Math teachers hear lots of questions and most will say that the most common question is “Why do I need to learn this?” The answers are many and it is just part of the tug of war that can happen in learning. In reality, I haven’t heard many great explanations of why we teach all of the stuff we do. Some could argue the importance of absolute value functions but he or she would not have the strongest argument. They are in the curriculum and a part of testing so they are a reality of what needs to be taught but in no way is the students’ question answered.

I studied math because I found it interesting and a challenge. The subject is based in problem solving and I am gravitated toward a system defined by this concept. My favorite teachers focused on the dialogue of math as a means to examine problems and present situations in different ways. I view much of math is a way to state the same thing in different terms so that a pattern is more visible. I think that’s cool stuff but not everyone does.

You know what else, I am a goofy white male. What does that have to do with anything? Nothing and a great deal. It should mean nothing but I find a ponderous number of math teachers to be goofy white males. While it is in no way true that all white males share the same cultural experiences it means that we likely do not share the experiences of non-white male students. That issue would not be a huge challenge for me if I taught at a suburban boys school but not in most any other school. The reason it is important is that a black female student may have a much easier time identifying a black lady teacher. The girl sees that her teacher is interested in math and decides to follow suit because of the model set by her teacher and not for whatever esoteric reasons we like to have.

Different population groups treat math differently. I have worked in schools where all of the advanced students were female and I have been in ones where few of the girls developed any competence in the subject. Women, in particular (funny to say for half of the population) have societal expectations that can drive or stunt their learning. If they are to marry early and tend to the house, then they don’t have the confidence or motivation to pursue their studies beyond a comfort zone. Oddly enough, many of the girls in lower socio-economic areas see that they will need to have strong careers to support themselves and their families and school is the most available avenue for them to achieve. Those who push for more women in STEM should be pursuing these groups for they are driven to work hard and overcome challenges to get a strong career.

I work to inspire all of my students but realize I can’t. The most challenging students lack confidence and I find trouble connecting with them in a way that they are motivated. I tend to feel that what they really need is a person they can identify with to aid them.

We can’t have a specific teacher for each student in the classroom but we have powerful and great resources outside of the classroom. As more teachers are using variations of the flipped class, we have in our hands a tool that can let each student learn from people they can identify with. Unfortunately, the mechanism is currently broken. Most of the math videos are made by goofy white men. There a few others but not nearly enough. Students and teachers need enough videos to find the best ones to deliver the subject to them.

The good news is that we are faced with a fairly easy problem to solve. Anyone who feels should add to the library of learning videos. It’s fun to do and watching others will help you analyze your own teaching. Present lessons based on your strengths. Show to others what math is to you. You don’t have to make videos, just take the time when searching through them to find different ones. Ones with different types of people and different types of presentations.

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By **Redmond**

# Fixing the broken.

Some years ago, my first year as a teacher in a real school, I had to take the Praxis exams to get into an education program. I needed to be registered and enrolled in the program to retain my Alternative Type II license. I needed my license to keep teaching. I needed the job because I had a family to support. Since I was eyeball deep in trying to make it through the first year and under more pressure than normal, this test was an added weight that I could barely handle. I hate taking tests and despise standardized ones even more even though I am a top level test taker. These tests help me out but I still can’t stand them. Nevertheless, I woke early one Saturday morning, made my lunch, and made the hour drive to the testing site. I had neglected to sign up for the test sooner so I had to go to Crowley’s Ridge Community College rather than the university two miles from my house. Crowley’s Ridge is a really interesting and beautiful place but out of the way for a testing center. It is a place different from the surrounding delta with it’s own plants and animals.

As I was leaving the city, I realized that I had forgotten a pencil and frantically found a convenience store that carried writing utensils. The corner store never has what you need when you need it and has made my life far less convenient with their promises of ease.

The sky looked and felt a touch ominous and spring has the most violent weather but I really needed to not worry and focus on the mater at hand of getting to the test site and taking the danged test.

Some time down the road I found the place, signed in, heard the never-ending instructions, and filled in bubbles. Filling bubbles, I can’t imagine how many kids fill in now days. I was only taking one test so I got to leave then; as I was heading to my truck, I could tell the weather was making a severe turn. Tornadoes were definitely in season. I checked my radio on the drive back and realized that I was just in front of a massive storm. It was to be a race and I didn’t feel like getting stuck somewhere outside of West Memphis is such a mess. I also wondered what was happening to all the folks taking the next test. A co-worker of mine who was there told me later that they had to spend some time in the shelter and wait till the tornado warnings lifted.

I did well on the test. I knew I did well when I left and was just waiting for the score so I could send it along. I had cleared one of the hurdles, though others would get in my way.

Everyone has tales of standarized tests and they will get more because they have permeated to all aspects of professional society since they are quick and easy to grade and a good test is a measure of skills/knowledge. Someone can do HVAC installation or sell fireplace inserts for years, he or she can know his trade but still not be promoted because of a test. These tests have real ramifications for those who take them.

We need to take standardized tests seriously as a skill that students need to learn. As far as evaluations and all the other drama created by NCLB, testing is part of the game and not a measure of our proficiency. We need to prepare kids for future tests so they can always have the skills to get past them. They need to separate their individual worth from test results and see them as part of the game to play. If they really want to change the rules then they will have to play the game before they can reach the point that they can do so. Teachers have been forced to focusing more effort than they should on tests because they are tied with individual and school evaluation.

I don’t know if you will find a more stressed out group than Algebra 1 teachers (at least in Tennessee). The Algebra 1 End of Course always creates the biggest problems and most of the schools at or near failing are doing so depending on that one test score. These are generally freshman and the fate of the school depends almost entirely on what the kids know coming in. I have seen all manner of creative efforts to get kids past the test. So much money, stress, work, time, and resources for one focus that is not the students. This test has been put in place as a major component of teacher evaluation and bad scores should lead to removal of bad teachers.

Somewhere, somehow, someone pushed across a narrative that schools are failing because they are filled with bad teachers. Where are they all? I haven’t found them. There are some teacher better than others but all the ones that I met work hard for their students. They teach and guide the kids in a meaningful way. Their students don’t score drastically higher or lower on the tests. If the scores are too low they are put on surplus and find another school. How can that be! Schools can’t get rid of all of the “bad” teachers because there aren’t enough to fill positions. That problem was somewhat addressed by the influx of Teach For America folks. You know which teachers are the worst? First year ones. They can turn into world class educators but they are most likely going to be chewed up and spit out by the system first. The evaluation system does more to get rid of good teachers than bad ones.

Teachers need to promote the professionalism of the trade. We do not have the money to change minds but we have the numbers and the good will of people. I know I am always proud to tell strangers that I am a teacher and they are always thankful. In a few minutes, I can help this person more deeply appreciate the field and instill more support for educators as a whole. That step is one we can all take pretty easily and one that can have a widespread impact. We have changed the discourse of what learning should look like in the schools, now we need to spread that discourse into the public. When the public dialogue shifts into favor of real reform, then we can face the challenge of doing so head on.

For now, do the best you can and work always on being a better teacher. Focus on what the students need to learn and how you best facilitate them. Tests are part of the game and help learners understand them as such and find ways to succeed on them.

# Testing for what.

I love teaching. Learning is fascinating and guiding through others in the process is a honest joy. Education should and could be one of the best parts of growing up but it is being suffocated by those who want to make it a controlled process rather than the inherently organic system it is. I asked the internet the other day “Why can’t I learn algebra?” and found enough interesting answers to get the rusty gears turning. People could not even explain why they did not understand the subject for it was so obtuse to their line of thinking. They had demonstrated problem solving through their writing or had successful careers and so they had the honest question of why they were forced to learn something that they never understood and so clearly aided little in their development. Math is a growing field and we have found a way to apply it everywhere we can because it provides evidence to our arguments and makes us more confident in our decisions; moreover, computers only make choices based on logical progressions, though those are designed by people. I have students who can grasp programming logic as if it is second nature but struggle mightily to solve an equation. They think they are bad at math but I see them demonstrating and making use of calculated and formal decision making, finding patterns, and solving a problem. If that ain’t math, then I don’t know what is. Really, what is math? My favorite thing to hear from a student is that, “It makes me think so hard.” I know I have had some measure of success and he or she is gaining the real key from the study (at the level I teach). I could have them make and design all kinds of cool stuff that allow them to have a purpose for thinking so hard. I could try to guide a discussion on infinite and nothingness and all that would entail, or try to make sense of transcendental numbers but I need to make them learn how to find out how long it will take John to paint a house if he and Bob usually can do it in 7 hours and Bob can do it in 28 hours. Why? It could be on a standardized tests. ETS, a non-profit, had a revenue of over $1,000,000,000 last year. The largest component of education reform is standardized tests. Kids in middle school are forced to stress out all year and then take two weeks of tests that determine their worth. Later on, they are forced to believe that the ACT/SAT is the most crucial element for getting into college because all those grades mean so little. There are many studies that don’t support this, Geiser and Santelices have an excellent study. Grades are the most important predictors of college success, followed by subject tests. Those tests are focused and allow the student to demonstrate his or her strengths. One going into the liberal arts needs to be able to write and one into STEM needs to be able to add. Grades show a greater body of work as well as a student’s work ethic.

At this time, we have models in other countries of better education systems that don’t use testing. The education communities are constantly proving that we need to foster real growth and learning for the best success but it falls on deaf ears. Your tax dollars pay for people with PhD’s to find the best solutions but those results are not use because someone else paid legislators for a different solution. I want the smartest people solving the hardest problems, not the richest.

I got lost somewhere in this train of thought but it is a question I am still working with. I believe all students can learn math and can find value in the subject but algebra is not the only math out there. Even as a math, it is a toolbox and not an end.

- to be continued.

# Problem solving, thinking, and some effort..

*This is an older post from my other blog. *

I work to try and help people build their schema of math understanding and develop that part of reasoning and symbolic logic. Unfortunately, I am not the most organized individual so I haven’t refined how to best reach my goal. My goal also involves changing others’ viewpoints and that just don’t always work so well. People are a stubborn species and teenagers are a stubborn subset of people. Getting them to buy into a different view of the world, particularly one based in the unexciting system of algebra, feels like a pointless fight at times. My only advantage is that the students do believe that something about the subject is good for them, even though they have to ask “When will I use this?” every class.

That supremely frustrating question is why I am shifting to a class more focused on problem-solving rather than mechanical-computational skills. I am also moving in that direction because modern students have astoundingly little space or need to recall facts and skills. They are absorbing information at peak levels. They know fifty times more bands that I did at the same age. So much of the working memory I need access to for traditional learning is used up on things far more interesting to teenagers. So….how do I teach a subject that requires a significant amount of this cognitive process? I have faith, perhaps too much, in my ability to present the information to people in way that makes sense but the same people often don’t store it. I am trying to circumvent this issue by using a problem-solving approach or a method in which I guide students to construct their understanding of mathematical systems. Both methods require thinking and both require effort from the students. Effort is a challenge in the subject. Student often ask for help as soon as they read a problem. He or she may work on it for a minute or two but rarely long enough to reason through it.

I have worked on a problem-solving approach for a few weeks in one of my classes and I am seeing signs of change, though. I hear fewer more questions and students seem to be following my guidelines to find some manner of solution. Next up, setting up the problems so that individuals develop an understanding of the rules of higher math. Until now, we were working mostly with old computational skills. I am adopting my system from Exeter’s Harkness math but I am confined to less time and with a broader range of students. I do know they are more engaged and that students who could care less for a lecture on solving quadratics enjoy they can find different ways to solve a problem. Watching people use different methods has taught me a great deal about math over the past few years. Ehh, I just got tired and stuff so I will continue this later…

# Art above all.

Art has never been a more important commodity. We live in a world focused on design and have reached a point where we are so immersed in information that creativity separates the wheat from the chafe.The application of art has spread into every facet of our lives and is integral in all fields, yet art has been marginalized in formal education. I remember feeling overwhelmed with a student load over 200 until I spoke with an elementary art teacher who had over 500 kids. As they limit art teachers and courses in many schools, we should take it on ourselves to fulfill this need and as an opportunity to expand our own classes. Not everyone needs to apply the arts but anyone who can should. I have enough confidence in visual arts to use them as a part of my classes and find those lessons are some of my very best. I wish I was a musician but it ain’t happening.

The core of visible thinking is using language to express thinking. Words are great but many people think more clearly through other sensory communication. They hear music, create visual solutions, organize a dance, create through food or otherwise find a way to communicate in a manner that is natural to them. Embrace these differences for we all enjoy the products of great music and fine food. Pay attention and you can recognize deep levels of problem solving while providing students an avenue they want to express themselves in.

Why art when you have so much else to worry about? Engagement, learning, fun, differentiation… perhaps I should expand.

Engagement is the most clear reason, and the linchpin of successful art projects in content area subjects. I have many students that will do everything in their power to not solve another equation but really want to make the cool drawing. Students who love art have a reason to focus on the math. Other students feel like they can’t draw can use the tools of math to create something they can be proud of. Everyone wants to make work they can show off (it’s also useful for your student displays and portfolios). I have the attention of the full cross section of students with a good art project and so the math element that I am most focused on becomes a lot easier.

One of the challenges in formal education systems is putting all of the pieces together. Students view algebra and geometry as different little problems focused on different things and not elements of a way of understanding issues. History is a study of interactions and reactions rather than a seemingly infinite list of dates and names. A good art project can pull the various elements and combine them so the student create something that demonstrates a cohesive comprehension of the subject.

Art provides another very special purpose in management. Students are people who faces struggles at home and in school. We try to provide the best space for their learning but sometimes they are too upset to focus. I like to keep some easy and engaging activities for students who need a way to relax. My handiest toy is an old spirograph. The child explores how it works and focuses intently on making awesome designs and I can focus on teaching the rest of the class. I know the student is learning and the simple allowance on my part shows that I care. He or she no longer views me as another disciplinarian and is willing to trust my guidance as an educator.

Art is outside of the box. It is freedom constrained by medium and nothing else. It is visible and design thinking in action. Anything so powerful has its challenges. Good projects are not easy to create nor are they always easy to implement. Students are often hesitant to embrace creativity in traditional classes. They get better with practice.

Projects take time so pay attention how to make them move more efficient but don’t be overly worried for a good project can cover a lot of content. Once you start one in the class, you are indebted to finishing so students have the final product. It could easily be the best part of the class all year and one that the students take the most from. When designing or finding projects, aim for a balance. The project should not be a paint-by-numbers but it needs enough guidelines or directions for the students to know where to go. Students most likely won’t be able to learn and apply a new concept in a profoundly creative way all at one but they can learn through the applications you have guided them through.

Art projects are fun. I love coming up with new ones and long for the day when I can teach all of algebra through art. Always go through the process yourself beforehand so you can identify potential pitfalls. Do they need to use a ruler? Is it digital music? A music video? You will face a thousand questions about the medium so figure out how to answer them beforehand. If you start of with these things early in the year then future projects move along nicely.

All this nice, how about an example? I am starting a new page on the site for projects and will include the one that led to the image up top. I invite anyone who has developed really good projects to send them along. Let’s keep working together and make schools far cooler.

# The art of the question.

The core of visible thinking is writing. One creates a complete thought when he or she formalizes it into tangible words but that is only half of the job of writing. The other role is to communicate the concept to a reader or listener. If the recipient understands the concept as the speaker formed it then the communication is successful. This dialogue is clear in directive statements such as when my wife asks me “Can you take out the trash?” and I do so in some hurried and confused mess at some point, though not always right then. Communication is less clear when we are working with concepts, stories, feelings, and all the other things that make it fascinating and frustrating to be human. We have art to help us through these challenges. I want to spend time on the non linguistic arts later but not today.

I teach math. The beauty and challenge of math is that it is a study of carefully ordered thinking. We have all kinds of rules, terms, properties, and muck but it is at its heart a system of logical reasoning and problem solving. Math is not “the” system but it is the most universally accepted one. As a math teacher, I answer lots of questions. So many questions. Too many questions.

Only teachers can understand the litany of questions students ask during a class. That blaze of interrogatives accosted in my first days in the classroom. As I look back, my steady need to respond was probably the most exhausting part of that first year. What harm is there in questions?

Not much but a great deal can come from the answers. People often ask before thinking. We also answer before doing so. If they do not immediately grasp the concept, they ask for further explanation. These questions are a cue that they need guidance. Embrace these for they are easier to work with then the other people who think they understand and don’t ask. Regardless, you know can now make thinking visible and, if done well, can bring the whole class along. Your motive is to direct the students to ask the right questions. You are the guide in a Socratic dialogue. Avoid responding with an explanation since that didn’t work so well the first time around. Try to ask simple questions that guide the student to respond with answers that build toward understanding. You may also want to ask other students to respond and have the process lead to a discussion. This process takes a profound understanding of the material and some flexibility. There are wrong answers but rarely is there one right one. I have learned a great deal about math by following a student’s line of reasoning. You can make quick assessments of understanding through this process and you can also find the elements that confuse students and lead to faulty reasoning. This second part is extremely useful moving forward because you can present material to address those errors.

However you present the material to your class, this dialogue is a powerful tool and one that inspires strong discussion within the class. It is only as good as the questions are so you may want to start the year discussing how important that all important element is. I am a big fan on this video from Veritasium. You could borrow from him or present the video but the focus is asking the right questions.

More to come one art as visible thinking…

# History let’s us tell a story!

History is a fundamentally interesting subject. It is our story told through so many different lenses and we learn more about our place in the world. The characters are fascinating and the more one learns, the more profound the study becomes. I was fortunate to take a wonderful survey course on the History of Science in college and left with a far greater appreciation for that realm of history. I wish I could have taken one on math history but the interwebs provides more resources than I can consume. As I read and watch lectures on the weird folks in mathematics and the profound discoveries of different societies, I have tried to process a way to teach mathematics through its history. Maths History is a fantastic resource to help one both better understand the progression of mathematical thinking and find ways to teach appropriate concepts within their historical context.

The first lesson demonstrates a unit I plan on doing in my 7th grade Prealgebra class that follows the history of the Pythagorean Theorem and how the Greeks understood the areas as numbers. Students can examine similar shapes as areas of triangles. The Pythagoreans focused on squares on the side of triangles but the theorem works for any similar shapes (semi-circles would be readily done). We can examine the relationships by constructing different triangles and comparing the area of the squares on the three sides. By changing the triangles, we can look at congruence and similarity, as well as the nature of right triangles. This part does not follow the Greek version but will allow us to work with measurements, fractions, and scaling. Once students come to a Pythagorean conclusion, we will move to studying the classic proof of the theorem and finally applications that can develop to the Distance Formula.

The Pythagoreans were an interesting group and they believed that the world should be explainable by rational numbers. They developed harmonies based on ratios of Natural Numbers. My own understanding of music is too slim but we could have a great small project examining harmonies as balances of rational numbers.

Rational numbers were great and wonderful until they studied a square. The ratio of the side to the diagonal of a square is the square root of two. The classic proof of this number’s irrationality is based on the fundamental theorem of arithmetic.As we study this proof, we can study prime factorization which leads to integer exponents, GCF’s, LCM’s and basic number theory. This element also leads to the most interesting part of the history, the story Hippasus. He was rumored to have either discovered the proof or revealed it to an outsider and so the Pythagoreans drowned him at sea.

The proofs are perhaps the most challenging element but it is important for students to see how the process works for proofs are the heart of mathematics. These are not the two column proofs of geometry class but real logical arguments based on reasoning that lead to new understanding. These proofs are challenging and provide wonderful avenues for visible thinking routines. If a student can explain these proofs and then apply them to a variety of problems then he or she will have built a strong foundation for mathematical reasoning.

The unit will take several weeks rather than the single lecture. How is that and why would it be interesting? A 7th grade prealgebra class really needs to cover common core standards for 7th and 8th grade. This unit addresses parts of 3 of the main objectives of 7th grade and 2 of the primary objectives for 8th grade.

Along the way, history provides a narrative arch and allows for many more avenues to explore and keep the material interesting.

For more resources: MacTutor is great. Cut-the-knot is wealth of information.

I will add more to this list and look forward to hearing from y’all.

Next up, history of numbers…