Peschel & Associates

37019 Northview Court,    Palmdale, CA    93550



"The Royal Road" Blog  --  Using Q&A Notes? --  Elementary Teacher Math Ed  --  Numbers & Base Ten  --  Grade 3  STARS -- CAHSEE Worked Examples -- Number Sense Strand  -- Grade 4 Word Problems -- Test



We argue that students need to supplement school math classes with after-school tutoring.  This is already done in all academically high-achieving countries where parents recognize that high school and college are the gateways to desirable, upper-middle-income professions.  And they realize that students will need extra help to enter the better colleges and to do well there.  The best private and public schools in America already offer this as part of their program.  Sadly, most public schools do not offer such help.


Experience and cognitive science teaches that the most cost-effective math teaching (or after-school tutoring) promotes individualized "mastery learning".  This implies a very small class size to include "one-to-one" tutoring.  The teaching should be spread out over the entire school year to take advantage of the spacing effect.  Simple class review, isolated homework help or quick prepping to pass entrance exams can be helpful but are not enough to build the needed foundations for serious college-level work. 


Our recommended process for "mastery learning" is very competitively priced and uses learning materials based on research in cognitive science. 

If you are an early middle school student we invite you to contact us.


To learn more please review and explore the Q&A Set below.


 +  How do we learn mathematics?


+  How well are California K-12 students doing  - particularly in mathematics?  



+  Why is mathematics so difficult for so many?  



+  How are high-achieving schools different?



 +  Is "math anxiety" real? Can it be overcome?



+  What math is needed for responsible citizenship - a highly technical career?  



+  Are there shortcuts to learning math - is there an easier "Royal Road"? 



+  What does cognitive science say about learning mathematics and best classroom practice? 



+  What is at stake financially? What can parents do?  


+  What are the costs of private tutoring? Are our costs competitive?  



+  When should mathematics "learning to mastery" begin?  



 + What are our qualifications to tutor or teach?



Q & A

The kinds of question we ask are as many as the kinds of things which we know..... These then are the questions we ask, and it is in the answers to these questions that our knowledge consists."    Greek Philosopher Aristotle - Posterior Analytics - Part 1 & 2 - First Sentences 

Thales was a Greek philosopher who lived between 624-546 BC in Miletus, Ancient Ionia. 

He is considered the father of Greek science, mathematics, and philosophy. He is the first person to have asked questions about the nature of the universe and considered the answers without thinking of gods, demons or myths.

This was a crucial step in scientific reasoning and led to an intellectual explosion which lasted hundreds of years in Greece.

He introduced Babylonian and Egyptian mathematics to Greece - and taught mankind the concept of "demonstration" - of demonstrating why something must be true. 

This changed our world.

Here are three questions he asked - and then answered:

How high is the pyramid?


How do I make a right angle at B to the line BC?





How far is the ship from the shore (without walking on water)?








+ History before Thales?   







“Coach, I have taught my team to rebound, but they just don’t learn. … Swen, you say you taught them, but you have not taught until they have learned.”     Basketball Coach John Wooden



"Not tomorrow, not the next day, but eventually a big gain is made. Don‘t ask for the big, quick improvement. Seek the small improvement one day at a time.  That‘s the only way it happens – and when it happens, it lasts.”   English Teacher & UCLA Basketball Coach Legend John Wooden




“If you want to learn to play an instrument or get better at playing an instrument there is a guaranteed way that you can achieve this: practice. Like with anything we decide to learn, the more we do something the better we get at doing it. This applies to everything we do in life. Its obvious that practice is a requirement of becoming a better musician, so you want to set up a good routine that is enjoyable and will keep you motivated to want to continue practicing.”  Music Practice & Motivation




"Musicians advocate that practice is of utmost importance in the development of any player. Jascha Heifetz, possibly the 20th Century's most amazing violinist, said, "If I don't practice one day, I know it; two days, the critics know it; three days, the public knows it."  Violinist Jasha Heifetz quoted in 'Perfect Practice Makes Perfect: Inspiring Players to Practice Their Art'




“The kinds of question we ask are as many as the kinds of things which we know..... These, then, are the … questions we ask, and it is in the answers to these questions that our knowledge consists.”  Greek Philosopher Aristotle - Posterior Analytics - Part 1 & 2 - First Sentences




“We are what we repeatedly do. Excellence, then, is not an act but a habit.”   Aristotle quoted at Brainy Quotes




I have described the elements of my program. I believe that they can be duplicated elsewhere with ease.  The key, for the teacher as well as for the student, is hard work.  Hard work makes the future.  When hard work is combined with love, humor and a recognition of the "ganas" the desire to learn, the ability to sacrifice, the wish to get ahead that burns in our young people, the stereotypes and the barriers begin to crumble. Master Mathematics Teacher Jaime Escalante