Easy to Make Fraction Wheels

Fractions are often a difficult concept for students to visualize.  Today I had the pleasure of working with a group of 5th grade students.  In order for them to show their knowledge of fractions, we spent 15 minutes creating fractions wheels. Below is a step-by-step guide on how to do this with your own students.

Materials needed: Cheap white paper plates, marker or crayon, scissors, ruler

First, have students completely color one of the plates.

Next, have them use the ruler to find the center point of both plates. This is a great time to introduce or review vocabulary having to do with circles such as "diameter", "radius", "chord", etc. 

When they have found the center of both plates, have them cut the radius of each plate.  

Finally, place one plate on top of the other and twist them so they interlock where they have been cut.  This will allow students to rotate the plates to create representations of different fractions. We spend a bit of time having them showing different fraction representations and explaining why they believed those representations to be accurate.

After spending some time exploring fraction concepts, students will use their creations to teach those concepts to other students around the world through short videos as part of the Distance Teaching Project. 

Learning, Teaching, Leading: Creating a Toolbox for Today

Over the past few months I've had the great experience of teaching a 3-credit equivilent graduate course for my school district to other teachers who were interested in learning ways they could teach 21st century skills.  Each of the 10 sessions was on a different aspect of teaching and learning, and each provided me with the opportunity to collaborate, teach, and learn.  We found many great resources along our journey, and since I'm sure there are others who could benefit from them, I'm sharing them here.  Embeded below is the Livebinder of the resources, videos, and information we used during the class.

Learning, Teaching, Leading: Creating a Toolbox for Today

Friday's Five - What We Should Be Teaching

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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.

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Ask a parent what qualities they want to see in their children when those children become adults.

Ask a teacher what qualities they want to see in their students when they graduate high school.

Ask your neighbor what qualities they want to see in the next generation of young adults that will be living in the neighborhood.

Ask a businessperson who is looking to hire someone to work at their business what qualities they want in their employees.

I can guarantee that none of them will respond that they are hoping young adults will be able to find the main idea of a passage, identify the author's purpose for writing a poem, or be able to calculate the mean of a series of random numbers without context.  With that in mind, today's post will focus on five subjects that are largely ignored in schools today due to the culture of standardized testing and the push for "accountability."  I don't think that one can argue that a focus on the following five areas in schools would not be beneficial to our students, our communities, our country, and the world as a whole.  If our students were "proficient" in these areas, everything else would take care of itself.

1. Innovation - We are robbing students of motivation and an understanding of what they are capable by forcing them to only perform tasks related to multiple choice questions on reading and math (those terms are used loosely) tests.  It used to be that "creating" meant that students would glue cutouts from a magazine in a shoe box.  Now, technology gives students the ability to share what they've learned in many ways instantly.  Their writing can be published instantly on a blog for the world to read.  Their videos can teach children thousands of miles away.  The possibilities are vast and numerous, and we need to take advantage of them.  
2. Empathy - The ability to understand others emotions and be compassionate is something that is rarely focused upon and is of paramount importance for our students.  There are numerous studies that show that empathy and success in business are closely linked.  A Google search for "empathy and success" produces over 770,000 results.  Even more important than business success, however, is the fact that being able to empathize makes one more able to help others.  
3. Service - In my experience, nothing gives a person a feeling of self-worth and a satisfaction of having filled one's purpose more than the act of helping others in need without expecting a reward in return.  We should be giving our students opportunities and encouraging them to find ways to provide service in areas about which they feel strongly.  
4. Critical Thinking - This crucial skill, which is closely related to innovation, is the one that has been most ignored due to our current standardized testing craze.  There is simply no way to truly measure the ability to problem solve and think critically on an easily scored multiple choice assessment.  Teachers don't demand critical thinking because they don't have time; they are forced to teach students to interpret test questions that measure low-level thinking skills instead.  Teaching critical thinking takes time, leads to unpredictable lessons, and puts students in control - all things which are frowned upon in many of our schools.
5. The Love of Learning - We have to stop using our schools as places where we fill students' heads with facts.  Unfortunately, most of what we teach can be Googled in less than 30 seconds on their phone, which too often we won't let them take out of their pocket.  Our students have figured this out and largely find school to be irrelevant.  I wish I could say that they are wrong.  We need to start using schools to show them the power of learning.  If we combine the above four subjects and teach our students to empathize with others, allow them to find ways to help others they can become passionate about, and give them opportunities to develop their creativity and critical thinking skills, what we will start to see is students who take control of their own learning.  They will learn without us asking them too.  How often do we hear complaints that students don't study?  What if they were so engaged and passionate about a topic that they didn't view learning outside of school as studying, but rather as necessary to fulfill a desire deep inside of themselves?  

Is what I describe above possible?  Yes, but not in a culture based on assessment and test scores.  It's being done right now in several amazing schools.  Unfortunately, those schools are the exception.  We need to change the culture of education so that this type of education is what is expected.  What if we defined success by the positive impact we have on others rather than by how many low-level thinking questions one answers on a once-per-year assessment? 

Now it's your turn.  Are there any important skills that you think we are not teaching our students?  What are some ways we could teach these topics in our schools?  Should schools be teaching the above qualities?  Would our society be better served if we left the development of these qualities to parents and continue focusing on reading and math?  Let us know your thoughts in the comment section below, and share the blog with friends and colleagues.  We'd love to hear their opinions as well!

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