Frozen Math

I saw this really cool GIF on FB the other day, showing a bubble freezing. As I watched it, you could see all these beautiful shapes emerging and eventually covering the whole bubble. (I of course wished that it was cold enough where I live for me to go out and try it myself, but alas….where I live seems to be having a no-snow winter this year.

Watch and see:

It looks like snowflakes appearing on the bubble, and snowflakes are fascinating. They are unique, they have amazing patterns that form naturally. Wouldn’t it be fun to explore snowflakes with students? Especially if you live in colder climates where there is actual snow to collect and study. How could we connect the beautiful patterns and unique qualities of snowflakes to mathematics? I set out to explore and found a few great resources for those of you who are interested in exploring frozen math. Yet another way to bring the real-world into the classroom and help students see the math that exists around them.  Even if you don’t live where snow may be, some of these resources provide some great tools for ‘creating’ snowflakes with students.

Here are some links:

  1. This is a nice site because it has several suggestions – from collecting real snowflakes to creating your own, to analyzing patterns and categorizing snowflakes. Great hands-on activities.
  2.  A wide variety of ‘frozen math’ activities here: including the Koch Curve/Snowflake, where students experience the iterative process to create a snowflake fractal.
  3. Some nice examples and how-to-make paper snowflakes:
  4. Some nice geometry connections and more paper-snowflake making here:
  5. This is a great math/science connection with a lot of further embedded links included:
  6. Vi Hart and Doodling is always fun to watch, and here she is doodling and folding with symmetry and fractions:

I am sure there are more options out there – these are just a few I stumbled upon in my searching. Don’t let the winter blues set in – get out there and collect some snowflakes and do some frozen math!

Fractions with a Calculator – Looking for Patterns

calculatorI have been working with teachers and using manipulatives, both physical and virtual, to help students think about fractions and develop conceptual understanding about fractional operations, versus just memorizing rules or tricks, as we so often do with students. There are fraction circles or fraction strips that work well as physical manipulatives, and there are several virtual manipulatives as well (i.e. for any Sketchpad users out there, and the National Library of Virtual Manipulatives to give just a couple resources).

Manipulatives are a valuable resource in math class as they allow students to visually represent numbers, manipulate them, get hands-on with the math, and make some connections before moving into just the numerical representation alone. When working with fraction manipulatives, from my own experiences and those I have had with students, the manipulatives can constrain the number of possible examples we can provide students (either because a teacher might not physically have enough for all students or the manipulatives themselves only go up to certain values). As an example, most physical fraction circle manipulatives allow you to work with a limited range of fractional values – halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths. Virtual manipulatives offer more options, which is nice because students should see more than just common fractional pieces or ‘nice’ fractions – sevenths, or elevenths or twenty-fifths as an example. Obviously, the idea of manipulatives is to provide that hands-on experience, visually see what’s happening, and then create conjectures.

Another tool that is often overlooked, particularly at the elementary level, is the calculator. Obviously, when dealing with fractions, you want a calculator that uses natural display, showing fractions in their numerator over denominator form so students recognize the fractional number. I realize many of you might be thinking that the calculator is a bad choice because it provides the answers….but that in fact is an advantage here when trying to help students recognize patterns and develop their own understanding of fractional operations.  We want students to recognize what seems to be happening – test it out on many examples before they come to a conclusion.  A calculator (like the fx-55Plus shown above) is a great way to do this.  If you don’t have manipulatives, you can actually use a calculator like the fx-55Plus to help students understand fractional operations.

Let’s take fraction addition. Obviously, we are going to start with adding fractions with like denominators.  You can put several different problems into the calculator and students can observe both the added fractions and the answers. Students can talk and share what they notice about the multitude of fractions they are adding (all with like denominators). They can make up their own addition problems and see if the pattern or things they notice hold true. Fraction and answers showing up quickly help them discern patterns because they can quickly see many examples, and use ‘funky’ fractions, not just the typical ones we tend to always rely on (i.e. halves, thirds, etc.). It’s even okay that the numerator might occasionally end up larger than the denominator – the pattern still holds true (i.e. the denominator remains the same, the numerators are added together).

With a calculator, you can use messy fractions with not your typical denominators and even numerators larger than the denominator. For addition, our focus is on what patterns do the students see with the numerator and denominator and do those patterns hold true no matter what fractions we are adding? We can get into simplifying the answers at some point, but at first, the focus is on the addition.

Once students have the idea that with a like denominator, you add the numerators, you can then switch it up. Let’s add fractions with unlike denominators.  You can encourage smaller numbers in the denominator and numerator to start, and then once students think they have the pattern, they can ‘test it out’ with some larger digits in the numerator and denominator. The thing here is the denominators are different and so how does the end result differ (if does) from when the denominators are the same? What might be happening? Test it out.

The beauty of the calculator (again, one like the fx-55plus that quickly and easily shows fractions in their natural display), is that students can create many examples to look for patterns and then quickly test their conjectures on different problems to see if it works. You are encouraging critical thinking, problem solving, and communication using a simple tool that provides much more diverse fraction examples than you can provide with manipulatives alone.

My point – when helping students develop number sense, especially with fractions, don’t rule the calculator out as a tool. You should use multiple tools with students to provide them with different ways to develop their own conceptual understanding. Calculators can be a tool, even at the elementary level.




The Language of Math – Consistency to Support Students

I’ve been teaching some courses at Drexel University, and in those course we really focus on the language of mathematics and using students prior knowledge to help them make connections and build on their mathematical understanding.

In a current course, we are exploring integer addition and using manipulative’s to provide both a visual and concrete connection to the idea of creating zero pairs, and then progressing to the more abstract addition of integers without manipulatives and how do you support students understanding and language. What has come up frequently is the terms ‘cancelling out’ and ‘disappearing’ and ‘opposites’ to explain or help students understand that creating zero pairs allows you to use the additive identity property.  But – what’s really happening, mathematically, is that we are using additive inverse (opposites) to create these zero pairs, which are NOT cancelling out or disappearing, but instead, are creating the quantity zero. And, once we have created this quantity zero, the remaining value can be added to those zeros using the additive identity property. Cancelling out or disappearing implies they don’t exist, which, when we expand the idea identity into multiplication where we use the multiplicative inverse to create a 1 (not a zero), cancelling out really seems confusing.

Seems like it shouldn’t make a difference, but think about it – if we use terms like opposites, cancel out, or disappear in one grade, and then the next grade or future courses, the same ideas are referred to as additive inverse, zero pairs, additive identity property, students will be confused and think they have never seen these concepts before. While it is very important to use language students understand to start the process, I think as mathematics teachers, no matter what grade level, we need to model proper mathematical language in conjunction with the ‘student-friendly’ terms we tend to rely on or fall back on. This way, students can relate, but then learn, build on, connect & utilize mathematical language so that we are all communicating on an equal playing field.

Something as simple as -3 being referred to as negative 3, minus 3, or even 3 negatives. That’s confusing. Let’s learn to be consistent with our mathematical language and use the correct vocabulary – both in our own teaching, but also in what we expect to hear from our students. So if a student says that the opposite of 3 is -3 (positive 3 and negative 3), let’s acknowledge that they are thinking correctly, and in mathematics we refer to that as an inverse. This will then help them make that connection when we talk about inverse operations.  Consistent mathematical language supports students understanding as they progress into more abstract mathematics.

We want our students to communicate mathematically with the language of mathematics and become proficient mathematicians. Let’s then make a conscientious effort to use and model correct mathematical language instead of the ‘short-cuts’ or ‘simplified language’ we tend to use.  Again – important to start out with this type of language to help students connect prior knowledge, but then more important to model using and support students use of, correct mathematical language. I think it would go a long way in preventing some confusion students experience as they move from grade to grade or course to course.

Multiple Representations on the Casio Graphing Calculators

One of the key things we try to help students with when studying functions is the idea of multiple representations – i.e. graphical, symbolic (equation) and table.  Ideally, we want students to be able to discern what the function represents or looks at no matter what representation they are given, and to be able to find patterns and important components about that functions from all representations.  Students should never learn about functions just through graphing, or just through symbolic manipulations or just through looking at data points in a table – they should be able to go back and forth and determine which representation is the most useful for the situation.

Unfortunately, too often, the emphasis is on one representation at a time, or at most 2. Let’s look at the graph and find the minimum, maximum, or intersection. Or, let’s find the roots of a quadratic by factoring, or symbolic manipulation. Or, here’s a table of points, where are the x-intercepts or the y-intercepts? Ideally, we want students to be able to look at all of these representations simultaneously so that they see the relationships between the representations and come to understand what the points represent in the table, in the equation, or in the graph.

Technology is one way to show all these representations at the same time, and then quickly manipulate and explore. There are obviously many technology tools out there, but as I have stated in previous posts, the most accessible technology tool for most students and teachers is the graphing calculator, not only because of it’s affordability, but because it is a tool most students have readily available.  It would be nice if all students had computers or tablets for daily classroom use, but that is still NOT the reality.

I have put together a quick video showing Casio’s three graphing calculators – the fx-9750GII, the fx-9860GII, and the CG10/20 or Casio Prizm, and how they can display the equation, graph and table representations of a function on one screen. No matter which model you have, you can achieve the same functionality, allowing students to work with multiple representations and explore relationships quickly and efficiently.

Check it out:

Changing Classroom Strategies – It Takes Practice and Commitment

downloadHappy New Year!  I hope everyone had a wonderful holiday season and is ready to start 2017 with a new outlook and determination to make this the best year yet, both personally and professionally.

My New Year’s post from last year, New Year’s Resolutions for the Classroomprovided a list of 5 things I use to do to rejuvenate my classroom each year – things I really tried to emphasize and focus on deliberately to help foster student engagement.  The list is still appropriate, so I am not going to repeat it here – read last year’s post if you are interested. Instead, this year, I wanted to focus on change – which is what a ‘resolution’ is after all.  And by change, I mean long term, sustained change, that becomes habit and routine, which, when we are talking about effective classroom strategies, these are the changes we want to be making in our instructional practices.

Change is hard, as we all know. It’s much easier to keep doing what we have been doing, even if we know it isn’t working.  That unfortunately has been the problem with education for a long time – change that will have lasting, positive impact doesn’t happen overnight, and therefore when results don’t manifest immediately on a test or in a classroom, we think the ‘change’ was a failure and move on to something else. (Hence the reason why education looks remarkably the same as it has for the last 100 years or more). Take the Common Core Standards – a very positive change if done right, but deemed a ‘failure’ when results on standardized tests didn’t dramatically change or show improvement immediately. It’s not the change – it’s that there wasn’t enough time – enough practice – enough support. Real change for the better, in anything you do, takes serious time, commitment, support and practice. And unfortunately, we do NOT give teachers enough of any of those things to really make significant changes in instructional practice.

According to Malcom Gladwell in the book Outliers, it takes 10,000 hours of practice to master a skill. Now, granted, there is some debate about that, but, the point here is it takes a lot of practice to get better at something or to make a change and become good at it. So, let’s say in a math classroom, we want teachers to change their practice and provide better questioning (i.e. critical thinking) practices. Let’s say this is your New Year’s resolution – you are going change how you ask questions of your students so that they are using problem solving and critical thinking versus just regurgitating answers or providing ‘correct’ or ‘incorrect’ solutions. This means that you must first learn what are some good questioning strategies and questions to ask that provoke thinking, and then you have to practice incorporating these into you current practices.  Deliberately incorporate, which is often very difficult, especially in a math class where it’s pretty easy to just ask what the answer is and move on when a student provides it. So – practice. Every day. On this one thing. If we calculate out 10,000 hours, and say you manage to do 2 hours a day of practicing good questioning strategies (that’s probably over-estimating, but we will give you the benefit of the doubt). So that’s 5,000 days.  Which….if we think about a typical school year of 180 days, it’s going to take 27 years to master the skill. Unrealistic, right? (Though…as someone who has been in education for 27 years, I would say my questioning skills are significantly better than they were when I started….but I still don’t think I have mastered it!)

27 years of practice to master a skill, or make a change that has an impact. Crazy. Let’s think about some changes teachers are asked to incorporate into their classroom, focusing just on math. There are new standards, so they have to change some of the things they have taught, the curriculum they use, the resources they have. There are recommended strategies – i.e. more collaborative learning experiences, incorporate more technology, foster more problem-solving and critical thinking, utilize questioning skills, focus on conceptual understanding not just skills, incorporate modeling….and the list goes on. Some of these are not new or changes for all teachers, but many are. And if each one of these ‘changes’ takes 10,000 hours to master, we definitely have a problem! Teachers are given usually a couple months to make these changes – if they are lucky, a couple of years, but then there are always new changes coming down the road, and there is NEVER enough time to practice any of the changes enough.

Obviously, no teacher is going to be given 27 years to practice something new. My point here – change in strategies imagesis important and necessary, and to change requires consistent practice over the long haul. You may not see the results right away, but don’t give up because it takes TIME and commitment!! Make those New Year’s resolutions to be a better teacher, to do better at questioning, to use technology more, to help your students think critically and to work collaboratively. But realize that it takes practice – lots of it – to make these changes have a real impact on student learning. Devote that time. Focus on one resolution/change at a time and just keep doing it – over and over – till you get better and until it becomes a habit. Practice truly does make perfect (or at least better) and your students will benefit. I don’t think it will take 10,000 hours to see positive results, but it won’t take a day or a week or a month either. It will take your commitment to practicing a little bit every day until it becomes routine and you continue to improve over time.

Keep practicing and Happy New Year! Let 2017 be the year of change!