New Year’s Resolutions for the Classroom

I hope everyone has had a nice holiday season and are planning to do something fun for New Yearsstock-illustration-80478039-happy-new-year-2016-background-for-your-christmas Eve. I myself am planning to have a potluck dinner with many of my neighbors and spend time talking, laughing and ringing in the new year. We have built a “New Years Eve Ball” out of chicken wire, lights, and 2×4, and are planning to do our own small-town ball drop from our friends apartment, which happens to be in the center of town. Hopefully we won’t get in trouble – I will be sure to post a picture!

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

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

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

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

  1. Use different, thought provoking questions in each class at least two times each day (questioning is a skill I still work on, so deliberately focusing on it helps make it become a habit) (just a few example below):
    • Ask “why do you think?”
    • Ask “what if”
    • Ask “what do you wonder?”
    • Ask “why?”
    • Ask “can anyone show a different way?
  2. Incorporate technology into a lesson at least twice a week as an EXPLORATION tool (not an answer tool)(if you already do it that often, then every day…something different).  This can mean calculators, software, apps, smartphones, videos – something that provides students with a chance to explore and ask questions that expand their learning/understanding and leads to more discoveries.
  3. Have students work in pairs 2-3 times a week (more if you are already doing this).
  4. Use exit passes every day.
  5. Start each class with a real-world application.

Obviously, you need to gear your resolutions to you – what is it you don’t do now or don’t do often enough that would increase student engagement and/or student understanding. And focus on 2-3 things that you are going to do regularly. Changing a little bit consistently makes it become natural, and once you have those in your repertoire, then you can add on some more. The key here – change something.

Hope you all have a wonderful New Year’s Eve.  Be safe. Be happy. And be motivated to make small changes that will help your students and you achieve great things in the new year.

Happy New Years!

Prime Numbers & Poetry

I will admit – it’s the holidays and I am in a quandary of what to write about. I’ve decided to cheat a little, and look for inspiration elsewhere – i.e., searching Ted Talks for a math-related topic that I find interesting or inspiring. In my perusing of Ted Talks, which I do every couple weeks since there are so many interesting topics and people to learn from, I found one that I had watched a couple months ago.  It struck me as a great one to share for two reasons. 1) It involves poetry & math, so it’s a lovely cross-curricular exercise if you were to use it with your students; and 2) the math poem, when you listen to his subtle innuendos and wording, teaches quite a lot about prime numbers. It’s really very clever.

I am sharing the whole video, though only the first poem is about math – prime numbers to be exact. I’d suggest listening to it a couple of times in order to really catch the very clever way in which Harry Baker brings in understanding about prime numbers through his love poem called “59”.  If it helps, you can use the transcript of the video to see the words of the poem.

I hope you enjoyed this fun poem. If you have never listened to a Ted Talk, I highly recommend doing so.  I used these with students because there are many out there that are relevant to many subjects and are engaging and thought provoking. They are great conversation starters or reinforcement or introductions to new ideas.

Fractions, Common Core & Calculators

I see so many examples of “common core math problems” that people are disparaging, whether from misunderstanding, a poorly asked question, poorly trained teacher, politics, mislabeling….the list is endless.  I thought it might be interesting to take a problem that is designed to support one of the CC standards, and show the many ways it could be approached to, in fact, show the spirit of the Common Core. In other words, show how students could analyze and try to solve it from different perspectives and multiple ways, which is really what the Common Core epitomizes – perseverance, multiple pathways, and justification of your thinking to deepen understanding and make connections.

Here’s the standard I’ve chosen. (Fractions always seems to be the bane of teachers and students, so it seemed a good one!):

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

This is a grade 5 standard. To put in context, students were introduced to fractions in 3rd grade, and have worked with adding, subtracting, multiplying, dividing, unit fractions, and  equivalency through 3rd and 4th grades. What this standard also asks is for students to create a story context for the equation – relate the mathematical skills to the context of a real-world situation to help them make sense of the problem. You can help this along by providing a real-world context.  A simple way to do this is to use recipes, and since we are in the holiday spirit, my context is going to be a chocolate chip recipe!

Recipe – Coconut Chocolate Chip Cookies (makes 1 dozen)

• 1-1/2 cups graham cracker crumbs
• 1/2 cup all purpose flour
• 2 tsp. baking powder
• 1-2/3 cups sweetened condensed milk
• 1/2 cup unsalted butter, softened
• 1-1/3 cups flaked coconut
• 3/4 lb. semisweet chocolate chips
• 1 cup chopped nuts

Preheat oven to 375°F. In a small bowl, mix graham cracker crumbs, flour and baking powder. In large mixer bowl, beat sweetened condensed milk and butter until smooth. Add graham cracker crumb mixture; mix well. Stir in coconut, chocolate chips and nuts. Drop by rounded tablespoonfuls onto ungreased cookie sheets. Bake 9-10 minutes, or until lightly browned.

Clearly, 1 dozen cookies is not enough! It’s fun to ask students how many they think would be appropriate for the class – how many cookies should each student get? How many should the teacher get? Based on these discussions, how many cookies are needed?  Which then leads to how many “recipes” do we need to make?

Let’s say there are 30 students, plus the 1 teacher, and it’s determined everyone gets 3 cookies. So – 31 x 3 = 93 cookies. How do we determine how many times we have to increase the recipe?  (Look at all the great math discussion we are having, in a real-world context, and just having some group discussion!)  Hopefully students then realize we need to divide the 93 by 12. (Oh no…doesn’t come out to a nice number.  Now what?!! Rounding? 7.75 recipes – what kind of math is that?) Ok – so, we have figured out that we should make the recipe 8 times – which will give us approximately 96 cookies. And now you actually get into the standard and have students multiplying the ingredients by 8 (but wait – you could have also done 7.75, changed it to its equivalent fraction form of 7  3/4 and used this too – maybe differentiate your students, and give some groups that to work with and others use the 8?)

If you’ve noticed, before we even start the actual math specified in the standard we chose, we have had in-context conversations, analyzed the problem, made mathematical decisions, rounded….so much math and Common Core related practices happening already (see Mathematical Practices). When we do focus on the standard, i.e. multiplication of the fractions by a whole number or a mixed number (depending on whether you chose 8 or 7.75), there is context and purpose. Set those students loose! Here’s where multiple pathways kicks in – let them decide how they are going to determine the amount of each ingredient needed. Give them the tools they might use – i.e. paper, pencil, manipulatives, dynamic software, calculator, etc.,  and let them make the decision on how they will approach solving the situation. You will get some visual models, some equations, a mixture of both maybe….that’s the idea behind a truly Common Core problem – students have the opportunity to approach the solution in the way that best fits THEIR understanding of the problem and their mathematical application. As long as they can justify their approach to the solution, it’s all perfectly okay. And if you have students share their approaches, it allows for rich conversation and the making of connections.

Possible examples (using 1-2/3 cups condensed milk ingredient):

  1. Equation: 1-2/3 x 8 = 5/3 x 8 = 5/3 x 8/1 = 40/3 = 13-1/3 cups
  2. Visual (drawing)IMG_2270
  3. Visual (dynamic math software for example)    2015-12-17_14-23-42
  4. Calculator (in this case, the fx-55Plus) I have included a quick video of how to use the calculator, which natural fraction number capability, to perform this calculation. Technology is another method students should be able to utilize when approaching problems. The important component here is to have students explain why they entered the calculation they did, what it means in context, and what the solution represents.

Hopefully you get the idea of how a simple problem is Common Core, not because of the problem itself, but due to the questions you ask, the discussions you encourage, the opportunities you provide for students to approach solutions in multiple ways, and the resources you provide to support their efforts. You need to be willing and able to accept multiple approaches and not expect every student to understand and solve in the same way. (THIS is where Common Core gets a bad rap – when ONE way is expected (because it’s in the textbook or because it’s easier to grade, for example), which is actually NOT Common Core! ) Do we eventually want students to be able to do mathematics in the most efficient way? Maybe – what I think is more important is that students have the chance to understand and make connections so that they can use the mathematics in different situations. So what if the way they understand it takes longer – math is NOT about speed, it’s about applying and using it to help solve a problem.

Anyway, hopefully you have some ideas. And – you now have a recipe too, just in time for the holidays! Happy baking!

Goodbye NCLB, Hello ESSA.

As of December 10, 2015, the No Child Left Behind Act of 2001 is no more and the Every Student Succeeds Act of 2015 is now in effect. There is, of course, a lot of work and details and implementation issues that will have to be worked out in the coming years, but I for one, am breathing a a little sigh of relief. The dreaded AYP (Adequate Yearly Progress) and HQT (Highly Qualified Teacher) mandates are no longer going to restrain what schools and teachers should be doing to support student learning.

I can only speak from my own experience, but as a teacher who taught before and after the NCLB Act was made into law, I know how much damage I saw in the schools I taught at, as a result of NCLB. In fact, the research paper I wrote on for my doctorate program application was all about how NCLB had ruined teaching.  (I tried to find it so I could quote some things, but I believe it may have been “recycled” in my most recent move.) What I do know, again, from my own experience, was that teaching changed. I taught in Virginia, and teachers and schools became so focused on state tests and reaching the magic NCLB % passing rate for their students, and making AYP, that teaching became all about the tests. All our classroom tests became multiple choice so that students were use to that when it came to the state test. Teaching had to focus on only the topics covered on the test, so “extra” stuff was frowned about. Memorization of facts and skills was focused on – no more focus on understanding or problem solving – just on the skills needed to pass the test. No more hands-on learning – we needed to teach test taking strategies.

As a teacher who strives to make mathematics engaging, hands-on, and technology rich, you can imagine my struggle. NCLB is in fact a major reason I left teaching in the classroom to go to Key Curriculum, an inquiry-based mathematics/technology publishing company. I wasn’t able to teach mathematics the way I believed it should be taught due to the standardized testing constraints and constant pressure to meet the magical AYP numbers and student passing percentages. I believed I could have more of an impact on mathematics education through supporting inquiry-based learning and technology integration via teacher professional development.

When the Common Core State Standards came along, I jumped for joy, because I saw this as a step back to true teaching. Relevant, real-world, engaging learning focused on understanding and applying mathematics. But – NCLB and the standardized culture we are immersed in has made the CCSS a difficult implementation, and unfortunately, a political tool.  The passing of the ESSA is  exciting because hopefully it will allow education to focus on learning, understanding, and applying rather than testing.

I’ve been researching different articles about what the ESSA will in fact change, fund, and mandate, as that will be a crucial factor in how states implement the new law and how schools/teachers/students are assessed. Assessment is still an important component of education – without it, how can we ensure students are learning and improve the ways in which we help them learn. The difference between ESSA and NCLB is, I hope, that assessment will be more formative versus punitive. The states have a lot more power and control – which could be a good thing, could be a bad thing. It’s obviously too early to tell.

I would suggest you read more about the ESSA on your own. A good summary can be found here. This article does a good comparison of the two acts. NCTM wrote a nice article explaining their support and what some of the ESSA initiatives are, so read that here.  There is also a government site that details the ESSA, which you can find here. I would especially look at the fact sheets posted here. (I will admit, some of the provisions are a little concerning to me.)  Here are a couple that stick out for me as either interesting or concerning:

  1. The one-size-fits all measure for accountability (AYP) is repealed, and states, not the Federal Government, will have power over measuring student and school performance.
  2. There are 69 programs that will be eliminated, and instead, a Local Academic Flexible Grant will allow states & school districts to allocate resources in a way that addresses their needs.
  3. States will determine and create their own strategies to improve failing schools.
  4. All states are free to opt out of the requirements under any program in the bill.

Obviously, this is a very short list – the provisions are numerous and complicated. Much better for you to read and compare on your own. As I said, I am a little concerned at some of the things I am reading (i.e. states opting out of everything, lack of funding, elimination of some great programs, etc.). But, only time will tell and hopefully, after the struggle we have had under NCLB, there can be a more positive approach to teaching and assessing students.

Histograms with ClassWiz & QR Codes

Students should engage and be hands-on with mathematics as much as possible. One of the activities I loved was the Sum of Two Dice, whether in my middle school classes or in my Algebra classes. That’s the great thing about mathematics – you can take an activity/concept and make it more or less rigorous depending on the questions you ask.

I am sure many of you have done this activity – I am choosing it for this post because it’s a nice way to 2015-12-10_11-23-28demonstrate how the Casio ClassWiz (fx-991EX) scientific calculator allows you to create frequency tables and with the QR code, see an online visualization of the data.

First thing – have students roll two dice and collect some data – i.e., the number on each die and then the sum of the two together. If you don’t have die (or don’t want to hear all the noise!) you can utilize the random integer option on the calculator to simulate rolling die. I usually had my students in pairs to do this – one record, one ‘roll’. Then have them tabulate the frequency of each sum and create a new table with possible sums & frequency.

2015-12-10_12-32-42The next step is to have them make a histogram of their own frequency table and then compare to other students.  In my example, students only rolled 24 times each, so everyone’s graph will look different and not be what we expected (more 7’s). Great discussions can come from these observations.

Hopefully, discussions lead to the idea that each group o2015-12-10_12-17-22nly collected a small sample of rolls, and that if we had more samples,
perhaps the distribution of data would be more what we expected – i.e. more sums of 7 appearing. Here’s where having a class emulator is a great tool – you can display one frequency table and collect the class data.  So – same sums, but combine each groups frequency to get a total frequency for the whole class.

Once you collect the class data, you can then create a new histogram. With the 2015-12-10_12-18-07ClassWiz you can easily do so just by creating a QR code of the table data, and, with the emulator, go directly to the visualization. You can also do this with the SmartPhone App Edu+ if you have that option. The nice thing about the emulator is you can immediately pop the visualization up and begin discussions and comparisons of whole class histogram versus individual groups. Being able to immediately see the visualization with only a scientific calculator, is powerful, especially as you can quickly compare between previous “one-group” histogram and current “whole-class”.

There are so many ways to use this activity – I use to use it with TinkerPlots, graphing calculators, students hand-drawing the graphs. The ClassWiz and its ability to create QR codes and online visualizations is another way to help students make meaning out of the math they are doing, especially when they only have a scientific calculator to use, as most middle school students do. Hopefully this gives you some more options.  I have included a short video clip on how to actually create the frequency table, QR code, and online graph using the ClassWiz fx-991EX. Try it!

Let’s Teach Probability & Statistics – We Need It!!

Let’s admit it – the world is chaotic and uncertain right now, and granted, a little scary. What I am finding even scarier is the statistics that are being constantly thrown at us, since so much of what we are seeing/hearing as ‘evidence’ is often distorted, misinterpreted or an outright lie.

I was listening to NPR at the gym the other day, where they were talking about lying (in particular the distorted truth in much of the politics today)(On The Media, Dec. 6, Lies, Lies, Lies).  At the same time, one of the TV’s hanging from the ceiling was reporting on Donald Trump vs. the media about who is distorting the truth more (now THAT’S a tough question). As I looked at the latest ‘poll’ results on the TV showing Donald Trump with a huge lead in the polls, I was also hearing on my radio one of the analysts on NPR talking about how deceiving these very polls were. According to Nate Silver :

“Donald Trump might be showing a 25-30% lead, but that’s 25-30% of the 25% of the voting population that identifies themselves as Republican, which is really only about 8% of the electorate.

Polls are very deceiving if you don’t understand the statistics behind them, such as who was polled, how many were polled, who did the poll, etc.

The NPR story had several other statistics from various politicians, though Donald Trump was the front-runner there. They had this statistical distortion as well:

Donald Trump stated: “We have 93 million people out of work, they look for jobs, and give up, and then all of a sudden, statistically, they are considered employed”.  The 93 million is in fact a statistical distortion (or lie). According to NPR, those 93 million”unemployed’ people includes: 38 million retired people, 10 million stay-at-home moms, approximately 6 million students, and 9 million on disability – so not really 93 million people out of work. But it sure sounds like a lot, and it scares people into believing that the unemployment statistics (at 5% unemployment) is a lie – a never-ending, vicious cycle.

This post is not about Donald Trump, of course – he just provides a lot of fodder for statistical distortion. This post is a call for a change in our math curriculum. We need probability & statistics education in our schools. As a mathematician and educator who has studied and used probability and statistics, I know how easy it is to distort the numbers/graphs to make them say what you want. I am perhaps a bit more skeptical of the statistics I see/hear from politicians/media/companies. What worries me is that so many people do not have this sense of skepticism, and are so willing to accept these “facts” and make poor decisions. According to the specialists, researchers and fact checkers on the NPR show, even when given the correct facts/statistics or shown that what they were told was a bold-faced lie, most people won’t change their minds about something. That’s SCARY!!

I think a large part of this inability to understand or question statistics is that most people did not take a probability or statistics course in K-12, and maybe not even in college, if they went. They don’t understand it, and therefore do not question the statistics they see/hear on a daily basis. The regular trajectory in K-12 is towards calculus, with HS Probability & Statistics often relegated to ‘advanced placement’ students. Most students never take it and their only exposure to Probability & Statistics is a section or two in lower grades or an algebra course. We do not put enough emphasis on the importance of Probability & Statistics, which is real-world mathematics we see every day in the world around us. The Common Core has definitely made Probability & Statistics more of a focus, which is great, and hopefully this will help reorganize math curriculum’s throughout the country so that all students are pushed to take Probability & Statistics in HS. This is my message for today – teach Probability & Statistics! And what better way to start doing so then to start exploring some of the statistics/polls that we see on a daily basis these days and help students understand how these ‘facts’ can be manipulated. Look at the statistics and explore the fact checkers and use mathematics to uncover the truth. Let’s help students be better informed and better able to understand and use mathematics to make decisions.

I found this great (and short) Ted Talk by Arthur Benjamin from 2009 that pretty much sums up why we need to teach Probability & Statistics.  If nothing else, so we can make more informed decisions about our world.

Math Questioning to Support the Mathematical Practices

In a post I wrote last month entitled Questioning In Math – NCTM Regionals Minneapolis I talked about the power of questioning in math to promote thinking, problem-solving and foster collaboration and communication among students so they can make mathematical connections.  I believe that if you change one simple thing – how you question in mathematics – your students will become better problem-solvers and better mathematicians.

In the Common Core Standards of Mathematical Practices, you will find that all 8 practices are designed to help develop in students this ability to think about mathematics, make connections, communicate their understandings and become problem-solvers. These practices are often overwhelming to teachers, who think they have to completely revamp all their instructional strategies. My advice is always to start small, change one thing at at time, and the best way to start is by changing how you question your students. That simple step can go a long way to addressing the skills & processes the practices set out to develop.

The obvious change is to never accept a students’ answer, right or wrong, but to always ask another question. This is a hard skill to develop, as it is easier to say “That’s right, good job!” or “Not quite – does someone else have an answer?” Something as simple as “Why?” or “How do you know?” or “Can you show me what you mean?” can go a long way in helping students learn to justify their answers and/or rethink their answers, and make connections or corrections in their approach.

Here are some questions to add to your repertoire,. to help foster in students the ability to think, connect, collaborate, justify, persevere and communicate – in short, to become confident problem-solvers. This is by no means an exhaustive list, and comes from many sources and years of experience, but it’s a start.

  1. Making Connections/Starting the Conversation
    • What do you think…..
    • What do you wonder about…
    • What do you know about….
    • Does this remind you of anything….
    • What can you tell me about…
    • Have you ever heard/seen….
  2. Exploring, analyzing and persevering
    • How do you know…
    • Does that always work….
    • Why did you do that or say that….
    • What would happen if ….
    • Can you think of another way….
    • Could something else happen….
    • Might there be another approach or possibility…
  3. Conjecturing, Collaborating & Justifying
    • Why do you think this is the right approach…
    • Are there other ways to show this….
    • What will happen next….
    • How does your way compare to …. how is it the same? Different?….
    • What would happen if….
    • Are there other answers/solutions/methods….
    • Why do you think that is….
    • Can you explain….
    • Why….
    • Who can explain what (name) did…..?
    • Is this like anything else you’ve done….
    • Is this similar to….how so?
    • How is this different from…how so?

I use to provide a laminated “book mark” for teachers to keep at the front of their room or on their desk with these type of question starters.  As they were teaching, just in case they couldn’t think of a good question on the fly, they had some to pull from.  Making a concerted effort to change your questioning takes time & practice. Cheat sheets are okay! Eventually it becomes part of you and your classroom expectations and the conversations where students are learning and engaging in mathematical discourse are worth the effort!