Learning and thinking have always been closely linked.  But sometimes, in our schools, we only achieve the appearance of learning--often accompanied by very little thinking.  This problem has been recognized by educators for a long time.  In recent years, however, we have been addressing this issue on various fronts.

       Let me begin by explaining what I mean by the term Appearance of Learning.  This is when students get good grades, but their actual learning is very low.  The grade to the students and to their parents makes it appear as if they have learned a lot, but the attainment of those grades may have been based partly on non-academic measures such as good behavior or mere compliance with rules such as turning in homework on time.  This is a problem because when students move on to more complicated coursework that requires previous knowledge, they struggle due to never actually learning the earlier content in the first place.

     Over the past few years, there has been a stronger attempt to balance the need for teaching skills (such as solving an equation) with the need for teaching conceptual understanding (such as using an equation to solve a problem).  This sort of "teaching" is different from when our students' parents were in school--a time in which just-getting-the-answer might have seemed good enough.  Today we know that this isn't good enough.  Students going to college and students going to work need skills and knowledge beyond merely doing what they are told to do.  Today's world demands more from our citizens.


       Our schools recognize this need and we are trying to make changes to address this challenge.  It's hard to change a system of 100,000 schools, 50,000,000 students, and 3,000,000 teachers.  But (indeed) this change is already happening.  What we used to call a "computer room" is now just about any classroom in the building.  Classrooms with students sitting in nice, neat, straight rows that discouraged collaboration among students have been replaced with classrooms with tables or desks arranged in groups to encourage students to work together and ask questions of each other and learn together.  Grades for non-academic behaviors (mentioned above) are strongly discouraged so that a student's grade can more accurately reflect his/her ability in the content.  Finally, student engagement strategies and growth mindset strategies are constantly being discussed and implemented.  These reflect our knowledge of the best way that students learn.  No longer do we expect everyone to learn strictly by listening to the teacher and taking notes.

       Indeed, the change to an educational system that requires more from our high school graduates is well underway.  The past is in the past and we're not going back.  Those that choose not to change risk creating a generation that is ill prepared for the challenges that they will face.  Learning, thinking, reasoning, and problem solving will be more and more integrated into our schools.

       You're welcome.

       My entire professional career has been in Secondary Mathematics.  As a middle school teacher, high school teachers, department chair, and supervisor; I have been immersed in the field of teaching and learning mathematics.  Whenever conversations arise about the instruction for very weak students or for very advanced students, there seems to always be special considerations for students when it comes to mathematics.  Principals, school counselors, and special educator alike all agree that the teaching and learning of mathematics seem to require very different skills compared to the teaching and learning of reading and history and (perhaps) other subjects.

       Keith Devlin has a relatively simple answer to this question which he explains in his paper written for the Mathematical Association of America titled, In Math You Have to Remember, In Other Subjects You Can Think About It.  Devlin explains that mathematics is often taught as a series of rules that you just have to memorize.  Whereas other subjects such as history and science are taught within a context that help the learner to make sense of the content.  Mathematics could be taught within a context (and, I would say SHOULD be taught within a context), but it usually isn't.  This makes mathematics seem like a secret language with tricks and complicated rules that sometimes work in some situations and don't work in other situations.  When mathematics is taught as merely a bunch of skills without any understanding of how and when to use these skills, many students struggle to be successful.


       In her book Mathematical Mindsets, Stanford professor Jo Boaler explains that all students can learn mathematics if teachers are equipped to help students to build understanding.  While it is relatively easy for trained secondary mathematics teachers to learn these techniques, most teachers have not received this training when they were in college.  Furthermore, most adults today (parents, school principals, school counselors, etc) grew up learning mathematics through a strictly procedural approach.  Because of this experience, most adults believe that mathematics should be taught as a bunch of skills.


       Fortunately, more and more teachers are mixing more and more conceptual understanding in their mathematics classrooms along with learning the skills.  There are places in which math is taught somewhat differently than it was taught in past generations, and this is a good thing in my opinion even if it makes some parents feel uncomfortable.  Math should not be taught as 100% skills without any connection to the real world.  For most students, this sort of instruction is just too abstract for them to understand.


       Mathematics is also a subject that builds on itself from school year to school year.  Students need to have a good understanding of the concepts so that they can build on previously learned content to understand the new content.  This idea of building from year to year is more prevalent in mathematics than it is in other subject areas.