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’:

 

A Math Nerd’s Dream Museum

img_3760I went to the National Museum of Mathematics (MoMath) today – what else would I do while in NYC?!!  If you were unaware, this is yet another img_3766great attraction to add to your to-do list next time you are in New York City. I was lucky enough to have a few hours today to myself and thoroughly enjoyed my hands-on experiences – me and several hundred school-age children.

The museum is focused on providing hands-on, interactive img_3782mathematical experiences so students can see, create, and play with mathematics.  There are games, art exhibits, bikes with square wheels to ride, cars to control around a mobius strip, img_3780angles, tessellations, fighting robots, logic puzzles….it was really fun, and there was a lot of ‘learning’ embeddedimg_3775 in all of the exhibits, though I did find I was the only one reading – the kids wanted to just ‘do’. But can you really blame them?

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One of my favorite exhibits when you walk into the museum is the wall of etchings done on metal plates. There are parabolic lights above them that move and due to the angles the metal etchings are at, it appears the whole display is moving and that the etches are 3D when in fact they are flat. The etchings themselves are beautiful – lots of mobius strips in there!!  I tried to capture it on video but it doesn’t do it justice.

Another favorite was the art exhibit showing the amazing geometric img_3770sculptures of Miguel Berrocal – famous for creating sculpture puzzles – i.e. sculptures built by pieces fitting together. There are numerous sculptures on display along with puzzle books showing the steps to build some of the sculptures. There are also two hands-on opportunities to try to build some of the sculptures. I tried my hand at the above sculpture, “portrait de Michele”, which they recreated the pieces using a 3D printer and then provide ‘directions’ to build.  My results are below….I was very proud of myself!

There was a little bit of everything – I made myself into a human fractal tree (that’s me as the trunk if you look really close). And then I made two 3D shapes (sphere and star) by putting together flat plates with 2D shapes (circles and triangles) in a layered order so that they end up looking 3D.  That was a challenge trying to piece the different sized shapes in the right order.

There was a lot more fun to be had – from the square tire bike to the shape challenges to building polyhedra. All in all, a fun-filled few hours doing some math and experiencing students enjoying doing math as well. If you ever get the chance to get to NYC, be sure to include the MoMath in your itinerary!

Visuals to Start Interesting Conversations & Problem-Solving

I realize most teachers and students in the U.S. are just beginning their summer vacations, so thinking about math and problem-solving is most likely the last thing on their minds. But – if any of you are like me, the summer was always a time to regroup, rejuvenate, and come up with new and brilliant ideas to utilize in math class starting in the fall.  I often spent my summers taking a class or finding projects to use/create, so always looking for ways to enliven my math instruction.

This morning, with all the news about UK voting to leave the EU, shocking news to be sure, I couldn’t help notice the many different visuals being bandied about to visually show how the votes were laid out.  It’s fascinating to look at these different representations, and then to just consider all the possible questions that arise.  Here are some examples of the visuals I have seen:

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The interesting thing with these visuals is they are all showing the same “results”, but from different perspectives or different ‘groupings’.  I love the map one – it clearly shows how the countries played out in the vote.  Now – this is NOT a post about the referendum – you will have to go to your news sources for information there.  But – from the math teacher side, all these visuals about the same results just got me thinking about how really great questions and problem solving could arise from the simple act of putting up a graph of some results and asking students “what do you think or wonder?” and letting them then investigate. For example, if we look at just the map, and don’t give them any numbers, they might wonder is it half blue/half yellow? How could they then determine the actual area of each colored portion of the graph?

Here’s a couple more pulled from the Prizm Resource Page:

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If you were to just throw these up on the screen at the beginning of class and ask the students to come up with some things they wanted to know about these visuals, it would lead to some student-generated questions that then would require the use of mathematics and possibly some background/related research, to find the answers.  If we are thinking about the mathematical practices, or habits of mind we are trying to instill in our students – such as analyzing, communicating, persevering, applying, arguing, critically-thinking, problem-solving, rather than giving them all the information and then asking them to ‘calculate’ the solution, why not let them find answers to questions that interest them? They would be applying mathematics in several ways, perhaps incorporating skills they have not yet learned but need – and in the process realizing that mathematics is useful and interesting.

Try it – find an interesting visual – graph, picture, etc. that spark in you some interesting questions that need math to solve. Put them in your “things to add to my class for the fall” and then get back to summer!

 

Math – Always Something New or Different

If you hadn’t heard, a group of Georgia Tech Mathematicians have proved the Kelmans-Seymour Conjecture, a 40-year old problem. Here is a an article that describes the conjecture and its proof in more detail for those of you interested: Georgia Tech Mathematicians Solve 40-year old Math Mystery” Now, I personally had no idea what this conjecture was till after reading the article – Graph Theory was not something I spent a lot of time on in college or in my teaching career.  What struck me was that this conjecture has been out there for 40 years with people trying to prove it, and it took a collaboration of over 39 years between six mathematicians to prove it:

“One made the conjecture. One tried for years to prove it and failed but passed on his insights. One advanced the mathematical basis for 10 more years. One helped that person solve part of the proof. And two more finally helped him complete the rest of the proof.”

Elapsed time: 39 years.” (Ben Brumfield | May 25, 2016)

Here’s what I love about this – it shows that math is a collaborative endeavor, that takes time and different approaches and insights and that something new can always be discovered or proved. Which is what we should be focusing on in K-12 math education, instead of the idea that there is one answer to a problem.  The standards for mathematical practice (part of the Common Core and based on NCTM Principles to Actions) are all about this collaboration, problem-solving, communication. It’s slow to take hold, and politics is working against it, but look at what can be accomplished when mathematicians, i.e. students, work together to problem-solve?

Math is not a single-solution, one-way only, or  learn-in-isolation. Let’s support the practices, let’s support teachers, let’s support students and create mathematical learning experiences that promote collaboration, real, relevant problem-solving.  It requires teachers being willing to accept multiple approaches and multiple methods of explanation (verbal, written, visual). It requires noise – collaboration is not sitting quietly at your desk.  It requires “mess” – using whatever tools or resources help students think about problems. It requires time.  But think about the new and different math that students will create and explore – and think about how much better prepared they will be for the mess that is the world.  That’s ‘college and career ready’ in my opinion.

Multiple Entry Points and Rich Math Tasks

I was reading this article the other day about how a strength-based approach to learning math (and learning in general) redefines who is “smart” and allows all students to succeed. In her article, Katrina Schwartz has some quotes and reflections from former students who learned math using the complex instruction method, and who were all successful.  They talked about how “math class made them feel safe, heard and able to express their ideas without fear”.  Wow – how often do you hear something like that?!!

Complex instruction is based on the idea that learning is collaborative, where students are learning using rich tasks with multiple entry points and pathways, and each student has a role and accountability. This isimage15 not a post about complex instruction however. What I was thinking about while reading the article was in fact about the Common Core Standards for Mathematical Practice and how they support what the students were expressing in the article – that math class “was a process and it required other people. It wasn’t just you and your work and not talking.”

If you actually read the 8 mathematical practices, you will notice over and over again words like communicate, justify, analyze, plan, make sense, look for entry points, reason, ask questions, make viable arguments, apply.  The practices are all about communicating and talking and finding multiple entry ways to solve problems. And working and talking with others to get there. Like complex instruction, these problems should be rich, where in fact, there are multiple entry points and possible solution pathways. Where each students strengths can support the process and help build the understanding of others. Learning is collaborative, NOT an isolating experience that a worksheet or a lecture so often create.

The Common Core gets a bad rap because so many publishers and testing companies have standardized it – by providing ‘common core problems/strategies’ that are in fact limiting and narrowly focused so they can be graded easily. When I see parents and students and teachers complaining about “common core problems”, I get so angry because what I am actually seeing are ‘forced entry points’ – meaning, rather than allowing all students to approach a problem from their mathematical strength and understanding, they are forced to choose between 1, 2 maybe 3, ways to solve a problem, which may NOT be understandable methods for them.  Therefore, NOT Common Core (or Complex Instruction). As it says in the practices: “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution”. It does not say mathematically proficient students are given specific entry points to solve a problem.

image16My hope is that with the changes in standardized testing on the horizon under the Every Student Succeeds Act, that math teachers and classrooms can truly begin to focus on students strengths and learning, not preparing for a test. To actually provide learning experiences focused on allowing students to work from their strengths. But it requires a willingness to have a noisy classroom with students talking and collaborating.  It requires  rich mathematical tasks, not standardized worksheets and drill and practice,  that truly provide multiple entry points. This in turn requires teachers who are willing to accept multiple solutions from students rather than the traditional one-way, algorithmic approach we tend to focus on. And it requires support for teachers – in resources, training, time, and expectations.

Read the article by Katrina Schwartz – it also has links to information about Complex Instruction and great feedback from San Francisco Unified School District who has made a concerted effort to teach mathematics this way. Read the Common Core Standards for Mathematical Practice. If you are not already doing so, try to incorporate some of these practices into your math instruction. It’s not what you teach, but HOW you teach, that has an impact on students.  Every student can learn math – it’s up to you to create a culture that helps them believe that.

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).

Modeling with Mathematics – Math Practice #4

Whether or not you teach in a state that has adopted the Common Core State Standards for Mathematics (or a modified version of them), the Common Core 8 Standards for Mathematical Practice should be something every math teacher fosters in their instructional practice. These practices are based on NCTM’s processing standards and NRC’s standards for mathematical proficiency in the Add It Up report. They are about helping students become problem-solvers, creative thinkers, communicators, users of multiple resources, and most importantly, able to apply what they know in multiple ways. That’s what teaching math should be about – helping students use what they learn in the world around them, now and in the future.

I am NOT going to get into a debate with anyone on the pros or cons of the CC Math Standards themselves.  That is a politically charged hot mess. Whether you are for or against these standards is irrelevant. How you teach and support student understanding – i.e. your TEACHING PRACTICES, is what makes the difference, NOT the standards you follow in your curriculum. No matter the content standards, they way you help students learn, understand and apply those standards is important and vital, and is what the practices are all about. Lack of how to incorporate effective practices is what I have found, from years of working with teachers, is one of the biggest deterrents in student learning. And, as evidenced in many articles and classrooms I have observed, there is a great deal of misunderstanding of the Standards for Mathematical Practice and how to incorporate them effectively.

In a previous post, I highlighted Mathematical Practice #5, Use Appropriate Tools Strategicallywhere I tried to explain what the practice meant and provide some examples. Today I’d like to do the same thing with Mathematical Practice #4, Model with Mathematics as I think this is one of the most misunderstood, or ‘misused’ practices. I myself, before doing an in-depth study of the practices, interpreted this practice wrong. In my mind, I thought it meant that I should be using manipulatives and ‘models’ (i.e. technology simulations, physical models, etc.) while teaching and I would therefore be modeling with mathematics.  That is part of the standard, but NOT the true purpose.  Remember, these standards for practice are what we, as teachers, are trying to foster in our students – meaning, we are trying to help our students model problem situations with mathematics to help them better understand it and/or solve it.

Let’s look at the actual standard.  I have highlighted key phrases that help clarify this standard:

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Notice the different ways “modeling” is represented: apply the mathematics; write an addition problem; apply proportional reasoning; use geometry; make assumptions and approximations; map relationships; use tools. This standard is not talking about physical models, but rather helping students look at a real situation (so context is important) and use mathematics to help understand it, explain it, make it simpler, relate it to something else, etc. Can they use physical manipulatives? Sure – geometry, tables, graphs, manipulatives – but all of those tools are to help them make sense and model the real world situation. As a teacher, what does this mean YOU should be doing to help foster students ability to model with mathematics? Well, for one thing, give them relevant and real problems to solve that are not “naked math” (see my previous post on this!) but contextual problems that force students to think, analyze and decide what mathematics will help them solve the problem. And then, provide them opportunities to collaborate, use a variety of tools, ask questions, and approach these problems in a variety of ways. Modeling with math means students apply mathematics and tools in ways that make sense to them so they can apply their mathematical understanding. That’s how YOU know they really understand the math, and that’s how THEY know math is relevant.

Obviously, the key here is making sure you provide relevant problems that “arise in every day situations” (as the standard emphasizes). Students learn and are more engaged if what they are doing is relevant.  And there are real problems every where.

lottery-abstract-illustration-dynamically-falling-balls-39580479So – here’s an every-day situation that arose just last night. There were 3 winners in the 1.6 billion dollar Power Ball lottery. Depending on the grade of your students, you could ask them a simple question – should these 3 winners take an annuity or take a lump sum? (Naturally – expect explanations and support for their answers)! In order to answer this question,  a lot of decisions need to be made concerning how to model the situation with mathematics. What, if any, equations would be helpful? Do they need a table to organize the data? What are the taxes and how will that impact the amounts?  These are big numbers – so could they make it simpler by using smaller numbers or proportions? Think of all the great mathematics that will happen and all the modeling with mathematics students will do within the context of this real-life problem. Think of the engagement. Think of the conversations. Hard to get that with a worksheet!