Pluto and the Mini-Planets.

“We recommend Edric Cane’s book, Teaching to Intuition, as an excellent source for how mathematics should be learned.”  NASA mathematician Fred Krogh (Principal Mathematician, Ret., Jet Propulsion Laboratory) was kind enough to recommend my book on the home page of the official site that made available to Universities and Industry the math and algorithms developed at JPL over 30 years of planetary exploration. I was taken by surprise and greatly appreciative of seeing a book that uses only examples from Elementary and Middle School math in such proximity with sophisticated,  pioneering mathematics. Krogh added: “This (Teaching to Intuition) is mainly for elementary mathematics, but the principles hold at all levels.”

Now I learn that one of the junior siblings of Pluto, one of those mini-planets or main-belt asteroids that, along with Pluto, do not qualify as official planets, has been named after Fred Krogh. The NASA page that gives the known characteristics of 5927 Krogh says of its eponym: “American mathematician Fred T. Krogh developed the accurate, flexible and fast numerical integration algorithm used to track and navigate NASA interplanetary spacecraft since the 1970s. His work is at the core of JPL planetary, asteroid and comet orbit solutions and ephemerides.”  

Most of us earthlings have little connection with Mercury, Mars, Jupiter, Saturn, Uranus or Neptune.  As for Venus, we can only wish. But Krogh was kind enough to look down on a book I wrote. Let me congratulate Fred Krogh on 5927 Krogh.

Grow Your Brain. Dedication

My latest book, Grow Your Brain, was written for use with 1st and 2nd grade children and for their benefit. But it was not written with only them in mind. This dedication makes it clear.


Grow Your Brain is dedicated to our 1st and 2nd grade children, their parents, and their teachers, but also to much older students, to the millions of College students throughout the US who were never given a chance to get a great start in math in their early Elementary School years.

In all our Community Colleges and many Universities, these students know the cost in time and money, in aggravation, in self-doubt and humiliation of having to take remedial classes on topics that should have been mastered years earlier. They know how close they were to giving up on a career in teaching, nursing, or the hundred different professions that make up the fabric of a thriving society because of their fear of Math and their doubt on their abilities. They know how, for some communities, it is not just a sizable proportion who are left on the wrong side of the achievement gap, even on the wrong side of basic functional numeracy, but a majority, and how costly that is for themselves and for society.  Grow Your Brain was written with them in mind, in the conviction that it doesn’t have to be so for the 1st and 2nd graders of today.

And to our teachers, I am in awe of what they do day after day. But, if it doesn’t achieve the desired goals for so many of our students, then, maybe the remedy is not that they should do what they already do well a little bit better, but use their talent and dedication to do things differently. On simple topics, in very concrete and practical ways, that is what Grow Your Brain attempts. For example, it uses a vertical number line instead of the traditional horizontal one: if this makes things just a little bit easier for children (and it does, much more and in more sustained ways than most can imagine), why not at least give it a try?  To move numbers around and group them, instead of a reliance on properties of operations, we use bubbles with no technical names attached to them: here again, a change in vision with multiple implications and benefits that stretch all the way to algebra years later. We use cards that focus on teaching a very limited number of facts instead of aiming to teach all that we want children to know, and then we build on that foundational knowledge.  Each element of this book has small details that focus more on promoting thinking and thoughtful connections than on the facts themselves.

I am always struck how the powerful jet engines that lift planes to 40,000 feet push the plane forward, not up. It is the speed acquired through the forward thrust that lifts the plane. Goals are not necessarily achieved by pursuing them head-on, exclusively, and at all cost. Knowledge itself is a thinking activity. Teach children to think, and the facts will come along for the ride.

Grow your Brain. Dialogs with 1st and 2nd graders

My new publication—Grow Your Brainprovides parents and teachers models of simple conversations with 1st and 2nd grade children. It focuses on a few simple tools that should allow children to take off in their understanding and knowledge of essential arithmetic. It gradually builds up a whole library of topics and strategies from which classroom teachers and parents home-schooling or just helping their child can choose and improvise short, daily, Grow Your Brain conversations.  It stresses mathematical knowledge as a thinking activity and as a network of connections —as opposed to an accumulation of isolated facts committed to memory. In response to prompts and questions, children respond and answer; they learn to speak about numbers.

The very first tool is a vertical number line to 10 with marks for each number but no numbers except for zero and 10. 5 is shown as a bolder mark. Why vertical? Because on a vertical line higher numbers are higher and lower numbers are lower. It’s a small detail among many small details implemented throughout the book that all aim to make learning facts and concepts just a little bit more intuitive. In fact, far from being a small detail, a complete switch from horizontal to vertical number lines has major beneficial consequences on children’s ability to make sense of essential concepts throughout their mathematical studies and beyond. Why no numbers except 0 and 10? If the marks are numbered, when a child is asked to point to 7, for instance, or to put a token on 7, the child just goes there without any need to think or learn. With no labels, identifying marks without having to count implies some thinking taking place and does not always come easily. A dialog then helps the child formulate that thinking. The parent/teacher’s side of the dialog could be as follows:

Show me 7 on the line.  Is 7 closer to 5 or to 10?  (To 5)  How far is 7 from 5?  (2 steps)  So what’s 5 + 2?  (7)  How far is 7 from 10?  (3)  So what’s 2 + 3?  (5)  And what’s 7 + 3?  (10)  Show me all the fingers of one hand.  How many is that?  (5)  Show me 2 more fingers of the other hand.  How many fingers are you showing now?  (7)  How many fingers are you hiding?  (3)  So what’s 7 + 3?  (10)  Show me all your fingers. Show me 7 again on the line. How far is 7 from 10?  (3)

Some children find such questions easy, and indeed they are just the first steps in what we do in Grow your Brain. But for others children, even those first steps will take significantly longer to master.  So some improvised version of this conversation will take place as long as needed at the same time as other approaches and other connections are blended in the daily dialogs.

Discussion cards giving different images of the 5+ numbers (5+1, 5+2, 5+3, 5+4) are used to confirm and expand the understanding experienced with the line to 10 (and later the line to 20 where 10 is used as a benchmark.) With 7 by then well known as 5 + 2, a child holding a card with some version of 7 (a 5/2 domino or Roman number VII, for instance) can be asked:

What’s 5 + 5?  (10)   What’s 2 + 2?  (4)  So what’s 7 +  7? (14)

The 4 of 7 + 7 = 14 is now connected with the 2 of 7 seen as 5 + 2. Knowledge of essential facts gradually builds up, not as isolated memorized phrases which are nothing more than skeleton knowledge, but as actual experiences which the child acts out and formulates and as meaningful elements in a network of connections.

The powerful engines used to lift a plane from ground level to 40,000 feet are not used to push the plane up but to thrust it forward. It is the forward thrust that is used to lift the plane. In the same way, Grow Your Brain focuses on helping children experience knowledge as connections with other facts and discover knowledge as a thinking activity. Knowledge is a consequence of the connections and the thinking.

For more on Grow your Brain, go to

What kangaroos can teach us about what it means to know math: A parable.

Let’s imagine students who know that 2 books + 3 books is 5 books, 2 cookies + 3 cookies is 5 cookies, and that 2 chairs + 3 chairs is 5 chairs. These students are taking a geography class on Australia. They learn a lot about that country and are told about an animal called a kangaroo. They are shown a video about kangaroos and learn  that it hops on powerful hind legs and that the mother keeps her baby (which they all call Joey) in a pouch on her stomach.  Yet, there is one thing that is true of kangaroos that these students are not told. In spite of the fact that Australia is a distant country with many strange creatures, they are not told that 2 kangaroos + 3 kangaroos is 5 kangaroos?  Why not?

Now let’s look at three students, student A, B, and C.  All three are asked to add 2/7 + 3/7. Student A searches inside his brain for a toolbox where he stores rules for operations on fractions.  In it, he identifies a rule that applies to the problem at hand. It says: “To add two fractions with the same denominator, you add the numerators and keep the denominator the same.”  He uses that rule to correctly add 2/7 + 3/7. Student B also reaches into his toolbox, but it is a little bit messy in there and he gets the answer wrong because he confuses the rule about adding with the rule about multiplying: he correctly adds the two numerators but also the two denominators. Student C doesn’t use a toolbox. She takes it for granted that 2 sevenths + 3 sevenths is 5 sevenths for the same reason that 2 kangaroos + 3 kangaroos is 5 kangaroos.

Of these three students, which one knows more math, the one who knows the rule and applies it correctly or the one who doesn’t need the rule?

More importantly, if we look at what takes place in classrooms and textbooks, which of these students are we seeking to create by our teaching?

And, whether we like it or not, if we look at the reality of what takes place and has taken place for a very long time, isn’t it obvious that in aiming to create student A we are also unavoidably creating student B?

The Common Core State Standards: The Pros and Cons

The Pros and Cons? That’s quite a controversial heading, isn’t it?  I’ve probably already lost all potential readers. Some, because how could there be any Pros? The others, because suggesting that there might be Cons automatically puts me in the category of those who just want to destroy those Standards. I see it quite differently. I share enough of the goals to want the Standards to take hold and succeed.  I see enough that I don’t like in them to fear that those goals may not be achieved. I see the Standards as a work in progress, as an opportunity to discuss, disagree, exchange ideas, and improve on them, all in the spirit of respect for the motives of those you disagree with. After all, we are not on Capitol Hill.

The subtitle to Teaching to Intuition is “Constructive Implementation of the Common Core Standards in Mathematics.” “Constructive” as in “constructive criticism,” and the details of my criticism are prominently expressed in the book.  So let me focus first on the Pros.

Sometimes, less is more. To best understand what the CCSS for Mathematics are really trying to achieve, as formulated in the Standards for Practice, I propose we stop half way through most sentences, that we read the verbs but skip the nouns they refer to. Let’s apply that reading strategy to the first paragraph of those standards:

“Make sense…persevere…make conjectures…evaluate…change course if necessary…explain…conceptualize…plan a solution pathway rather than simply jumping to a solution attempt…check answers…use a different method…continually ask: ‘Does that make sense?’…understand…identify correspondences between different approaches…”

Revealing, isn’t it? Essentially, as I see it, the CCSS Mathematics want to change an approach to learning/teaching/knowing math from a passive regurgitation of facts and procedures to an activity of the mind. The Standards recognize that knowing math does not consist in Mathematical truths stored in memory that students have the ability to blurt out when Pavlov rings the appropriate bell or when the test produces the appropriate question. I was thrilled when I realized that the Standards are really about math as an activity instead of an accumulation of knowledge—about discovering, working things out, understanding, formulating, communicating, modeling and applying. If done well, this should make mathematics more approachable and practical; both easier and more useful; a routine characteristic of an educated nation.

This approach to mathematical knowledge echoes and amplifies what professional organizations have long recommended—it wasn’t born in a vacuum–, but it represents a dramatic change to what actually happens in most classrooms. Mathematical knowledge seen as an activity of the brain instead of as passive and easily testable facts would represent a change in culture, and that is never easy to achieve. Change is threatening and moves at the speed of a Greenland glacier before climate change. So I welcome Standards that encourage concrete steps in that direction. Significant change is urgently needed. The alternative cannot be tacit satisfaction with the status quo, or some vague assumption that we can keep on doing the same thing just a little bit better. The future economic standing and technological leadership of this nation is at stake. That is something that makes us all stakeholders: parents, teachers, leaders in all fields of activities, and of course, students and the adults they become. The CCSS are the best realistic opportunity for seeing significant change happening, and some early tentative indications show encouraging first steps in that direction.

But it is not a given that the aims of the CCSS will be achieved, not only because of opposition from the outside, or because of all the peripheral but essentially unrelated agendas such as the uses and misuses of tests, but also from the flaws in the Standards themselves. A cultural change of this magnitude cannot be formulated by a few and delivered to a grateful world in a few in-service sessions. It needs to be offered as a work in progress worthy of gradual implementation and experimentation and extensive long term interactive training, where teachers learn, but also communicate their perplexities and offer their suggestions. In fact, to put it bluntly, the Standards for Content convince me that the writers of those Standards themselves do not have all the answers as to how to implement the goals set forth in the Standards for Practice.

A case in point is the heavy emphasis put on the commutative and associative properties of operations offered as a strategy for adding in the Standards for grades 1 to 5.  Those properties of operations are a clumsy and inefficient way of moving numbers around in a series of additions and subtractions —in particular because they don’t apply to subtractions. But they could be interpreted as offering an explanation, so the deeply ingrained procedural mindset still present in all of us takes over and the thinking goes: Providing an explanation is what we want when we say that students should understand, those properties are some kind of an explanation, so let’s offer those properties. They allow us to check the box that we have met the requirement of asking students to understand. The deeply ingrained procedural approach to mathematical knowledge, instead of being cast aside, is transferred to meeting the requirements of the new goals, and essentially, on this topic, nothing has changed.

So, if not the properties of operations, what can guide us as we seek to move numbers around in a series of additions and subtractions?  Not all groupings are acceptable, so how can we tell? As shoppers, we all know the answer: As we put a $4.00 box of cereal in our shopping cart, we all know that this adds $4.00 to our final cost. As we use a $1.00 coupon, we all know that this reduces our cost by $1.00. As shoppers, we all act from knowledge that the order in which purchases and coupons are scanned is irrelevant. So students can be helped to think about their shopping experience and draw from it the mathematical understanding that it represents. Instead of looking at a series of additions and subtractions in terms of those properties of operations, where the signs are permanently stuck between two numbers and where those numbers can switch position (commute) if the sign is an addition but cannot if it is a subtractions, our students will recognize that they already practice a different understanding, one where each number is permanently attached to the operation sign in front of it. As we look at the label on the box of cereal, we don’t just have 4, we have a 4 that is added. As we cut out the discount coupon, we don’t just think: “$1.00.”  We think of one dollar that is saved. It’s the sign and number combinations that matter and that can be moved around and grouped freely in our shopping cart, at the check-out counter, and in a sequence of additions and subtractions.

If we all know this, why don’t we teach it? 1st graders can re-enact their understanding with independent price tags and discount coupons that they manipulate and group as strategies for adding and subtracting. They can transfer their understanding to a series of additions and subtractions by drawing bubbles around sign and number, and then move those bubbles around or group them by crossing out {+ 7} and {+ 3} anywhere in the sequence to make 10, or {+ 14) and {– 4}, or even {– 8} and {– 2} to make {– 10}. Try those empowering strategies with the properties of operations! I have seen 1st graders who can calculate 9 + 7 – 4 + 1 – 6 + 3 before they know how to add 9 + 7.

Quite a few grades later students are taught operations on integers. This remains for many a significant hurdle to a smooth transition into algebra that, for some, lasts all the way to remedial classes in Community Colleges and Universities. The perspective learned from those shoppers can change a major obstacle into something almost as intuitive as operations on counting numbers, without any need to process the understanding in terms of cumbersome rules, additive inverse, or other technical terms. For years, students saw an addition and added; they saw a subtraction and subtracted. How could it be any different?  And then they face -237 + 148. Seen as an operation, -237 + 148 is an addition that we evaluate by performing a subtraction. There is no longer a complete identity between operation (an addition) and calculation (a subtraction.) Hence the confusion and the rules that some master and that many don’t.

With the shoppers’ perspective, and a vertical number line as the standard model in our mind, we can intuitively perceive {-237} as moving from zero a total of 237 units down into negative territory. {+ 148} then moves us back up 148 units, clearly short of making it all the way back to zero and into positive territory. Both the need to subtract the distances and the negative nature of the answer can be intuitively perceived. 6th graders get the point based on their previous experience of what we have called the shoppers’ perspective and their customary use of a vertical number line. There is no longer any need to drag the confusion and frustration from 6th grade all the way to College remedial classes. On this topic, the simple introspection introduced in 1st grade could have a statistically significant impact on the number of students who never satisfactorily make it through algebra.

This is just one example fleshed out in more details in Teaching to Intuition. On this and other entirely different topics the link with real life and the urge to understand that are at the heart of the Common Core Standards can be recognized by students in the mental structures and thinking patterns of their own mind before they are translated and generalized into the language of Mathematics. Mathematics then becomes for our students something that comes from within instead of something imposed from the outside. The truths of math are recognized instead of just being learned. The Common Core Standards may not quite implement their goals in this way, but I see an invitation to do so in their emphasis on understanding, on modeling real life, on Mathematical knowledge presented as an activity of the brain instead of a passive accumulation of facts and rules. But to achieve their goal, these Standards have to be accepted as a work in progress; one that will benefit and grow from criticism and contributions on one pedagogical topic after another until Math becomes for the next generation a natural and integral dimension of everyone’s perception of the world we live in.



“All teachers through grade school at least should study your book (Teaching to Intuition).  It would make a huge difference in how students could learn and enjoy math.  Have you managed to get this book used in college math education classes?”

I received this comment from Fred Krogh who also put a link to my book on his website, that said::

“We recommend Edric Cane’s book, Teaching to Intuition, as an excellent source for how mathematics should be learned.   This is mainly for elementary mathematics, but the principles hold for all levels!”

Now Fred Krogh has no business reading Teaching to Intuition. Mathematically speaking, he is at the other end of the spectrum.  He is a professional mathematician, not just someone who knows a lot of Math and applies it to new problems, but one who creates new Mathematics that others may learn and use. He worked for 30 years at NASA’s Jet Propulsion Laboratory where he was part of the team that designed and sent into space planetary probes, including the Viking probe, now the first man-made object to have left our solar system.  Within a large team of engineer and physicists, he was Principal Mathematician, the pure mathematician that others could consult and rely upon.

In contrast, all the Math included in Teaching to Intuition is taken from the Elementary and Middle School curriculum, what everyone should know.  I deleted from an early draft everything that might go beyond the most elementary pre-algebra topics. I took out an excursion into Trigonometry as possibly leading readers back to some long forgotten, no longer familiar territory. So Fred Krogh has no business reading my book. But his wife bought a copy, and he laid his hands on it. From his detailed comments, it is clear he ended up reading much more of it than could be expected. He e-mailed me: “I am very impressed… I could make a list of all that I like in your book, but then this email would probably be too long to send.”

Fred is the gatekeeper to much of the pioneering mathematics that was done at JPL, often by himself, during those years of active planetary exploration. His website makes the topics available to other researchers and practitioners. Reading the list of mathematical algorithms and software available on the site is like trying to read a foreign language (Sample: Eigenvalues and Eigenvector values of a Hermitian complex matrix.) But he felt strongly enough about the kind of perspective described in Teaching to Intuition, and the need for it at all levels of mathematical endeavors, that he put a link to it on the home page of his site, the only outside link on it.

I am sorry that I cannot reciprocate and recommend ‘Hermitian complex matrices’ to readers of Teaching to Intuition. Like most of us ordinary humans, I just don’t have a clue.


I’ve published a second book this year, Making Friends with Numbers. Subtitle: Creative worksheets for Multiplication Facts. Let learning or reviewing multiplication facts teach you math. Its for 3rd and 4th grade students just beginning with multiplication and also for students in higher grades—yes, even High School—who don’t know their facts. There’s a Middle School near here planning to use the workbook in a 6 week remedial class 3 days a week for students from different 6th, 7th and 8th grades that need the review—for many, it’s more than a review. So it seeks to meet a very specific need. Check where a few sample pages can be seen.

Bay Area Math Project (BAMP) Institute for teachers

For an author, early feedback on his book is always eagerly awaited. Based on the material in Teaching to Intuition, I was invited to be one of the instructors at a (San Francisco) Bay Area Math Project (BAMP) Summer Institute for teachers. My fellow instructors were some of the top Math educators at UC Berkeley’s Lawrence Hall of Science and San Mateo County Office of Education’s STEM Center (Science, Technology, Engineering, and Mathematics), generously funded by Oracle and the Heising-Simons Foundation.  For my first presentation, I shared the morning with Harold Asturias, UC Berkeley’s director of the Center for Mathematics Excellence and Equity (CeMEE). (Let me put it bluntly: As a member of the AAAAAIE (*), I am impressed with all these acronyms). Feedback from participants was particularly encouraging and I had my first book-signing experience.  I am including here material from the Institute’s flyer:

From July 30 through August 8th, the Bay Area
Mathematics Project (BAMP) and the San Mateo County
Office of Education (SMCOE) invite teachers from grades K
through 8 to participate in the Fractions from a Common
Core Perspective Institute. The California Mathematics
Project (CMP) Task Force has produced wonderful
professional development materials about fractions on the
number line, a representation that the math Common Core
(CCSSM) emphasizes.
April Cherrington, a member of that task force, will be one
of the presenters. Other presenters include Edric Cane,
author of the new book Teaching to Intuition; Lew Douglas,
BAMP Co-Director; Emiliano Gomez, UC Berkeley MDTP
Director and teacher of pre-service courses on campus;
and Keith Terry and Linda Akiyama sharing wonderful
kinesthetic activities that will be easily available. Videos will
be shown and discussed to illustrate effective pedagogical
practices called for by the eight Standards for Mathematical
Since 1984, BAMP has supported innovative teaching
methods and provided content and leadership development
for thousands of Bay Area teachers.

(*) The AAAAAIE, or Aie! Aie! Aie! Aie! Aie!, of which I am the founder and probably the only member, is the American Association Against the Abuse of Acronyms In Education.

Place Value

Some things are presented to students as if they needed to be taught, learned, and tested, when the teacher could instead find strategies that allow children to experience them, become familiar with them, and ultimately take them for granted.  That alternative could be called teaching number sense. When that has happened, a time may come when children can be helped to formulate what they already know and practice in words that could be seen as rules.  A determination to expect the knowledge to be imparted by the rules, as opposed to trusting that it can be experienced and taken for granted, can be counter-productive. All teachers practice the two approaches to some degree, but the emphasis is seldom as heavily weighted as it could be on handing knowledge over to a child’s intuitive understanding.  Let me briefly look at Place Value as an example of this perspective.

Place Value represents an essential number sense understanding, indispensable just about any time children and adults manipulate numbers. It is also one of the topics that can take hold quite naturally instead of being allowed to build-up into some major hurdle, from the early implementation with addition and subtraction, to its application to decimals, multiplication of decimals, and connections to fractions and percentages.

I just said that place value was something that could take hold quite naturally. That is said from the perspective of the child, not the teacher. From the teacher’s perspective, fostering in the child that natural understanding may take careful and detailed planning and a determination not to miss any opportunity, however small. It may imply devising a number of strategies that are not just implemented on a few occasions when that is the topic of the hour, but on a constant basis as it is a constant concern in the mind of the teacher, including extensive practice adding and subtracting dollars and cents. Let me expand on two of them.

Of course, we want students to align decimal points when adding. We also want them to see the symmetry as we move left or right that allows them to read 2100 as 2 thousand 1 hundred and 0.012 as representing 1 hundredth and 2 thousandths. But the decimal point is not the axis of symmetry of a number with decimals: to the left of the decimal point we have the ones, and then the tens; to the right we have the tenths and then the hundredths. That is at the heart of a child’s confusion when interpreting hundredths, for instance. In terms of place value, the ones’ digit is the axis of symmetry of a number with decimals. Children can be helped to acquire a vision of the correct axis of symmetry by having them color the ones’ column or circle the ones’ digit of a single number as they learn to interpret the place value of decimals. Now, tens and tenths, hundreds and hundredths are equally distant from the perceived axis. We may also devise simple graphic ways of illustrating the symmetrical connection.

An axis of symmetry belongs equally to both sides. With the unit digit perceived as the axis, the next step is to encourage students to begin with the unit digit as they seek to correctly identify the decimals, in preparation for a world of science that uses them constantly.

  • 5,634.92?  The mind focuses on “4 hundred and 92 hundredths”, which of course, interpreted as the improper fraction that it is, is 4 units and 92 hundredths, with the two decimals now easily identified as hundredths.
  • 267.532?  Focusing on the decimals but including the unit digit, we think: “7 thousand, 5 hundred and 32 thousandths.” The mind just needs to drop the 7 of the unit place value to correctly interpret the decimals without any need to count the tenths, hundredths, and thousandths.

A second concern applies to a child’s very first experience with place value. When standards limit the range of numbers a first grade student should practice adding and subtracting to 100 or 120, I immediately conclude that the limit does not represent a concern for teaching place value. Children like large numbers. As soon as they learn to add ‘5 books plus 3 books’, they know how to add ‘5 hundred plus 3 hundred’ or ‘5 thousand plus 3 thousand’. They take great pride in adding such numbers, and one student is very likely to suggest expanding the understanding to millions. Limiting the range of numbers to 120 is essentially asking students to learn first the most difficult numbers, the ones where an understanding of place value is not reflected in the all important oral version of the numbers.

There is a reason why the verb “to be”, the most frequently used word in the English language, is also the most irregular: I am, he is, we are, she was, they were, been, being. It’s a marvel that we recognize all these entirely different words as different versions of the same verb. Such irregularity would be unsustainable with a verb that was used only on occasion. Who would bother to remember all that? The same is true with numbers where the smaller numbers, the ones most frequently used, are also those that can afford to be the most irregular. In their oral version and its spelling, the sequence of numbers up to eleven is purely arbitrary: we cannot imagine the next one from knowledge of the previous ones. ‘Twelve’ has a very thin connection to the ‘two’ of 10 + 2 just as ‘twenty, thirty’ have a very discreet connection with ‘two’ and ‘three’.  The logical patterns emerge systematically only with 100 and larger numbers.

By starting with the oral version of large numbers, and gradually moving back to digit numbers and to numbers smaller than 100, we are only following the basic strategy of moving from what is easy to what is less so. We are in the process transferring the ease and understanding, not just the facts themselves.  I would want to do this in the early grades without ever referring to ‘place value.’ Putting numbers larger than 100 or 120 off-limit in first grade robs us of one strategy for allowing children to grow into a natural understanding of place value.

On this and other topics we may be satisfied with letting children become very familiar with implementations that are very easy and almost intuitive, and finding strategies to do so. We may then find that acquiring the knowledge of the more difficult applications no longer represents such of a challenge.

Teacher Appreciation Day

Jerry Becker sends out to math teachers his selection of articles and opinions of interest to the profession. I received through his good offices an article posted by Bill Henk on the Marquette Educator blog (May 9th, regretting that National Teacher Appreciation Day had not made much of a splash this year. Teachers deserve better. I couldn’t agree more, but submitted a comment that, if encouragements and kudos are deserved and needed, why don’t we teachers practice more of a culture of recognition among ourselves? I noted how seldom I had sent a note of appreciation to authors of articles in our professional journals or to colleagues whose presentation I had benefited from and used with my students. I contrasted that attitude to the culture that we see displayed by basketball players, where even a missed free throw may be met by an encouraging pat on the back.

Bill Henk, Dean of the College of Education, added his own comment to my comment on his post, and it deserves to be read. So, after a few lines of my own comment, here it is.

billhenk commented on What’s Wrong With Our National Teacher Appreciations?.

in response to Edric Cane:

I fully agree with Bill Henk in his comments on National Teacher Appreciation Day: “The teachers we know are smart, talented, dedicated, capable, passionate, caring, and hard-working educators.” But then, have we, teachers, told any one of them? Have we sent a two line e-mail in the past 10 years to a presenter at a […]

Hello Edric. Thanks for writing such a thoughtful response to my post about appreciating teachers. You’ve taken the conversation to another level with your ideas.

I actually read your response as soon as we received it, and was immediately impressed. But I have been so busy with gearing up for the end of the semester, including commencement ceremonies, that I haven’t had time until now to respond.

As someone who’s done a great deal of presenting and a fair amount of publishing in my career, I can tell you that I never tire of hearing that my work had value for someone — as information, as food for thought, and even for entertainment. It’s really not about ego gratification for me (my ego is plenty big as it is!), but rather just knowing that I’ve made some difference for others and that they appreciate it enough to make an acknowledgement. We humans need and deserve affirmation now and then, and it takes so little effort to provide it. I think you’re right that we take it for granted that others are doing it, that we’re too shy or fearful to make the comments, or that we’re just too self-absorbed to make the gesture.

This topic is a timely one for me in another respect. After our College of Education graduation event a few days ago, I was walking out of the venue with my cap and gown thrown over my shoulder in my carrying bag. I was exhausted because the dean has a very large speaking role in the ceremony, and frankly, it’s nerve-wracking because our audience now exceeds 1100 people. Anyway, a little girl, seemingly all by herself and maybe all of 10 years old or so, recognized me and tapped me on the shoulder to stop me. I was caught somewhat offguard, but swung around to give her my full attention. She then said, in a way that belied her years, “Excuse me. I just wanted to tell you that I thought you did a really great job today.” Wow, I didn’t see that coming.

Even so, I said to her “Thank you so much. You have no idea how much that means to me coming from you. You just made my day, and I hope that our paths cross again one day soon.”

Well, she lit up like a Christmas tree and walked back to her group. Her courage in capturing my attention had been rewarded so to speak. Most importantly, she felt good about validating someone else, a true gesture of humanity. And I can tell you that the exchange certainly felt rewarding to me, too.

Thanks also for the shout out to my former colleague, Jerry Becker. Jerry has been a truly extraordinary filter, conduit, and disseminator of valuable information to an enormous number of educators in the decade or so I’ve known him, and we all owe him a debt of gratitude.