Math Test Prep – It’s That Time of Year Where We Bore Our Students Into Failure

I know when I was teaching in the k-12 classroom, this time of year was always so frustrating as a teacher and even more frustrating and anxiety-ridden for my students. This is the time of year when standardized testing is occurring or about to occur, in the majority of states. This can mean state-tests or national tests such as the AP exams, SAT and ACT. For me, the biggest ‘anxiety inducer’ was the mandatory End-of-Course tests that all my math students were required to take and pass with a 70% or better in order to earn the credits needed to graduate. No pressure there…..

Things have changed a bit as we move into the new era of ESSA, with many states changing the standardized testing requirements, but there is definitely a lot of pressure on students to perform, and on teachers to get their students to achieve at specific levels. This impacts teacher evaluations, school evaluations, etc. I’ve always hated that these ‘one-point-in-time’ tests have such dire impacts on teachers and schools, considering they do not reflect student growth over time or other impacting factors such as absenteeism.

But – regardless, tests are out there, happening now, and causing teachers and students undo stress. I know, for me, part of the frustration was the inordinate amount of time we were ‘required’ to prep students for the test. This included days specifically set aside to practice for the tests instead of teaching, and a ridiculous number of ‘practice tests’ and test taking prep.  Boring, stress-inducing, and really kind of pointless in my opinion. I felt we spent entirely too much time preparing for tests instead of actually teaching our content and letting students continue to learn. It was as if ‘learning’ stopped and the whole school went into ‘test-prep’ mode, and we forgot what school should be about – engaging students in learning and understanding, not preparing them to take a standardized test. My thoughts were these prep times only increased students anxiety about the tests and often, the long, drawn-out, constant test prep led to student burn-out, apathy, and failure. For many students, they got so tired and bored of ‘practicing’ that when the real test(s) came along, they made beautiful designs on their bubble sheets instead of actually focusing on answering the questions. (Yep – that really happens).

What are my suggestions? Keep teaching. And not teaching to the test or for the test, but teaching. Teach new things. Teach applications of things that might be on the test but  NOT through standardized-test questions, but with real questions, real problems, and real applications of the things students should know for the test. Worksheets with multiple choice answers are NOT teaching, or learning, or engaging. Technology with “practice” problems and right/wrong answers is NOT teaching or learning. Do something with the knowledge students should be able to use and do on these tests. Create interesting learning experiences, where students have to problem-solve and apply the knowledge and talk to each other. Example: instead of 20 solve these ‘systems of equations’ problems on a worksheet, provide real-world problems where a systems of equations is needed to find the solution. Where students have to work together to create the equations and come up with the solutions. Where they get to decide the most appropriate method to solve the system. Way more interesting and much more insightful into what students know and can do.

It’s not that you shouldn’t prepare students for tests. It’s that you should do it in a way where students are applying their knowledge and engaged in applications of that knowledge. It’s not about worksheets and test-taking strategies. It’s about understanding and applying the concepts. Tests suck. Don’t feed the anxiety and the boredom and the apathy towards tests by creating rote, mundane, drill-and-kill test prep. Make it about engaging students in applying their knowledge in interesting, relevant ways. There are many resources out there that can provide excellent ‘test prep’ ideas and problems in a much more exciting way than a worksheet with 40 multiple choice problems. (Bleh).

Some fun #math sites with challenging application problems to use for ‘test-prep’:

 

Global Warming? Let’s Look at Some Data

I realize that I am most likely among the minority of folks when I say I miss snow. I have lived in the Philadelphia area going on 3 1/2 years now, and this ‘winter’ has to be one of the most disappointing ones so far.  I think we’ve seen maybe 3 days of snow – less than 3 inches, and all gone in a couple hours.  I haven’t even had to shovel or scrape the car but one time…. There has been a lot of rain. It’s raining today, and suppose to get to 60. Yep – sounds like spring to me, NOT winter! Where’s my snow? Where’s the sledding?

I grew up in Virginia and spent most of my life in Virginia, where we got a lot of snow – I remember some pretty amazing snow storms and tobogganing down the driveway with my brothers and sister. I then moved to Houston, TX for five years back in 2008 and basically lost any hope of seeing snow or even seasons. There is no real winter….no real spring…definitely no change of seasons in Houston, though it is definitely as hot as people say. When we moved back east to the Philly area 3 1/2 years ago, I was so excited to experience a fall again, and my first winter here we had so much snow, we were actually tunneling our way out.  It was great! Sledding at the castle, power outages forcing us to hunker down at the local bars – snowstorms were fun – even the shoveling brought out the neighborhood and a lot of goodwill!

 

The lack of snow this year, and the weird warm temperatures this winter, where it has felt more like spring than winter, has me thinking about whether this is a normal pattern for the area or is it ‘global warming'(which according to our illustrious leader is a hoax), or is it something else? I think it would be an interesting and relevant real-world investigation for students to look at and analyze and make some conclusions and even some predictions, no matter where they live. My guess is lots of you are experiencing some weird weather patterns this ‘winter’ – i.e. Utah & California for example.  I know the kids around here are disappointed there have been no snow days, so they’d probably love the chance to study the numbers and see if this is an expected pattern and hopefully find a chance of snow still exists.

No matter where you live, weather patterns are a great way to analyze data and apply mathematical concepts. Most countries, states, cities and town keep a historical record of weather data – by year, by month, by day.  There are lots of different measures taken into account – temperature (lows & highs), precipitation (rain and snow), barometer pressure, wind, etc. This data is relatively easy to find as well just by doing a simple internet search. Many sites provide customization, where you can specify month, year and other data that you are interested in looking at. I did a relatively simple search for Philadelphia historical data, and compared the month of January from 2013 to 2017 – here are the numbers:

Granted, a little hard to see, but just in a quick glance, students might note that this past January 2017 we had about 5.59 inches of snow fall compared to 19.41 inches in 2016 (all in one day?!!), 3.9 in 2015, 25.86 in 2014, and 3.75 in 2013. Based on this, maybe it’s every other year that we get a lot of snow? Maybe this has nothing to do with global warming? Is there enough data to make these conclusions? Should we be looking at more months or more years? What about the average high or the average lows for each month? Does that make a difference? There are so many interesting questions and comparisons that students could explore with weather data. As a teacher, you could be applying a lot of things like ratio, proportion, measures of central tendency, different types graphical displays, fractions, decimals, algebra.  It’s a font of real-world data that could be used in so many different ways and in so many different math courses. And students would be interested, especially if you are using data from where they live.  Maybe compare the data to other similar cities or other very dissimilar cities. Do a cross-curriculum investigation – i.e. science, language arts, history.

Depending where you live, you can use weather to help students relate mathematics to their own world and explore their environment while doing math. In CA, as an example, you’ve received a tremendous amount of rain this winter – is it enough to end the drought? How long would that take and how much rain? Interesting and relevant questions students would love to investigate. In Utah, how has all the snow impacted the skiing and tourist dollars coming into the state? In Louisiana, South Carolina, Georgia, Florida – how common are tornadoes in ‘winter’?

Lot’s of questions. Lot’s of data out there ready to explore.

One last question – will there be a big snow storm in the Philly area in the next few weeks? I hope the answer is yes…I need a snow day!

Frozen Math

I saw this really cool GIF on FB the other day, showing a bubble freezing. As I watched it, you could see all these beautiful shapes emerging and eventually covering the whole bubble. (I of course wished that it was cold enough where I live for me to go out and try it myself, but alas….where I live seems to be having a no-snow winter this year.

Watch and see:

It looks like snowflakes appearing on the bubble, and snowflakes are fascinating. They are unique, they have amazing patterns that form naturally. Wouldn’t it be fun to explore snowflakes with students? Especially if you live in colder climates where there is actual snow to collect and study. How could we connect the beautiful patterns and unique qualities of snowflakes to mathematics? I set out to explore and found a few great resources for those of you who are interested in exploring frozen math. Yet another way to bring the real-world into the classroom and help students see the math that exists around them.  Even if you don’t live where snow may be, some of these resources provide some great tools for ‘creating’ snowflakes with students.

Here are some links:

  1. http://www.educationworld.com/a_curr/mathchat/mathchat015.shtml This is a nice site because it has several suggestions – from collecting real snowflakes to creating your own, to analyzing patterns and categorizing snowflakes. Great hands-on activities.
  2.  A wide variety of ‘frozen math’ activities here: http://mathwire.com/seasonal/winter05.html including the Koch Curve/Snowflake, where students experience the iterative process to create a snowflake fractal.
  3. Some nice examples and how-to-make paper snowflakes: http://mathcraft.wonderhowto.com/how-to/make-6-sided-kirigami-snowflakes-0131796/
  4. Some nice geometry connections and more paper-snowflake making here: http://playfullearning.net/2015/02/snowflake-math/
  5. This is a great math/science connection with a lot of further embedded links included: http://beyondpenguins.ehe.osu.edu/teaching-about-snowflakes-a-flurry-of-ideas-for-science-and-math-integration
  6. Vi Hart and Doodling is always fun to watch, and here she is doodling and folding with symmetry and fractions: https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/vi-cool-stuff/v/snowflakes-starflakes-and-swirlflakes

I am sure there are more options out there – these are just a few I stumbled upon in my searching. Don’t let the winter blues set in – get out there and collect some snowflakes and do some frozen math!

The STEM Around Us

NCTM Innov8, the new team-based conference that NCTM is sponsoring, is going on right now in St. Louis, Missouri. Our team is there of hqdefaultcourse, supporting math teachers with our technology and a great team-building session based on the Wheel of Fortune and the probabilities of winning (session is Friday, November 18 at 10:45 am in Room 265/266). St. Louis brings to mind the very famous St. Louis Gateway Arch, something math teachers attending will probably be exploring and trying to mathematically represent – is it a parabola? (In fact, it is NOT a parabola, but rather a flattened catenary). (Cool 3D mathematical model here).

This idea of looking at real objects and connecting mathematics to them is something math teachers do often. It makes complete sense, and, as I have been teaching a geometry course for Drexel these last several weeks, I have really deepened my appreciation for this idea of looking at our constructed world to find the mathematical connections and relationships. What I think we tend not to do with students, and what we should do much more of, is go beyond the obvious “shape” explorations and function fitting to explore the STEM connections.

What I mean is after we identify the inherent shapes and/or functions in ‘real-world’ objects, start asking questions that get students thinking about the why behind those shapes. The why questions lead to investigation and research by students into science, technology, engineering, and math applications that would take them much deeper into understanding the world around them. And, I wager, this type of questioning will engage students in learning and applying what they learn in a much more relevant and interesting way.  Giving them purpose for learning. And, as a result, we might have more students going into STEM fields.

Some examples:

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Why, for example, are most buildings polygon shapes, particularly triangles and rectangles? Why don’t we see more circular or cylindrical shapes for buildings, besides the grain silos or water towers? Is there a reason? This is where engineering would come into play – are certain shapes stronger from an engineering perspective?

science-test-tubes-all-the-ladies-in-washington-zsrurj-clipart

 

Why are science and medical tubes cylindrical? Is their a scientific reason for these shapes/containers? Why not use a prism shape, so then you could set the vials down on a table versus having to store them in special holders so they don’t roll away? Is the shape somehow connected to the way molecules or blood cells behave – i.e. science factors that might determine the tools used.  2791136-image-of-the-motherboard-without-a-pc-processor-closeup

Look at all the different shapes on a computer motherboard – there are cylinders, rectangles, squares, networks of curves/lines of wires, prisms…so many things going on. Students could ask whether certain shapes provide better conductivity? Or heat control? How does the height of a component impact it (notice the different heights of the cylindrical components). I don’t even know the questions to ask here, but this is a great example of where technology comes into play.

I feel that if we allowed students to explore beyond simple things like fitting a function to a curve or identifying shapes in a picture, and really focused on STEM applications and reasons behind the use of those specific shapes, we would be encouraging students creativity, curiosity, and developing research capabilities in order to find solutions. It would be so engaging and really get students interested in those STEM careers, but more importantly, a better understanding of the STEM around them.

 

Casio vs. TI – Finding Max/Min Points of Functions

DispCap1In this weeks’ comparison of the Casio vs. TI calculators, I demonstrate how to find a max/min of a given function, using the Casio Prizm and the TI-84+ CE. In my example I use a cubic function because it allowed me to show both a maximum point and a minimum point on the curve. Why might students be asked to find a max/min point of a function you ask?  Well, besides the obvious ‘on a standardized test’ question, what we really want students to be able to do and understand is what the max/min points mean in the context of the problem/situation. In a real-world application, how does that max/min point help us understand what is going on in the problem? The short answer is it provides specific points where ‘something’ has happened, and finding these points provides insight, allowing students to ask different questions or analyze the situation.

Here is a common example: A quadratic function might be used to model the path of a ball as it is thrown or hit, with x representing time and y representing its’ height.  So the max point in this case would be the maximum height of the ball at a given point in time before it begins its descent back to earth. We want students to be able to find that max/min point, in context, so they can answer questions or make conjectures about the ball. For instance, in this ball example, is it possible to change the angle the ball is thrown/hit to increase the max height, but keep the time the same? Being able to quickly find these max/min points so that interesting questions and conjectures can be made and students can apply mathematics in challenging and deeper ways is one benefit of using technology, such as graphing calculators. The max/min points can be a starting point for deeper exploration.

Below is a quick video on how to find a max/min point of a function (using a cubic as the example, since it has an example of both a maximum and minimum point).

A Picture’s Worth More Than Memorization – Prizm Pictures

DispCap7With all the talk of learning math in context and making mathematics “real-world” and relevant to students lives so that they see and connect what they are learning to a purpose, it is often amazing to me how often there is no attempt to help make those connections. Especially as students move into more abstract concepts. From personal teaching experience, whenever I could bring in something – a physical object, a picture, a story, a technology simulation (loved Sketchpad!) that helped provide my students with a connection to the concepts we were learning, the more my studentDispCap4s engaged and remembered. And more importantly – were able to apply it to different situations because they hadn’t just memorized isolated skills/steps, but learned in context, and therefore had a connection that they could pull from easily.

One way to bring in context and relevancy is to use pictures of real things that can then be used to apply mathematical concepts, or used to solve interesting problems. If you have technology, where students can actually use the pictures with plotting points or graphs, then it becomes a powerful learning tool. There are lots of ways to do this – I for one used Sketchpad as much as possible in the classroom. What I have found as I work with schools throughout the country is that access to technology with math students on a consistent basis is very limited – laptops shared with 20 teachers, or one computer lab shared with multiple departments.  But – graphing calculators are something a majority of DispCap3students do have access to, and Casio Prizm is one that has built in pictures that can be used in a variety of ways and in a variety of math subjects.  Hands-on, real-world, and easily accessible. DispCap6

Here’s the basics way to access the built-in pictures in the Casio Prizm:

  1. Choose “I” from the menu (or arrow down to “I”). Hit EXE.
  2. Click EXE to open the CASIO folder
  3. Use the arrow key to choose the g3b or g3p folders.  Hit EXE.
  4. Use the arrow keys to scroll through the picture choices to find one that fits – there are so many choices!  Hit EXE
  5. Picture appears, and voila – start exploring!  Choose OPTION button to begin.

Obviously, there’s more to it than this – go here to find out more, but what I wanted to share in this post is the built-in pictures already in DispCap2existence on the Casio Prizm that could create some powerful context and applications for mathematical concepts you might be exploring with your students. Bring the math into students world.

Here’s link to the Casio Prizm Quick Start Guide that is also very helpful.

 

Power bills as sources of math questions.

I’ve been thinking a lot about graphs lately, and how in general, many people are deceived by graphs because they don’t understand numbers, scale, sampling size, etc.  In this very contentious political time, it seems many people are fooled by the statistics they “see” graphically.  In my last post, I quoted Dan Finkel’s line “when we are not comfortable with math, we don’t question the authority of numbers”, specifically referencing people’s willingness to believe statistics they see or hear because they don’t really understand where these numbers came from or what they represent.

We can help our students get a better sense of statistics and numbers by providing them as many opportunities to explore, in context, graphs and statistics and ask questions and make sense of these. That could mean exploring all the statistics and poll results currently happening with the presidential election.  Or looking at weather predictions. As I looked at my power bill yesterday, I realized how easy this type of access to real numbers can be, as I stared at the graphical representation of my gas and electric over the past 13 months. (There is also a numerical table showing daily use of kilowat hours (kWh) and 100 (C) cubic feet volume of gas (Ccf). There alone is a whole bunch of mathematical calculation/conversions/ratios).  What I love about my graphical representations is there is a 13 month trend – so I can see where my usage was last year at the same month, and then see how my usage has changed throughout the year.  Below are my December & January graphical representations for both gas and electric usage.

ELECTRIC:

December Electric

December Electric

January electric

January Electric

 

GAS:

December Gas

December Gas

January Gas

January Gas

 

Just from these graphs, there are a lot of assumptions that can be made, and questions that could be asked, that would then lead to more exploration.  For example, December electric from 2014 and 2015 is about the same, but January 2015 is significantly less than January 2016 – why is that? (hint: my children are home for break, so we use more electricity). Gas use in December of 2015 was much lower – was this because it was warmer in December? Are we having a warmer winter than last year? The gas bills seem to show that – but, we could then go look at the weather temperatures for the same time frames in the area I live and see if there is a correlation between temperature and gas usage (i.e. heat). Why is the electric so much more in the spring/summer months and gas is lower? There are so many questions, and, if we brought in the tables of daily usage, cost of kWh and Ccf (volume) we could be doing math calculations, comparing costs, etc. Maybe compare bills from last year to this year and see if the price in oil/gas has had an impact on the overall monthly charge. I like the idea of bringing in the weather and comparing to the electric/gas usage. You can get average weather for the area you live in pretty easily, but it would be even better for students to collect actual temperatures over time and make their own graphs and comparisons.

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Average Climate Chart

The point I am making here is that a simple thing like a power bill can be a powerful tool for visualizing math, doing math, making connections, and asking questions. Or try looking at some statistics from car sales or stocks or polls on the presidential election. It leads kids to ask interesting questions, explore mathematics they care about, and opens them to the real-world aspect of mathematics and how numbers can be used to inform, deceive, and help make decisions.  These types of explorations are interesting and help students become involved in the world around them as well and better prepared for the realities of things like gas bills! Anyway, just another suggestion on how to bring some context into your math instruction in a relatively easy way.