Category Archives: Trends in Math Education

Blood Pressure

Imagine you’re a doctor. As a doctor, your responsibility is to improve the health of your patients, and so you sometimes administer some routine, standard tests to find benchmarks of your patients’ health. One such test tells you the systolic and diastolic blood pressure of your patients. It takes about 30 seconds to administer and can tell you early if there are warning signs that a patient isn’t thriving, or whether a treatment you prescribed is helping.

One year, you walk into an interesting new reality. All doctors must administer the same blood pressure test at the same time of year, as part of the accountability system for clinics, doctors, and healthcare administrators. You tell all of your patients to come to the clinic at the same time of day for the test. The test takes a total of 15 hours. Some clinics administer the whole suite of blood pressure assessments over the course of one week, where other clinics spread the process out over two or more weeks. All patients must be tested, and so make-up blood pressure tests tie up the clinic for another week after the tests are done. You’re very curious about this new blood pressure test, but you had no part in creating it, and even though you are actively administering the test, you’re forbidden from looking at the testing instruments or the patients’ results as they are in process. You take this rule very seriously as you know someone who lost her job practicing medicine for explaining part of the test to a patient. In fact, you need to remove anything from your office that would give any help to patients in understanding their blood pressure test, and if any patient discusses the test inside or outside the office, their result is invalidated. Their invalid result is then sorted with the rest of your patients’ scores.

It would be great to know the results of the blood pressure test right away, so you could make some changes to your treatment plans for some patients. Alas, you get the results after four months of waiting anxiously for this important benchmark of your medical practice. With quivering hands you open the results file and get two numbers for each patient you tested. You sort and analyze the data.

It’s pretty interesting stuff. As a rule, your patients had improvements in their blood pressure over last year. You wonder if other doctors saw similar improvements. One clinic across town is still on an improvement plan for having consistently unhealthy blood pressure readings for years. Ninety percent of the staff turned over and they got a new administrator. Blood pressure readings are improving slightly, but you hear the workload and stress have been unbelievable. The stakes are so high that you’ve heard rumors of clinics cheating and falsifying patient blood pressure data, major scandals that paint an ugly picture of your profession.

Although you are encouraged by your patients’ overall results, the clinic administrator wants you to meet with a coach to craft treatment plans for patients whose readings were unhealthy. Does our clinic need a nutrition plan or should we prescribe beta blockers for all? You wonder about individual patients. This patient probably just didn’t try very hard on the blood pressure test. Motivation has been an issue all year and you suspect there are emotional issues related to a recent divorce. That makes it hard to focus on a blood pressure test for 15 hours. Another patient would probably thrive with diet and exercise, but you wonder if there are genetic issues at play or a hormonal issue. You just don’t have enough information to create a treatment plan, so the session with your coach is difficult. She wants answers you can’t give her because you only have blood pressure numbers. It seems a little silly to have tested patients for 15 hours and get no actionable or diagnostic information, but this apparently is the new set of rules for doctors.

You don’t want to seem as if you’re against doctor accountability, but the process is frustrating. Blood pressure readings used to give you a quick snapshot of information. You would like to be able to use the blood pressure reading to guide further questions for coming to a diagnosis. It blows your mind that the blood pressure reading is now seen as an outcome instead of a data point in the picture of a patient’s overall health. That you and your administrators are actually making diagnoses, treatment plans, clinic personnel assignments, and long-lasting community health decisions based on a blood pressure reading. That a simple snapshot of overall health takes 15 hours over the course of days of secrecy and stress. If I’m testing a patient for 15 hours, couldn’t I have diagnosed and treated their actual problem instead of just getting one indicator? How can anyone be expected to practice medicine in this kind of environment?

Wait. I didn’t actually mean “doctors”, “clinics”, “patients”, and “blood pressure readings”. What I meant was “teachers”, “schools”, “children”, and “math and reading achievement scores.”

This is a picture of my classroom almost ready for standardized testing time. I’m not done covering every poster. I think it’s OK to share this because there are no testing items in the room.

A classroom almost ready for several days of testing

A classroom almost ready for several days of testing

It’s time for a teacher-led manifesto for what we really want and need in order to improve the educational experience for our kids. Some people may look back on this era and wonder why our educational system didn’t rocket to the top of the international charts after putting these reforms into place.  But if you have boots on the ground, you will not be one bit surprised. We work in a baffling world, friends.


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Posted by on May 3, 2014 in Trends in Math Education


Project-Based Learning via Javascript

I’m joining the maker movement!! I’ve been feeling tugged for a long time to teach mathematics using computer science – learning programming as a means to accomplish math goals.  Some of the factors at work are:

– The current math curriculum has been around since long before the computer, and the nature of mathematics in society has changed to become more computer-based.
– Millions of computer science jobs may go unfilled due to a lack of skilled programmers
– In our school district, in a tech hub of northern Colorado, a child can actually choose, in the year 2013, to never take a programming course beyond their sixth-grade tech class.  Ever.  They can actually opt out of learning to program.
– Yet students cannot opt out of learning to solve systems of inequalities with a pencil.  This seems backwards to me.
– My own understanding of mathematics deepened when I learned to program computers.

I made a promise to myself to incorporate programming into my common-core-based seventh-grade math class this year.  This is a heavy promise.  I am not teaching programming as an end in itself (even though I believe that is worthwhile).  I need to integrate the common core standards and teach programming with a purpose.  It is not easy.  Yet I was really proud of the first unit we did and the first unit project my students pulled off.

Our first unit was on congruence transformations.  I gave students the task of creating a computer program that created a design with symmetry.  It had to include two transformations, and they could choose from translations, rotations, and reflections.  They had to describe their transformations in a write-up and explain how they created them.

The platform I chose was Khan Academy’s Javascript. Any programming environment would work.  I considered Scratch, but eventually chose Javascript because of Khan Academy’s easy integration with Google accounts and the class management tools given to teachers.

Here are a few examples of the incredible artwork my students produced with computer programs.  It took several days but was WELL WORTH IT.  I was so impressed with the quality of their writing and their ease with using the difficult math vocabulary.  They really had to think hard about transformations to make their projects work, and the results were nothing short of amazing.

Student work from our programming project on congruence transformations!

Student work from our programming project on congruence transformations!

I didn’t expect the results to be as good as they were, frankly, but it turned out to be a great project not just as a way of introducing programming, but also to really understand congruence transformations deeply.

Here is the rubric I used for the project.

Khan Academy Programming Project

I would really encourage you to try it.  I’ve been emboldened to do more with computer science in math – the key being to hold on to the mathematical purpose behind it.


Every Choice Has a Cost

I attended some personal financial literacy training last weekend, presented by the Colorado Council for Economic Education.  In Colorado, we’ve adopted the Common Core State Standards (CCSS) mostly, but with added standards for Personal Financial Literacy… which show up in our standards tagged with (PFL).  A couple of co-workers and I thought the training would be a good introduction to what we need to teach in the new standards, and we weren’t disappointed!

If there’s one enduring understanding I could take away from personal financial literacy, it’s that “every choice has a cost”.  The ability to analyze and make informed judgments on the costs of your choices could make a big impact on your future success.

In sixth grade, the standards include saving for goals, using percents to solve saving and investing problems, and explaining the difference between saving and investing.

In seventh grade, we move on to understanding taxes – computing taxes, describing the role of taxes in society, and demonstrating the impact of taxes on your income and spending.  In addition, students should be able to compute interest and use unit rates to make purchasing decisions.

In eighth grade, students should understand how credit and debt impact their life goals, and describe the components of credit history.

There is a lot there, and we don’t tend to cover financial literacy well now.  However, it helped me to think of the big understanding, that the thread woven through many units of study must be helping students to analyze the cost of choices and make good judgments based on that analysis.

I want to put in a plug for the Council for Economic Education and the Financial Fitness for Life books.  The books have kid-friendly resources that cover finance problems, and put them in context of analyzing choices.  You can view some activities on the companion web site for the middle school books – click on a lesson and get to some information about the lesson and web resources.

I’m working on adding this analysis to the conversation as we work on our units on percents, probability, and linear algebra.


Gender Equity… Dance vs. STEM

In my spare time, okay, not that I have spare time, but in the couple of hours a week that I carve out for being responsible for my own health, I take clogging lessons.  I got interested in it while helping my daughters practice, and when their studio offered an adult class, I jumped on it and haven’t looked back.  I like making exercise and rhythm part of my life.  I’m well aware of how beneficial the arts are to a well-rounded education, and I feel it’s made me a more creative, thoughtful, energetic, and resourceful person.

The gender gap in dance is one that is obvious, and glaring, and hasn’t budged for decades.  My adult dance class is one of the few with a male dancer in it.  Most classes have no boys at all.  The hip-hop dance classes are the least unbalanced, but that means there will be two or three boys in the class with seven or so girls.

For me, it leaves a question in the air.  Why don’t we care?

The gender gap in STEM education and tech jobs spurs a call to action.  We create girls’ computer camps.  We create science field trips just for girls.  We analyze our data and fret over the unfulfilled potential of our girls.  Why is there no similar call to action to get more boys involved in dance?

I did just a tiny bit of research today online to find out about the gender gap in dance.  The few tidbits I found out are that the gender gap in dance is wide and hasn’t budged in decades, and that interestingly, and these pieces of info were fascinating, the wage gap in the performing arts favors men, and boys tend to get more attention and are called on more often in dance class.

This contrasts to the STEM gender gap, in which the boys outnumber the girls, especially in computer science and physics, but are still paid more and get more attention and are called on more often.  Or at least, that’s the way it used to be.  Is it still true?

We don’t apply the same level of urgency to getting boys into the arts as we do to getting girls into STEM.  Presumably, it’s because the job market in the arts isn’t perceived as growing as quickly or having as much earning potential.  Does that mean we’re right to apply no energy into getting boys to enjoy dance?

As a point of reflection, I’ve done a project a couple of times when I felt my classes (and I) were in need of a movement break.  Instead of doing math warmups, we planned a flash mob.  We’d find a funny dance on youtube and practice it for ten minutes a day, and then plan a secret day and time to play the music and have the math classes coalesce and just start dancing.  The students LOVED it – boys and girls alike.  It helped create community, a sense of purpose, got us some much-needed physical activity, and made us laugh.  I have no data to inform whether this is a good educational practice or not – so is it?  How important is a dance break for a student’s educational well-being?

I welcome your thoughts!


Computer Science and Math

Our school has a 40-minute intervention/enrichment period at the end of every day, and students can take a different class each quarter.  This quarter, we made a decision for me to teach a computer science enrichment.  We invited students to join it, with the intent of targeting certain needs.  We wanted to target students who had potential to grow in math if they learned it in a little different way.  The invitation list consisted of some partially-proficient math students who might make a big leap if they learned math in a non-traditional class, and some gifted students who might enjoy an extension to what they knew.  We created the class to be balanced between girls and boys.

I’m enjoying writing a curriculum that gives kids an intro to programming but also weaves in math topics. I am making things up a little as I go, based on what kids want to learn next.  Here is what we have done so far, and I’ll include the links to the demo programs I made.

1)  Introduction to drawing.  Students learned to create shapes on the screen, and how to plot objects, thinking of the screen as a coordinate plane. We also explored RGB color – how colors on a screen are a combination of Red, Green, and Blue lights, and by turning on the lights in different proportions, you can make any color of the rainbow.

2)  Variables and Expressions.  Students learned how to “anchor” a small drawing at certain x,y coordinates, and how to create the rest of the shape around those anchor points using variables.  We discussed the relationship between these expressions and algebra expressions, and how the computer uses substitution to replace a variable with a value.

3) Binary Numbers.  I really appreciated place value when I learned and understood the world of binary, so we spent a day and a half working on building that understanding.  I mainly used worksheets from Computer Science Unplugged.  The worksheet on this webpage is great and very kid-friendly.
I demonstrated the conversions with a Javascript program, although Javascript doesn’t have simple ways of representing binary numbers.  I haven’t used this in class yet, because I wanted to get into data types and functions before getting here.

4)  Functions.  In elementary school, students get used to a representation of a function as an “in/out” machine.  In middle school, we extend that to equations that include variables, and they are related to these old “in/out” rules of the past.  In computer science, a function is also an in/out machine.  It takes parameters as inputs, does something to them, and outputs something new.  I showed the students how they could make their little character from Lesson 2 appear on the screen multiple times by wrapping it in a function.  The computer again uses substitution to use the input parameters.  For advanced students, they learned how to make their drawing appear randomly by substituting a random number instead of a fixed one.

Next, we will work on creating algebra expressions with our variables to make our drawings animate.  We’ll also learn about “if” statements and how they can be used to make decisions.

Loads of fun!  I’m loving it.



Is Algebra Necessary?

I got sucked into the online buzz generated by this July 28th New York Times Op-Ed by Andrew Hacker, titled “Is Algebra Necessary?”  It struck a nerve.  Many nerves.  Hacker’s main premises were: 1) Algebra is too hard, and is responsible for many a talented student’s academic failure – backed up by statistics, and 2) Algebra is not necessary for most careers, backed up by flimsier statistics.

Of course Algebra is necessary, we cried!  Algebra teaches you to make real-life situations abstract, to simplify them to their core.  It teaches you to play with the variables involved and to quantify cause-and-effect.  It teaches you to argue logically and reasonably.  It teaches you the life skills of discipline and hard work. It teaches you organized thinking and linearity.  It’s good exercise for your brain.

These are noble goals.  Everyone should learn these skills.  These are not taught in high school mathematics.  Some students learn them.  But look at a high school Algebra 1 or Algebra 2 exit exam and tell me where the teacher is assessing the student’s ability to abstract real-life situations and argue logically.  You’ll find a couple of word problems in the problem set on systems of equations, and I can almost guarantee that will be the extent of the assessment on abstracting real-life situations.  There’s an assessment here, and also here, and here’s a standardized practice exam for Algebra II here.  The students are required to do very little other than notice the structure of a problem and follow the right algorithm.  If you’re required to show work, you can show evidence of organized, linear thinking… or, you might just show evidence of being very very good at memorizing a sequence of steps.  Students who are already good at pattern-finding and generalizing may excel at this, as they’ll recognize the structure of a problem and know right away it’s, say, a trinomial with integer coefficients they can easily factor.  Will this teach a student to become good at pattern-finding?  Students for whom this does not come easily will sweat over the same trinomial, trying to recall the steps needed to get the problem into a form the teacher will like.  They’ll probably mix up one or two of the steps and arrive at an answer that would make no sense in context, but there is no context around the problem, so one answer looks just as meaningful as another.  Students learn a set of rules to apply for a certain structure of problem.  Then another set of rules to apply for another structure of a problem.  The noble, higher purposes of Algebra – the abstract reasoning, the beauty of a logical argument, the modeling of a complex situation into a symbolic form that can be manipulated to apply to other situations or prove other truths – are not actually ever taught.

There’s more.  All of this algorithm-crunching and polymonial-factoring and imaginary-number-figuring by hand would make a lot of sense to keep in the high school curriculum if this were how anybody actually used mathematics. But here is the dirty little secret:  nobody does math this way in the real world.  Who solves a quadratic or a double-integral by hand?  Who shades in those cartesian graphs of systems of inequalities?  Who converts a cartesian graph to a polar graph with a pencil, or finds the inverse of a radical expression on paper?  NOBODY.  It’s time-consuming and error-prone.  Computers are better at it, and we have better things to do with our time.  You know how long we’ve been teaching algebra by hand, with paper and pencil?  The earliest evidence I could find was from 1923.  The application of math in the real world has changed a little since the 1920’s.

Inevitably, you’ll be teaching the simplification of expressions using complex numbers, and a student will raise her hand and ask when we’re ever going to use this. You’ll be grumpy about this, because they might use it someday, if the student becomes a theoretical physicist or a mathematician, and anyway, that isn’t the point.  The point is that when a solution can’t be found in the set of real numbers, look, there is a whole other infinite set of numbers that are imaginary that have limitless possibilities.  It’s beautiful and expands the horizons of what your brain can do.

It’s too bad students don’t actually ever discover that. They learn a set of rules for simplifying expressions if they contain the letter i.

So then you’ll be having coffee with some parents, and a parent will say Johnny has been having trouble with complex numbers, and will ask you honestly when their child might ever be expected to use this.  And you’ll say the kid might, if they become an engineer, and that math is important if a kid wants to get into a STEM career.  Algebra II success is a great predictor of college success.  Then you’ll change the subject.

You know you’ve done this.  I have!

I have a story about the way you use math in engineering.  I would like to share it so you understand why we so badly do need to completely restructure mathematics education.

I worked as an engineer in an electronic test and measurement company, during the late 1990’s and early 2000’s.  I was a software engineer and I had to study a lot of math to get there, because engineers use a lot of math.  Once, I was on a project where my job was to write the drivers that took measurements on the power output of a fiber-optic amplifier.  I remember there being a function that described the power output in terms of another variable, and although I didn’t need to solve it because it was already done, I solved it anyway to prove to myself that the equation worked.  I remember the equation contained a log function which I struggled to remember how to “undo” (is it a base 10 if there’s no base specified, or is that a natural log??), but once I refreshed my memory using the information superhighway, I was good to go.  I balanced the equation carefully and tested it with a couple of pairs of variables, and indeed the equation was derived correctly just as I received it.  I was satisfied.  I spent about an hour on this task, if you also include the time I spent searching online for cool information and simulations of logarithmic and exponential functions.  And watching a funny cat video.

So I spent an hour using all that precious knowledge I had gained in the many, many years of math I took in high school and college.  Next I had to write the program that retrieved the output from a spectrum analyzer and calculated the power from it.  The spectrum analyzer produces a curved trace, which you receive as an array of numbers.  I actually had to calculate the area under the trace.  I didn’t integrate it by hand… I used a computer program to calculate the area of a lot of tiny rectangles under the trace and add the areas together.  With a few lines of code, this took a very short time and really just used seventh-grade math.  I was going back to the very basics of what calculus is all about, and it was humbling to see I spent so much time studying integrals and here I was, a professional engineer, adding rectangles together.  But in the digital world, this is what you do.  Almost nothing is really curved in the computer age.

I finished the software and then had some decisions to make.  The more little rectangles I measured, the longer the program took.  Did the customer need the result in milliseconds, or did they want a more precise answer that took several seconds to get?  How precise was the spectrum analyzer and could I make it more precise?  Was one measurement enough?  Should I take three traces and average all my answers? Should I take ten traces, drop any outliers, and average the rest?  How would I know if a trace were an outlier?   What result should I be expecting?  How could I independently verify the results I got?  Under what extreme conditions did I need to test this?  In what format did the customer want the data saved?  What was a pass versus a fail?  These kinds of questions filled the rest of the development cycle.

The project lasted several months.  I spent one hour of that time doing algebra.  The rest of the time I spent puzzling over what information I needed, what my results meant, and how to make them presentable and understandable.  Since that project, it has never been lost on me that I spent years and years of my life manipulating equations by hand, and I used that knowledge in my career once.  And I was an engineer in a bona fide STEM career.

Think about that, and think about how we teach algebra.  A STEM career involves making lots of decisions about messy data with unknown sources of error.  Understanding how to abstract and idealize a real-life situation is certainly an important part of math.  This is what algebra is supposed to teach us to do, and it doesn’t even do that well.  We teach symbolic manipulation via a bunch of algorithms instead.  We rarely get to having students truly abstract a real-life situation, and we completely neglect teaching communcation, challenging the reasonableness of answers, sources of error, estimation, precision, tradeoffs, and general problem-solving.

I have some half-baked ideas, of course, of what a 21st century math education actually looks like.  But I’d like to hear from you:  If you were king, how would you structure secondary math?

For some great inspiration, enjoy this TED talk by Conrad Wolfram.  You’ll start to get the idea on where I stand.

For more inspiration, read a piece that has challenged me from my early days of becoming a teacher:

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Posted by on August 15, 2012 in Trends in Math Education