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

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Classpad.net Version 1 – Just In Time for School!

Welcome back to a new ‘school year’ (for some anyway). I’ve been on a bit of a hiatus the last couple months, working hard and doing a bit of travel. But, time to get back to it and what better way to start things off but with the launching of Version 1 of Classpad.net.

I posted about Classpad.net back in May, in my post Classpad.net – My Math Love-Affair Continues, This time I want to actually delve much more into what Classpad.net is and share some activities and images to give you a sense of the power of this web-based software. We’ve been in Beta-mode, where we’ve been fixing bugs, working on functionality improvements, and other things while teachers and students have been playing around with the software. Big shout out to all of you who’ve been giving us feedback – we’ve been updating and making changes and fixing bugs in large part to your input. Today is the launch of Version 1, so no longer in ‘test-mode’. Does that mean it’s done? Absolutely not! The beautiful thing about web-based software is that we are constantly improving and updating and adding features. It’s really in its infancy, with so much more growth and functionality and improvement on the horizon, which makes it even more exciting knowing this is only the beginning.

What is Classpad.net?

Great question. At it’s heart, it’s FREE (yes…forever) web-based, dynamic, math software. We call it ‘digital-scratch-paper’ because you can pretty much do whatever you might do when you pull out a piece of paper – i.e. write some notes, do a calculation, make a graph, create a table, draw a picture, measure something. As we know, there are lots of math software and tools out there – but most have specific purposes (i.e. only do statistics, only graph, only do calculations, etc.), so we end up having to use one tool to make graphs, another tool to create geometry constructs, yet another one to do some statistical analysis. And then, if we want to create an assignment for students, we have to use yet another tool to copy-cut-paste our various tables, graphs, constructs, and directions into a usable document. Classpad.net allows you to do all of that on one ‘paper’, which can then be printed (PDF), or shared (unique URL), or saved.  You can send this to students via URL (email or post on your website), students can make their own copy and do their work and send it back to you. It’s all there on one page – and, the beauty is, you can arrange and rearrange things on that paper as you want. To the right is a snapshot of a ‘paper’ showing all the stickies – i.e. text, calculate, graph, geometry, table/statistical plot. You have unlimited scroll and vertical space, and all objects are moveable – arrange and rearrange to your hearts content. You can title the pages and change the banner color to help sort and group content areas.

What Are The Components of Classpad.net?

You can pretty much do all the mathematics you need with Classpad.net for all K-12 curriculum content areas, including Calculus and AP Stats. There are some features that as of today are behind a ‘paywall” (i.e. nominal fee for the add-on app feature), but these are features that most K-12 teachers would NOT want students to have or necessarily need (re: CAS ability, allowing for solving equations or factoring polynomials, as an example; handwriting recognition, and a few others as we add in functionality).  But, here are the general components of Classpad.net, and with each there is a quick GIF showing some aspect of each component:

TEXT – text is just that – you can pull up a text sticky to write directions (for student homework/tests) or descriptions. You can also type in mathematical expressions/equations/terms in the text. Text stickies can be moved and resized as needed, color changes, and you can set a sticky for students to respond to (or students can add their own text sticky to write in answers and reflections as they work on things.

 

 

 

 

CALCULATE – as you would expect, calculate does calculations and so much more. You can define functions and lists, and use them later in graphs and statistical tables. Due to natural display, you can get exact answers. You can use function notation and shortcuts (see the ? at top right of Classpad.net for the function list). And, as with all the stickies, you can move the calculation stickies wherever you need them to be or pull them up whenever needed – all on the same paper.

 

 

 

GRAPH – again, you can graph anything – equations, defined functions, inequalities, integrals, etc. You can create sliders to move graphs and compare functions. You can find area under the curve, click on the graph to see key points, add moveable points to a function plot, look at the table of values, or plot from a table a values, make moveable lines for lines of fit. Comparing graphs is easy too – you can put graphs together or pull them apart to look at things separately. You can have multiple graphs on your paper – either merged or separate. You can add pictures to your graphs as well.

 

 

 

GEOMETRY – Yep, you can even add geometry to your page. We are still building out the geometry component, but right now you can do what you would expect with a geometry tool – i.e. create geometric constructs and specific constraints (perpendiculars, parallels, etc.), measure (area, length, angles, etc.), transformations including dilation, with features that are also unique (so you can construct conics, you can draw free-hand and then ‘adjust’ shapes and objects to have particular constraints. There’s the ability to create a rotational slider. You can create Hide/Show buttons and functions and expressions, and of course typical things like hide objects and change size, colors, etc. I am excited about geometry because I know it’s only the beginning and there’s so much more we are going to be adding.

 

 

STATISTICS – So much to do already, and still so much more to come with statistics. But, what’s the most fantastic part is you don’t have to go get a ‘statistics’ tool for students to be able to collect data, record it in a table, and then analyze that data. This could mean measures of central tendency, or standard deviation, or making different statistical plots to represent the data. Normal distributions, many types of regressions, box-plots, dot plots, histograms…so much there already and we are adding more in the future. As you would expect, we have a spreadsheet that can do calculations or use pre-defined lists (see calculate). You can then add functions to your statistical plots – so everything is all in one place for students to explore and connect.

 

 

As you can see – there’s so much to do, all one one page and one platform (#one-stop-shopping) and it’s free! It’s designed to be usable on touch-screen devices and mobile-devices as well as laptops and PC’s. The perfect tool as you are preparing for this school year, or are just starting your school year (or maybe you are already in-deep to your school year….it’s never too late). Go explore and give it a try and make sure you are letting your student know about this tool. We are also building out our ready-to-use lessons and our video library of support, as we continue to add and improve functionality, so stay tuned. Check out our social media sites for updates and support and we would LOVE to hear from you – share what you and/or your students are creating!!

Check Us Out and Share Your Papers and Experiences:

1. Classpad.net Youtube
2. Twitter (@classpadnet)
3. Facebook
4. Our website – subscribe so you can start saving and sharing your work with others! Classpad.net

 

 

 

#TheMathContest – Supporting Student Problem-Solving

My last several years in high school, I was a ‘roving’ teacher, meaning I didn’t have a classroom of my own, but switched classes just like the students. This made for a very challenging prep experience, and required me to be super-organized and self-contained on my little rolling cart. The rooms I ‘borrowed’ for my classes did allow me to keep an area for my students (to turn in homework and pick up missing work, etc.). In each class, my students had a portfolio (i.e. file folder), where they kept their work, one of which was the daily ‘warm-up’ problems.  These were basically a set of 5-6 problem-solving activities – some applications, some skills practice, some real-world scenarios, some puzzles, etc.  Students were expected to pick this up daily and work on these in the first few minutes of class, which gave me time to: a) get there; b) check homework; and c) set up for the class/lesson, etc.  It gave everyone a chance to ‘settle’. Students had a choice – they could do some or all of the problems by the end of the week, and I just checked portfolios and work at end of week. We would always discuss possible solutions the following week (and also they earned points for their efforts).

Needless to say, since I was providing these problem-solving experiences daily, I had to find lots of different resources for these problems, especially those that were more application and thought-provoking. Can’t tell you how many problem-solving books I purchased! There were other sources, such as The Math Forum P.O.W. (now no longer in existence, though their P.O.W. “s do still live on at NCTM), and even my textbooks had some great problems if you looked for them. The point is, it took a lot of effort to provide these challenges for my students. Obviously, I could have done it once a week instead, but for me, it served that duel purpose of focusing my students every day while I was en-route. My goal, and something I think all teachers should be striving for, is to provide students some challenges and problem-solving experiences on a regular basis – ones that may utilize prior knowledge or challenges them in different ways of thinking with new skills.

For those of you looking for such challenges, there is a new resource available from Ole Miss’ School of Education called #TheMathContest. It is actually a reboot of something Ole Miss did in the past, but it’s been revamped and improved, and now is sponsored by Casio Education and encourages the use of the new, FREE, online math software, Classpad.net that I talked about in my last post. Basically, new problems are posted each Monday, and each user can submit one answer per hour. Correct solutions earn points and you can view rankings on the website. Go to the link above to get more details on the contest. There are monthly rankings and annual rankings, which you can view online. How points are awarded is explained here.

This would be a great way to engage students and get them doing some challenging math, not to mention trying out the new software as well! If I were still in the classroom, I think I might add this as extra credit for students (for trying) and then maybe have a collaborative problem-solving time where we discuss possible approaches to the solutions after the previous weeks problem has ‘expired’.  Or maybe group students in ‘teams’ where they submit as a team? In any case, it would be nice to have a problem challenge already done for me each week, that’s for sure!

One thing Classpad.net is doing is posting video solutions to past P.O.W.’s which you can find on our Youtube Channel  Here is an example from May 7, 2018’s Problem of The week:

 The Problem:  Find the 1-millionth term ins the sequence {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, ……}

 

There is still time to try this weeks #TheMathContest Problem of the Week for May 14, 2018!  And check out the rankings – you will see students from countries all over the world who are participating.

ClassPad.net – My Math Love-Affair Continues….

I am a lucky woman.

For my almost 30 years in education, I have loved what I do. Teaching math, helping others teach math, finding amazing tools and resources that make learning math engaging and exciting – my ‘work’ is a labor of love. My love-affair with mathematics and teaching has been influenced by many experiences and people and has led me to yet a another new adventure in my quest to help others love and appreciate the beauty of mathematics – Classpad.net,  a free, web-based software that I have been directly involved in, from conception, to development, and now, to public release and hopefully, viral usage!

Some of my Key family

It’s been a weird path of growth, with connections leading to new opportunities, and more connections, and more opportunities. As a new teacher, and also working on my masters at VCU in VA, I worked under John Van De Walle, who started me on the path of making mathematics hands-on and visual and based on problem-solving. This quest led me to look for resources and share my love of math at conferences – sparking my professional development/training itch.

DG5 Groupies!

My search for visualization and hands-on resources led me to a closet in our math department, where I found Discovering Geometry and Sketchpad. And as I used these resources to present at conferences, I got to know and LOVE Key Curriculum and become, I admit, a groupie. This led to getting to know the Key sales folks and being asked to become a Key consultant. All this PD experience led to an administrator job, where, miracle of miracle, all the Discovery books from Key were just being adopted, so I was part of this implementation, which led to meeting Key’s PD trainer, Tim Pope. As a result – lo and behold, this groupie is working for Key!

It was a dream come true! The Key family, one full of former math educators all trying to share the love of mathematics and create inquiry-based, engaging math through great problem-solving and dynamic math technology tools, was amazing. Then – the dream burst, the family split up, and the books went to Kendall Hunt (with Tim), and the technology to MHE (with me).

Heartbreak.

Casio Family

Time to open a new door: I decided to finish my doctorate and branch into the unknown world of education consulting. And that Key family? They are still there – sending connections and opportunities, which is why I now teach at Drexel, work with Casio, travel the world for The Dana Center and Department of Defense Education Activities, among many other experiences.

At this moment in time, my worlds have collided. My Casio family, which is a group of math educators trying to share the love of math and teaching and learning math through dynamic visualization, is inspirational. We’ve worked as a collaborative team, with Casio‘s incredible R&D team in Japan, to create a tool that is going to revolutionize mathematics. It’s everything math teachers want on one page, and it’s just in it’s baby-phase right now with potential for growth that is exciting.

The guys behind booth magic!

Classpad.net has a partnership with Kendall Hunt just recently announced. Those very Discovering Mathematics books I so love will be adding to their power of inquiry by providing our tool as the discovery math tool embedded in the ebooks. My new family is joining with my old family….(and Tim and I are reunited) (and we have a podcast too – 180days Podcast)(shameless plug)!

Right now? It feels like I’ve connected many parts of my life – where many of my previous ‘experiences’ and worlds have joined together. Not sure if this is the circle of life, or a Mobius strip, or maybe an example of a network with many nodes. But whatever it is, it feels right, it feels exciting and it feels limitless.

So, what is Classpad.net?

It is something that makes me proud to be a part of because it is a web-based software, freely available to teachers and students, that encompasses all the things I wished for as a teacher, and it’s all in one place instead of several different tools that don’t communicate with each other. My doctorate dissertation was on edtech, and how teachers have so many technology tools forced upon them (hardware, software, apps, tablets, PC’s, interactive whiteboards, student response systems, etc) and none of them talk to each other, and each require separate training and support. Instead of using any of these tools effectively, teachers use the ones they are comfortable with, and often not the tool that makes the most sense for helping students learn. Or worse, no tools at all.

Classpad.net solves that problem by being a tool where you never have to leave the page – you can do geometry on the same page you are doing statistics. You can add a calculation, you can make a graph – all from one place. You can dynamically show mathematics and students can explore math and make their own discoveries on a table or a laptop or a phone – with the touch of a finger. There is a complete CAS (computer algebra system) engine behind this software, so it’s capabilities and functionality are incredibly robust. We are just in the ‘beta’ stage of release, which is even more exciting because we are really seeking input and feedback from users – what’s not working for you? what do you want? And, just like a start-up tech company, our team is responding quickly and changing based on what teachers and students want and need. The possibilities are endless because we have Casio’s 60 years of worldwide technology expertise and the experiences and input of math teachers building something that can be what teachers and students really need, want, and use – all in one place.

We have a Classpad.net Youtube Channel that we are just starting to build out, but here’s a quick overview of Classpad.net

It’s only the beginning – so check it out. But, as someone who has had a long-standing love affair with math and math technology, this is going to be a fun ride with so much more to come!! Join the fun and start creating with math and sharing your love of math as well on Twitter and Facebook!

Stephen Hawking – Understanding the Universe

Pi Day is a day math and science enthusiasts love, not only for all the fun Pi Day activities, but also because it was Albert Einstein’s birthday, one of the most iconic mathematicians/scientists/physicists, who developed the theory of relativity. This year’s Pi Day, ironically, the world lost another renowned physicist, Stephen Hawking. Whether you never heard of Stephen Hawking until the movie The Theory of Everything (2014) or whether you only know him from his voice on the Simpsons or Star Trek, The Next Generation or The Big Bang Theory, his death is a loss to the world of science and math.

It seems silly for me to list all that Mr. Hawking contributed, when his website does such a terrific job of that already. Here is the link to his site, where you can read about Stephen Hawking, see the many publications of his, the books he wrote, the lectures he gave (transcripts), the movies about him and with him, and videos he made. This is a great resource for teachers and students to explore.

I want to leave you with a video of Stephen Hawking talking about his love of science and as he said “I did my work because I wanted to understand the universe”.

Rest in Peace and thank you for all your contributions that have helped the rest of us understand our universe a little better.

 

Pi Day 2018

I know there are many math teachers prepping for Pi Day (March 14, 2018), so I wanted to provide some links to resources that might help support your efforts.

One thing I use to do with my students – middle and high school alike – was have everyone bring in ’round’ food – i.e. Moon Pies, Little Debbie Snack, pies, cookies, etc.  We would verify ‘pi’ by using string to measure the circumference, and rulers to measure the diameters of all the items brought in before anyone was allowed to eat. We’d have a contest on who could recite the most digits of pi – that was always a hoot. There was always some history about pi and I would where one of my Pi t-shirts (hey, math teacher – so yes, I have Pi T-shirts!!) – my favorite being the pi symbol made of skittles that said ‘Sweety Pi”.

Below are some links to activities and historical facts about Pi for those of you searching for things to do with students on Pi Day.

1. Did you know Albert Einstein was born on Pi Day? This article provides some history about Pi, such as how it got its name. It wasn’t from the Greeks, surprisingly!! http://time.com/4699479/pi-day-2017-history-origins/
2. The Exploratorium has a bunch of resources, from history, to activities, to the numbers of Pi, and if you live in San Francisco, admission is free on Pi Day – https://www.exploratorium.edu/pi  and the Pi Day event http://sf.funcheap.com/annual-pi-day-exploratorium/
3. There is an actual PiDay website – all things pi for March 14.  Tons of ideas and resources here http://www.piday.org/
4. Did you know some stores and restaurants have Pi Day specials? (Whole Foods, Blaze Pizza) – http://www.wral.com/pi-day-deals-wednesday-march-14/17396508/
5. This link has some history, some activity suggestions https://www.wincalendar.com/Pi-Day
6. Another site with activity ideas and fun facts about Pi – including the Pi song (see below) and a Pi video http://www.chiff.com/home_life/holiday/pi-day.htm
7. If you live in NYC, the Museum of Math has free admission on Pi Day and are serving pie! https://momath.org/about/upcoming-events/
8. If you live in or near Princeton, NJ, the entire town celebrates Pi Day – probably because Einstein lived there. https://princetontourcompany.com/activities/pi-day/
9. NASA’s Pi In the Sky challenges for Pi Day https://www.jpl.nasa.gov/news/news.php?feature=7074
10. 25 Ways to celebrate Pi Day https://holidappy.com/holidays/25-Best-Ways-to-Celebrate-Pi-Day-314

There’s a Pi Song?!!!

Enjoy you Pi-day preparations!!!

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.