Let’s Explore with Geometry and Start the School Year Off Right! (New Features with ClassPad.net)

I admit it. I am a geometry nut.  It is my favorite subject to teach, which I have been doing for the past 30 years (wow….said that out loud!!). Geometry to me is all about logic and connections and relationships of shapes. It should be hands-on, it should be visual, and with technology, is should be dynamic – meaning you can see and discover relationships through movement and manipulation. There are many good resources out there (for those of you looking for a ‘textbook’, Discovering Geometry has always been my go to – it’s all about learning geometry through hands-on discovery and connections. It’s on it’s 5th edition, and the ebook has dynamic investigation using ClassPad.net (formerly used Geogebra), and ClassPad.net has made huge strides in advancing it’s geometry functionality, which is what this post is focused on. My goal over the next few posts is to focus on specific geometry explorations using some of ClassPad.net’s geometry functionality, but today’s post is an overview of what’s new.

ClassPad.net has all the tools you would expect a geometry software to have – i.e. points, straight-edge tools, polygon tools, display tools, expressions, equations, etc. It has some others don’t have – i.e. tools for conics for example. Below is a list of some of the added features as we continue to improve the functionality of the software (which is FREE, btw!!)

Quick List of New Functionality:

  1. Compass Tool
  2. Ability to add in images and use them as part of your geometry explorations
  3. Ability to create sliders for transformations (dilations, rotations, translations, reflections)
  4. Trace feature
  5. Multiple Grids, including isometric
  6. Ability to lock constructs
  7. Ability to create a rigid polygon (meaning it won’t change shape once constructed)
  8. Ability to add tick marks to sides and angles
  9. Ability to change the style of points – i.e. dot, square, x
  10. Ability to measure exterior angles explicitly and create angles 0-360
  11. Ability to construct a specific regular polygon (n-gon) by constructing one side and choosing n (number of sides)
  12. Ability to duplicate constructs without have to ‘reconstruct’ them.

I will be creating videos on each of these features and how to use them for future postings, but today, I wanted to show you where you can find the different new features. Be sure to visit ClassPad.net and sign up for an account (so you can save any work you do). Both the Free and Basic accounts are completely free and have everything you could need for a classroom (don’t forget there is calculations, graphing, statistics, financial tools, and text as well as geometry!). Below is a quick how-to on finding where all the new features for geometry are – stay tuned for future how-to’s on using the specific features. Meanwhile, why not try and explore things on your own? Have fun!!

 

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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:

 

Slow at Math ≠ Bad at Math

*Note: This is a recycled post from my personal blog.

“Speed ISN’T important in math. What is important is to deeply understand mathematical ideas and connections. Whether you are fast or slow isn’t really relevant.” – Laurent Schwartz, mathematician

If you haven’t seen the video by Jo Boaler and some of her Stanford students entitled “How to Learn Math: Four Key Messages”, you definitely need to. Besides the four powerful messages (which I will list below), it has some great stories and quotes, one of which is the one I have above.  Jo Boaler has done powerful research and written some terrific books on mathematics and learning math (one of my favorites being “What’s Math Got to Do with It?” and the video about these four key messages in math is so interesting.

Here are the four key messages about learning math (I highly recommend you watch the video to clarify and define each message a bit more):

  1. Everyone can learn math at high levels
  2. Believe in yourself (your beliefs about your abilities actually changes the way your brain learns)
  3. Struggle and mistakes are really important in learning math
  4. Speed is NOT important
All of these speak directly to the way we still, sadly, often teach and learn mathematics. One that really struck out for me was #4, speed is not important. I remember my own daughters struggling with the timed math tests – i.e. you have a minute to try and solve 100 times tables, or complete as many addition problems as possible. Very stressful, very ridiculous, and to top it off, they were penalized with poor grades if they couldn’t reach the arbitrary goal of “x amount of problems in 1 minute”. It still goes on and students memorize and stress over these timed math drills. Why? It’s ridiculous. If we continue to do this to students, then they begin to believe they are bad at math (see #2 above), which leads to them thinking they can’t learn math (see #1), and therefore leads to them giving up when problems get tough (see #3). A self-fulfilling prophecy.
So – I ask those math teachers out there who continue to put pressure on students to perform mathematical skills in a timed matter, where speed is important – stop. Just stop. Focus on what mathematics should be – understanding why those calculations matter, what they are related to, how they help us solve real-world problems. Help students make connections.
I know I keep coming back to it – but the Common Core Mathematical Practices seem to embody these four key messages. No where in there does it say students have to be able to do ___calculations in _____ minutes. Math is NOT about speed – it’s about the struggle, perseverance, conjectures, connections, and applications that help students solve relevant, real-world problems and see the beauty and need for mathematics.
Check out the video here

Applying Prior Knowledge Is About Precise Mathematical Language

In the course I am currently teaching at Drexel University, we have been focused on the importance of addition and multiplication properties as students progress through mathematics. Particularly the idea of inverse numbers (additive inverses and multiplicative inverses),the additive and multiplicative identities, and the commutative and associative properties of addition and multiplication. A strong foundation in these concepts, which starts in elementary school and builds as students progress to more abstract and complex math concepts such as proportional reasoning, solving equations, composition of functions, and working with matrices, is really important. In fact, if we spent more time using precise language and justifying our reasoning with properties consistently, as we model and help students learn and discover, there would be a lot less confusion and much more connection of prior knowledge to ‘new’ concepts. Instead, we often provide a short-cut, or a ‘trick’ (with a cute acronym like KSP (keep, switch, flip) or ‘Cross-multiply-divide’ with no basis in the true mathematics. Students focus on memorizing isolated rules versus connecting new concepts and seeing learning as just an extension of prior knowledge.

Let me try to explain what I mean by providing a sense of prior knowledge and how it connects to more abstract concepts:

Prior knowledge:

  • Additive identity: 0 and the Additive Identity Property:  a + 0 = a (5th/6th grade)
  • Additive inverses create the additive identity – so -b + b = 0 or -c + c = 0 (5th/6th grade)
  • Multiplicative identity: 1 and the Multiplicative Identity Property: b*1=b or 1*b=b (5th/6th grade)
  • Multiplicative inverses create the multiplicative identity (i.e. a 1) = d * 1/d = 1 or -1/f *-f = 1 (5th/6th grade)
  • Addition and multiplication are commutative (switch the order and you get the same solution) (1st/2nd grade) and associative (switch the grouping and you get the same solution) (3rd grade)(this explains why we want to change subtraction to addition of the additive inverse number, and why we change division to multiplication of the multiplicative inverse (reciprocal) – so we can USE THE PROPERTIES!!!

Understanding the above, then makes solving equations easier – and we don’t need to avoid equations with fractions or decimals, because the properties apply to these rational numbers as well.

Example:  -5 = (1/3) x – 8

  1. Change the problem to addition of the additive inverse: -5 = (1/3)x + (-8)
  2. Add 8 to both sides (commutative property – can add in any order) because adding additive inverses (8 and -8) make zero (additive identity property)  -5 + 8 = (1/3)x + (-8) + 8
  3.  Group the inverses (associative property) and solve:  -5 + 8 = (1/3)x + (-8 + 8) which is equivalent to 3 = (1/3)x + 0  equivalent to 3 = (1/3)x
  4. Use the multiplicative inverse property (multiplying by the reciprocal will create a 1) and multiply by 3/1 on both sides:  (3/1)*3 = (3/1) *(1/3)x (commutative property allows us to multiply in either order on both sides).
  5. The multiplicative identity property says 1 times any number is itself, so we end up with 9/1 = 1 *x or 9 = x

*Note – we did not use subtraction or division at all – we used the understanding of inverses, identities, and addition/multiplication properties to explain. No tricks, and working with actual numbers (so fractions and integers) with justification for all steps.

Example: Solve the proportion  3/16 = x/20

  1. This is really an equation where the quantity x is being multiplied by 1/20. Understanding that I can use the multiplicative inverse to multiply by the reciprocal to make a 1, I multiply both sides by 20/1:
    • 20/1 * 11/12 = x/20 * 20/1 (commutative property lets me multiply in either order on both sides)
    • I can even decompose my multiplication and think about making ones through the same understanding: 4*5*3 /4*4= 1*x
    • 15/4 = x/1 or x = 3.75 (multiplicative identity)
  2. Note – the trick we often tell students to memorize is ‘cross-multiply and divide’, but if instead we focused on just applying their understanding of multiplicative inverse and making those 1 pairs, there would be less confusion, less forgetting the ‘trick’, and less applying that trick to other problems where it is in appropriate. 

Obviously I can’t demonstrate a whole course of study in one blog post – what I am really emphasizing here is how important consistent mathematical vocabulary and use of properties is, instead of acronyms, short-cuts, tricks, mnemonics, etc. that we often give students with no basis in understanding. Instead of seeing math as a connected whole, building on to prior knowledge as they move through the grades and topics, we treat it as isolated topics with no connection. It’s no wonder students think every year they are learning something new. If last year when they worked with division of fractions their teacher taught them to “Keep, Change, Flip”, and this year the teacher is talking about Ketchup Covers Fries or KSF….no wonder they are confused. None of these are grounded in the properties and vocabulary of mathematics.

What we should be doing instead is focus on applying properties and using the mathematical language/vocabulary/properties right from the very beginning and ALL THE TIME. So instead of disconnected acronyms of KSF or KCF,  they focus on extending their understanding of additive inverse, inverse operations with the inverse number and division of fractions ends up being just an extension of what they did with subtraction of integers – i.e. use your inverse operation with the inverse number. So dividing with rational numbers is just multiplication (inverse operation) by the multiplicative inverse (i.e. reciprocal), similar to subtraction being addition (inverse operations) with the additive inverse (opposite signed number) – same general idea, same vocabulary, and just building on prior knowledge.

Let’s stop dumbing down mathematics and use the words and properties that truly allow students to connect and look for those patterns and develop their own understandings and rules. Let’s get away from tricks and mnemonics as our ‘teaching’ method – instead, let students figure that out themselves through the use of precise math language and application of properties. Let’s start in elementary school. Use precise mathematical language (along with clarifying words of course, but always with (not instead of) proper mathematical language/vocabulary/properties).

Think about it – we wouldn’t change the Spanish word for grandmother (abuela) or the French word for bread (pain) to other words, because then how would we communicate and be understood by others speaking those languages? Why is it okay to change the words or use different words or tricks, instead of the using the math language and properties? No wonder students are often so confused or why teachers think they have to ‘reteach’ things every year – if we are not consistent with students in using mathematical language, we are in fact talking a different language to them. No wonder they so often seem lost and frustrated.

New Year’s Observations: Supporting Educational Change & Teachers

I read an article the other day in Edweek about a recent study of teachers regarding the many educational reforms/changes they have seen and been asked to implement in the last couple of years. The article, Majority of Teachers Say Reforms Have Been Too Much” by Leana Loewis, reports on results from a survey done by the Edweek Research Center. I won’t repeat all the findings, as you can read the article and look at the results yourself, but the gist is there have been a crazy amount of education reforms teachers have been asked to make, from standards, to pedagogy, to assessment, to evaluation, and, frankly, it’s exhausting and they are getting tired. And often these changes happen all at once with results expected immediately. A quote from the article that says it all: “Teachers are incredible. They keep up with it because they have to.”  But – at some point, somethings gotta give. In large part, what teachers need is time and support, and this made me think back to something I wrote in my personal blog about change and how educational leaders can support these teachers who are struggling with so many reforms. I’d like to share my 3 suggestions for supporting teachers and change/reform as we begin this New Year.

Observation 1: CHANGE IS EMOTIONAL – change is hard NOT because we don’t want to change (often assumed of teachers who resist change), but because there is often a lot of emotion behind the change. Teachers may want to embrace new curriculum, or learn new roles and new skills, however…they may have LOVED what they used do use or do still want to do that – and it’s emotionally wrenching to have that taken away or altered. In a sense, teachers may be mourning for what is gone and nostalgic about how perfect it was (which it most likely wasn’t). There may be an emotional road block to educational reforms…one that can be overcome, but it will definitely take time, support, and understanding from leaders, students, parents and other teachers, as well as commitment on a teachers part to persevere.  So leaders – remember this about your teachers when it comes to implementing new educational reforms- it may be an emotional reason vs. fear of new or different resources/strategies. Try to address the emotion and provide relevance and reasoning for change and time and support.

Observation 2: RESISTANCE/RELUCTANCE TO CHANGE IS MULTIDIMENSIONAL – It’s easy to tell someone that if they learn a new skill or strategy, that things will be fine or be better. But learning that new skill/strategy or knowledge might not be the true road block – it could be that they don’t understand the relevancy to what they do, or they have preconceived notions or beliefs that cause resistance, or they are missing some necessary background experience/knowledge.What matters here is again, time to learn, but more importantly, dissemination of background, relevance, and connection to what they do and how these new or different skills/resources/strategies will make things better. Without a reason, a purpose, a connection, learning the how-to won’t ever change the internal beliefs and therefore never change behavior in a lasting, effective way.

Observation 3: SOME CHANGES MAY NOT BE FOR EVERYONE – it’s hard to accept, but not everyone can, will, or needs to change, whether that be a skill, strategy, or knowledge base.  What is important is to understand this, try to provide all the time, information, and support to push change along, but in the end, accept that some folks are not going to change and be prepared to deal with it. Whether this means encouraging them to find another place that fits their needs and interests, providing alternatives or simply accepting status quo, forcing those who are not ready, willing or able to change does NOT lead to success.

In education, we tend to introduce education changes, with little training and little time and expect miraculous results quickly. Real change, with long-term benefits is not quick – so let’s take this new year to really look at what we are expecting from our education reforms and assessing whether we have provided that time, addressed those emotional needs, provided reasoning and support. If you want success, you have to work at it.

Equation App (Pt 2 in series) – Solving Equations – Why Use a Calculator?

Solving equations is a large part of the mathematics curriculum as students move into those upper-level concepts. If we look at the Common Core Standards, students start solving one-step equations for one variable in grade 6, adding on to the complexity as they move into higher mathematics where they have multiple variables and simultaneous equations and complex functions. It is important to help students understand what solving equations really represents – i.e. determining the values of unknown quantities and to help them solve them in a variety of ways (i.e. graphically, using a table, using symbolic manipulation, and yes….using technology such as a graphing calculator). And connecting those unknown quantities to real-world contexts is a big part of this as well. Students should solve in multiple ways and express their solutions in multiple ways so that they really understand the inter-connectedness of the multiple representations (graphs, tables, symbolic) and what all these quantities mean in context.

That said, many teachers are reluctant to use the equation solver that is often part of a graphing calculator because, as I have heard multiple times, it does the work for the students and just gives them the answer. True. But – there are ways to utilize the equation solver so that it supports the learning, not just ‘gives the solution’. The obvious way, and probably the most frequent way, is to have students solve the equation (s) by hand, showing all their inverse operations/work, maybe even sketching a graph of the solutions, and then using the graphing calculator to check their solution. Very valid way for students to both do the work, show their steps, and verify their solutions. But – the reverse is also a great way to try to help students learn HOW to solve equations. Working backwards, so to speak.

By this, I mean, use the equation solver to give students the answer first, and then see if they can figure out how to use symbolic manipulation and inverse operations to reach that outcome. As an example, start with a simple linear equation, such as 2x – 5 = 31. Have students plug this into the equation solver and get the solution of 18. Then, in pairs or small groups, have students look at the original problem and try to figure out how they can manipulate the coefficients and constants using inverse operations to get to that solution of 18. So maybe, plug the 18 in for the x.  What would they have to do to the other numbers in order to isolate that 18?  This forces students to use inverse operations to try to ‘undo’ the problem and end up with 18. In doing so, they are discovering the idea that to isolate a variable, you have to undo all the things that happened to it.  Give them a harder problem. Same process….and let them get to a point where they try to solve using their ‘understanding’ of inverse, and then they use the calculator to ‘check’.  The idea here is students are figuring it out by starting with the solution and working backwards to understand the process for solving equations. And they develop the process themselves versus memorizing it.

Rather than thinking of the calculator as a solution tool, think of it as another way to help students discover where those solutions come from.

Here’s a quick video on using the Equation App (solver) on the CG50. The process is the same on Casio’s other graphing calculators. This is another installment in the app exploration series, started last week with the Physium App.

Origami – The Math Behind the Paper Folding

I am about to start teaching an online geometry course, and it has me missing some of the things I use to do with my students to help them discover relationships, and work with angles and symmetry, which was origami. Origami is the art of paper-folding – and using it in geometry is a great hands-on and visual tool to help students discover angle relationships, symmetry, linear relationships.

Origami is something I am sure most of you are familiar with and maybe have even attempted to create some origami art yourself. I have two friends who are origami wizards and often post their creations on FB – and it’s pretty amazing the shapes they create. When I recently went to the Museum of Math in NYC there was a whole exhibit devoted to Origami.

In my class, obviously, we did relatively simple constructs – folding one piece of paper into things like cubes, birds, shapes. The focus being on the folding and shapes created from each fold and looking at the angles and relationships that developed after each fold. But – as I have discovered, there is some really complex math behind origami, and really complex shapes that are created all from one sheet of paper that are simply astounding. I just found this Ted Talk from 2008 by Robert Langdon that discusses the mathematics behind Origami and how because of mathematics, folds that before were impossible are now possible, allowing for origami constructions that are astounding. Those of you who teach geometry, I think this will be very interesting to you, though I think other math subjects as well will find some applications. At the end of the video there is also a link to some templates for folding some more intricate origami constructs.