I’m in the middle of teaching my girls how to conceptualize proportions. They can solve them with no problem, but they don’t really understand what they mean…and that all lies in their (in)ability to truly understand fractions.

Typically, fractions literacy begins in grade 3 ( according to the online version of NCTM’s Principles and Standards ) and continues to develop until grade 6. However, as any middle grades ( or high school or college ) math teacher can tell you, students never fully grasp the concept of fractions as comparing a part of something to a whole ( or in the case of rates/ratios comparing a part to a part, part to a whole, etc ).

I discovered just how bad off my students were in this regard when I tried ( unsuccessfully ) to show them the visual representation of proportional relationships.

I was using an activity involving scaling down recipes to illustrate the physical aspects of proportional reasoning….and fractions. The activity required us to take 3 cups of flour (required for the original recipe) and divide them each in half, thus leaving us with 6 piles of flour, each measuring a half of a cup. The questions that followed included:

How many 1/2 cup piles have you created ? ( 6 )
How many of the 1/2 cup piles are need for the scaled down recipe? (3)
How many scaled down versions of the original recipe can you make? (2)

I thought the process would be much smoother than it actually was, but the students had trouble understanding the visual representation of each pile being divided in half. Even more difficult, was understanding that the concept of half is relative to what is considered a whole. Essentially, we couldn’t get though the discussion of dividing the piles of flour into 6 halves and that the 6 half cup piles are equivalent to the 3 whole cup piles….or that the 3 half cup piles of flour is the same as 1 and 1/2 cups of flour.

But…they could do all of the above with numbers…and I was stumped, frustrated, and slightly annoyed.

How do you teach students who have internalized a process to understand a concept rooted in physical reality ?

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In recent years textbook publishers have been offering digital and online supplements to justify the exorbitant prices they charge for physical books. As long as print is the primary means of communication in primary, secondary, and higher education the major publishers will continue to monopolize the market. Amazon’s Kindle and other e-readers are making it possible for smaller publishers to get their content into the hands of students. However, the processing and display limitations of e-readers is preventing their use from becoming widespread. Also, learning is becoming increasingly student centered, interactive, and media driven. As such, single display e-readers without color are becoming antiques before they even hit the market. The best possible solution would be a netbook sized device with dual displays and some sort of stylus input. One display would be a dedicated e-reader and the other would be dedicated to traditional computing…all of it opensource of course!

I wonder how soon we can get something like that to market.

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I was a bit harsh on textbooks in my last post. In a perfect world all teachers would be creators of the content that they use in their classrooms. This isn’t a perfect world. Modern classroom teachers are often responsible for so many other things these days that writing the content to be used in your class is impossible. Time simply doesn’t allow it.

However, there is still a way to use the textbooks and get the most out of them without having to (completely) reinvent the wheel.

The following is a typical textbook question/problem:

James is taking a cab from the airport. The cab company charges a pick up fee of $2.75 and also charges $2.25 per mile driven for all passengers. Write an algebraic equation that represent this situation where X is the number of miles driven and Y is the total paid to the cab driver.

While there isn’t anything wrong with this question, per se, it isn’t very intellectually interesting to the student, nor does it provide students with an opportunity to see the interconnectedness of the various strands of mathematics. Classroom teacher can (and should) take the simple, single skill driven textbook questions and modify them to be more complex and intellectually interesting to the students.  See the modified question below:

Tiffany Smart received a cell phone as a birthday gift. She promptly downloaded the Ubertwitter and Facebook applications for her new phone. Her phone company charges her a monthly fee of $39.99 for her plan minutes and a fee of $0.05 for every MB of data she uses via her mobile internet.

a. What are all of the variables in this problem solving situation? Identify them as either rates of change, constants, independent and/or dependent variables. Be sure to explain how you have chosen to categorize each variable.

b. How can she use an algebraic equation to determine the cost of her cell phone plan each month if her data usage fluctuates each month?

c. Tiffany uses 100MB of data in January and her data usage increased by 25MB each month after January for three months. Represent her data usage in a table and in a graph.

d. As a result of the increase in her data usage her cell phone bill does not remain constant from month to month. Use a table to represent her cell phone bill is she uses 0, 50, 100, 125, or 200 MB of data per month.

e. What relationships (if any) exist between the variables in this problem?

The modified question is considerably longer. However, the additional sub-questions direct the student to see the verbal, visual, and symbolic representations of the problem solving situation. Whether or not the question is truly more intellectually stimulating than the first is up for debate, but it is definitely more rigorous. Hopefully my (and your) attempts to make questions relevant to students is well received  and not seen as a weak attempt to be “cool”.

*Notes: This tactic ( which I use fairly often because I don’t have the time to write ALL of my own content ) still isn’t as good a teacher as experiential learning. The nature of any teacher/textbook created question is to be teacher centered with a specific goal in mind. The questions themselves, by definition, constrain the students’ thinking to a single box. However, by layering questions it is possible to do more with less with your students.

How do you modify YOUR content ?

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The structure of the typical middle grades math textbook follows this basic outline:

1. Vocabulary review

2. Brief explanation of concept.

3. 2-5 very specific examples of the concept in isolation.

4. 1-2 very specific examples of the concept in a problem solving situation.

5. 20-50 problems that mirror the examples.

The problem with this structure is that if teachers simply “taught the text(test)” students would have almost no opportunities to see the interconnectedness of mathematics concepts. Nor would they have the opportunity to think creatively to solve any problems since they would just be parroting back a technique rather than applying said technique in new or novel situations.

A better lesson/textbook structure would be:

1. Teacher presents students with a rigorous and relevant problem that they can think about but cannot solve without specific mathematics concepts that the teacher will introduce later.

2. Discussion should take place among the students about the best possible strategy to solve the problem. Teachers should participate in the discussion and ask questions that guide students toward the realization that they need certain mathematics concepts to achieve their goal.

3. The students and teacher work together to come up with a solution to the problem.

4. After the problem is solved through exploration the teacher bridges the gap between the students’ problem solving process and the specific skills/goals the teacher had in mind when the lesson began. Essentially, the teacher begins teaching only after the students have been given a change to struggle with the problems at hand.

The benefit to my process is that it gives students a chance to be interested in the concept first. It also gives the students a need for the mathematics that the teacher plans to introduce. Teaching based on curiosity and need is harder to do and requires more planning, more time for students to make mistakes, and more time for questions…but students learn from questions…they don’t learn from lectures (or textbooks).

The only example I can think of where this is taking place in any organized fashion ( because it is hardly organized when I do it! Like I said, it’s pretty difficult to pull off and often takes months to get the students used to learning like this) is here.

All of this being said….my open-source text has been scrapped and I’m starting from scratch…to somehow constrain what is essentially experiential/collaborative/free form learning to a .pdf file. Wish me luck.

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The day soldiers stop bringing you their problems is the day you have stopped leading them. They have either lost confidence that you can help them or concluded that you do not care. Either case is a failure of leadership.

-Colin Powell

I took an educational leadership class once…until I dropped it. The class was taught by a professor who has never worked as a school leader in K12. The course was supposed to be about the basic requirements of school leadership and it somehow ended up being about issues of racism, professional development, and making presentations. If this is any indication of how our current crop of educational leaders are prepared we shouldn’t be surprised that our schools are failing.

I’m in the process of working toward become a local school leader and I think there are several keys do’s and don’ts when working toward becoming a succesful school leader.

Do:

1. Start out as a teacher leader in a core content area. I have nothing against coaches/band teachers who become Principals and Assistant Principals but I think it best that you know the pressure of high stakes testing and being evaluated based on the academic performance of your students before evaluating someone else on those same guidelines. Experience is a hell of a teacher.

2. Get yourself a battle tested mentor. You don’t, and won’t, have all of the answers. An older, wiser ed leader can help you steer clear of the mistakes that they’ve made in their careers. Listen twice as much as you talk.

3. Work in business or take a business management class before trying to step into the position of an ed leader. Educators are taught to manage children. Individuals who work in business are taught to manage adults.  Ed leaders have to manage both.

4. Lead by example. If your team sees that you are willing to do what you are asking them to do they’re more likely to trust your judgment and honor your requests.

Don’t

1. Try to run before you crawl. Learn the ins and outs of school leadership from every angle. Trying to step into an ed leadership position before learning all of the angles can leave you looking like a jerk. Trust me, I’ve seen it happen.

2. Do anything without a plan. Too many ed leaders make it up as they go. The kids suffer. Morale suffers. And ultimately you lose support from your people.

3. Fail to recognize the work of your people. Most often, people will work hard for you if they know you appreciate what they do. If you fail to recognize hard work morale will die.

4. Forget that all ed leaders started out as classroom teachers. Remember your roots and stay grounded.

5. Stretch the truth about your areas of expertise. Given enough time the embellishments will start to show. Again, your people will instantly lose respect for you and that’s a hard thing to regain once it’s gone.

I hope I can follow my own advice when the time comes.

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Wicked men obey for fear, but the good for love.
– Aristotle.

Inspired, once again, by TheJLV.

When I told my parents I was going to become a teacher neither of them told me they disapproved of my decision…they both just hit me with blank stares. See, I was supposed to be one of the best and brightest and the best and the brightest don’t waste their lives as teachers. After all, teachers don’t make any money.

I didn’t know how to convince them, or anyone else in my circle at the time, that what I was doing was worth more than some change in my pocket. What I wanted to do was make school a place where a kid like me was acceptable.

I’m a young Black man raised on Hip Hop, rampant consumer consumption, pop culture, and the internet. I, and those like me, are the present and the future of this country. And we have a lot of people scared to death.

They don’t know what to make of how we embrace, use and abuse our love of technology, pop culture, and all things improper to make math and all of school make sense to the most unlikely candidates. My teaching style in particular has made me enemies and caused me all sorts of grief at work. I could easily tow the line and do as the Romans do but then my students wouldn’t be as engaged in class as they are. I would become one of THEM instead of being one of ME.

My close friend and co-worker Mike Jones ( name changed to protect the innocent ) tells me all the time that the way we do things is intimidating to the old school and to those that subscribe to the old school mentality. He says things like “Jo, they’ve never seen it done the way we do it…and that scares them.” Does he mean that its too Black, too young, or just too different ? I don’t know…and most of the time I don’t care.

I do what I do because I love to do it. I’m good at what I do because I love to do it. How can you not be on board with that?!

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I’ve wrapped up my 1st algebra unit with my 6th graders and I have to say it was an overwhelming success. Student growth from the pre-test about 6 weeks ago to the post-test a few days ago shows a range of 10-50 percentage point gains with an average gain of roughly 30 percentage points.

For many of them, this was their first time actually digging deep into algebra and using it to solve problems. They were able to represent problems verbally, visually, and symbolically. I started on error analysis with them toward the end of the unit. That proved to be a little more difficult than I had hoped, but we still have plenty of time in the year for that.

The culminating activity for them involved having them learn to write secret codes with algebraic equations and functions as an encryption device.

Before jumping into the activity I showed a short clip from “A Beautiful Mind” It was the pentagon scene in which John Nash stares at the wall of code and deciphers it mentally. That isn’t exactly where I was going with the activity but it quickly got me some buy in from a classroom full of pre-teen girls who had never heard of cryptography or used the word encrypt or decrypt a day in their lives.

We began this exercise using simple substitution to “hide” messages. A = 1, B = 2 and so on. They grabbed onto that rather quickly and some of them even made the leap that this was essentially the equation y = x with X being the input position of the letter and y being the encrypted position of the number.

Next, we used one step equation to encrypt and decrypt secret messages. Essentially, if A = 1, B =2, C = 3…we would use the equation 5x = y to encrypt the message. A = 1 became A = 5 and so on. We toyed around with this simple encryption for a day or so and eventually moved up to multi-step equations as encryption and decryption devices.

Finally, we moved into the realm of non-linear functions as encryption devices and the use of tables/grids to decrypt secret messages. This proved to be a little more difficult because developing an equation for the system we used may be about a year away for my 6th graders. This was the most time consuming of all the activities and I witnessed a loss of interest and some burn out on this activity.

I like the idea of using real world application of mathematics beyond using equation to figure out how many people can come to your party, etc.

I have been receiving jokes and little messages from my students in the form of encrypted numeric messages since we concluded the activity and I love it! One disappointment took place yesterday though. We were completing our second review of the last 6 weeks when I witnessed one girl passing a note to another. I was overjoyed for a second because I had hoped that they were using a code to communicate in class. Unfortunately, it was in English and I was able to read it quite easily.

Sigh.

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The film with the same title as this post is probably one of my favorites because it begs the question…”Is it ever too late to change?”

I often ask my colleagues at what age or stage in life does a student’s refusal to work to their full potential simply turn into that student just being average or below average. At what age does potential simply give way to actual performance? The answers always vary.

My conclusion is that there is no age or stage of life when the potential of another should ever be given up on. Redemption, change, growth are all continuous variables and as such have no ending point.

Everyone has a 25th hour.

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Torture numbers, and they’ll confess to anything. ~Gregg Easterbrook

Statistics and data have always been incredibly vital to public education. However, since the NCLB mandate and the Bush era that defines research as being replicable, generalizable, and quantitatively based. As such, school leaders LOVE data, graphs, and Microsoft Excel. I’m all for data and statistics in schools under certain conditions.

1. Schools employ a pre-test, treatment, post test design: I’ve worked in schools where only benchmark assessments were given. Student and teacher performance were only weighed against absolute standards. Growth wasn’t tracked at all.

2. Assessments must be frequent…but not too frequent: I’m a fan of monthly pre-post test assessment as well as quarterly benchmark assessments. Monthly pre-post tests may seem excessive, however teachers should be assessing AT LEAST that often anyway…

3. The data MUST be used to change instruction: If schools gather data just to have graphs on the walls or in their data rooms the process of gathering the data is pointless. If teachers and schools are not using data to change or improve instruction then they’re essentially trying to make a cow fat simply by weighing it.

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1. The students.
2. The students.
3. The students.
4. The students.
5. Summer vacation.
6. The students.
7. The students.
8. The stories that the students tell me.
9. Spring break.
10. The students.

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