Mathematics for the Intuitive Learner

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Don't be surprised if your young person comes home and demands you go with them to buy pencils (preferably mechanical pencils with .9 mm lead), erasers, loose-leaf college-ruled paper, a ruler and a protractor.
Why?  Because they are going to lose points if they don't have them.  Pencils are required for tests as is loose-leaf paper.  We don't want ragged edge paper torn quickly from a notebook.  And we don't want to wait five minutes for a student to neatly tear off that ragged strip from their "clean-tear" paper.  Nor do we have time to search for a good pencil sharpener.  So either they use a mechanical pencil with a thick lead or they have a good supply of wooden pencils -- just as they would on an ACT or SAT test.
And get a nice large pink eraser, because they really work and because students really do make mistakes they need to erase.

As for protractors and rulers, these are the tools of the trade.  Don't get expense protractors with fancy attachments -- just reliable, easy-to-read basic protractors.

Welcome to all my students -- past, present, or future!

Where possible my classes will be taught somewhat like the guilds in years past where students went to learn arts and trades.  The guild model was probably much more successful then than the current traditional education model is today.
When apprentices learned within a guild, they learned in sequence by observation, imitation, supervised practice, and personal creating.  The master teacher would look for where guidance was needed and the student would take over his or her own education to the point where intervention was needed.  Progress, not grades, was the point.  Evaluation was simply helping a student develop his or her own self-monitoring skills as compared with those of the master teacher.
Contemporary schools, more than ever, focus not on growth and progress but instead on grades and avoidance of failure.
In the guild model, "failure" is a necessary rut on the road to success.  Students who don't "fail" on a fairly regular basis simply aren't trying.  The point is to enrich each student's intuition so that the student can deal better and better with the craft being mastered.

(Mostly from the teacher's point-of-view)


            This means to work a problem on the whiteboard with a smooth running commentary.  Avoid long explanations.  The idea at this point of the lesson is not to teach the process, but to expose the student to the process.  As with learning language, the student’s first acquaintance with a new word or phrase is to hear or see it in context.  From the student’s perspective he or she is “shadowing” the teacher.

            The demonstration should not be interrupted, not even by the student.  The student should be watching and not worrying him- or herself about understanding it.

            The student's desktop should be clear and the student should not be taking notes nor conversing with anyone else.

            At the end of the demonstration, the teacher should avoid such questions as “did you get that” or “did you understand?”  The reason for this is simple:  Most students will not be aware of any change in their understanding and will be upset by the implication that they should now understand the topic.  Similarly, the floor should NOT be open to questions.

            The STUDENT’S RESPONSIBILITY during this component is to be fully attentive to what the teacher is demonstrating.



            In this process, the students will be copying problems as they are worked.  Students should not be working ahead.

            The teacher should regularly sound out both individual students and the class as a whole with questions such as “what do you think the answer to this step is?” or “what do you think the next step is?”

            Avoid asking a student to give a sequence of steps or an explanation of the process.  Also avoid asking the student to compute anything.  The point here is to stimulate and search for an intuitive response.  We want to find out who is “getting it” already and to show that these micro-steps can be picked up quickly.

            The STUDENT’S RESPONSIBILITY is to volunteer answers, to guess his or her own answers when the teacher is calling on another student, and to try to intuit the single short answer to the teacher’s question.



            At this point, the students should be attempting some problems on their own.  This is where the student becomes an actual and the major worker.

            Student and teacher share mutual goals:

1.  To identify the problems the students already know how to do.

2.  To identify the problems with which the students need help.

3.  To identify the presentation techniques (how the problem is presented on the paper, which includes neatness, order, heading, skipping lines, writing fractions appropriately, content, etc) that need improvement.

            It is now the STUDENT’S RESPONSIBILITY to

a.  attempt a sufficient number of problems that all or almost all of the areas where the student has difficulty (classroom/monitored practice).

b.  work a sufficient number of problems in each area that he or she no longer needs to think about those problems (homework/non-monitored practice);

c.  remember that this process is not about getting the problems perfect and not feel bad about errors.

            It is the TEACHER’S RESPONSIBILITY to observe the student’s work both as the student requests and randomly and to explain steps the student has difficulty with and to ensure that the presentation practice matches what would be expected on a test or in a higher level course.  Explanations should be short and directly address the student’s immediate needs.  This is not the time for meta-lessons.

            The teacher should probably be either walking around the room from desk to desk or having students come up to his or her desk and lining up for assistance and quick checks.

            Generally speaking, students should NOT be helping one another.

            NOTE!  There is a difference between monitored practice and unmonitored practice such as homework.  Students should never be assigned homework or unmonitored practice that they are unable to do – practicing a lot of problems incorrectly only serves to make that error intuitive.  The purpose of homework is to move that which the student can do with conscious thought from the self-conscious part of his brain function to the intuitive part.



            Once the student is pretty much working intuitively, the teacher can then (as time allows) tie in the processes that have been mastered into a larger scheme of things, into techniques for solving word problems (what the book calls “problems”), into other subjects (both in and outside the domain of mathematics).


SUMMARY:  The natural learning processes moves through the following stages:



            Taking Ownership


In terms of the teacher-student dynamic this can be expressed as

            Teacher does – Student sees

            Teacher does – Student imitates

            Student does – Teacher assists

            Student does – Teacher sees

To My Students:
All true learning is intuitive learning.  This means you have learned to do something without having to think about the process.  This is how we speak our native language.
If you do learn something intuitively, you do not have to worry about tests.  You do not have to worry about forgetting it.  You do not have to worry about getting it wrong.  You will "own" it forever.
Learning something intuitively is not hard, but it does take a lot of time.  There is a reward for spending all this time learning something, however; ultimately you will find that it saves you time ... and makes you powerful.
  • To learn something intuitively, you must drill a lot.  There are three kinds of drills:
    • Slow Drills are drills where you practice something in order to figure out the steps.  You are only working on a few steps at a time.  When you work a problem, you always want to check the answer.  If you get something wrong, find out why (it's usually best to ask the teacher).
    • Mimicking Drills are drills where you imitate someone else (like copying what a teacher writes on the board).  You work at the same speed as the person you are copying.  You can think about the steps as long as it doesn't slow you down.  You can ask the person you are imitating to stop and explain something if you want.  If the person says no, this is not a good time, then just keep copying him.  You are not taking notes; most of what you copy, you should probably throw away.
    • Speed Drills are drills where you try to work as fast as you can.  It is okay to make careless mistakes as long as you understand the process.  You are trying to get yourself comfortable with a procedure so that you don't have to remember how to do it.  Check these problems only after you have worked a bunch of them.  But be careful not to work at a speed so fast that you just can't work the problems at all.  You should be picking up a little bit of speed each time you repeat the drill.
  • A good way to practice is to follow a slow drill or a mimicking drill with a fast drill.

You might enjoy visiting this site.

On "memorizing"
Too many of our MYP students aren't ready to be MYP students.  Some of them are coming from a watered-down elementary school in which they were taught a lot of concepts, encouraged to use a calculator, and given a good grade for their efforts.
Unfortunately, none of these skills puts the student on a college-bound track, nor do they move the student toward genuine academic success (which is one reason why the local public schools have had dismal showings on the math sections of the state assessments year after year).
I know this is a difficult thing to be told, but the simple fact is that if a student does not know his or her addition and multiplication facts, then he or she will perform at about a 20% level on the higher math functions.  In assessment terms, that's the difference between an A and an F!
Any student here can memorize.  They have already memorized their names, their phone numbers, street addresses, favorite music, the primary colors, the taste and names of their favorite foods, and how to ask if they can borrow $50.  In other words, their brain has memorized hundreds of things (objects, ideas, activities, and the names of all these things) every week.
(It doesn't help that elementary school teachers have worsened the problem by buying into the idea that a child can not learn that 6 times 8 is 48 but can learn three verses, word- and note-perfect, of their favorite song.  These teachers place tables and calculators in the hands of these youngsters, teach them tricks to "get around" knowing their facts, and enable them to dodge all the learning dependent on those facts.  The ultimate damage is extensive.)
But it is not too late to remedy a lot of that damage.  It takes drills, some traditional and some not so traditional.
Good drills involve distractions, such as bouncing or throwing-and-catching a ball during the drill, in order to keep the student from "figuring out" the answer.  The idea is to encourage the student to trust the intuitive area of their brain.
Some of the best drills involve the students saying the facts (three plus four equals seven) over and over instead of providing answers to questions.  That's because language mastery moves from observation through repetition to mastery.  "Five plus nine is fourteen" is no different than "sugar is sweet" or "most apples are red" or "the Spanish for orange is naranjo."
Saying -- not querying -- the facts while driving to or from school is an easy drill (boring, but easy) and requires nothing other than attending to what is being said.
[Note:  By the way, I am always looking for successful methods of teaching the tables used by others.  If something has worked well in your household, please let me know.  The best ideas will be shared here.]