Thursday, March 10, 2016

A lot of thoughts on the big picture of math instruction

The topic that has been eating away at me since the day I committed to becoming a math teacher is "why isn't there more project based learning (PBL) in math?".  I had seen a handful of awesome schools, but most of them did PBL everywhere except in math, that unfortunate subject that just seemed to get in the way.

Over the past couple of years, I had been moving away from my belief that it wasn't done because nobody knew how, shifting towards a different view that math was in fact something different that needed a different form of instruction.  To be clear, this different form does not resemble what we do in most math classrooms now, but it isn't PBL either.  My visits to High Tech High and Summit, along with my discussion with Amir at DPEA, solidified this view.  I will do my best to explain what I think math instruction should be striving for.  <!--Begin opinion-->

At its core, math is the study of patterns.  You can look at patterns of growth and decay in a function when you discover an underlying mechanism between steps such as add 1 each time (counting, leading to a y=x equation), add 3 each time (skip counting / adding, leading to a linear equation with a slope y=3x), or double each time (multiply by 2, leading to an exponential equation y=2^x).  Inverses of these core functions include logarithmic, radical, and inverse functions.  Other functions include absolute value, step functions, sine waves, and many other patterns to describe how one thing changes in relation to another.

Patterns are also prevalent in Geometry.  Intersections of lines create congruent angles in the same places every time.  Parallel lines produce a number of identical intersections in a predictable way.  Closed shapes grow the sum of their angles in a predictable, drawable way.  Similarity, which is calculated through proportions, really just comes back to linear functions and the patterns discussed above.

Patterns also show up in the way you evaluate expressions.  For example, 1x + 2x is similar to 1√(2) + 2√(2) and 1x² + 2x².  A key rule that we rarely discuss when teaching these skills (that you must first factor the common component out, then evaluate the numberic addition) would take longer to do, but would clarify why expressions like 1√(2) + 2√(5) or 1x + 2y cannot be further simplified.

Some of the best math teaching ideas, such as 3-act lessons ( started by Dan Meyer and visual patterns ( started by Fawn Nguyen, start by giving students a chance to understand a concrete situation.  However, they do not stop there, instead pushing students to figure out what happens next.  This may start with wild guesses, but with practice, becomes a process of figuring out the underlying mechanism of change.  From here, students can test their mechanism by building out the next few steps to see if it produces what they expect.  Finally, students use the growth mechanism to write an equation (such as turning "it doubles each time" into "y=2^x").  This is where observations becomes what we traditionally recognize as math, and students are armed with a powerful tool to predict far beyond what they can model in front of them (how many pennies are needed to cover the earth?  or how many boxes are in step 43?).  Mathematicians and people in many fields use modeling to make similarly complex predictions that they cannot easily imagine or model.

One step from this towards PBL is teachers who ask students to estimate crazy-huge numbers such as "how long of a thread would be needed to roll up into a ball of yarn the size of the earth".  In these kinds of problems, students figure out some base measure, such as taking known lengths of yarn and making a ball, and taking measurements.  They also look at the growth rate, so in this example, the volume grows at a rate proportional to the cube of the length.  Using the starting points and growth function, along with known reference numbers such as they radius of the earth (after using dimensional analysis to convert all units to be the same), they can make their prediction in a way that they could justify and explain to others.

Another branch in the high school math picture is logic.  This shows up most clearly when working with proofs in Geometry, as it is one of the few places we ask students to use generalized rules in an explicit way to make a new generalized rule.  Boolean logic, the use of AND, OR, and NOT, is another place where we ask students to apply general rules to determine an outcome.  When connected with probability, we can calculate the chances of one or more events occurring.  Even more mathemagic happens when you introduce randomness, study the patterns present in random events, and then apply it to data collection through statistical inference.

One piece of mathematics instruction is teaching the process used to do difficult mathematics, something embodied in the idea of "productive struggle".  Students need enough background to get a start in the problem and enough confidence to play with a few ideas.  This is also where groupwork is highly effective, as students can bounce ideas off of each other as they try new methods of moving forward.  This is different than simply plowing into a new problem and discussing when they get stuck -- it is instead a collaborative planning and experimentation process.

In my visit to Summit, I saw that one group's presentation for math class focused on this kind of group scenario.  The teacher asked the group to explain the problem to the class, explain how they approached it, and walk the class through the solution (the last part looking more like a demonstration speech).  After that, the group had to present on why the problem was interesting and challenging, where they struggled and how they overcame it, and what they got out of working on that problem.  The problem was heavily tied to the math content they were studying, but was an application problem that reached into physics.

More generally, if you want to see how math aught to be, just go to the expert: Jo Boaler.  I liked her stuff since I first came across it, but I appreciate and understand it more every day.

Projects come into play when math is being applied in a more open-ended way.  Ideally, this happens outside of the math classroom (science, computer science, and engineering are great places).  Why?  Most of the math teachers I talked to, including me, find that their students are not getting enough practice and fluency in core math skills when they spend a disproportionate amount of time on projects, causing more harm than good.

Many of the better math projects that I remember from college and witnessed this week are rooted in science or social science contexts.  For example, you could study a simulation of an owl population and try to find patterns to predict the next steps.  You could model a billiards ball on a computer and program in the equations that govern the system.  You could calculate torque requirements for a robot.  There are a host of cool things you could do, but most of the interesting ones are either covered in the "pattern recognition / estimation" above (generally a small amount of class time to complete a problem) or belong embedded in content from another subject.


And then there is what I do now.  It is a small percentage of what is above.  Our curriculum is organized topically, such as by quadratics and radicals, since that seems like an appropriate way to lay out one class of problems at a time.  Given the procedural instruction method (Madeline Hunter's I do, we do, you do), it makes sense.  Video-based instruction is a nice improvement: the "I do" phase is now delivered at the time and pace the student needs.  Mastery-based quizzing is a fantastic improvement for this system as well: the "you do" phase is checked and repeated until the student can actually do it, rather than moving on aimlessly when the teacher decides it is time.  Students working on practice in groups or collaboratively on the whiteboard makes the "we do" phase something that a teacher can monitor but students can struggle with more effectively.

I'm not sure what the place is for efficient algorithms that can obscure deep understanding.  For example, I can multiply any two digit number by 11 in my head by adding the digits and making in the middle number, so 72 * 11 = 7_(7+2)_2, or 792.  There are reasons why this works, but when I use this trick, I don't think about them, I just execute the algorithm.  We do this more commonly with students when we use the quadratic equation, factoring tricks, "combining" like terms, and strict rules (such as "never cross out terms through addition in a fraction").  All of this helps me do math faster, and when I take an ACT, I can burn through thanks to all the algorithms I was taught over the years.  (For helpful examples, see Tina Cardone's Nix the Tricks).

The balance we play as math teachers is when to ask students to generalize and derive, and when to hand out the shortcut.  If you go to my YouTube channel, you will find over 500 examples of me ruining a good learning opportunity for students as I provide them with a recipe on exactly what to do next.  That said, I learned with examples and procedural instruction through all of my math career.  I made lots of connections along the way on my own and have a very rich understanding of mathematics as a result.  Without some procedural fluency from the drill-and-kill I received, I may have never been "good enough" at math to advance to interesting new challenges or become a math teacher.


Given all of this reflection, this is what I am imagining for 9th grade math:

The baseline is rooted in patterns.  It would give all students a chance to play in the physical world, look for growth mechanisms (and distinguish the many different underlying functions), and then predict much larger numbers in a way they can explain.  Tasks for students need to have context (as in nearly all word problems), and answers should be right with justification, not just black and white.  3-acts, Visualpatterns, estimation problems, and Desmos lessons would all fit very natually in this framework.

Building on patterns is language around functions.  We describe graphs with terms like "intercept", "zeros", "concave down", "region of increase", "increases as x goes to infinity", "axis of symmetry", "vertex", and others.  These terms can be introduced as relevant questions are asked ("find the peak height of the projectile, which is called a 'vertex'") and further generalized with more questions that use the term differently.  They could also be taught through more traditional video lectures and students could memorize them as baseline knowledge.

Algebraic manipulation through explicit form change and student's justification would form the second pillar.  Students need to be able to distribute and factor linearly with extreme comfort, as it forms the basis of all that we do in algebra.  x^2 + 4x^2 + 3x can be rewritten as (1 + 4)x^2 + (3)x, making the grouping rules clear.  We can do the same with fractions: 1/4 + 2/4 can be factored out as (1 + 2)(1/4), making the answer of 3/4 painfully obvious rather than something potentially tricky (trying to remember the "rule" of whether or not to add the denominators).  Equation-based thinking about equivalence needs to be practiced in many different forms as well, and notation needs to adapt with it.  When you "add to both sides", the new term should be on the far right of each side, not lined up with the "like term", or it can lead to the appearance of new rules.  Subtraction and division should be banned, instead simplifying to make all operations negative addition or fractional multiplication.  Practicing this with one variable, multiple variables, multiple powers of a variable, and radicals in the same unit enable connections to be formed and skills strengthened.

Some skills, such as trinomial factoring, offer little return on investment at a young age.  As students work toward Algebra 2 and advanced math, more in-depth time can be spent playing with the relationship between factoring and binomial distribution to analyze how form change can lead to new insights about an expression.  We don't inspire this kind of mathematical curiosity right now since most students have no interest.  I think this is not worth fighting for a while.  Quadratics can be solved graphically or with a computer with a focus on purpose rather than calculation.

Throughout the course, students would have smaller numbers of problems to work on with higher levels of rigor.  Communication of understanding could come through group presentations, conferences with the teacher, or written justification of work.  The course would need to also assess students individually in the applied use of vocab and math notation.

I would love to see clarifying questions and push-back with everything I am saying here.  The path forward is still muddled in my mind, and though it keeps getting clearer, I need a lot of help from anyone who took the time to read all of this.  Please keep poking holes so I can continue to grow.  Thank you!

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