tied to equity and equality. I almost didn’t separate it out, but I do think it is a key component behind why many students do NOT get equitable education opportunities. The goal of providing quality education to all students means we are providing them with equitable access to resources and learning opportunities – i.e. students with learning disabilities are getting the extra services and supports they need to be able to learn; students from low-income areas are getting the technology and materials and qualified teachers needed to address their instructional needs; students who excel at math or science are provided with technology and resources that allow them to explore and expand their understandings; students who are artistically or musically inclined are provided with teachers and courses that let them learn and create.

It was hard to find a ‘definition’ for access, because it’s really a process of ensuring students get what they need. I found this nice summation of access on the Glossary of Education Reform that I am going to use to inform our discussion going forward:

 “The term access typically refers to the ways in which educational institutions and policies ensure—or at least strive to ensure—that students have equal and equitable opportunities to take full advantage of their education. Increasing access generally requires schools to provide additional services or remove any actual or potential barriers that might prevent some students from equitable participation in certain courses or academic programs”.

As you can see, all these terms and ideas are related, and it is often hard to think of them in isolation. Hopefully now you have a better understanding of each, and in our follow-up posts, we will explore issues surrounding these using our common understanding.

Education Growth Mindset – So Important for Teachers and Students

I just came back from Kaiserslautern, Germany, where I was working with Department of Defense Education Activities (DoDEA) math teachers as part of the DoDEA/UT Dana Center College and Career Ready Standards Initiative. Our focus this summer, which kicks off the next year of continued support and training, was on helping teachers create a classroom culture of student discourse and a growth mindset that allows students to develop deeper mathematical understanding and become problem-solvers and confident mathematicians. It was a fabulous two days, and the teachers, some who had never explored this idea of ‘growth mindset’, really had some powerful conversations around this idea of providing students productive struggle opportunities and helping them develop this sense that they can solve problems, and they can improve mathematically, and they can learn. It was rather eye opening for many.  How many of us educators have come across those students who give up without even trying because they think they can’t do it? Or they have been so ingrained in the idea that they are ‘bad at math’, so they don’t even try? That’s what this idea is about.

Carol Dweck is a leader is this field of Growth Mindset, and how to motivate and help support this idea of a growth mindset. In fact, the teachers I worked with as part of our workshop, read an article by Dweck that provided some insight into what we as both teachers and parents, inadvertently sometimes do that prevents students/children from having a growth mindset. Something as simple as the way we praise can actually interfere with this growth mindset. More here.

Many of you may be unfamiliar with what a growth mindset is, so I found a great TedTalk from Carol Dweck that explains the idea behind it. As educators, this is something to really think about because we want to develop in our students the willingness to persevere and solve problems that may seem difficult.

 

Rethinking Summer School – Equity & Promoting Student Learning

Summer school – I know that it conjures up bad thoughts in most of our minds. Having to go to summer school usually means you failed a course or a grade and you have to make it up.  But – do only the ‘failures’ or the ‘bad kids’ need to go to summer school? Is that what summer school is for? This is what most of us think of when we consider summer school, when in reality, summer school should be a place where all students could go to keep on track, get ahead, or learn some new things. Research shows that the 3-month summer break is often a huge learning set-back for many students, particularly minority students and students living in poverty, causing a widening of the achievement gap, in part because these students are often denied opportunities for summer ‘enrichment’ courses or camps. Summer school options are usually focused on remediation and failures, and not very enticing for students to attend voluntarily, and so we have most students taking a 3 month break from any learning. But what if we approached summer school differently? What if it weren’t a punishment, but rather a place where students were motivated by other students or college student mentors and were engaged in new and interesting topics that kept them learning?

I found this really motivating TedTalk by Karim Abouelnaga, who from his own experiences with school, decided to try to change the way we rethink summer school. It’s not too late, even for this year, for those of you educators out there getting ready for this years summer school to consider making some changes that would make summer school a learning opportunity for all students.