When I started teaching, the math textbooks at the high school level had a great variety of problems to do, from the easy to the challenging (some were even hard for me as the teacher!).  The books were laid out so everyone could easily see where the "easy" questions were and where the cut off was to the more challenging then the third cut off to the most challenging.  I miss those books.  They offered lots of fodder to help students practice at their level of ability.  One way they lacked was in asking students to explain their thinking (at least it did not formally get asked, it was an implied expectation).  Most, if not all, teachers expected students to show their work to justify their answers which is a great way to know the thought process used.  The criticism was that the answers became about the algorithm and not the thinking that may have been used to get the answer.

As I continue to work with the new textbooks and support materials, I struggle to find many, varied challenge questions.  A recent review of a text intended to meet the Secondary Math I (SM1) that was written “from the ground up” not just an older text reworked to fit the CCSS, was limited to maybe 15 questions per section and 10 of those questions were multiple choice with only 5 open-response problems to solve.  There were some great examples for students to read through and try to understand what the concept was, but it was mostly examples.  Why do we not use “key concept” boxes or “concept summary” boxes to highlight the formal math along with the examples?  I am not convinced that exploratory learning is the only way we should be delivering material.  I believe that exploratory learning is one way to deliver concepts, but there are many, varied ways to help students understand topics.

Perhaps my rant is based on my work as an advocate for the gifted & talented.  G/T math students tend to pick up on concepts quickly.  They do not always want to stop and explain what makes great sense to them.  I believe we need to work with them to do such explanations (as simple as showing their thought process).  At the same time, asking them to “discover” the right process can be very frustrating for them.  Why not tell them the formal definition or algorithm and, when needed, also give them the time to explore these definitions and algorithms?

One other frustration I have as a math teacher is how we have to all use the same textbook for all levels of students.  Perhaps this is not the case in your school district, but in mine (and I’d hazard a guess in many) the idea of one teacher using one resource and another teacher using a different resource is not acceptable.  Just as students need different ways to understand the concepts, so do teachers need different ways to deliver the content.  My g/t students would do well with a more wordy book and higher level questions, but the middle or SPED students could use more manipulatives and mid-level problem solving (like using whole numbers instead of fractions when discussing geometry topics).

The one constant in my teaching has been the basic topics.  All students should know how to manipulate numbers and variable quantities with all the operations.  All students should know how to find the perimeter, area, and volume of objects in the real world.  All students should be exposed to real world situations and how the math they are learning helps them to determine a solution.  These ideas have remained the same over the course of my career.  I dare say my approach has changed somewhat in that now I encourage students to approach the problem in any way that will make them successful, but to try and do it whatever way is currently being presented.  I have learned new ways to think about solutions and I am more accepting of alternate solutions than I would have been back in the 80’s.

I used to teach the way I was taught.  I figured that if it was good enough for me, it would be good enough for my students.  In class, I would model how to do a problem (to show how super intelligent I was) with minimal student input  and then assign 10 - 20 problems that were similar with different numbers (the more difficult problems probably used fractions rather than whole numbers).  We would waste a good portion of the next class period going over the homework (whether we needed to or not) and then I would demonstrate a new concept (masterfully, of course), assign the requisite homework, and the cycle would continue.

I used to think the best classes were quiet classes where students sat in neat rows, raised their hands with alacrity, and waited patiently to be called on.  They would always give the correct answer (or at least the one I was looking for) and everyone else would nod their heads sagely in agreement.

All that has changed.  The more research I do and the more I learn, the more mortified I am about by initial forays into teaching mathematics for understanding.  My classroom is messy (no more neat rows) and noisy as students discuss and critique each others' thinking.  I try never to say anything a student can say for him or herself.  I allow students to make mistakes and then I ask them questions so they can self-correct and achieve a sense of personal empowerment.  I listen for all answers - even if they are not the ones I had in mind and I find I learn a lot from my students.  I assign a lot less homework and through using ASSISTments, I can pinpoint the assignments I need to bring up in class to allay any confusion and save time for more hands-on mathematics.

It is a lot harder to teach this way (I never quite know what to expect and I am a bit of a control freak) but I sense that the students who are willing to put in the effort learn, truly learn, the material as they make it their own.  I am still working on a strategy to motivate those students who would rather not put themselves at risk and try.