Friday's Five - Assessment: Our Students Deserve Better

I've been thinking a lot about assessment lately.  Not "assessment" in the narrow term meaning the high-stakes tests to which we are forced to subject our students, but assessment in the more global sense: how we determine what our students know and change our teaching to make sure they are learning.  That's what assessment is for, isn't it?  Despite the nonsense that is being thrown around by non-educators about the need to test kids to identify bad schools, bad teachers, bad kids, bad administrators, etc., assessment is really about identifying how we can help kids learn more.  The rest is all political mumbo-jumbo that's hurting our kids because it takes the focus off where it should be - student learning.



There are a few reasons that assessment has been at the forefront of my mind lately.  The most obvious is that the last month of school has been a fragmented mess of teachers struggling to promote real learning in the wake of schedule changes, lost teaching time, and stressed students due to mandatory state "assessments."  I've also spent a lot of time reflecting on my own assessment practices as part of my PAEMST application (one of the more grueling and beneficial experiences I've done as a teacher), which is due on May 1st.  Finally, this is the time of year that we place students into their courses for the next year - a practice that has increasingly become dependent on "data" instead of teacher recommendation.

In order to use assessment properly, to increase student learning, here are five things we need to keep in mind:

1. Use the right tool for the right job.  Often we are told as teachers to "use assessment data to drive instruction."  The problem is that by "assessment data", those making this demand are talking about state assessments, benchmarks, or diagnostics.  You can't make day to day changes that benefit students based on this data.  Learning that one of my students scored low in the "geometry category" five months ago on a state assessment is worthless to me compared with the exit card that showed me that he/she didn't understand that area was a two-dimensional measurement.  The latter allows me to correct the misunderstanding immediately, thus leading to greater learning.
2. I've heard Chris Lehman say before that educational technology should be like oxygen - imperceptible, ubiquitous, and necessary.  The same can be said for assessment.  We need it and should be using it all the time as a way to guide our students, but if our students are stressed about how they are being graded, ranked, sorted, or judged, they aren't focused on learning.  And learning should be our goal. 
3. "Assessment" and "Grading" are not interchangeable terms.  Often they are used that way because we tend to want to make everything measureable.  Data doesn't have to be numbers to be useful.  Again, learning should be our focus, not ranking or judging students.  Tests and quizzes will, for better or worse, always likely have a place in schools.  What is more beneficial for students, though:  giving them a 30 on a quiz in which they got 7 out of 10 questions incorrect, or sitting down with that student to discuss their confusion and helping them identify ways to learn what they haven't yet?  "Grading" is something that is done for the benefit of teachers, parents, colleges, and others.  Good "assessment" is done for students. 
4. Standardized tests, benchmarks, and diagnostic tests are not bad assessments unless we use them in ways for which they were not designed.  When we start using data from a benchmark or diagnostic tests to determine a student's placement in basic or advanced math classes or data from student standardized test scores to judge teacher efficacy and school quality we fail our students.  Arguments that my car got great gas mileage because it goes from 0 to 60 in 2.5 seconds or that my brother is a great basketball player because he has can punt a football 60 yards would be dismissed as absurd because those aren't valid metrics to use to judge such things.  Why aren't the conclusions we are erroneously drawing from bad metrics in education being dismissed as absurd?  I believe, as Joe Bower put it so well, we can't measure what's important, so we are putting importance on what we can measure.  It needs to stop for the sake of our children.  They deserve better.
5. We need to do a serious cost/benefit analysis of how we assess students.  The assessments that are given the most importance in schools right now are also the most costly in terms of time and money that have ever been given in schools before.  We spend billions of dollars as a country each year on the tests themselves, test prep materials, and resources to meet the logistics of administering the tests.  We spend weeks of time that could be spent on learning critical thinking and innovation demanding that kids learn test taking skills and low-level thinking facts so that they can pass the tests.  And what do we get?  Lousy data.  Data that is far, far inferior to the formative assessment data I could have collected in much less time and that could have been used immediately to teach students. 

Some will say, "but scores have gone up since we started testing kids, so there must be some benefit to all this testing."  While scores on state tests have gone up, this argument is totally false.  We, the public are being manipulated.  Politicians have made the tests easier over the years to show how wonderful they (the politicians) are at "improving education."  Anyone who has compared state tests from 7 or 8 years ago to current tests can see this easily.  Our students score almost exactly the same on international tests as they did before we implemented high-stakes testing.  We've spent trillions of dollars and countless hours of time that could have been spent on real learning for nothing.  Actually, it hasn't been for nothing.  We've spent it to make politicians look good and to help their buddies who own stock in companies that produce testing materials make a buck.  We could have gotten so much more for so much less.  Maybe it's time to let educators determine how to educate our kids. 

Friday's Five - I'm Not a Trained Monkey! (and other thoughts)

Some Fridays it's hard to come up with a topic about which to write.  Others it's hard to choose one topic because there are so many ideas I have floating around in my head.  Today is the latter.  I guess that means I should have blogged more during the week.  In any case, I'm going to share five thoughts that I've had the past couple of days.

Photo Credit: C. Frank Starmer

1. I'm not a trained monkey.  It's state assessment week(s) here in Pennsylvania.  The majority of my time in school has been spent watching students fill in bubbles with a #2 pencil.  Any trained monkey could do this.  I want to teach.  I want my students to learn.  The purpose of assessment is to guide teaching so that students learn more.  I won't get the results of this assessment until these students have moved on from my classroom.  It's a political shell game that doesn't benefit my students, and all the free snacks in the world won't convince them differently.  I'm a teacher, not a trained monkey.  Let me teach.  Let my students learn.
2. About those free snacks during state testing time - If research shows that kids' brains work better when they are well fed, have snacks, etc., shouldn't we be giving them the snacks during the learning and not during the assessment?  Funny how something as simple as a snack can illustrate so perfectly how out-of-whack our priorities have become.
3. Yesterday in the faculty room, someone was complaining that our elementary school pedagogy is too driven by the demands of colleges.  When talking about being more innovative with how we assess, teach, and organize schools, the counter-argument is often, "But what will happen when they get to college?  They'll be expected to listen to lectures and learn on their own."  Here's the thing: sticking 50-200 people in a room, lecturing at them (whether you use a PowerPoint presentation or not), and telling them to read textbooks in order to find additional information is not good teaching.  It's not the best way for people to learn.  I don't care how much people pay to subject themselves and their kids to that nonsense, it's still lousy pedagogy.  If colleges really cared about student learning and not their profit statements, they'd tailor their pedagogy to be more like kindergarten.  More play.  More investigation.  More collaboration.  More learning.
4. The difference in the restlessness of elementary students after changing the clocks for Daylight Savings Time in the spring is stark.  It's like they know they should be outside now.  After hearing John Medina explain during his ISTE keynote last summer how the human brain performs optimally outside, while the body is in motion, and in changing meteorological conditions, this restlessness makes a whole lot more sense.  
5. I've been lucky enough to be nominated for the Presidential Award for Excellence in Math and Science Teaching (PAEMST) this year, and a whole lot of my energy has been spent the past few weeks preparing my application.  The application is extensive and overwhelming, but I'm benefitting a great deal from the reflection and introspection into my practices that is required.  Part of that reflection has made me re-realize how much I benefit from all of you out there in my PLN - on Plurk, Twitter, Facebook, and those who I connect with in the blogosphere.  I am sincerely grateful to all of you for helping me better myself and my teaching.   

But, What Does it Look Like in My Classroom?

"But, what does it look like in my classroom?"

When discussing Project/Problem Based Learning (PBL) and other pedagogical practices this question sometimes pops up.  Teachers new to this type of learning often understand the theory, but have picturing the application of the ideas in their classrooms.

Today, I had my students watch a 25 min. video on how Disney Imagineers use levers and pulleys when designing attractions for Disney theme parks around the world.  I then told them that they had to design a new Disney attraction or restaurant with a story and theme in which pulleys and/or lever would be used.  After having lunch to think about their ideas, they were given 40 minutes in the afternoon to design a model or concept art of their idea to pitch to the class.  Tomorrow they will make their presentations and we will put their ideas into a Voicethread, which will be embedded on our class wikispace.

As my students (both regular and special education students) were totally engrossed in their work, having great discussions about their designs, and producing amazing visual descriptions of their ideas, I came up with my answer to the above question.

It looks a lot like me walking around my classroom looking for someone to help and nobody needing me because they are fully engaged, collaborating, and using technology to solve their own problems.  

The Arrogance of American Public Education

Today a member of my PLN posted this picture on Facebook:

The sad fact is that our schools are designed for students who listen well, comply with authority, have no urge to think for themselves, don't question the bias of information being given to them, have no desire to experiment, and learn at exactly the pace that we teach. 

If they learn too quickly, we force them to sit quietly (often bored out of their minds) while the rest catch up with them. 

If they learn too slowly, refuse to buy into the factory model of the 1900's that we force upon them, are interested in the arts more than standardized test prep, or prefer to learn while doing instead of listening, we label them as "learning disabled."  Often we medicate them to make them more compliant to our methodology.

Maybe we need to start teaching differently.   

Maybe our students have realized something that we have yet to discover.

Maybe we have a collective teaching disability and are too arrogant to change ourselves. 

After all, if our job is to educate and our students aren't learning, are we doing our jobs?  

Friday's Five - How to Talk About Math

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Friday's Five is a feature every week where I pick a new topic and list five items that I think fit best.  Then I ask you, my readers, to share your thoughts in the comment section.  For an archive of past topics, check the Friday's Five Page.  If you'd like to make suggestions about future topics or discuss topics I bring up on the blog with others, make sure you click the "like" button on the right hand side of the page to join A Teacher's Life for Me on Facebook.  Don't be shy about sharing the blog and Facebook Page with others.  Each post has a "Tweet" button on top and buttons on the bottom that allow you to share in several ways, including e-mail, Facebook, and Twitter.

Duke TIP/ Flickr

I spent a lot of time thinking about teaching math this week.  Our district is struggling to develop an action plan for transitioning to the Common Core Curriculum from the current Pennsylvania Standards, and I've been reading a few books on elementary math teaching.  While some of those thoughts are fresh in my mind, I wanted to focus this week on ways teachers (especially elementary teachers) can promote mathematical discussion in their classrooms.

Too many times our classrooms resemble ping-pong tables.  We ask a question. (Ping)  A student gives and answer. (Pong)  We tell them whether they are correct or not. (Ping)  Then we repeat the cycle over and over again.  There's a lack of in-depth discussion about math.

After all, math is not about numbers or right answers.  We've got calculators and computers for that.  Math is about thinking and solving problems.

The books I've been reading are from a company called Math Solutions (Full disclosure - Math Solutions sent me these materials for free, but I do not currently have, nor have had in the past any kind of financial  agreement with them.)  Their mission is to help K-8 teachers teach math in a way that promotes understanding rather than focusing on using procedures to get a correct answer.  I really like what I've learned about them so far and the fact that they aren't a textbook company or a program that's being touted as the answer to your standardized test score problems.  Some of five suggestions below were influenced by what I've read.

1. Stop being scared of math. - The biggest thing we can do to promote mathematical discussion and understanding in our classrooms is to stop treating math as if it's something hard to understand and difficult for most adults.  It's not if it's taught correctly.  Why is it acceptable for an elementary teacher to say that they couldn't possibly pass an 8th grade math test, but not acceptable for that same teacher to claim that they can't read on an 8th grade level?  They are both unacceptable.  
2. Focus on the meaning, not on the procedure. - For example, when teaching division, stop trying to get kids to memorize that they should divide, multiply, subtract and bring down.  They're going to forget it anyway (ask any 3rd, 4th, 5th, or 6th grade teacher and they'll back me up on this).  Instead focus on what it means to divide.  Explain that you are putting things into groups.  Ask them to come up with a story that fits the division problem. Those discussions will promote your students' understanding infinitely more than a cute story about how Dead Monkeys Smell Bad, or any other memorization of procedure you push upon them.  
3. Give students problems that don't have a simple answer. -  "What's the healthiest vegetable?" is a much better problem than "Carrots have x calories per serving.  How many calories are there in 8 servings?"  The first problem promotes a lot more thinking, discussion, research, and debate.  It also will lead to a whole heck of a lot more math, and it's a lot more interesting.  You won't find questions like that in a textbook, which brings me to #4.
4. Get away from your textbook. - Textbooks are often a crutch that holds us back in elementary schools.  There are a plethora of real problems out there in the world that both require math and have relevance to our students.  Work towards solutions to those problems.  Discuss them and demand that your students discuss, debate, and research them.  Your students will gain skills they'll need in life, motivation, and a sense of purpose in addition to learning math. 
5. Make the question "why?" the one you ask all the time. - I wrote a whole post on this, so I'll keep it brief here.  Students must know that they will have to understand math well enough to explain their thinking to others and not just spit back a correct answer.  When our classroom expectations rise to this level, students will rise to meet them.  There is no more important question to ask.

Now it's your turn.  How do you promote math discussion in your classroom?  There are certainly more ways than the five I listed.  What barriers do you see in implementing the ideas I'm suggesting?  Why do elementary teachers seem to view math so negatively?  How can we get them to change that view?  Please let me know your thoughts in the comment section below, and pass the post on to others who feel strongly about math.  I'd love to hear their thoughts as well.  

First Reflections

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flickr/faungg

At the conclusion of a school year, it's natural and professional to look back at what you did right, what you did wrong, and what can be improved.  An honest reflection of our practices is the one of the most powerful tools we have in improving our craft as teachers.  Below are some things I've thought about in the past few days, now that I'm on summer vacation.

Here's what the standardized testing data says:  84% (16 out of 19) of my students passed the state reading test.  One student missed passing by one question.  The same student had passed in 4th grade by one question.  100% of my students passed the state writing test.  I was slightly surprised and very happy about that.  It's not a surprise that all of my math students passed the math test, but 2 students fell from "advanced" in 4th grade to "proficient" in 5th grade.

I've been pretty vocal (or whatever the blog equivalent is) about the evils of standardized testing.  The way we use tests to evaluate teachers, judge schools, and drive every aspect of our school day from recess to pedagogy has been devastating to our educational system.  However, standardized tests do have a small purpose in education.  Using the data generated to see students' strengths and weaknesses, and then help those students overcome their weaknesses has been done successfully for decades.  State assessments don't measure what's most important, but the data they generate shouldn't be totally ignored.

This year I think I did a good job at expanding my use of technology to effectively teach collaboration and creativity.  Our class wiki received over 15,000 hits during this school year alone and now has had visitors from 122 countries.  Most of those hits came from people searching Google or Bing for information and getting it from the content my students created.  That's a pretty powerful thing for a bunch of 10 and 11 year olds in a tiny town in Pennsylvania.

I demanded more critical thinking from my math class and saw more learning from this group than any other math class I've taught.  The results on the math final exam were excellent (all but one student scored a 91 or higher), but what really exited me was the one question I gave them after the final was over.  It basically asked, "Joe Smith ran his best mile in 6 minutes.  Later that month he ran his first 26 mile marathon.  How long did the marathon take him?"  Almost every one of my students resisted the urge to multiply 6 by 26 and added on time to account for fatigue.  They thought about the problem instead of just manipulating numbers.  (Thanks to Dan Meyer for that problem.)

As the Head Teacher in the building, I'm proud of the way our discipline program has continued to evolve and the leadership role that I've been allowed to take.  Our office discipline referrals fell 30.1% from last year.

I'm extremely proud of some successes that I've had with students in tough situations and personal triumphs that I've seen in some of our students in different aspects of their lives that I can't mention here.  If you are a teacher, you can imagine the types of situations to which I refer.

I continue to feel unsatisfied, however, with my pedagogical practices when I'm with my reading class.  I love teaching math and American history.  I can't say the same for reading, and I think that comes across to my students more than it should.  In math, I have the confidence to let the textbooks gather dust while I focus on good teaching and learning.  In reading, I don't have that same confidence.  I'm hesitant to stray from the textbook, although at the end of the year I used our wiki to create book clubs based on interest that went very, very well.  You can see the results here.

Since I am well aware that the best way to teach reading is in the content areas, I'm a bit disappointed in myself that I did less direct reading instruction this year within American History that I've done in the past.  Those subjects had a lot more separation between them than I would have liked.

I need to get better with passing control in the classroom to my students.  This is difficult for me.  I want my students to be able to work the way adults do.  Right now I'm typing in a comfy chair with my feet up.  Most of the time when I read it's while relaxing on the couch or lying on the floor.  When I do my taxes, I usually have a snack and a can of diet soda next to me.  But in my classroom, I demand that my students sit in those ridiculously uncomfortable blue plastic chairs for much too long.  They may have the snack they brought only at 9:30.  I think my students would learn more if I gave them more freedom, but I've found that hard to do.

Friday's Five: Ways to Use Formative Assessment

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Friday's Five is a feature every week where I pick a new topic and list five items that I think fit best.  Then I ask you, my readers, to share your thoughts in the comment section.  For an archive of past topics, check the Friday's Five page.  If you'd like to make suggestions about future topics or discuss topics I bring up on the blog with other readers, make sure you click the "like" button on the right hand side of this page to join A Teacher's Life for Me on Facebook.  Don't be shy about sharing the blog and Facebook page with others.  Each post has buttons on the bottom that allow you to share several ways, including e-mail, Facebook, and Twitter.


We've all had students that try blend into the background.  They are the students who never raise their hand, sit there silently when you call on them until you choose someone else, and go through the school year determined to participate as little as possible.

We've also all been in the situation where several of our students have failed a test when we thought they knew the content.  Even though we recognize that they haven't learned what they were supposed to, we often feel we have to move on because of the amount of material that needs to be taught in a school year.

Is there an easy way that we can make sure all of our students, even the ones who try not to participate, are learning?  Is there an easy way to determine whether our students really know content before testing them?

The answer to both of those questions is a unequivocal "YES!"  There are several types of formative assessment - long term (like semester or end-of-year), medium term (end-of-unit, etc), and immediate (in-lesson).  It's the daily, in-lesson, formative assessment that is most important to increase learning for all of your students.  It also allows you to diagnose where you need to change your instruction so that your students learn what they are supposed to before they are tested.

Formative assessment is a way to diagnose the patient, instead of waiting for the autopsy.

Here are five simple ways to start using formative assessment to ensure all of your students are learning:

1. Get a set of individual white boards and have your students use them.  Have your students constantly show you that they understand what you are teaching them by showing you on their white board.  A quick glance around the room will tell you who understands and who doesn't.  Make sure the students that need more help get it.  
2. Stop having your students raise their hand to answer a question.  When that happens, you only get an answer from one student.  Instead, have every student write the answer to the question in their notebook or on an individual white board.  Maybe have them share their answer with a partner, and let the partner write it down, or have each student record an answer in a VoiceThread or Blabber.  It doesn't matter how they answer, just make sure that every student is responsible for giving an answer and justifying it.
3. Use exit cards.  At the end of a lesson, pose one short question to the class that deals with the day's lesson.  Have them answer the question on an index card and hand it to you before they leave.  Take a quick look at the cards.  If all the students knew the material move on to something else the next day.  If many couldn't answer the question correctly, start the next day's lesson with a review.  If some of the students need more help, build in an intervention into the next day's lesson.
4. Demand that students tell you when they don't understand.  I've found that colored cups work well for this.  I stack a green, yellow, and red cup on each student's desk.  If they understand what I am teaching, they show a green cup.  If the cup is yellow, they need me to slow down.  If the cup is red, I need to stop and re-teach something.  How do I make sure that they are telling me the truth? If someone has a red cup, I choose someone with a green cup to do the re-teaching. 
5. Let them give you a "thumbs-up."  During your lesson, ask a few yes-or-no questions of the class. Have them respond with a thumbs-up or thumbs-down.  You'll be able to quickly assess which students are not understanding your lesson.  

Lately we've heard a lot about "Data Driven Instruction."  Too often it means sifting through standardized test data.  The data we need to be using to drive our instruction is the data we get from formative assessment techniques like the ones listed above.  That data is easy to understand quickly and allows us to make changes immediately to ensure that our students learn.  Dylan Wiliam had a great way of putting it when I saw him speak.  His quote went something like this:  "Teachers who do not use formative assessment and then wonder why their students failed a test are like pilots that never make course corrections and then wonder why they ended up in Cleveland instead of Miami."

Now it's your turn to share.  Do you use formative assessment in your classroom?  How?  Do you have other techniques or ideas to share?  Leave your thoughts on these questions, or anything else you want to add in the comment section below.

Different Expectations

I'm sure most teachers have had some variation of this experience:

It's time to review a homework assignment from the previous night.  One student has nothing but a blank paper.  When you ask that student for his/her homework, he/she says, "I didn't know how to do this."  You then respond with something like, "I can't give you credit because you made no attempt.  I expect you to at least try."

We expect our students to try things that are difficult.  We understand that learning takes effort at times, is sometimes difficult, and requires a certain level of perseverance.

Other times, we hear this:

I didn't have time to do my homework last night because I had to ___________________ (go to soccer practice, go shopping with mom, wash my hair, etc.)

We expect our students to make their job as a student a priority.  Spending time on other activities instead of homework is unacceptable.

Do we have the same expectations of ourselves?  

Most teachers acknowledge that today's student will graduate into a world where information is stored and accessed on-line.  Most agree that our student's jobs in the future will require the use of new technologies to video conference, collaborate with others in distant locations, quickly judge the validity of a great deal of information in short periods of time, and perform many other "21st century tasks."  Those who don't recognize these facts are either egregiously uninformed or delusional that the 1950's are going to make a comeback.  

Alek Shresta/tigweb.org

If we are truly preparing our students for the world in which they will live, our schools should incorporate the same technologies mentioned above.  Too often, they don't.  Even when students are able to use a computer in our classrooms, too often it is for standardized test preparation or as some form of digital babysitting where they spend a period playing an "educational" game that requires little thinking.  A look at a 2009 study by the National Center for Educational Statistics, Teachers’ Use of Educational Technology in U.S. Public Schools, shows how lacking we are.  85% of our high school classrooms never use videoconferencing.  66% of our high school teachers don't even have their students use a computer on a regular basis.  

I've heard two common responses from teachers when they discuss why they don't learn to use technology to teach the 21st century skills our students will need in their classrooms:  "I don't know how" and "There's no time."  If we don't accept these excuses from our students, why do we accept them from ourselves?  Isn't it our job to learn?

Math is Not about Numbers

Today, I came across an article today in Education Week entitled "Researchers Probe Causes of Math Anxiety."  It was a decent article.  There were a few insights I found interesting.

When I finished the article, I read the comments.  Michael P. Goldenberg, a math coach in Ann Arbor, Michigan said this:

The way most US teachers present the subject in K-12, it's about only or primarily the following: calculation, arithmetic, and speed (with accuracy, of course). None of those things are particularly what mathematicians deal with. No mathematician is judged by speed of calculations - arithmetic or otherwise. Calculation may not even be a particular strength of a professional mathematician. Mathematicians by and large deal with abstractions, patterns, connections. Of course, some deal with applications of mathematics to sciences and engineering and other "real world" problems and situations. But when it comes to pure calculation, it's hard to beat a computer for speed and accuracy. What the computer won't give is insight, leaps of heuristic thinking that connects seemingly unrelated ideas in two or more areas of mathematics, the recognition of underlying structural similarities, etc. Computers don't think.

I had been thinking of writing a post entitled "Math is Not about Numbers" for a while.  I actually started this post a week ago, and saved it as a half-completed draft.  I don't know, however, that I can say it any better than Michael P. Goldenberg did. 

In a few years all of my fifth grade students will be using a calculator and/or computer to do their calculations.  I refuse to spend an entire school year teaching them procedures to calculate.  I'm going to spend the majority of the time in my math classes teaching them to make connections, recognize patterns, and make predictions.  I'm going to teach them to do what computers can't - think.  I'm going to teach them math.

Thinking: It's Not Just for the Smart Kids

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Today I had the pleasure of spending the day at the Northeastern Intermediate Unit at a math training for elementary teachers.  The two Math Curriculum Specialists at NEIU, Karla and Leeta, do a great job and today was no exception.  There was lots of great discussion on how to structure lessons in ways that make math more rigorous and build conceptual understanding for students.

At one point in the training we were asked to classify different mathematical problems as requiring either "low-level thinking" or "high-level thinking."  At the conclusion of this we discussed why we chose the categories for each problem.  This conversation concerned me.

Many times other participants reasoned that a problem required high-level thinking for reasons such as these:

• "None of my students could do that"
• "That problem is really hard"
• "There was more than one step"

To explain why they thought problems were low-level thinking problems, these explanations were given:

• "My second graders can do that, so it can't require high level thinking"
• "Everybody in my class could do this"
• "That's something that's taught in the earlier grades"

What worries me about the above comments is that they imply that any multi-step problem makes students think at a higher level, that younger students shouldn't be required to think at higher levels, and that only our best students can reason mathematically.  These assumptions are simply false.

False assumption #1 - All multi step problems require higher thinking

It's dangerous to confuse the number of steps in a problem with how rigorous it is.  Higher-level thinking is analyzing, synthesizing, and evaluating.  A bunch of steps that require nothing more than computation still allow a child to know nothing more than memorized procedures.  Little mathematical understanding is required.

We shouldn't confuse how difficult a problem is with how rigorous it is.  Students can struggle with problems for a variety of reasons: lack of vocabulary knowledge, reading problems, etc.  If you asked me to find the square root of (45x - 3/π), I would have a great deal of difficulty.  That doesn't mean that it requires analysis, synthesis, or evaluation.

False assumption #2 - High-level thinking should only be done in the upper grades

If we don't teach our students to think when they are in the lower grades, what makes us think that they will know how to do it when they get older?  One of the hardest things for me as a teacher is to get my students used to having to analyze and evaluate their thinking when I ask them "why?"  Students should be required to think at all times.

Too often we only give first grade students questions such as, 4+3=?

We should spend more time asking, "How many different ways can you make the number seven?"

The latter question is just as grade-level appropriate, and it allows students to think instead of just manipulating numbers to find the right answer.  It allows for learning, and not just memorizing. 

Don't think I'm saying that knowing math facts is not important.  It is.  But understanding math is important, also, and often ignored.

False assumption #3 - Only the smart kids are capable of mathematical reasoning

Again, I think this misunderstanding comes from the confusion between difficulty and rigor.  I would think that even struggling 5th grade students would be able to come up with several answers to the second question above with little difficulty.  That doesn't change the fact that it requires a different level of thinking than basic recall and/or calculation problems.  

Many elementary teachers struggle with math.  Many would even admit that they are not great at math.  We'll ignore the fact for right now that we'd find an elementary teacher who claims to not be able to pass an 8th grade reading test to be completely incompetent, but an elementary teacher who claims to not be able to do 8th grade math completely normal.  I'm sure that will be the subject of a future blog post.

Because of their own background, the way they were taught, and/or their perception that math is about numbers and getting correct answers (it's not), the belief is out there that our "smartest" kids are the only ones who are capable of being good at math. 

Math is not about numbers.  Being able to multiply numerators and denominators may get you the correct answer to a fraction multiplication problem, but it does not show that you understand what multiplying fractions really is, or what problems you face in your life that may require you to use this skill.  Being able to Divide, Multiply, Subtract, and Bring Down may get you the correct answer to a division problem, but it won't show that you know that division is putting items into equal groups. 

These memorized procedures aren't math, but in the United States we rarely require more than that from our students.  This was shown pretty clearly in the 2007 TIMMS study in which math and science teaching and learning were examined in countries around the globe.  The kids who aren't good at memorizing these procedures that we often teach out of context and without relevance are not bad at math. 

All students are capable of higher-level thinking.  Some are capable of doing this higher-level thinking with more difficult problems, but all students should be required to think.

The alternative is that we develop a generation of students who can't. 

Photo Credit - flickr.com: Sidereal

The Need for Narrative

Our students think we're boring.

There's lots of reasons why that we have no control over.  Students are bombarded with stimuli from every direction from the second they step out of our classrooms to the second they enter in the morning.  It seems like they're playing games on portable devices, texting, watching TV, playing video games, listening to music, or all of the above just about every minute they are awake.  This is a product of the culture we live in.  We can argue about whether that's good or bad, but it won't change it, and it won't help us teach any better.

There is one thing that can't be argued, though.  Bored students are emotionally unengaged, and emotionally unengaged students don't learn.

Our real dilemma is that there's nothing especially exciting about most of the material we teach.

When I teach narrative writing and how to create plot to my students I often start with this short story:

One sunny day I went to the store.  The store was three blocks from my house.  When I got to the store I decided that I really wanted a pretzel stick.  The pretzel sticks cost 10 cents each.  I reached into my pocket, took out a dime and bought one.  I ate it.  Then I walked home.

I then ask them what they thought of my story.  They always tell me it's terrible.  When I ask why, they tell me it's because nothing happens.  It doesn't have a plot.  There's no problem, and thus, nothing in which to get interested.  It's impossible to make an emotional connection to the main character.

Too much of what we teach resembles that pretzel story.  Nothing interesting happens.

In his book, Lies My Teacher Told Me: Everything Your American History Textbook Got Wrong, James Loewen explains that textbook companies, in their quest to sell as many social studies textbooks as possible, have eliminated many of the stories that make history interesting.  Two that come to my mind quickly are the real story of Thanksgiving and Christopher Columbus's actions, but there are many, many more accounts of events in textbooks that range between skewed and downright false.  There's little doubt that real history would be less politically popular, tougher to sell for those textbook companies, and much more interesting to our students.

When we eliminate the controversy the plot is destroyed, and the narrative of our history is reduced to a series of boring facts.  Students are uninterested, and emotionally disconnected.

Math class needs narrative just as much.  Too often we give students procedures and hints to memorize along with pages of calculations, but no real reason to connect with the material.  It's not real to them.  Even the "word problems" in textbooks and on standardized tests are absurd failed attempts to make math "real."  Take the following problem that was on a recent state standardized test I administered:

You go to the store and buy a block of cheese that is a perfect cube.  How many faces, vertices, and edges does the cheese have?

Are we really expecting students to believe that this would happen in their lives?  Or that they would ever care how many faces, vertices, or edges their cheese would have?  It's what Jo Boaler describes as psuedocontext, and it's doing our students a disservice.

Let's stop giving students a multitude of calculations and stop giving them ridiculous problems that lead them to believe that math is something only used for obscure, unrealistic situations.  Like the great story, let's give them narratives where the main character, which in this case is them, has to overcome an interesting problem.  In the process there's a good chance we'll make math relevant to our students.

In the past few years, I've tried to make an effort to build more of these types of problems into my math class with varying degrees of success.  Some, like our Pi Day activity this year, have gone great.  Others, like a shipping problem I created, haven't gone nearly as well.  With each attempt, though, I've gotten better at forming the type of problem that both allows for great understanding of the topics we're learning and gives students a sense of purpose and emotional connection to the math.  There's no doubt from my experiences that students are more engaged with these types of challenges.

Student engagement in other subjects can be increased using many of the same ideas.  In reading, we often ask students to read informational and persuasive pieces in which they have no interest and no emotional investment.  Why not tell them a story, and then give them a problem they have to solve through research instead?  After all, there are only two real reasons we read outside of schools:  because we enjoy the material, and because we need to in order to get information for some real reason.  Shouldn't our reading classes in school start to reflect that?

The most interesting parts of life usually make great stories.  The most interesting teachers are usually great storytellers.  The most interested students are usually those who are most engaged with the material they are learning.  Let's bring the great narratives that make our lives interesting into our classrooms, and watch our students flourish.

The Most Important Word to Use in Your Classroom

One of the toughest jobs that a teacher has right now is to overcome the culture of standardized test prep.  The pressure to pass state tests has lead us to a place where more importance is placed on memorization of factual nuggets, learning test taking tricks, and following memorized procedures than real critical thinking.  For many teachers, it's tough to remember that we entered this profession to inspire the next generation to greatness when we spend the majority of our time filling our students' heads with unrelated facts.  The term "problem solving" used to mean the ability to actually come up with practical solutions to real problems.  It has evolved to mean "coming up with the right answer to a math problem that is written with words." 

That's not problem solving.  That's not critical thinking.  That's the ability to read and make a calculation.

It doesn't take real thinking.  It won't help you figure out how to solve the problems that will face you in life.

When it comes to reading and social studies, our demands on students are no better.  We still ask for the main idea of a passage that students have no interest in reading, but never insist they read something they feel strongly about and give them a change to motivate their fellow students to action.  We ask for the date of the American Revolution and the cause of the Civil War, but never insist that they find parallels to current world events. 

It's as if those in power want us to pump out automatons that blindly follow orders instead of innovators who can mold the future. 

Can we rise above these pressures to inspire students, demand critical thinking, and at the same time prepare our students for state tests? Is there something that can be done which doesn't take a ton of professional development, training, or an overhaul of current practices?  Is there something we can do right now?

Yes.  Start asking "Why?"

When your students tell you the answer they came up with on your math assignment, don't tell them whether it's right or wrong.  Ask them why that's their answer.  When your students tell you that the main character in the story was "friendly", don't let them off the hook.  Ask them to prove why he was friendly.  When your students tell you that the Japanese bombed Pearl Harbor, ask them "why?"  Make the use of "why" so ubiquitous in your classroom that your students know it's coming before you even ask.

Your students will hate it at first, and for a while it will probably be uncomfortable for you.  They'll look at you with blank stares, speechless at the fact that you are making them think.  After all, they've been convinced by our standardized testing culture that this type of real thinking is unnecessary.  They'll hope that staying silent for long enough will convince you to give them the answer.  They'll tell you "because the textbook says so."  They'll hope that telling you, "I don't know" will shut you up.

Don't let them get away with it.  Demand an answer that shows real understanding. Do this until the culture in your classroom is one where critical thinking is expected.

Do this, and your students will be able to do more than just pass "the test."  They'll start to evaluate and judge.  They'll start to wonder and debate.

Something else might happen, too.  You may start to remember "why" you chose the profession that creates all others.  Along with inspiring your students, you may find renewed inspiration yourself.