# Geometry and the Holidays

The holidays are upon us, so of course it makes complete sense to look for geometrical connections. Or maybe that’s just me?

As a geometry teacher (just finishing up a Geometry & Spatial Reasoning course), I am seeing geometry connections everywhere. From the wrapped presents, to the origami ornaments, to the snowflake patterns, I am constantly looking for those real-world connections and easy (and cheap), ways to get students working hands-on with math.

We are all familiar with ‘holiday math’ problems that connect to wrapping presents – i.e. how much wrapping paper do you need, how much ribbon, etc. Area, surface area, linear length connections all very obvious. But, as a geometry teacher, I am also curious about the gift boxes themselves. I know it is often difficult to find 3D models for learning, so boxes provide a cheap way to provide students hands-on explorations of nets, area, surface area, volume. So – teachers – get your students to bring in boxes after the holidays – so much you can do with these!!

Another thought – origami. This time of year, teachers often create holiday decorations with their students with paper-folding, which is fun, obviously, but can also be a great way to apply many math concepts. Shapes, fractions, and transformations for example. Take the following two origami designs – a star and a tree. As you are folding, you could be having students think about the individual shapes, but also the dimensions, the fractional parts after making a fold, what types of transformation have occurred – even congruence and corresponding parts.

I

For example, in the star above, after folds #1, what fraction of the square does each smaller square represent? When we fold that triangle in #2, what type of triangle is it? What fraction of the original square is represented in that yellow triangle?  What type of transformation does each fold represent? Are the triangles in #3 and #4 congruent? How do you know?

Again, looking at the tree folding above, what shapes do you see in #1? What fraction of the whole paper is each shape (so squares and triangles)? How about in #2? And which shapes are congruent? How do you know? Lots of great math, that you could really explore with students while they are also doing a fun hands-on activity.

Hopefully you can use some of these ideas with your students. Have a wonderful holiday season!!

Advertisements

# Financial App (Pt 3 in series) – Let’s Talk About Money

With the holiday season upon us, and people often spending beyond their means, it seems appropriate to continue the CG50 (and all Casio Graphing calculators) app exploration with the Financial App.

One thing we do not spend enough time on in K-12 education is financial literacy. I know there are some states that are trying to address this, but it is not enough. This lack of understanding about money, savings, taxes, interest, debt, etc. is a huge contributor to our enormous debt crisis. Take our current political focus on the ‘tax reform’ bill that’s up for a vote soon – most people do not understand the ramifications of this because they don’t really understand anything about finances and how taxes work. We do not in this country teach the basics of financial literacy, which is why we have so many people drowning in debt, losing their homes, barely surviving month-to-month on what they make, and forget about having the ability to save for the future. How many students really understand about saving money? Or how taxes impact their hourly wages (i.e. \$10/hour is not that great when you factor in all the taxes taken out)? Or how not paying of your credit card monthly can make that \$300 dollar purchase become a \$400 or \$500 dollar purchase?

When I taught in Virginia, they started a Personal Finance course ‘elective’ (only for those students technically not on the college prep track – which was silly, as ALL students should take a course on Personal Finance). I was lucky enough to be the pilot teacher in my school, so I could pretty much create the course. My goal was to help students understand the importance of financial planning so they could survive and thrive in the world, no matter where their path took them. We started with learning about different career options they were thinking of, and what a typical annual salary might be (so plumber, electrician, hair dresser, doctor, lawyer, teacher, etc). They learned to fill out job applications, and write resumes, and then we ‘pretended’ they had been hired and were receiving biweekly payments (I actually gave them ‘checks’). We learned about payments, investing, taxes, rent, credit cards, insurance, amortization,balancing a check book (the class had a ‘bank’), etc. They had to determine where they would live, whether they would get a car, how much they could spend on food, entertainment, etc. based on their salary. What they quickly learned is that their wages, after taxes, were often NOT enough to do much else – no fancy apartment and having to make tough choices (i.e. gas or food, no car, no expensive smartphone, taking bus, walking, no movies every week, no fast food, etc.) When a student comes to you all excited about their \$9/hr job and all the things they will buy, and then realize after their first paycheck that it’s going to take months to have enough, it’s eye opening. And scary.

What I learned is that we do not talk to students about real-world, practical mathematics enough –  simple things like saving money, calculating tips, balancing a checkbook, interest, credit card debt, etc. This is math they need in their everyday life. This is math that has purpose. This is math that will help them make smarter decisions about their future. Maybe if we did, we wouldn’t have so many people struggling to survive or believing every unrealistic promise they hear in the news..

My message – let’s get some Financial Literacy into K-12 mathematics programs!

With that said, here is a quick video on the Financial App that is available on the Casio graphing calculators. This video uses the CG50.

# Graphing How-To – Hyperbola & Asymptotes on the CG50 Prizm

I realized I haven’t posted a ‘how-to’ video in a while, and, with this being the end of summer, there might be some students and/or teachers out there trying to learn some new skills before the school year starts up, hence this post. Why hyperbolas? No reason other than it helps highlight the Conics menu and I love the way it looks! My plan is to try to do some more how-to’s on a monthly or bi-weekly basis, especially as school starts up, so if there is a specific topic you want to explore, please let me know, as well as a specific Casio Calculator. There are several how-to videos out there already on my YouTube channel and Casio’s YouTube Channel, but if you have a specific content/calculator in mind, I will do my best!

The video below models one of the problems in our free Quick Start Guide for the CG50 (our newest version of our colored graphing calculator) on how to graph a conic section, in this case a hyperbola, using its equation, and then from that, finding its asymptotes, coordinates of the vertex, and the coordinates of the foci. This will be a nice example of how to use the Conic Menu option on the CG50 graphing calculator as well as our other graphing calculators, since the steps are the same.

Enjoy!

Problem: Construct the graph of the conic section given by this equation: . Once graphed, find the asymptotes, the coordinates of the vertices and the coordinates of the foci.

# Access & Equity in the Classroom – A Teachers Role (Equity, Equality, and Access to Quality Education -Part 3)

This is the 3rd installment in my 3-part series on equity, equality and access to quality education. Here are links to Part-1 and Part-2, where I first define these terms and then I talk about funding issues that impact access and equity. As noted in Part 2, funding is a huge component of why schools and districts don’t provide equitable access to support student needs, and why low-economic areas tend to have inequitable education experiences and poor access to the supports and resources needed to help all students learn and achieve, based on their individual needs.

As a teacher, school funding is out of our hands for the most part (except for the personal funds we all spend to make sure the students in our classroom have resources and support). Parents and community leaders need to take a really close look at the money teachers spend out of their own pockets to address some of the inequities within their own classroom and school – it’s not right, it’s not fair and there needs to be more push-back on education policy and more support from local businesses, community advocates, and state and local school boards to ensure that schools that need funding and resources are getting those in an equitable fashion (remember, not equal, but equitable – all schools do not need the same). Teachers will spend their own money, even when they have very little, because they care about their students and what happens in their classroom, but they shouldn’t have to.

But, I digress.

What I want to talk about in this post is what teachers can do in their classrooms to address equity and access to quality education. Teachers, even without adequate funding, resources and support, are the most able to provide equity and access for the students in their classroom because that is where the learning happens. And it’s the learning, it’s the teaching strategies, it’s those interactions and learning experiences that can provide equity and access for all students. Let’s remind ourselves about what equity and access means – it means each student getting what THEY need to learn, meaning they have access to rich learning experiences and teaching that provides them with the support they need to understand the content, to think, to make connections, to apply that learning, and to achieve to their potential. To learn, despite their gender, their race, their socio-economic status, or their disabilities.

I can only speak from what I know, so I am going to take a mathematical approach to equity and access in the math classroom, but even if you are not a math teacher, these ideas and processes work in your classrooms as well, with the only difference being in the content.

NCTM (National Council of Teachers of Mathematics) has a position for what it means to have equity and access in the math classroom, so I am including it here (this links to the full article):

Creating, supporting, and sustaining a culture of access and equity require being responsive to students’ backgrounds, experiences, cultural perspectives, traditions, and knowledge when designing and implementing a mathematics program and assessing its effectiveness. Acknowledging and addressing factors that contribute to differential outcomes among groups of students are critical to ensuring that all students routinely have opportunities to experience high-quality mathematics instruction, learn challenging mathematics content, and receive the support necessary to be successful. Addressing equity and access includes both ensuring that all students attain mathematics proficiency and increasing the numbers of students from all racial, ethnic, linguistic, gender, and socioeconomic groups who attain the highest levels of mathematics achievement.

This means that all students should be engaged in real-world learning, problem-solving experiences, and applications of the content. These types of learning experiences are not just for those ‘advanced’ students. This means providing opportunities for students to engage in collaborative learning, where they are communicating their thoughts and ideas with others, where they are taught and allowed multiple approaches and multiple solutions, where they have supports (i.e. questioning by the teacher, partnering with others, hands-on materials, technology/visuals, etc.) that might help them make connections or get to that next ‘aha’ moment.  Lower-performing students shouldn’t be relegated to doing drill & kill worksheets and ‘remedial’ math classes where the focus is on test-taking strategies and memorization, but rather should be exposed to the same challenging problem-based, inquiry approaches as the high performing students, but with different supports to help address their needs (so scaffolded questions, or suggestions on strategies, or working with a partner, etc.).

A large part of this equity and access means teachers need to BELIEVE that ALL students can achieve and learn, with the difference being that some need more supports than others. I can’t tell you how many times I hear, “well, my lower-level students can’t do that” or “my students won’t talk or show me different approaches” or “my students will just wait for the ‘smart’ ones to do all the work’ or “my students have a hard time reading so we don’t do word problems” or “my students will just give up or just ask me to show them the answer”. I could go on, but I think you get the point (and have perhaps made those same comments yourself). It becomes a self-fulfilling prophecy if you think this way, try something once and it ‘fails’, and therefore you don’t do it again – and then you and the students believe they can’t learn, or they can’t talk, or they can’t solve problems, etc. This is where inequity becomes a huge issue in classrooms – because we then resort to teaching students the ‘one way’ to do things (i.e. often the ‘way that’s on the test), and those students who need a different approach or who can’t memorize, can’t ‘perform’ or ‘achieve’ because they are NOT getting what they need to learn, and the cycle continues. To promote equity and access within your own class, you need to do some planning, some hard work up front, and be consistent – but it can change how you teach and how students learn so that all your students are getting what THEY need to learn. As a teacher, this is your responsibility within your own classroom.

Here are some suggestions:

1. Starting day one, begin creating a classroom culture that promotes communication, collaboration, and respect. Students need to ‘learn’ how to talk with each other and listen to each other – so practice getting them in and out of groups, sharing ideas (start with non-academic sharing first, like ‘what’s the best movie you saw this summer and why”), working with partners and presenting their thoughts. Practice respectful listening. Practice and model appropriate responses when someone might make a mistake (mistakes should be accepted as part of the learning). There are several places to go to help you learn some collaborative teaching strategies – this is a nice list of articles with good tips.
2. Learn to ask questions instead of giving answers or telling students they are right/wrong or yes/no. Simple questioning skills force students to start thinking, communicating, making connections, asking their own questions. Again, many resources out there to support questioning skills and provide some sample questions (“Why” is always a good one, or “Can you explain?”). Here’s one resource.
3. Set high expectations and be consistent with those from day one. Expect students to not only show their work, but to explain their thinking (write out in words or draw pictures or explain verbally). Model this when you teach or show things to students (think-out-loud is a great way to model this type of behavior in mathematics class). Consistency is important!
4. Provide problem-solving strategies from the beginning so that students realize that they have multiple ways to approach an unknown problem or situation. These are great strategies to incorporate in those first couple weeks of school and then to reference as they come up the rest of the year. And yes – even elementary students need problem solving skills.  (Notice & Wonder should become a habit of mind for all students, no matter the age because it provides that ‘think time’ and that ability to try and connect to prior knowledge and use what you know). The Math Forum is a wonderful resource for learning about the strategies and for getting problems to use in class.
5. Expect and allow for multiple ways to approach math problems. As long as students can justify what they did and it is mathematically sound reasoning/thinking, it should be okay. This is probably the single most important piece to equity in the math classroom – allowing students to solve problems multiple ways, using the strategies and methods that work for them, and allowing for multiple solutions/solution pathways. This is the hardest thing for teachers i think because we ‘know’ the ‘right’ way – but the right way is not the only way, and some students may never get the ‘right’ way, but they have a way and it gets them there and that should be okay AS LONG AS THEY EXPLAIN THEIR THINKING (see #3). To make this work, see #4.
6. Provide interesting learning experiences that promote thinking, multiple pathways to a solution, even multiple solutions. You will not get students working and communicating if you give them a worksheet with 30 process/skill based problems. You need to find interesting, relevant, problem-solving experiences that engage all students, that allow all students, no matter their ‘ability level’, a way to start thinking about solving. These types of problems should require previous math content knowledge and/or applications of new math content, require some analysis…..so think rich tasks.  There are many resources for interesting problems out there – content-related too – (Math Forum, Mathalicious, YummyMath, Illuminations, links to other resources)
7. Less lecture, more inquiry, student-based learning. Hands-on, visualizations, student questioning, student explanation. This does not mean you need to have a different activity for every student – that would be exhausting. You need to find learning experiences that address your content that allow all students a way to ‘enter’ the learning from whatever level they are at.

Teaching one way and expecting the ‘same’ approach for all students, no matter the level, will always leave some students behind and others stagnating.Our teaching should always be focused on the standards and content, with the way we structure the learning and the way we allow students to demonstrate their understandings providing the differentiation that will let all students achieve – those who are ‘behind’ learning to catch up and those stagnating able to move ahead and explore. The more students can connect with, engage in, and explain mathematics using what they know  and building on this knowledge, with the teacher guiding them to deeper understanding through questioning, modeling, and supports as needed, the more equitable the learning becomes.

# The Access Formula – (Equity, Equality, and Access to Quality Education – Part 2)

In last weeks’ part 1 of this series on equity, equality, and access to quality education, I defined access to quality education as “the ways in which educational institutions and policies ensure—or at least strive to ensure—that students have equal and equitable opportunities to take full advantage of their education. Increasing access generally requires schools to provide additional services or remove any actual or potential barriers that might prevent some students from equitable participation in certain courses or academic programs.” Access then encompasses many aspects of education, from funding, to resources, to programs and services that help ensure that all students are getting an equitable education (what they need to support learning).

Obviously, all school districts and schools strive to provide access to needed services and supports for their students. There are federal laws in place designed to ensure that all students are getting access to equitable education and getting supports they need. The Individuals with Disabilities Act (IDEA) is a law that all public schools must adhere to, which ensures that students with disabilities (including learning disabilities) have access to the least restrictive educational setting, have rights to ensure they get the services they need to support their learning, and parents can have a say in the educational decisions made regarding their students. An example of this from my own teaching is having an interpreter in my math classes that signed for the deaf students in my classes, or a student with an IEP (individual education plan) who needed copies of my notes because they had a learning disability that interfered with their ability to take their own notes. There is also Title I Laws and Funding, specifically designed to address low-income and disadvantaged students and ensure that schools that serve these students are getting the funds they need to support achievement, through things like extra academic supports for reading and writing, pre-school and after-school programs, with the goal to improve achievement on state standardized testing. I won’t go into all the details (link provided gives more information), but the idea here is to provide additional academic supports to low-income& disadvantaged students who are struggling academically. There are no federal laws that pertain to gifted students needs, though there are individual states and local schools that provide resources and supports for gifted education, but it varies by state.

As you might surmise, there is definitely an attempt to provide access to equitable education. But what’s the reality?

From my own experiences, access is NOT equitable. I would wager in most school districts, there is a huge disparity between what resources are available and the quality of education received at various schools within the same district and between districts within the same state. I have worked in many urban school districts where one middle school has low-achieving students at computers every day in math class, working on computer programs designed to support their mathematical skill development, and the other middle school down the street barely has enough rulers for the 42 students in the class, with a teacher who just has one computer and projector and a room full of ELL students speaking 5 different languages. Same school district. Not equitable access to resources. Not equitable learning environments or supports.

This is NOT an isolated situation, as I am sure many of you have experienced similar situations personally, whether as an educator or as a parent. Why is there such disparity when there are laws designed to ensure access to equitable resources and education opportunities?

The obvious answer is funding, which is a huge factor in access discrepancies. While everyone thinks funding for schools comes from federal money, federal funding makes up only 8% of public schools funding, so those TitleI funds and IDEA funds only accounting for a very small portion of education funding overall.  92% of funding for public schools comes from the states themselves, from taxes, lottery receipts and other sources, and a large portion from local property taxes. This means two school districts next to each other, one with a lower-economic base and less property tax, and one with a higher-economic base and more property tax, are going to have vastly different educational funding and resources. High-poverty schools spend less per student, and often have a more difficult time keeping quality teachers because of the disparity in available resources and services. It’s a vicious cycle. Many of the problems come from the way states/school districts apportion school funding, using different funding formulas that from the outset are ‘unfair and unequal’:

In a separate report, “Is School Funding Fair? A National Report Card,” the Education Law Center answered its own question with a resounding “No.”  Among the findings:

• Fourteen states, including Texas, Pennsylvania and Illinois, have “regressive” school funding, defined as providing less money to schools with higher concentrations of students from low-income families.
• In 19 states—including California, Florida, Colorado and Washington—the funding systems are defined as “flat,” meaning they “fail to provide any appreciable increase in funding to address the needs of students in high poverty districts.”
• Only four states have school-funding systems that earned “fair” ratings: Minnesota, Massachusetts, New Jersey, and Delaware. But “these states have a sufficient overall level of funding and provide significantly higher amounts of funding to high poverty school districts,” according to the report. (from https://www.theatlantic.com/education/archive/2015/06/how-funding-inequalities-push-poor-students-further-behind/395348/)

If we go back to what equitable education means – students should be getting the resources, supports and instruction that supports their individual needs. With the type of funding described above, this is clearly not what is happening (nice graphic here by state showing the difference by state in funding for poor school districts vs. wealthy school districts). Poverty and all that it entails, seems to be a huge factor in the access to equitable quality education. It impacts parental support (hard to be there to help with homework or pay for tutors if you are working two jobs to make ends meet), access to educators and schools support personnel such as school counselors and nurses, access to technology, education supplies (paper, manipulatives, textbooks, calculators, etc.), school safety, and all the myriad of other components that make up quality education.

When thinking about education and how to provide access to the resources every student needs, we need to restructure how states and school districts distribute school funding and resources. Districts need to do a true evaluation of what each school in their district has in terms of education resources (materials, technology, personnel), and most importantly, what they need to support those students in those specific schools. If the two schools down the street from each other in the same school district have math classes in one school of 20, with students at computers, and the other school with math classes of 42 and no computers and not even enough seats or rulers for students, there is clearly a problem. The first step is to really do an inventory of what schools have, what class rooms and sizes look like, what personnel resources are and really identify the glaring discrepancies. You can’t fix something if you don’t even know where it’s broken – once we see the vast difference in access within school districts, then we can take that next step of thinking about ways to reapportion some of those resources in a more equitable way.

# Equity, Equality, and Access to Quality Education – Part 1

Back-to-school is already upon some, and for many will be starting up in the next few weeks. With that in mind, and especially with the very public conversation around school choice and ESSA and accountability for schools, I’ve decided to do a 3-part series on equity, equality, access and quality education. These are ‘buzz’ words that are thrown about in news stories and education settings, but I think often times these words or terms are used incorrectly, or interchangeably, with many people not really understanding what is really being said or what the meaning behind these terms actually might be. With that said, this first part in my series is going to focus on defining these three terms so that we are all on the same page and have a common understanding in which to move forward.

Quality Education

This term is loaded. Everyone wants a quality education for their child and schools and states strive to provide quality education for all their students. But what does this mean? What does this look like? I am going to define it here and in later follow-up posts we will dive more deeply into this.

There are many definitions out there for what quality education means. I actually had a hard time finding an ‘official’ definition, but found the term ‘quality education’ used frequently in vision/mission statements from many education organizations and school districts. Which is interesting – we use the term, yet we don’t define it, so how are we ensuring that students are indeed getting a quality education?

Here is a definition of Quality Education from ASCD (Association of Supervisors of Curriculum Development) and EI (Education International) which I think provides a strong common understanding that will connect to equity, equality and access.

A quality education is one that focuses on the whole child—the social, emotional, mental, physical, and cognitive development of each student regardless of gender, race, ethnicity, socioeconomic status, or geographic location. It prepares the child for life, not just for testing.

A quality education provides resources and directs policy to ensure that each child enters school healthy and learns about and practices a healthy lifestyle; learns in an environment that is physically and emotionally safe for students and adults; is actively engaged in learning and is connected to the school and broader community; has access to personalized learning and is supported by qualified, caring adults; and is challenged academically and prepared for success in college or further study and for employment and participation in a global environment.

A quality education provides the outcomes needed for individuals, communities, and societies to prosper. It allows schools to align and integrate fully with their communities and access a range of services across sectors designed to support the educational development of their students.

A quality education is supported by three key pillars: ensuring access to quality teachers; providing use of quality learning tools and professional development; and the establishment of safe and supportive quality learning environments. (retrieved from http://www.huffingtonpost.com/sean-slade/what-do-we-mean-by-a-qual_b_9284130.html)

Equity and Equality

The definition of equity in the dictionary is “the state or quality of being just or fair”. The definition of equality is “the state of being equal, especially in status, rights and opportunities”. So what does this mean in terms of education, especially as these two terms are often used interchangeably, when they are very different when it comes to education? Let’s look at each separately in terms of education.

Equality in education would mean that all students are treated the same and are exposed to the same opportunities and experiences and resources. This is deemed as fair because everyone is getting the same instruction, the same assessments, the same resources, the same access to teachers. However, if students are coming into a classroom with different capabilities and different backgrounds – which is the reality no matter where you are – (this means educational knowledge, socio-economic status, family support, etc.), then treating them equally is going to disadvantage most students. No one will get what they truly need to learn – most will not get the appropriate supports and opportunities they need to be successful and to learn to their full potential (as examples, those with special needs would not get the additional supports needed and ‘gifted’ students would not be exposed to more challenging learning experiences they might need).  Everyone gets the same and so everyone suffers to some extent.

Equity in education means that all students get what they need from education, meaning instruction, assessments, resources are distributed so that every students individual needs are met in a fair way so all students can be successful. This relates to the statement above, under quality education, that students have access to personalized learning so that their educational needs are supported, allowing them to be prepared for future success, whether that be a career, college or some other aspiration. So unlike equality in education, equity in education is not the same for everyone, rather it supports everyone with what they need. A students socio-economic status, gender, race, or ability level do not prevent their access to education resources and opportunities. Equity does NOT mean equal. Equity implies an education for each child that meets their specific needs,  both pedagogically and developmentally, so they can be successful in their future endeavors no matter where they live or what their economic status might be.

Access

Access to education is closely tied to equity and equality. I almost didn’t separate it out, but I do think it is a key component behind why many students do NOT get equitable education opportunities. The goal of providing quality education to all students means we are providing them with equitable access to resources and learning opportunities – i.e. students with learning disabilities are getting the extra services and supports they need to be able to learn; students from low-income areas are getting the technology and materials and qualified teachers needed to address their instructional needs; students who excel at math or science are provided with technology and resources that allow them to explore and expand their understandings; students who are artistically or musically inclined are provided with teachers and courses that let them learn and create.

It was hard to find a ‘definition’ for access, because it’s really a process of ensuring students get what they need. I found this nice summation of access on the Glossary of Education Reform that I am going to use to inform our discussion going forward:

“The term access typically refers to the ways in which educational institutions and policies ensure—or at least strive to ensure—that students have equal and equitable opportunities to take full advantage of their education. Increasing access generally requires schools to provide additional services or remove any actual or potential barriers that might prevent some students from equitable participation in certain courses or academic programs”.

As you can see, all these terms and ideas are related, and it is often hard to think of them in isolation. Hopefully now you have a better understanding of each, and in our follow-up posts, we will explore issues surrounding these using our common understanding.

# 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