#TheMathContest – Supporting Student Problem-Solving

My last several years in high school, I was a ‘roving’ teacher, meaning I didn’t have a classroom of my own, but switched classes just like the students. This made for a very challenging prep experience, and required me to be super-organized and self-contained on my little rolling cart. The rooms I ‘borrowed’ for my classes did allow me to keep an area for my students (to turn in homework and pick up missing work, etc.). In each class, my students had a portfolio (i.e. file folder), where they kept their work, one of which was the daily ‘warm-up’ problems.  These were basically a set of 5-6 problem-solving activities – some applications, some skills practice, some real-world scenarios, some puzzles, etc.  Students were expected to pick this up daily and work on these in the first few minutes of class, which gave me time to: a) get there; b) check homework; and c) set up for the class/lesson, etc.  It gave everyone a chance to ‘settle’. Students had a choice – they could do some or all of the problems by the end of the week, and I just checked portfolios and work at end of week. We would always discuss possible solutions the following week (and also they earned points for their efforts).

Needless to say, since I was providing these problem-solving experiences daily, I had to find lots of different resources for these problems, especially those that were more application and thought-provoking. Can’t tell you how many problem-solving books I purchased! There were other sources, such as The Math Forum P.O.W. (now no longer in existence, though their P.O.W. “s do still live on at NCTM), and even my textbooks had some great problems if you looked for them. The point is, it took a lot of effort to provide these challenges for my students. Obviously, I could have done it once a week instead, but for me, it served that duel purpose of focusing my students every day while I was en-route. My goal, and something I think all teachers should be striving for, is to provide students some challenges and problem-solving experiences on a regular basis – ones that may utilize prior knowledge or challenges them in different ways of thinking with new skills.

For those of you looking for such challenges, there is a new resource available from Ole Miss’ School of Education called #TheMathContest. It is actually a reboot of something Ole Miss did in the past, but it’s been revamped and improved, and now is sponsored by Casio Education and encourages the use of the new, FREE, online math software, Classpad.net that I talked about in my last post. Basically, new problems are posted each Monday, and each user can submit one answer per hour. Correct solutions earn points and you can view rankings on the website. Go to the link above to get more details on the contest. There are monthly rankings and annual rankings, which you can view online. How points are awarded is explained here.

This would be a great way to engage students and get them doing some challenging math, not to mention trying out the new software as well! If I were still in the classroom, I think I might add this as extra credit for students (for trying) and then maybe have a collaborative problem-solving time where we discuss possible approaches to the solutions after the previous weeks problem has ‘expired’.  Or maybe group students in ‘teams’ where they submit as a team? In any case, it would be nice to have a problem challenge already done for me each week, that’s for sure!

One thing Classpad.net is doing is posting video solutions to past P.O.W.’s which you can find on our Youtube Channel  Here is an example from May 7, 2018’s Problem of The week:

 The Problem:  Find the 1-millionth term ins the sequence {1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, ……}

 

There is still time to try this weeks #TheMathContest Problem of the Week for May 14, 2018!  And check out the rankings – you will see students from countries all over the world who are participating.

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Casio Scientific Calculator QR Code – The Power of Visualization

I was recently asked on my YouTube video channel if Casio’s graphing calculators also have QR code capabilities like the Casio FX991 ClassWiz Scientific Calculator. It was a great question – and my response was the graphing calculators don’t need that QR code because they already have the power of visualization. The purpose of a QR (Quick Response) code is to get information quickly, whether that’s an audio or a visual or data (usually on your mobile device). With graphing calculators, that is part of the calculator – we can enter data in many forms and see multiple representations of that data very quickly – a graph, a table, a function, specific points, etc.within the graphing calculator itself, making a QR code unnecessary. And, if you are using the graphing software/emulators, you can put these graphs and multiple representations up very quickly.

Why does the Classwiz then have a QR code? This is a scientific calculator, which is incredibly inexpensive (from $15-19), so what’s the reasoning behind including QR code capabilities? The answer – to add the power of visualization and make this calculator have ‘graphing’ capabilities at a fraction of the cost. You can enter data in the form of functions, tables, spreadsheets, and then have the ability to see graphical representations of this data with the QR code.

Here’s a short video that talks about the differences in the graphing calculator versus the scientific calculator and demonstrates the QR code. You will also see a comparison of the tables and graphs represented on both calculators.

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

pexels-photo-164501With 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 stock-photo-working-coffee-phone-work-check-budget-finances-personal-finance-e841754e-765d-426e-af94-4b6a4ce9891fthose 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.

pexels-photo-164527What 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.

Equation App (Pt 2 in series) – Solving Equations – Why Use a Calculator?

Solving equations is a large part of the mathematics curriculum as students move into those upper-level concepts. If we look at the Common Core Standards, students start solving one-step equations for one variable in grade 6, adding on to the complexity as they move into higher mathematics where they have multiple variables and simultaneous equations and complex functions. It is important to help students understand what solving equations really represents – i.e. determining the values of unknown quantities and to help them solve them in a variety of ways (i.e. graphically, using a table, using symbolic manipulation, and yes….using technology such as a graphing calculator). And connecting those unknown quantities to real-world contexts is a big part of this as well. Students should solve in multiple ways and express their solutions in multiple ways so that they really understand the inter-connectedness of the multiple representations (graphs, tables, symbolic) and what all these quantities mean in context.

That said, many teachers are reluctant to use the equation solver that is often part of a graphing calculator because, as I have heard multiple times, it does the work for the students and just gives them the answer. True. But – there are ways to utilize the equation solver so that it supports the learning, not just ‘gives the solution’. The obvious way, and probably the most frequent way, is to have students solve the equation (s) by hand, showing all their inverse operations/work, maybe even sketching a graph of the solutions, and then using the graphing calculator to check their solution. Very valid way for students to both do the work, show their steps, and verify their solutions. But – the reverse is also a great way to try to help students learn HOW to solve equations. Working backwards, so to speak.

By this, I mean, use the equation solver to give students the answer first, and then see if they can figure out how to use symbolic manipulation and inverse operations to reach that outcome. As an example, start with a simple linear equation, such as 2x – 5 = 31. Have students plug this into the equation solver and get the solution of 18. Then, in pairs or small groups, have students look at the original problem and try to figure out how they can manipulate the coefficients and constants using inverse operations to get to that solution of 18. So maybe, plug the 18 in for the x.  What would they have to do to the other numbers in order to isolate that 18?  This forces students to use inverse operations to try to ‘undo’ the problem and end up with 18. In doing so, they are discovering the idea that to isolate a variable, you have to undo all the things that happened to it.  Give them a harder problem. Same process….and let them get to a point where they try to solve using their ‘understanding’ of inverse, and then they use the calculator to ‘check’.  The idea here is students are figuring it out by starting with the solution and working backwards to understand the process for solving equations. And they develop the process themselves versus memorizing it.

Rather than thinking of the calculator as a solution tool, think of it as another way to help students discover where those solutions come from.

Here’s a quick video on using the Equation App (solver) on the CG50. The process is the same on Casio’s other graphing calculators. This is another installment in the app exploration series, started last week with the Physium App.

CG50 – What Are All Those Apps?

As many of you know, I post quick videos in the blog to show different things about the Casio calculators or math or teaching. Many of these are posted on my YouTube Channel. I will occasionally get comments from viewers asking questions, and I do my best to answer them. If I can’t answer the question, I find someone who can, or research until I do have a response. Just the other day, when I was asked “how do you use the constants on the CG-50 calculator”, I was not quite sure what was being asked, since I tend to use the calculator from a mathematics teaching perspective, and hadn’t explored using constants (from a science perspective) and wasn’t even sure what was meant by the ‘constants’ in this particular question (as it could mean the constants in a given equation).  Turns out the viewer was asking about the Physium Menu/App on the calculator, and how to get the constants from these tables and values into calculations. This is something I have honestly never used because I am not a science teacher and therefore rarely, if ever, have need for this app. But – it got me curious and seeking out an answer (which I did find and explore so I could give a reasonable answer).

In my ignorance, I realized that there are many apps on the CG50 (and other Casio graphing calculators) that I have never really explored, not just the Physium App. Mostly I focus on the most-used menu items – Run Matrix (to do calculations), Graph (to work with functions and graphs), Table (functions using table representations), Equation (solving equations), and Picture Plot. But there are a lot of other menu items that I need to explore and learn to utilize since they all are useful for different contexts and applications. This is now a goal of mine – to try to learn and explore the basics of the other menu items (apps) of the CG50 (and other) graphing calculator, starting with the Physium Menu/app. Here’s what I have discovered:

The Physium application has the following capabilities (so science teachers, take note!!)

Periodic Table of Elements

  • You can display the periodic table of elements
  • The table shows the elements atomic number, atomic symbol, atomic weight and other info
  • Elements can be searched for by element name, atomic symbol, atomic number or atomic weight

Fundamental Physical Constants

  • You can display fundamental physical constants, grouped by category to make it easier
  • You can edit the physical constants and save them as required
  • You can store physical constants in the Alpha memory and use these saved constants in calculations in the RUN-MAT menu/application

Now, I am still not a science teacher, so this would not be a menu item I will use often, but I wanted to do a quick video of what I discovered in my own exploration.  And – there is a link to the how-to guide for the Physium Menu/App for those of you interested in exploring more. If you have a CG10 or other graphing calculator from Casio and don’t have the Physium menu/app, you can download it here.

 

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 kids-girl-pencil-drawing-159823experiences, 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.

cute-children-drawing-teacher-preschool-class-little-40195392Here 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.

Learning from Webinars

Online learning takes many forms, of which webinars are one. For teachers, webinars are nice because they are usually content or instructional strategy focused and they are relatively short in length, so you can fit them into your busy schedule. Webinars are often live – meaning happening in real time, and you register and sign on at the designated date & time and can interact with the presenter (usually via a chat forum).  This is nice because if you have questions, you can ask them right away. But – the disadvantage here is you have to have the time to sit in on the webinar, which is often not the case, considering our crowded school and personal schedules. If you can participate ‘live’, I highly recommend it. When I worked at Key Curriculum and hosted our weekly webinars, I know the live interaction was a very positive aspect of the learning.

Let’s face it – the reality is it is not always easy to get to a scheduled ‘webinar’, even if it is the most interesting topic in the world. Which is where the beauty of technology helps out – because most webinars are recorded and archived for on-demand viewing (much like our on-demand television binge watching craze!). Most education companies or organizations that host webinars will have them archived somewhere because they WANT you to log on and watch – it’s good for business. There are a couple of good sites listed below that have some educational archived webinars that might spark your interest:

There are more out there, but this should be a great start. And – not to let Casio be outdone, we have many archived webinars as well that focus on integrating Casio technology and math content. These are free and accessible on our Casio YouTube site  We have short how-to videos on this site as well, so the way to determine a longer, content focused (and/or technology focused) video is to look at the time stamps – those that are 20 minutes or longer tend to be the webinars. I’ve included one below on Proportional Reasoning, since this is such a huge issue with students of all ages, and the presenter, Jennifer N. Morris is one of my favorite people and math teachers. Enjoy!!