About MathLaboratory.com

READ FIRST   Click this text to explore our FAQ

Click on any paragraph text marked with a down arrow … For “retrieval practice” keep toggling the paragraph question to self-quiz yourself – until you no longer need or wish to.

Who are our postings here aimed at?

We offer here guidance and a self-directed learning for all students wishing to become proficient in mathematics – whether already in a career or college bound.

Mature Students seeking a personalized, self-directed math learning program

– Parents wishing to responsibly guide their children’s math studies for rewarding careers

– Tutors seeking research-basaed guidance to help give expert advce, re-teach lessons or design effective  retrieval practice.

– Teachers needing refreshment of concepts in teaching, re-teaching and intervention.

Why  supplement math class work?

The gap between high school math achievement and college math readiness continues to widen.

Why take expensive remedial math courses in college?

Is math class supplementing needed for top tier colleges or  STEM careers?

Yes – if you want to enter top tier colleges or avoid college remedial math courses.

STEM (Science, Technology, Engineering and Mathematics) courses require even more supplementing and preparation,

By themselves the current Common Core Mathematics Standards are not designed for such careers – nor for entrance into top tier colleges.

More advanced mathematical study is needed.  Since math skills build on each other they must be learned through mastery – not through the typical annual “spiral exposure”.

STEM career preparation requires serious parental responsibility for academic dedication and coaching support.  This should be assumed in preparing for all professional sports and music careers.

Are there best practices for learning & teaching mathematics?

YES.  Recent cognitive research has identified which practices are better than others to either teach or learn mathematics.

We explore here those BEST PRACTICES that help students become independent learners of mathematics.

This includes students still IN school – or later in their careers.

So in summary – how tn should mathematics be best learned?

There are six steps to learn a math skill.  Our learning materials supports this recommended process.

-1-   Develop a genuine, enthusiastic curiosity –  become a true mental explorer. Develop this enthusiasm so you will persist in the mental effort to focus attention, understand connections, avoid confusion,  get help for difficulties and overcome inevitable discouragements.

-2-   Keep formulating inquiry questions – to check on what you know and what you don’t know.  Persist in seeking answers to these questions. Use the 5W2H question stems:  Who -, What -, When -, Where =, Why -, How -, How much -?

-3-   Connect your new learning material items to integrate them with the old material items you already learned. Master the prerequisites first.

-4-   Adapt/create mental models to summarize thr new knowledge, contrast similarities and differences and connect knowledge items. Create these models as schema for effective comprehension and improved recall.

-5-   Identify and interleave review topics which focus on summary understanding and of differences and similarities.  Distribute practice in using active recall of these schema, summaries, concepts and skills.

-6-   Practice fluent recall of concepts, summaries, connections and skills.  Do this through repeated, directed self-quizzing of your own or our prepared materials.

Why teach new skills to supplement math class work?

Our ultimate goal is to help students become motivated, self-directed and more independent math learners.

This requires studies that supplement regular the regular classroom curriculum.

We explore, reflect and muse on those best practices that supplement student class work.

Learning math to mastery requires effort and discipline. Isn’t there an easier “royal road”? 

The only real “royal road” respects the latest discoveries in neuro-science.

But NO – learning is not effortless.

We do advocate several methods.

Rapid learning of concepts, skills and reasoning practices.

We emphasize development of fluent skills.

And finally we advocate continual, distributed review practice to ensure confident and reliable LONG TERM retention of the math needed.

These learning skills are now newly recognized in support of continuing education – driven by need and desire – especially beyond school.

And so our fundamental aim is to help every student become a self-directed,  independent learner.

Such learners – with some guidance – can then teach themselves (“learn”) the math skills they need – whether still in school or later in their careers.

In time we will provide lesson content here to guide parents, prepare teachers (& tutors) and inspire students to rapidly learn the math they need – and to retain that learning  We will include supplemental lessons on math standards, special worked examples for test topics and blogs on effective study skills.

These may help students (and parents) explore different ways to look at their formal school lessons and become more self-responsible independent learners.

 “Retrieval Practice” is very effective. How does it work?

Our guided Q&A exploration here illustrates how we promote & then satisfy our natural curiosity “to know” and “to learn” – in this case to learn a little about MathLaboratory.com.

This page is a list of Frequently Asked Questions (FAQ) with our responses. As noted the answer to any question is revealed by clicking (/touching) it.  Only one answer is revealed at a time.

Here is how to try out this discipline

  1. Test to see if you have an interest in the answer to the question – then ask yourself if you can answer before revealing ours.
  2. Explore each answer by revealing them and reflecting on our reasoning.
  3. Self-Quiz yourself for retention by repeatedly toggling the Q&A items.
  4. Reflect on your experience by asking yourself: … ASK:  Did I learn something using retrieval practice?  …. What can I take away today?

In doing this you have experimented in a small way with learning through “retrieval practice”.

Why should one prepare for college?   

It is crucial if your goal is to prepare for a recognized career, particularly in STEM (Science/ Technology /Engineering /Mathematics) field.

However,  there are both benefits and costs in getting a college education.  Not everyone is suited to such academic goals.

Here is what parents of middle grade students should look at. The issue is  a simple one – DO YOU NEED OFFICIAL CREDENTIALS.

Without college credentials students cannot pursue lucrative careers.

Recognize that degrees in different fields also reward differently. Consider the following:

Remember: Going to school, excelling and then going on to college or an advanced degree takes time, mental training, discipline and money. Then finding a good starting job is always hard and takes much time and effort. This has been particularly difficult in our current economy.

Your parental choice for a student in Middle School may be the difference between for him/her to simply getting a series of jobs or pursuing a rewarding life-time career.  On the other hand many adults decide to learn math later to get the formal school credits to gain access to these careers. It is possible but just harder if you must work at the same time.

If a student builds interest, develops talent and maintains discipline – but there is no money – there are scholarships, work programs and student loans. Many students – like myself – could work through high school and college – and borrowing only very modestly. Frankly, that wasn’t “much fun” and took a few years of earnings to become financially fit.

Please note that employers don’t only look at student grades. They look at the student’s ability to enter a good school and at their EFFORT AND DISCIPLINE in finishing a difficult program. After all that is what will be needed for career success as well.

When should I learn mathematics?

In school, if you can!

Become “proficient” in middle school to become “college-ready”.

Mathematics is a cumulative body of knowledge and skills that must be learned in steps. The problem is that if you don’t start early and stay with it you won’t have the background for pre-requisite courses for admissions or advanced placement courses in high school for getting more college credits.

If you are an adult and you haven’t learned much math in high school or even gone to college – and you now need a math skill at work – you can teach yourself what you will need.

This is harder – much harder – since you will need to spend at least 20 hours practicing to become each new skill. Be mindful that you will not have “credentials” until you get it from an accredited school.

Know also that one needs to start becoming math proficient in middle grade school.  It is then that you will create and build on critical foundations for high school and college.You will need to explore accelerated class path ways since Common Core Math is not designed for STEM careers.

Choose to avoid having to take expensive remedial courses in college. Never become overly indebted with student loans – seriously consider getting scholarship help pr working  while pursuing evening studies.

The only way to avoid Minimum Wage Unemployment  (as in are you really worth $15/hr?) is to offer to do “independent contract work”.

Remember the physicist Albert Einstein: He needed to express his thoughts using a newly created math discipline – tensor calculus. So he hired a mathematician to tutor him so he could become proficient.

It is NEVER too late to teach yourself to learn what you need. It just takes longer – sometimes a lot longer.

Do top-earning careers really need math skills?  

Yes – if you want to become a mathematician or math teacher you certainly do need them to become credentialed – and teach. It is also very clear why STEM science, engineering and all technical fields need varying degrees of advanced math.

But what about lawyers, for example?

The top 15 highest earning college careers all require considerable math skills. Without math skills these careers will not be open to students.

Does public high school prepare you for college work? 

Generally not. For example, sixty per cent of students entering California colleges need remedial work in math or English. California is not the only problem,  Recently we learned that only 22% of New York City high school students were “ready” for college.

It is perhaps not yet well understood that the new Common Core Mathematics Standards are not designed to prepare studemts for STEM careers or acceptance at premier colleges.

In the 2011–12 fall enrollment period the U.S. average per student expenditure for public elementary and secondary schools was $10,834.  The state with the highest expenditure was New York ($18,616), the lowest was Utah ($6,849) and California was at ($9,053).  .

In California – unless supplemented – mathematics education  appears to be failing badl. In the 2006 PISA Benchmark 14 nations outranked USA top-ranked-state Massachusetts in “advanced math achievement level”. Internationally, 19 nations outranked California while it ranked 16th below the U.S. average – behind Israel but ahead of Portugal.

In 2009 as many as 24% of all California high school students didn’t pass the basic English tests or an eighth-grade-level math test. Worse yet, of the 40,000 students entering the California University System 60% or 24,000 needed expensive remedial math or English courses or both.  And yes – these are actually “B grade-point-average college prep students” that “did well” in high school.

Very sobering is the fact that in the last two generations we tripled public school funding — without improving student math achievement.

Genrally, most public school students in America are not learning the math needed to prepare for their careers. For many students the remedial courses at college really cost substantial time & money. That is why parents supplement public or private school classes: ensure college admission; avoid the  need for remedial classes; and provide the background math assumed by many college classes.

How would a “virtual laboratory” for math classes be useful?  

All the top schools support their science classes with regularly attended laboratory classes for credit.  Even honor students seek informal lab support.  All students need to “chew” on the abstract concepts they learned in class.

It is important to make abstract concepts very concrete and meaningful to you.

Once learned students can help themselves and classmates through re-teaching and sharing practice in study groups.  You understand much better if you learn to explain concepts and skills to yourself and then to others.

Math lab support in elementary and secondary schools is becoming ever more necessary with adoption of the Common Core Standards for Mathematics. The standards as such are not the problem. They simply need to be supported with ongoing, continual review and then be supplemented for accelerated class programs.

There is a continuing problem with promotion of new progressive teaching practices involving group work and explorations in “student discovery learning”.  What do 6th graders and their parents do when called “little mathematicians” with discovery insights – who are not able to do simple arithmetic needed for advanced study?


We truly honor all hard-working mathematics teachers – whether in public schools, private schools or simply mentoring-tutoring students. But we also believe teachers must teach until students learn.

Parents need to take responsibility for choosing the teachers that educate their children.

And students must do their part – they need to learn to make the effort to be the best they can be.

In this economic environment and expansion of class sizes this is a big challenge.

I experienced good teaching practice as a college freshman in 1957 at Columbia University. Our physics professor, Dr. Polykarp Kusch, had recently been awarded the Nobel Prize in physics.  He simply demanded that his several physics lectures each week be supplemented with a single weekly mandatory laboratory class. He knew that only a lab class would help us keep up with his lectures. We needed that time to “chew” and “learn” what was abstractly presented to us.  I much better in this class simply because of that extra study time.

Teaching public school classes is really not different. Only the best teachers take the time to encourage to “chew” and reflect on what is to be learned.

We recommend that ALL students be informally tutored after-school to supplement their formal math classes. Even the best students need mentoring and guidance to reinforce what they are learning. In addition, everyone needs skillful guidance in applying and extending their new “understandings” to new topics.

Teachers with a 25+ student-teacher ratio cannot do it this effectively.

Supplementing class work is also the norm in academically-high-achieving school districts (or nations). Parents recognize that high school and college are the gateways to desirable, upper-middle-income professions. They realize that students need extra help to truly learn, enter the better colleges and – more importantly – continue to do well there.

Many of the best American private and public schools offer some supplementation of their class program. In many of the better-ranked public school districts teachers “moonlight” by regular tutoring for a fee.

MathLaboratory.com: A “Virtual Math Lab”?

Our purpose here is to help teachers, tutors, parents and students each do their part in helping students learn these skills.

Our concept is of a Virtual Math Lab where teachers, parents, tutors and students come here regularly to explore math teaching and learning.

We hope to inspire confidence in parents, help teachers improve through preparation, support tutors in lesson development and help students to really “learn to learn”,

We correlate our lessons and reviews to state, national and international curriculum standards.  Because textbook lessons also correlate to these standards we can guide exploring the concepts of this week’s lessons, review previous lessons to connect with prior knowledge, be carefully guided in practicing skills and thus come to be better prepared for the next lesson or test.

In this way, we could provide resources here to help students preview and prepare for their next school lesson.

Experience and cognitive science teaches that the most cost-effective math teaching (or after-school tutoring) is individualized “mastery learning”. This implies a very small class size and would include “one-to-one” tutoring. We know that simple class reviews, isolated homework help or quick prepping to pass entrance exams are not enough. Although often costly they may be quite helpful prepare for college entry. However, they don’t really to build the needed foundations for serious college-level work.

We can help students at any level master all the mathematics they need – beginning in the very-important third/fourth grades – or much later when college remedial math is needed.

It may take time and effort by all since we believe our teaching and carefully guided practice must be spread out over an entire school year to take advantage of the “spacing effect”.

What teaching practices do we emulate?  

Montessori for promoting a young student’s curiosity to explore.  

Kumon for developing a daily written work discipline for mastery.  

Escalante for following a focused curriculum using the AP Math exam qualification

Kaufman for rapid skill acquisition/re-acquisition.

Brown et al for successful retention of learning through retrieval practice.

We start with review of the fundamentals for proficiency on the NAEP Grade 4 national assessment. We supplement the new Common Core math standards with the old California math standards for proficiency on the NAEP Grade 8 and 12 national assessments.  These will have been aligned with the TIMMS and PISA international standards of excellence for Grade 4 and Grade 8.  In this we support the joy of exploratory learning as inspired by Montessori school methods – but then insist on daily written math practice beginning with Grade 6.

We privatize a middle and secondary school supplemental program as advocated in Jaime Escalante’s famous advanced placement calculus program at Garfield High School in the Los Angeles area.  We want to ensure that by late middle school there is readiness for a path from Algebra 1 to advanced placement calculus by 12th grade of high school. In this we teach a virtual form of self-directed daily student practice so parents and students can monitor progress of daily mastery steps.

Jaime Escalante considered it a major academic achievement simply to qualify and sit for The Advanced Placement Calculus Examination. We do so also.

Student success in each of these classes depends on many factors: family finances; interest; motivation; the right kind and amount of deliberate practice; and, his/her willingness to “learn-how-to-learn”.

When more of our online teaching materials are available we will actively seek donors willing to sponsor supplemental math lab work for able students with lean family finances.

Do we accept new students?

We are currently looking for a small number of middle school students from families willing and financially able to commit to a disciplined learning program.  (Think simply of of music instruction.)

Our continuous program offering begins in Grades 3-4 to establish readiness for Algebra 1 by Grade 8.  We also want to make sure that our students’ achievements in Grade 4 and Grade 8 match the NAEP and TIMMS international standards.

We then continue with the all-important middle school (Grades 6-8 Program).

We conclude with our secondary math program for the California curriculum that includes SAT and AP Test Prep.

We use the Common Core Mathematics Standards for California – supplemented with the old California, NAEP and international standards.

We correlate these standards with now  affordable used textbooks based on the previous outstanding California mathematics standards.

Are our costs competitive?

There are no written contracts.

For local students in California’s Antelope Valley the very short term in-the-home hourly fee is $60 per hour.

For rapid skill learning we recommend goals to acquire each skill in thirty days or less. This would be equivalent to twenty hours of study, averaging sixty to ninety minutes of guided practice each day.

Home-school or mastery math re-learning usually costs $3,000 a year for a full home school math program.  For late-middle and high school-level students we average 2 hrs/week of instruction with five-days-a-week of 15 min/day  homework practice.  More daily lessons with practice may be needed for any catch-up subjects.

Finally, please consider this: Our annual cost is very competitive and compares favorably with that for master music instruction – where typically 30 min/week lessons usually requires many hours of daily practice.

For a free online trial lesson contact: alpeschel@caa.columbia.edu

Where do we teach?

We teach in a studio classroom located  in Quartz Hill, a small, special Lancaster community in California’s Antelope Valley.

It is within a 10 minute walking distance from Joe Walker Middle School or the Quartz Hill High School. Our classroom is within 5 miles of Antelope Valley College.

We conduct lessons in home, at a library, in a coffeeshop or online.  We specialize teaching at distance online – using a PC and 4+ inch screen smartphones.

What is unique about MathLaboratory.com?  

We focus on developing a student’s conceptual faculty for learning and for using math. Our goal is to guide self-directed, deliberate practice to gain math skills RAPIDLY.

Our methods of instruction emphasize working in an individualized and small group context with the emphasis on learning rapidly and to mastery.

Our methods and materials are based on the latest research in cognitive science.

We use the latest technology in online teaching and the monitoring of student practice.

We help students to become more disciplined in using the many tools available from Internet providers.- so they can learn more independently.  In particular, we link to many excellent Khan Academy math video lessons.

Our main focus will be on teaching students how to think, “learn how-to-learn”, develop mental discipline and explore mathematics lesson content with true enthusiasm.