# Creativity of Students – Provide Opportunities for Expression

I was straightening up my office – something I realized I do not do enough. I found a file of student projects from when I was teaching Geometry over 15 years ago. We had done some geometry poems for Valentines day – i.e. write a poem that utilizes mathematics vocabulary (getting that ELA and creativity flowing in my students), and I had clearly saved a few of my favorites.  There were other files of student projects – scale drawings of bedrooms and furniture (so students could ‘rearrange’ their rooms using a scale model), dilation pictures, transformation sketches from Sketchpad, problem-solving portfolios, and designing an aerial view of a city using geometric shapes and properties. As I walked through memory lane, looking at student work from years ago and remembering specific students, it really made me miss those classroom experiences. And what I had forgotten is how incredibly creative and thoughtful students are when given the chance to express themselves – you learn so much about them if you let them, what they know about mathematics, what they think, and what they don’t know if you provide opportunities to approach mathematics creatively.

I’d completely forgotten about the problem-solving portfolios I did with both middle and high school students in all my courses. They were given a choice of problems connected in some way to the math content we were learning or applications of prior knowledge, etc., and they were to choose from several. They had to complete one per unit and put it in their portfolio as examples of their problem-solving and learning/application of mathematics. This was way before the ‘Common Core’, but as I look at my expectations, it was very Common Core like. The idea behind was really very much centered around helping students to persevere and think critically about problems, use problem-solving strategies, and explain their interpretation of a problem, plan out a solution path, justifying their thinking, and showing multiple ways to approach a problem, and analyze their solutions to see if they made sense.  Here are the ‘steps’ they needed to go through and demonstrate in their problem-solving:

1. Restate the problem in your own words, writing out any questions or wondering you have about the problem.
2. Create a solution plan – what do you think about the problem  and why (is it hard, easy, does it seem similar to something you have seen or done before), what math might be needed, what problem-solving approach will you start with and why do you think this might be a good approach? What do you think might be the solution, before you begin?
3. Work through the problem – include everything, especially if you changed your original plan and why. Write down everything that comes to mind and what you did to think through things.
4. What is your solution and why do you think this is a reasonable solution?
5. Analysis of your problem solving – What did you think of the problem after working through it? What did you learn from doing the problem, either about yourself or about math, or both!?

In reading through some of these (I’ve posted some samples below from several different portfolios), you can ‘hear’ students personalities coming out, you can immediately see if they might have a misconception about what the problem is asking or an interesting approach to a solution, or identify those who really needed some extra support because their art work was more substantial then their mathematical work! It gives great insight into who might need some extra support or who might warrant some extra challenges. But mostly – the freedom to choose, think on their own and be creative and work through their problems provided students and ability to express their learning in a different way than an answer on a test. I remember at the time I was considered a rather eccentric MS/HS teacher because I did all these ‘strange’ things like keep math portfolios and journals, use manipulatives, used technology (Sketchpad) and projects instead of tests to demonstrate learning. But – in looking back on the past, and looking at what we want from students today in mathematics, with College and Career Ready Standards and Mathematical Practices, I think it’s the right path. Provide students opportunities to think, choose, be creative, find multiple solutions, justify their answers and question their results. It brings out their creativity and they learn to express themselves as mathematicians.

# Failure is Key to Learning & Perseverance – ISTE Keynotes

I was unable to attend one of my favorite conferences this year, the International Society for Technology in Education Annual Conference (ISTE), which was held in San Antonio, TX June 25-27 a couple weeks ago. I was doing some training in Austin, TX so could not make it. But – because it is a technology conference, they video many of the presentations, especially the keynotes, and make them available on their ISTE Youtube Channel

This years completed videos are not up yet, but there are some ‘teasers’ of the three keynote presentations. I particularly like the closing Keynote speaker, Reshma Saujani, the CEO and founder of Girls Who Code, on coding and how coding is all about the iterative process and failure – i.e. learning from failure and doing things over and over until you get things to work. This is how we develop that creative thinking, that problem-solving, and that critical thinking in students – letting them fail, learn from that failure, and try again. It speaks to the mathematical practice of making sense of problems and perseverance. Here is the excerpt that is currently posted on the ISTE website – short but worth a listen:

Both of these speak to something I have been focused on this summer in a course I am teaching – how to get students to persevere in problem-solving and be okay with ‘failure’. Instead of giving up, to have that drive to find another path or look for another solution. I am sorry I missed the conference, but it’s nice to get a peek into some of what was focused on, and I think it’s something educators really need to think about as we use the summer to plan for next year – how can you support productive struggle and learning from failure and perseverance in students?

# Using Connections to Build Understanding

I am teaching a Geometry & Spatial Reasoning course for Drexel this semester for their math masters program for teachers. Absolutely love it because I am learning so much from my students/peers, but because it really is bringing home the importance of prior knowledge to help build connections and real-world connections in helping students learn versus memorize, and construct and reconstruct based on their ability to make connections.

My students, who are a mix of very new math teachers, experienced teachers, and even some career-switchers still in the early stages of teaching, are having this great discussions on the importance of using prior knowledge to help student make their own connections. Some have been doing this all along, but others, as they themselves struggle with some of the geometric concepts we are ‘learning’ (relearning in some cases), are coming to understand the value in helping students use what they know to build on and connect to new information. Makes it easier to recall, and builds a confidence in students that when faced with an unknown situation/problem, they have the skills and confidence to look at it, break it down or add in things to make the unknown familiar and then look for and make use of structure (see what I did there….Common Core Math Practice #7!) to help reach a solution or develop a new conjecture/conclusion.

As an example, we’ve been doing a lot of work with inscribed angles in circles and how do you help students use prior knowledge to build the idea that an inscribed angle is half the measure of it’s intercepted arc if you don’t want students just memorizing formulas? Basically, the conversations revolve around constantly using prior knowledge to make connections, which might mean you need to add in an auxiliary line to a given shape to ‘see’ something familiar (i.e. a linear pair or a triangle, as examples). A strategy that really helps students look for and make use of the structures they are familiar with to help them make sense of a problem.  Here’s an example of just one way to explore inscribed angles, using previously knowledge about triangles:

• In Fig 1, we have an inscribed angle and its intercepted arc a. How could you show that angle 1 (the inscribed angle) is half the measure of it’s intercepted arc? Here’s where students need to make sense of this structure – what prior knowledge can they use to help them?
• In Fig 2, they add in a radius (auxiliary line), because they know all radii in this circle (any circle are equal – doesn’t change the original inscribed angle….but now – we have a triangle and a central angle (angle 2).  What do they already know? Well, they know the central angle 2 is the same as the measure of the intercepted arc, which is the same intercepted arc as angle 1 (inscribed angle).
• In Fig 3, students are looking at the triangle created and using prior knowledge – we can mark the two radii equal, making this triangle an isosceles triangle, which they already know from prior knowledge has two base angles that measure the same (angle 1 & 3). Angle 2 is an exterior angle to the triangle, and angles 1 & 3 are remote interior, which they know from prior knowledge sum to the measure of angle 2. Since angle 1 & 2 are equal (isosceles triangle), that makes them each half of angle 2 (Sum divided by 2). Angle 2 is equal in measure to the intercepted arc, so angles 1 & 2 are each half of that, so the inscribed angle 1 is half the intercepted arc.
• Fig 4 shows that the relationship holds true even if you change the size of the inscribed angle.

This is of course just one example for an inscribed angle, but they can then use this to show that inscribed angles that are not going through the center of the circle have the same relationships – ie add in auxiliary lines, use linear pairs, or triangles or other known things to help make sense and show new things. Prior knowledge, connections – they really matter.

As teachers, it is our duty to make sure we are modeling and helping students use what they know to build these connections and see the relationships. It takes deliberateness on our part, it requires modeling, it requires setting expectations for students till it becomes a habit (habits of mind) to look for and make sense by pulling in previous knowledge.

Another thing we need to do is make connections to real-world. My students are sometimes struggling with this idea of relating prior knowledge and new ideas to real-world applications, but if you get in the habit, its not so hard to do. Since I am focused on circles and the lines that intersect them now with my class, I pulled up a ready-to-use lesson from Casio’s lesson library that is a great example of a real-world connection to circle concepts that would force the use of previous knowledge.  The lesson is briefly described below:

• – Use coordinate geometry to represent and examine the properties of geometric shapes.
• – Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

This activity uses the Prizm Graphing calculator and picture capability to help build understanding.

The kinds of questions and connections to prior knowledge that can be asked of students just by looking at the image are pretty endless. What relationships do you see (i.e. lots of diameters, or straight angles, lots of central angles, all the angles are 360, are their auxiliary lines we could add to find the areas or relationships or angle measures, etc.).

If you look around, you can probably find a real-world example of most math concepts your are working on with your students. Show them pictures, show them real objects they can get their hands on. Start asking questions. Ask them what they recognize or think they already know. Ask them if they could add something or take away something to see a familiar object/concept. How does that help them? What relationships and connections help them get to something new or interesting?

My Drexel course and student are reemphasizing for me (and them) the importance of prior knowledge to help build connections on a continuous basis, all the time, every day. It helps students think mathematically and consistently use vocabulary and math concepts to deepen and create new understanding and relationships. It also promotes logical reasoning and problem solving – win-win!

# Solving Equations with A Scientific Calculator

Solving  equations is a skill that students are expected to be able to do in pre-algebra and beyond. If we look at the Common Core State Standards, these skills actually come into play starting as early as 6th grade, with students expected to solve one-step equations and progressing to systems of equations by 8th grade. An important aspect of solving equations is connecting a real-world context to these and understanding what the ‘solution (s)’ mean in terms of that context.

The use of calculators or technology to help students solve equations is a controversial one at best, and as a math teacher, I do believe that students need to know the processes to solving equations without the use of technology first. But – when we get down to real-world application and problem-solving, the technology becomes a tool that allows students to go beyond just “getting the solution” and to making meaning out of those solutions, and using their solutions to make decisions – which is the ultimate purpose of finding those solutions, right? In these cases, I firmly believe that the use of technology, (more often than not a calculator), is a necessary tool so that students deepen their understanding and are not bogged down in the process of the calculation. Part of the practices – “use appropriate tools strategically”.

As an example, let’s consider a simple real-world context that involves solving a system of equations, something required by the time students reach 8th grade (see Common Core Standards). Let’s say a scientist is mixing a saline solution and has one solutions that is 10% saline and the other 25%. He needs to make a 85 ml bottle that is 15% saline. How much of each of the two solutions should he mix to create the 85 ml bottle of 15% saline? This requires our two equations, with x = the amount of 10% solution and y= the amount of 25% solution.

• x + y = 90 ml
• .1x + .25y = 12.75 (15% of the 85 mL saline)

Perhaps students are actually in science class doing a lab and creating this new solution. While it would be reasonable to do this by hand using substitution, if this is part of an experiment, then using a calculator to get the answer quickly and therefore get on with the experiment might be a more logical step, especially when time is of the essence in classes. I am going to demonstrate on the fx-991Ex how to solve this problem.  I am using a scientific calculator because in middle school, students are more than likely going to have access to these versus a graphing calculator. This video shows how you can quickly solve the simultaneous equations, and also, with the QR code capabilities, also see a graphical representation of the solution.

If a scientific calculator is all your students have access to, remember that they can do a lot more than you might think.  I will explore more features of the ClassWiz in later posts as we continue to explore mathematics and using technology to support learning.

# Creating a Classroom Culture That Encourages Student Discourse

I just spent two days earlier this week working with middle school math teachers. Our focus was on the 6 – 8 Common Core Geometry standards, and how they build on elementary geometric concepts and continue to build that understanding that students need when they get into high school geometry. As part of our work, we also focused on the Math Practices, because it’s the intentional alignment of practices and content to create engaging mathematics activities that really help students develop the deeper understandings. By that I mean you shouldn’t be teaching the content standards in isolation – they should be combined with helping students make sense of the problems, choosing appropriate tools to explore and apply the standards, and really explaining and justifying the conclusions they make.  Practices and content go hand-in-hand.

In our many collaborative discussions these past two days, as we really dived into both practices and content, what was very apparent was how important it is to create a collaborative, safe, classroom. Mathematics classrooms should constantly focus on vocabulary use (by both teacher and students), modeling, discussing your thought process (in many ways – spoken, written, pictorally), explaining and clarifying your thinking, asking questions, and really focusing on all types of communication. Mistakes or misconceptions that students have should be expressed freely, without fear of embarrassment, and students should be free to try multiple pathways to solutions and multiples ways to express their understanding. Students are not going to talk about mathematics if they feel they will be laughed at or considered ‘stupid’ – and that requires a classroom culture that fosters real communication between students that involves listening, ‘arguing’ against someones responses in a constructive, polite way, and a sense that it’s okay to make mistakes because we are all in this together, learning.

What the teachers expressed as their “ah-ha” from our days together was that in order to create these types of classrooms and math learning, you have to start the process right at the beginning of the year, during those first weeks of school, when students are new and class is unknown, and math concepts are relatively familiar since we are starting, in theory, where we left off at the summer. Those first couple weeks of school are the perfect time to create that collaborative classroom culture.

Do a little bit each day – change the groups up, do it whole class, do it with partners. The idea is that you are helping students learn to talk to each other constructively so that when you get into the real learning of new math concepts, they are already comfortable with each other, with some of the learning tools they will be using, and understand that in this math class, we work together and listen to each other and support each other.

Learning is not an isolated activity – we, as teachers, are there to facilitate learning and help students become active, productive, problem-solvers. This happens in classrooms where it is okay to communicate, it is okay to make mistakes, it is okay to have your own approaches to problems but that requires justification of those approaches so others can learn from them.  The more you create this type of learning environment, the more your students will persevere in tackling those tough learning moments.

# Random Numbers to Spark Student Thinking

In my last post, I mentioned attending a session presented by Jennifer N. Morris on making math meaningful. (She is presenting two times at #NCTMRegionals Nashville – Session #244 for 9-12 and Session #275 for 3-5, which is the Origami/Fraction/Random Number session – BE SURE TO CHECK HER OUT – She is AWESOME!). One of the activities in her session incorporated the fx-55plus calculator and using the random number generator to spark engagement, problem-solving, numerical thinking and communication with students. Simply by hitting the random # key, which creates infinite random fractions (many complex!) and have students (participants) determine if that random number was acceptable if it represented the part of a cookie they would receive led to amazing thinking. What is acceptable? What are target fractions? How are students making their estimates and decisions? How do you know yours is bigger or smaller than the person next to you? Participants were asked to line themselves up in order, least to greatest, using their random fraction, which sparked great discussion and comparison. They checked their lineup by converting the fraction very quickly to a decimal, so equivalency and number sense.

During the discussion about all the different concepts students could be focusing on (number sense, fractions, estimation, equivalence, conjectures, probability, etc.) from this simple random number generation, teachers in the session offered several suggestions for using the random number generator on the calculator.  Here are just a few:

1. Use Random Integer to simulate the roll of a die for data collection (you could use two calculators to simulate two die).
2. Assign every student a number, and then use Random Int constrained to the numbers in the class (i.e. 1-20).  Use Random Int to pick a number, and that student is the one called on
3. In Collaborative Groups, assign each group member a number and use random number generator to determine who in group shares, or leads

Jennifer used the fx-55Plus because she loved how easy it was to generate the random numbers. Someone in the group asked about the scientific calculator and graphing calculators, and did they also have the random number generator. The answer was yes, but it was a bit more involved. So – realizing that using random numbers is useful no matter the grade you teach, I thought I would show a quick video on how to generate random numbers using the Casio calculators. The great thing about Casio is that calculators with the same face-plate layout have the same steps. I’ve listed below the calculators I am demonstrating and then some other calculators that would have similar steps to generate random numbers:

• Fx-55Plus
• Fx-350EsPlus, fx – 300ESPlus, FX115-ESPlus, fx-991EX
•  fx-9860GII, fx-9750GII, fx-CG10Prizm

Go be random!!

# Questioning In Math – #NCTMRegionals Minneapolis Observations

John Diehl #NCTMRegionals

Had a nice time in Minneapolis these past two days at the #NCTMRegional. It must get ridiculously cold here in the winter since they built an entire interconnected-Skywalk throughout the city. I think I only went outside twice the entire time I was here – getting in and out of the taxi! (Which was pretty terrific as the first two days were rainy).

I went to a few sessions this time that really got me thinking about the importance of questioning in mathematics. Even when utilizing technology or hands-on manipulatives/resources, the questions we ask the students are vital in order to deepen their understanding and encourage discourse and exploration. Questioning to me is the most important skill a teacher can develop to help their students – more important than any resources or technology that might be available, because it is only through questioning that we  foster rigor and develop deeper thinking to help students understand and make connections in mathematics.

Teachers talking statistics!