# 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

# Education Growth Mindset – So Important for Teachers and Students

I just came back from Kaiserslautern, Germany, where I was working with Department of Defense Education Activities (DoDEA) math teachers as part of the DoDEA/UT Dana Center College and Career Ready Standards Initiative. Our focus this summer, which kicks off the next year of continued support and training, was on helping teachers create a classroom culture of student discourse and a growth mindset that allows students to develop deeper mathematical understanding and become problem-solvers and confident mathematicians. It was a fabulous two days, and the teachers, some who had never explored this idea of ‘growth mindset’, really had some powerful conversations around this idea of providing students productive struggle opportunities and helping them develop this sense that they can solve problems, and they can improve mathematically, and they can learn. It was rather eye opening for many.  How many of us educators have come across those students who give up without even trying because they think they can’t do it? Or they have been so ingrained in the idea that they are ‘bad at math’, so they don’t even try? That’s what this idea is about.

Carol Dweck is a leader is this field of Growth Mindset, and how to motivate and help support this idea of a growth mindset. In fact, the teachers I worked with as part of our workshop, read an article by Dweck that provided some insight into what we as both teachers and parents, inadvertently sometimes do that prevents students/children from having a growth mindset. Something as simple as the way we praise can actually interfere with this growth mindset. More here.

Many of you may be unfamiliar with what a growth mindset is, so I found a great TedTalk from Carol Dweck that explains the idea behind it. As educators, this is something to really think about because we want to develop in our students the willingness to persevere and solve problems that may seem difficult.

# Thinking Ahead – Planning for Next Year’s Classroom Culture

I was in Austin all last week training for UT Dana Center (@UTdanacenter) International Fellows
(#UTDCIFF) and Department of Education Activities (@DoDEA) College and Career Ready Initiative teacher workshops happening this summer. A major focus for the week was on classroom culture and how important this is to mathematical learning and student discourse. Everyone at this training was either a current math teacher, a supervisor, mentor, coach, professional development provider, etc., so naturally, as part of the conversation, the following questions/concerns arose:

1. What is classroom culture and why does it matter?
2. How do you get students to talk to each other and engage in productive learning?
3. How do you respond to teachers who say things like, “well, this would never work with my students” or “I can’t get my students to talk about math when we are in groups”…

You get the picture, and I am sure you have either thought these things or heard these from teachers you work with.

The short answer – it takes planning, training, and consistency. If a teacher thinks that they can just put students into groups, give them a problem, and they are going to immediately start talking and working together, they are very quickly in for a big surprise. Especially that first time, and especially if you have never done these types of collaborative learning with your students. Which brings us back to classroom culture.  What is it and why does it matter?

There are many definitions out there of classroom culture. I will give you my perspective. Classroom culture is a classroom environment where students feel safe making mistakes, they are comfortable sharing their thinking process with other students and with the teacher, and all ideas are entertained and acknowledged. Everyone’s voice is heard, everyone gets a chance to participate, and there is respectful conversations and debate about the work being done.  This matters because then students are given permission to persevere in problem solving situations where they may not know the answer, or may have a different approach then someone else or may have a question about something another student or the teacher has shared. It ties into those mathematical practices (#1 & #3, just to name a couple):

• Make sense of problems and persevere in solving them
• Construct viable arguments and critique the reasoning of others

But, this type of engagement, discourse and collaboration with and among students doesn’t just happen. Here are what I consider the three basic elements:

1. Planning

Planning entails thinking about the structures you want to use with students (so pairs, small groups, whole class) and the types of discussions and work you want to students to engage in. There is more to it than this, but some things to think about are

• What task are students working on and what is the goal (a worksheet of 40 problems is NOT going to promote student discussion). Provide a rich task that fosters critical thinking, questioning, problem-solving.
• How do you want students to engage? Are they talking in pairs first and then sharing with the small group? Does each pair/group need to show some product (i.e. their work, their thinking, the end result).
• How will you bring the whole class together at the end? Will each group share out? Will you hang work and have a ‘gallery walk’ and come together to share?
• How will you know that students have learned or reached the goal? What should students be able to do?

You need to think of these things ahead of time, most importantly because without an engaging, rich, though provoking problem, the conversations students have won’t be productive (and can lead to all the issues mentioned previously).

2. Training

How do you get students to talk about math (or any subject?) How do you get students to work in pairs or small groups and stay focused on a task? How do you get students to listen to each other and to provide critiques without insult (i.e. no ‘that’s stupid’ or “you’re an idiot”). It takes training.  I mean that literally. You have to show and model what it is you expect of them and practice, practice, practice.  Again, there is more to this than what I am listing, but here are some ideas:

• Start those first few days/weeks of school with non-content related activities that are non-threatening, fun, and where everyone feels comfortable sharing (so talk about ‘the best horror movie’ or argue for/against a ‘beach is the best place to vacation’)
• Set up group norms – i.e. if someone is talking, everyone else is listening; everyone makes mistakes, and that’s okay, you can support them and provide alternatives, but never insult them; everyone must contribute one idea; everyone’s idea should be heard; you can disagree but must provide a reason why; etc.
• Show them how to get into small groups (so physically moving desks back and forth – it’s fun to do this a timed game); show them and practice how to talk with elbow partners, or face-partners, or the people next to them.  Practice sharing talk-time (a time works here).
• Show them and practice group ‘roles’ – i.e. timer, recorder, controller, group spokesperson, etc. Switch roles up.
• Practice different ways of calling on students (so they know they are all responsible at any time) – so person in the group/pair with the shortest hair, or the darkest colored shirt, or blue eyes….really anything works.

There are obviously lots more ways to set up these collaborative processes, but the idea behind training is that there are some expectations for talking, sharing, and working together, and if we practice these and adhere to these, then our time learning is going to be more positive and productive. Practice, practice, practice.  Which leads to consistency.

3. Consistency

I know teachers here this all the time – if you set boundaries for your classroom, you need to be consistent or students will not follow them (heck, this is true for parents as well!). Again – those first few days and weeks of school are where you set these boundaries up and start practicing with students and modeling both behaviors and actions. More importantly, follow through on any consequences. For classroom culture, this means if you have an expectation that students listen when others are talking, whether that be student or teacher, then be consistent.  If you are talking and they are not listening, stop – call it out – and then talk again. Same thing for students talking. Acknowledge when something is not adhering to expectations and call it out and then refer back to your expectations. Students very quickly learn what is expected, and if they realize that you are going to consistently hold them to these expectations, such as listening, allowing for mistakes, everyone’s ideas matter, etc., then they are going to feel comfortable speaking up and sharing their questions and their solutions/ideas. It becomes a classroom where learning is up front and center and ‘we are in this together’ becomes the norm.

CHALLENGE

I plan to do some more specific posts about classroom culture and provide some resources connected to planning and training. For now, I brought this idea of classroom culture up at the end of a school year because as teachers, you are about to embark on a summer of rest and relaxation. For most teachers I know, it is also a time where we do some personal learning and planning for next year. I would like to challenge all of you to really think about how you want your classroom culture to be next year. You need to start on day one of school creating this classroom culture, so spend some time this summer planning for that. What structures do you feel you could incorporate (i.e. pair work, small groups, etc.) and learn about those structures. What are rich tasks and go find some that would work for the content you teach. What do you want students doing when they are learning together? Go find some tips and ideas for how to create those collaborative discussions and problem-solving environments.

Only YOU can change the classroom culture in your own classroom – so think about what you want that to look like and sound like, and spend some of your summer learning and finding ways to foster this culture in your classroom when school starts in September (or August).

# Fractions with a Calculator – Looking for Patterns

I 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. DynamicNumber.org 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.

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.

# Math Magic or Calculators?

I was perusing my news feed trying to find something of interest to write about, and came across an article entitled “The Common High School Tool That is Banned in College” i.e. the calculator. It’s an interesting article, worth a read,  basically comparing the high school perspective on the use of calculators to the college perspective or non-use of calculators. There is no right or wrong answer – I think it depends on the math content, what you want students to do (i.e. basic algorithms to solve problems or using mathematics to solve deeper problems).  Depending on your goals, the use of calculators and technology differs. As with any technology, calculators are a resource that needs to be used appropriately, and we need to be teaching that.  Common Core Mathematical Practice #5 – Using Appropriate Tools Strategically is all about this. Calculators have their place and are important to help explore and expand mathematical understanding, but we have to help students understand when their use is necessary and not a ‘crutch’, as stated in the article.

This was on my mind obviously, when I then ran across a tweet post by Go!Math Videos @gomathvideos that shared a TedX talk by Arthur Benjamin entitled “Faster than a Calculator”, which naturally sparked my interest and seemed related to the question of should we be using calculators. In the video, Arthur Benjamin has members of the audience use calculators while he does calculations in his head. He then goes on to wow everyone with his math ‘tricks’ (what he calls mathemagics). He ends by doing a 5-digit square calculation by thinking out loud as he ‘solves’ a problem. It’s fascinating – he changes numbers to words to help him solve – he is definitely using his own ‘algorithm’. The video does not answer the question should we be using calculators – but it definitely shows that calculators are just one way to get a solution and it may not always be the fastest. Anyway – just some fun for this last post of 2016. Enjoy!

Wishing everyone a Happy and Safe New Years!

# The Last Five Minutes of Class

You teachers out there know that those last five minutes of class – when students are ‘packing up’ even if you are not quite finished with the class activities, or, you’ve finished and they are suppose to be working on their assignments or reviewing – are often  a ‘wasted’ five minutes. In my many years at schools, I saw teachers use that time in various ways – but more often than not, it was simply time to get ready to leave, basically chat time and get your stuff together and wait time. Not productive learning time at all.

It’s easy enough to make these moments into fun, engaging, mathematics problem-solving that students, believe it or not, actually come to enjoy and request. I use to have a few different things that I would pull out – focusing on either logical thinking or number-sense or puzzles. Here are just a couple of things:

1. I had the 24-game – several different versions.  I(If you have never played this or seen this, you should explore it). So, in those last 5 minutes, I would pull out a card, write the 4 numbers on the board and students would try to reach the target of 24. As an example: 2, 3, 4, 4 and you can add, subtract, multiply or divide using each number one time, to make 24. I often had candy for anyone who could come up with a strategy.
2. If you don’t have actual cards, you can create your own version of ‘reach the target’.  So, pick 4 random numbers using a calculator, and give students a target number to try to reach (so 24). Or, choose 2, 3, or 4 random numbers with a calculator (or have students give you numbers) and ask students to use all the operations and come up with the smallest outcome and/or the largest outcome.  This is a lot of fun – you get some interesting problems and students have to explain their answers and defend their solutions.
3. Give the students a logic puzzle.  I actually purchased several logic books, and so would read one out to the students or draw/show the picture on the screen and they could work in pairs/small groups to try to come up with a solution. Great critical thinking and collaboration going on here – and if we couldn’t get the solution before the bell rang, we would take it up the next day with most of them working on it overnight. Here are some good resources for logic puzzles:
4. Read a story.  Yep. Even with my high school students, I would read stories.  Math related of course. You would be amazed at how they actually enjoyed listening, and of course, once the ‘story’ was finished (which might take a couple days depending on our actual time at the end of each class) we’d discuss the ‘math’. Some of my favorite books:

Students loved the challenge of these last 5 minutes (sometimes it would be more). It was a very competitive yet non-threatening time where they could test their math skills or thinking skills, work together, and have fun with numbers and logic. That time was no longer wasted – it became a time students actually looked forward to and often requested.

As you are nearing the winter break, there is probably a bit more time to spare or a bit more time needed to keep students attention.  Use that time in an engaging way that allows for some critical thinking, collaboration and a game-like atmosphere that challenges students and keeps those last five minutes productive.

# Math and The Electoral College

With the election looming, and yet another Presidential Debate this evening (anyone else dreading it?) and more polls than you can shake a stick at, it seems appropriate to think about the math behind the Electoral College. I admit to really not understanding this whole system – and I know I am not alone. With the rampant conspiracy theories about the November 8 election, and a ‘rigged election’ and cries to eliminate the Electoral College and go to a popular vote only, it had me diving into “what does the Electoral College mean, why do we vote this way, and is it fair?” I think this is a GREAT conversation and critical thinking activity to have with students, especially in classes like statistics where you can actually study and do ‘mock votes’ and see what the outcomes are with or without the Electoral College.

A quick summary of what the Electoral College is – and please note, I still am a little iffy about whether I truly get it. In 1787 the delegates of the Constitutional Congress made the decision to do this indirect way of voting for the President of the USA. It was a compromise between those who wanted a) individual citizens to vote for President (1 person, 1 vote, majority wins); b) letting State legislators choose the President; or c) letting congress choose the next President. The idea was to create a method where the best candidate was chosen. Individuals in a state vote for President – the winner in that state gets all the states electoral votes (though some split the electoral votes now), and the electors (who are elected by voters),  put in the final vote for President. The person who gets the majority of Electoral Votes (270 or more) wins. Still confused? How is this fair?  Bear with me….I am hoping I can figure that out myself!

If you look at the image above, which outlines the number of Electoral votes per state, you can see a huge difference – some states have an enormous number of Electoral Votes, and some very few. As you can see – size might matter (CA, TX, FL), but not always – VA is relatively small and yet has 13 Electoral Votes compared to say Montana, a larger state with only 3 Electoral Votes. So – how is the number of Electoral Votes determined? Hawaii gets 4 vs the very large state of Alaska only 3. So – it must have something to do with population numbers, which in fact is the case. The number of Electoral Votes is the number of state representatives in Congress (both Senate and House of Representatives), which are based on the population of the state. Every state will have at least 3 Electoral Votes (2 Senators, 1 Congressman). Obviously you now see why winning states like CA, TX, FL, PA, and NY are so crucial because of their large populations and large number of Electoral Votes.

I have been reading a lot and searching for good websites that might be helpful for teachers wanting to figure this out with their students. There are several sites that talk about the Electoral College and what it is – I didn’t find these too helpful from a teaching perspective, but they may be of interest to some of you from a historical, “why do we do this” perspective.

1. Nice interactive map – http://www.270towin.com/
2. Article about the ‘fair or unfair’ aspect of Electoral College and the funding – not sure it answers the questions but makes you think: https://blogs.scientificamerican.com/guest-blog/the-funky-math-of-the-electoral-college/
3. This is a nice site with lots of historical perspective and answers to questions, like does my vote count? http://www.learnnc.org/lp/media/lessons/davidwalbert7232004-02/electoralcollege.html

Here are some sites I found that would be helpful for doing some simulations and having interesting conversations with students. Many of these are interactive, with the ability to create election results (or simulated) to get a better understanding of how the Electoral College system works, and hopefully make a determination about whether it is fair or not.

1. This was my favorite – be sure to check out the “Play Presidential Politics” link, as it has a simulation vote where you can create your own populations for states.  Would be great for students. http://www.sciencebuzz.org/topics/electoral-college-math
2. Lots of information (some of which I used) in kid-friendly language: http://www.congressforkids.net/Elections_electoralcollege.htm
3. Nice lesson here – some interaction/lesson plan info as well: http://www.scholastic.com/teachers/article/math-majority-rules
4. Yummy Math – nice lesson here (using the picture above!) http://www.yummymath.com/2016/electoral-college-vs-the-popular-vote/
5. From NCTM – a lesson on the “fairness” of the elections – Love Illuminations! https://illuminations.nctm.org/lesson.aspx?id=2825

So – is it fair or not fair? Does your vote count? I am not sure I can give you a definitive answer. It probably depends who you are, who you want to win, and where you live. But, in my readings looking at several charts that compared the ‘weights’ of individual votes toward the outcome (i.e. Does your vote count?), I think my personal opinion is yes, your vote does count, and yes, it is fair.

Notice Alaska, with a smaller population, has a much more weighted vote compared to CA. This may not seem fair – but, Alaska, a huge state with many diverse needs and interests but with a small population, deserves an equal representation in the government, which may not happen in a one-vote majority rule election, if we look at populations sizes of CA or NY, with their enormous populations. This is why the Electoral College was created in the first place – so every state gets a fair share of representation for their interests in the outcome, no matter their populations size, and those ‘states’ with larger populations don’t end up  deciding everything. NY’s interests are vastly different than Alaska’s, after all. A popular vote would be unfair because those larger states, who lean more one way or another, would control the outcome, leaving those states with fewer people, left out of the equation, and their interests not accounted for or lost in the process. My personal conclusion – I am actually for the Electoral College, after all my reading and my still foggy, but much better, understanding of the system.

One final note – look at the overall United States (in chart above) – the total population, the total Electoral votes, and the weight of each individual vote.  It’s 1.

So YES – your vote counts – get out and VOTE!  (And make it an educated vote, based on candidates proposed policies and plans – not based on emotion).