# Pee In the Pool and Other Summer Problems – Problem Solving Resources

As part of my daily brush-up-on education news, I read over my Twitter feed to see what interesting articles or problems the many great educators and educational resource companies I follow might have shared. I laughed so hard when I saw the Tweet from @YummyMath asking how much pee was in the water, with a picture of a large pool and many people in it. Come on – let’s admit it, we have all asked that question at one time or another (especially if you are a parent!!)  It’s a great question. And now I am curious. Where to start? My thoughts are I’d probably need to do some research on the average amount of pee found in a pool and then go from there. The great thing here – Brian Marks from @YummyMath has done that work for me, and even has an engaging ‘lesson starter’ video to go along with the lesson (link to the lesson). So – this would be a really fun problem to start out with that first day of school – funny, lots to notice and wonder about, getting ideas from students on where to begin, what information they might need, etc.

In an early post this summer, Summer Vacation – Use Your Experiences to Create Engaging Lesson Ideas, I talked about how your own summer experiences could raise questions and interesting problem-solving experiences to bring back to the classroom. But – as the tweet from Brian Marks @yummyMath reminded me, there are other amazing educators and resources out there who are already thinking of these questions and even creating the lessons for you. No need to reinvent the wheel, as they say – if there are some interesting questions and resources already being posed and shared, then use them. Saves time, maybe provides some ideas you hadn’t thought of before, or maybe it takes something you did think of and provides some questions or links that you hadn’t found yourself. As educators, we need to really learn to collaborate and share our expertise so that we are not individuals trying to support just our students, but we are educators trying to work together to improve instructional practices and student achievement. Isn’t that what we try to stress within our own classrooms – i.e. working together, communicating, and sharing ideas because this leads to better understandings and new approaches? Same goes for our teaching practices and strategies.

Here are some fun problem-solving resources, with lots of different types of problems, but definitely some ‘summer-related’ things already started for you!

1. YummyMath – (check out the ‘costco-size’ beach towel activity….that’s funny!)
2. Mathalicious – (Check out the ‘License to Ill’ lesson – relevant to todays’ debate on Health Care & Insurance)
3. Tuva|Data Literacy (Check out their lessons and their technology for graphing and analyzing data, and their data sets – so much here!)
4. RealWorldMath
5. TheMathForum
6. Illuminations
7. Center of Math
8. MakeMathMore.com
9. MashUpMath

# 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.

# Math – Always Something New or Different

If you hadn’t heard, a group of Georgia Tech Mathematicians have proved the Kelmans-Seymour Conjecture, a 40-year old problem. Here is a an article that describes the conjecture and its proof in more detail for those of you interested: Georgia Tech Mathematicians Solve 40-year old Math Mystery” Now, I personally had no idea what this conjecture was till after reading the article – Graph Theory was not something I spent a lot of time on in college or in my teaching career.  What struck me was that this conjecture has been out there for 40 years with people trying to prove it, and it took a collaboration of over 39 years between six mathematicians to prove it:

“One made the conjecture. One tried for years to prove it and failed but passed on his insights. One advanced the mathematical basis for 10 more years. One helped that person solve part of the proof. And two more finally helped him complete the rest of the proof.”

Elapsed time: 39 years.” (Ben Brumfield | May 25, 2016)

Here’s what I love about this – it shows that math is a collaborative endeavor, that takes time and different approaches and insights and that something new can always be discovered or proved. Which is what we should be focusing on in K-12 math education, instead of the idea that there is one answer to a problem.  The standards for mathematical practice (part of the Common Core and based on NCTM Principles to Actions) are all about this collaboration, problem-solving, communication. It’s slow to take hold, and politics is working against it, but look at what can be accomplished when mathematicians, i.e. students, work together to problem-solve?

Math is not a single-solution, one-way only, or  learn-in-isolation. Let’s support the practices, let’s support teachers, let’s support students and create mathematical learning experiences that promote collaboration, real, relevant problem-solving.  It requires teachers being willing to accept multiple approaches and multiple methods of explanation (verbal, written, visual). It requires noise – collaboration is not sitting quietly at your desk.  It requires “mess” – using whatever tools or resources help students think about problems. It requires time.  But think about the new and different math that students will create and explore – and think about how much better prepared they will be for the mess that is the world.  That’s ‘college and career ready’ in my opinion.

# Teachers & #Edtech – Ready-to-Use Lessons Can Be A Support

I am a little obsessed with edtech and integrating technology into math classrooms. It’s what I have been doing for the last 16 years of my educational career, first within my own school and district, and then, throughout the country through my work with Key Curriculum, McGraw-Hill, Kendall Hunt and Casio. I read a lot about the infusion of technology in schools these days, but my reality, when I go to schools and districts throughout the country, is that the use of technology in mathematics education is actually very, very limited. There are of course countless reasons for this – a big one being funding. Most schools I work with have 1-2 computer labs that math teachers rarely get to use, or they have a laptop cart shared between 15 math teachers. They have calculators – sometime – most of which are broken, have no batteries, or they honestly don’t know how to use. There are also the instances where there is a lot of technology available, but the teachers don’t know how to use it, don’t have resources to support it or they haven’t had a chance to find a place where it would support their curriculum.

The reasons for not using technology are many. But – in my own personal research, one of the biggest deterrents to integrating technology is lack of training and support. A recent survey of teachers by Samsung shows teachers do not feel prepared to use technology in classrooms.  Not a surprise. Unfortunately, the majority of professional development is still the one-stop workshop, where new technology/apps/ are bought and teachers are trained for a few hours on the tool, with little or no emphasis on teaching with the tool, which is the most important aspect of technology integration. Technology is only a tool – and when used appropriately, can enhance and expand learning. This involves more than learning how the tool works. It involves looking at the curriculum and instructional goals, determining what tools (of which technology is only one) are going to provide the best fit, and then creating instruction that incorporates the tool as part of the learning, not as an add on, not as something extra we do after we learn.  This is what is missing most of the time – helping teachers make technology fit into their instruction as part of the learning, not as something extra.

One of the things I found in my research is that if teachers are provided with pre-made, ready-to-use lessons that can replace current lessons and use the new technology, they are more likely to start using it, especially in the beginning stages of learning. Lack of confidence is a huge reason teachers don’t use, or continue using, new technology – this is helped if they are given a push, especially in the first stages of learning, that allows them to use technology without too much stress – i.e. the lesson is ready to go, there are teacher notes/guidelines, and it FITS INTO THEIR CURRICULUM. In the Samsung survey, 80 percent of teachers said it would be helpful to have pre-existing lesson plans that help them easily integrate technology. I found this was one of the strongest indicators of continued integration of technology in my research.  It’s one of the things Key Curriculum provided for Sketchpad, it’s one of the things Casio provides for their calculators.  If teachers are given new technology and ready-to-use lessons that show them and students how to work with the technology while learning required content, they are much more likely to use the technology.  And – the more they use, they more confident they become with it, the more likely there will be continued implementation.

To go along with ready-to-use technology lessons, ones that scaffold learning for both teachers and students, is collaborative lesson planning. Teachers should have the chance to work together to plan lessons to incorporate technology. Again, in my own research, teachers expressed how the monthly collaborations with other teachers from around the district, as well as the online sharing community, really helped support their own efforts to integrate technology and gave them new ideas. Sharing ideas, planning for where technology is appropriate, learning from each other – all of this is powerful in helping teachers be more confident in using technology in instruction. There is no reason for teachers to reinvent the wheel for every lesson – if there is a premade lesson out there, or a lesson another teacher has tried, that will support others integrating technology, there should be sharing and collaboration.  Teaching is a profession, not an isolated, individual endeavor – we should be working together to improve and help students learn and help each other learn.

# New Year’s Resolutions for the Classroom

I hope everyone has had a nice holiday season and are planning to do something fun for New Years Eve. I myself am planning to have a potluck dinner with many of my neighbors and spend time talking, laughing and ringing in the new year. We have built a “New Years Eve Ball” out of chicken wire, lights, and 2×4, and are planning to do our own small-town ball drop from our friends apartment, which happens to be in the center of town. Hopefully we won’t get in trouble – I will be sure to post a picture!

Anyway, as the year draws to an end, and as all of you who are teachers see the end of your winter break draw to an end, I thought it would be good to share something I would do while I was teaching in the classroom. It’s so easy to get ‘tired’ this time of year – the school year is not quite half-way through, you might be going back to face midterm exams, and after a lovely vacation, the thought of going back and facing your 30 *(or your 150 students or more) seems exhausting. But, this is a time to think of ways to rejuvenate not only yourself, but your students and your classroom. Look upon the new year as a way to make some small changes in how you teach or structure your classroom, and you will find that it keeps the energy and relaxed feeling you had during vacation going.

I always made myself some New Year’s Resolutions for my math class.  Usually only about 3-4 things I wanted to start doing differently or more often in my classroom starting that first week back after winter break. It was a challenge to myself and I found it made me more excited about facing the next semester.

What do I mean by math classroom resolutions? It can be something very simple – like adding a different question into your teaching, or incorporating technology into class twice a week if you haven’t before.  The key here is to choose some things that you don’t do currently, or know you don’t do well, and focus on doing these things on a daily/weekly basis.  Little things that can make big changes.

Here is a list of some things I use to do:

1. Use different, thought provoking questions in each class at least two times each day (questioning is a skill I still work on, so deliberately focusing on it helps make it become a habit) (just a few example below):
• Ask “why do you think?”