Last year I was awarded the Balomenos award for math educator of the year by ATMNE the six State New England region of the NCTM. Here is the Powerpoint presentation that accompanied the lecture which is below; my longest blog ever.

Maths: the
Science of Pattern and the Art of Making Sense

Including a
visual presentation;

Maths: the
Science of Pattern and the Art of Using

Cents

Balomenos Lecture

ATMNE Conference

Killington, Vermont

October 24, 2014

Tim
Whiteford PhD

(1) First, I
would like to thank the VCTM nominating committee and the ATMNE board for
presenting me with this wonderful award. The more I have learned about Richard
Balomenos the more I am humbled by the task ahead of me but I also feel a
kindred spirit through my boundless enthusiasm for teaching math to young
children and my tendency to speak my mind which characterized Richard’s
approach to life.

My life as a
teacher began in 1972 at Sefton Park primary school on the edge of inner city
Bristol in the UK. To one side of the school, on the hill, was a white middle
class neighborhood, to the other, down the hill, was St. Pauls, the part of the
city settled by Indian, Pakistani and West Indian immigrants. My first class of
34 fourth graders was composed equally of students from these four ethnic
backgrounds so my awareness of diversity
in the classroom began the moment I started to teach. If this wasn’t enough of
a challenge, the headmaster walked into my classroom 15 minutes before class
started on the first day of the first semester of my life as a teacher and,
with a wry smile, said to me “ Do you
realize that if all 34 of those students you are about to face decide, as a
group, to do nothing you ask or tell
them to do there is absolutely nothing you can do about it”. Knees shaking, I
began my teaching career determined that the students in my class would be
motivated by their interests in what they were doing and learning and not by the fact that I was their teacher. **(2)** **Making sense in maths is a function of your motivation. **

**(3)** The images of pennies** **on a wall** **you will see throughout this presentation are** **taken from the hallway wall outside my office. Last semester, one
of my students completed her eMath notebook assignment on the Math of Pennies.
It was so well done that I made a paper copy of the Powerpoint presentation and
put it up on the wall outside my office. Each time I walked past it I kept
seeing more and more ways that pennies could be used to stimulate interest in
the patterns inherent in the study of math. I began with this Sierpinski
triangle and it has just kept going. My goal was to convey to my students that
elementary school math can have an aesthetic component and can be more than
arithmetic. I call the display Maths; The Science of Pattern and the Art of
Using Cents. The response has been quite amazing.

My first
couple of years of teaching were characterized by coming to terms with the fact
that New Math **(4)** was just not
working. It had been around in the UK for several years through the School
Mathematics Project (SMP) and had been something I had studied as an
undergraduate but it was clear that set theory notation and the lack of
computing and calculating skill development were too radical even for the most
ardent supporters such as a newbie teacher like me. This was my first
experience of a math program designed to help children make sense of the math
they were learning but it was just too radical for the conservative world of
the classroom; a revolution instead of an evolution. I have always found it odd how New Math has
been held up as a warning every time we have tried to institute change in math
education while ITA (Initial Teaching Alphabet), a disastrous form of new
spelling instruction, died quietly and was never heard of again **(5).** **Making sense in maths is a function of making meaningful connections.**

I loved the
five years I taught fourth grade at Sefton Park Primary school, the scale 3
post I obtained for developing the math curriculum in the school, and the time
one of the parents called me aside one parent night and asked if I knew what
was growing amidst the grasses in the student science projects. I told him that
we had planted packets of birdseed purchased from a local store to observe and
record the growth of the grasses. This one, he said, cradling a complicated
leaf pattern in his hand is a grass of a very different sort. “This is marijuana”,
he said. I thanked him, he smiled quietly, and I carefully removed the thirty
or so offending grass plants from the student “gardens” in the classroom **(6).** **Making sense is a function of overcoming one’s ignorance.**

In 1977 I
took a one year leave of absence from teaching fourth grade to spend a year in
the US in graduate study at the University of Illinois, a leave that turned
into the rest of my life, so far. As a T.A. my job, at first, was primarily to
teach undergraduates. During my first class I came face to face with what
George Bernard Shaw referred to when he said the UK and US were two countries
separated by a common language when I asked students if they had seen the
rubber so I could clean the chalk board. There were no computers or SMARTboards
at that time. My focus again in my
graduate studies was math education and I met up with a remarkable person, one
of my doctoral advisors by the name of Harold Lerch, known to everyone as Fuzzy
(and no, he was not the origin of the disparaging term “fuzzy math” used so
frequently to describe “new math”). I spent many hours collecting data in
schools in Kankakee in Illinois for the math education text he was in the
process of writing for publication. It was here, I learned the value of
observing children while they are engaged in maths activities. I watched second
graders bounce in their chairs, or count ceiling tiles to count when they
didn’t know their add facts. I also learned that it was not cool to carry your
bat to first base in intramural softball games, a habit I had learned while
playing cricket in the UK. I even had
Max Beberman’s son in a graduate math ed course I taught in 1981. Max Beberman
was a University of Illinois professor and to many, the “father“ of new maths **(7).** **Making sense in
maths is a function of your personal
experiences in life.**

And so to
Trinity College in Vermont in 1982 with my newly minted PhD in Elementary
Teacher Education. I continued to advocate for a math education based on
Richard Skemp’s **(8)** concept of
relational understanding versus instrumental. Skemp’s identification of these
two types of understanding provides us with a theoretical model for making
sense of mathematics. We can learn instrumentally through tricks such as “change
the sign and flip the second fraction” or mnemonic devices such as FOIL; or we
can learn through making connections, constructing schema and seeing patterns
as he suggested in his definition of relational understanding. The former is
fragile, the latter robust. **(9)** It
has also seemed to me to be incredibly important that we make a clear
distinction between knowledge of symbols and procedures; procedural knowledge, and
knowledge of ideas and concepts; conceptual knowledge, as defined by James
Hiebert. To confuse one for the other can inhibit our ability to make sense.
For example, it is a mistake to think that the procedural knowledge involved in
calculating 34 + 29 is teaching the conceptual problem solving concepts of
addition; ideas such as part-part- whole, joining or comparing. The algorithm,
34 + 29 is a procedure based on the ideas of place value and base ten**. (10)** **Making sense in maths is a function of understanding the epistemology
of maths.**

As a
cognitivist I focused more and more on how children interacted with maths and
how prospective teachers interacted with children interacting with math. It
began to seem to me that the private universes children were developing as they
developed their mathematical understandings could harbor all sorts of weird and
wonderful ideas; ideas that we needed to access if we were going to help them
develop strong and realistic mathematical ideas. I shall never forget working
with a young student teacher who was completely puzzled by a confident
kindergartner’s assertion that 6 – 6 = 5. She had been developing the idea of 0
with the students by giving them some small cubes and then asking them to give
them all away so that they had none left, or 0. Remember, this was before the
time when we developed the idea of 0 as the starting point for counting numbers
and was probably a left-over strategy from the new math programs, the idea of
the empty set. Anyway, the three of us sat down one morning in the kindergarten
classroom to have a chat about 0. I gave the student 6 Unifix cubes and asked
him to show me how 6 – 6 = 5. So, very quickly, he placed the Unifix cubes on
six of his fingers, five on one hand and the sixth on the first finger of the
other hand. He then counted the 6 cubes and said “1,2,3,4,5,6” . He then took away his right hand with the one Unifix cube on the
first finger and said “See, 6 take away 6 is 5” showing us the 5 remaining
cubes on his left hand. He was in that ‘no-man’s’ land so clearly and
eloquently identified by Bob Wright in the Math Recovery materials where
children are partly using nominal numbers and partly using cardinal numbers to
count with **(11).** **Making sense in maths is a function of where
you are in the conceptual progression of an idea. **

But it
wasn’t until I returned to working in public schools (Trinity College was
clearly not surviving) as a math professional development specialist in 1999
that I began to see just how much we needed to acknowledge that there was
something about math in the context of students from different cultures that
we, as a profession, were not addressing.

As a math
coach in South Burlington I worked with a third-grade student from the Congo
who couldn’t count orally beyond 8 and was diagnosed as having difficulty in
math because of her limited English. Curious,
I interviewed her in French and found she could not count past ‘huit’ and had
no cardinal sense of number at all. Upon further research I discovered she had
received no formal instruction in maths before coming to the US. On another
occasion, in the same school, a parent from India asked me for advice with her
fifth grade son who, she said, just couldn’t grasp the idea of division. When I
asked her for an example she said “he just cannot understand how 12 goes into 4
three times”. Also confused, I asked her to demonstrate what she meant. So she
quickly shared 12 pencils between four imaginary children 3 times. Her
idiosyncratic use of English, probably a function of the process of
translation, made perfect sense to her and her son but not to her son’s teacher
**(12).** **Making
sense in maths is a function of the language we use in mathematics.**

A year later, in the Burlington school
district, I was excited to discover a student from Bosnia doing subtraction in
exactly the same way I had learned it when I was a child growing up in the UK. **(13)** She even used the same language, “borrow one
and pay it back” putting a small 1 next to the top number in the ones place and
another small 1 next to the number in the tens place in the bottom number (12).
The teacher was completely bewildered by the fact the student used a completely
different procedure and yet got the correct answer each time. We talked about
how this equal addition method had been used in the US up until the mid 1940s
and how it was still used in many parts of Europe. We decided that as long as
the student knew what she was doing it was perfectly OK for her to continue
using this method as opposed to the standard decomposition method now required
in the Common Core Math standards **(14).**
**Making sense in maths is a function of
the cultural math of the learning context. **

This same
teacher, Lillian was her name, also made
special 10 x 10 squares like a chutes and ladders board that started in the
bottom left hand corner so that her ELL students could see that 53 really was “‘higher”
than 21 **(15).** **Making sense in maths is a function of the instructional materials we
use. (16) Making sense in maths is also a function of how much those around you
care about you making sense. **

In 2004, I had
the opportunity to work with a group of
newly arrived Somali Bantu students in the Burlington School District helping
to teach 26 K – 6^{th} grade students the task of developing number
sense in English. We started with the numbers 1 – 20 then went on to the decade
names and how to count to 100. First, we gave the students large numbers of
link cubes to make different numbers. Within five minutes they had all made
guns out of the link cubes. We told the students they could not make guns and
so they quickly made cell phones and started talking to each other in Maay Maay. After several days of continued practice they
appeared to have mastered the teen numbers and the decade numbers so we asked
them all to count by tens to 100 as a group activity. They began, “ten”,
“twenty” and so on but when they got to “ninety” they all followed it with “twenty”. They hadn’t heard the ‘n’ sound at the end of
the teen numbers and so thought we had the same words for 15 and 50 and so on.
They must have thought we were crazy having the same number name for such
different numbers of things. **(17)** **Making sense in maths is a function of what
you hear.**

For a year I
worked with teachers with students from all over the world (over 45 languages are
spoken in the Burlington School District) come to terms with the demands of
learning and making sense of US mathematics. I watched students from Bosnia
recognize the importance of the number 3 and children from Asian countries worry
when dealing with the number 4, a number associated with death in many Asian
cultures. I watched children from different countries have a difficult time
understanding the significance of the numbers used in many of the sayings we
have in English and started wondering if other cultures shared in the same
mathematical cultural characteristics that we have; do they have special names
for 12 like a dozen?. Do they identify odd and even numbers? do they write the
date the same way we do? (there is no Pi day in the UK because it is 14/3/13).
So I started collecting information about the different maths around the world
and compiling it into a web resource to which anyone teaching a student from
another culture would have access. **(18)**
http://academics.smcvt.edu/twhiteford/Math/Cultural%20Math/Their%20Math.htm

I also started to realize that some
student were coming from countries where girls and boys were educated
separately, where boys were educated but girls were not. I also learned that in
some countries such as Singapore students with special needs are not expected
to attend school at all and if they do they have to go to special schools,
something we abolished in the US some 40
years ago as unethical. It still amazes me how Singapore is so revered for
coming top in the TIMMS report in math scores every four years when the school
system there appears to be so elitist and based on different ethical values
from those that underpin education in the US. Also, children in Singapore are
sorted by exams at age 10 into those who are successful in math and those who
are not; a practice that was terminated in the UK in the 70s **(19).** **Making sense in maths is a function of the opportunities or
restrictions in diverse national education systems .**
I returned
to higher education at St. Mike’s in the Fall of 2005, refreshed and renewed by
my public school experiences. And as I was watching student from other
countries trying to make sense of our mathematics I was also watching my own
son Andrew, who has Down Syndrome, struggle to complete math activities with
any sense of meaning. For years he would come home with pages of addition and
subtraction algorithms in which he had learned how to plug in the correct
numbers so that he received check marks or a smiley face at the top of the
paper. This went on through upper elementary and well into middle school until
I finally suggested that this excessive use of hand calculating really wasn’t
going anywhere. I suggested many
interactive websites where he could learn some fundamental math concepts and
begin to recognize mathematical patterns that would help him navigate his way
through the mathematical and quantitative aspects of life. It seemed to me so
much more important that he learn how to use a calculator to complete the
arithmetic while focusing on the concepts of joining, separating, part-part-
whole and comparison as outlined in Thomas Carpenter’s definitive work **(20).** **Making sense of maths is a function of the maths we select and expect
children to learn. **

Like Conrad Wolfram I even began to wonder why we still spend so much time
teaching children hand calculations : some say it takes up 80% of the time
children spend studying maths at the elementary school level. How much better might it be to spend that time in elementary school teaching
number sense and numerical relationships through equations, estimating…………. and,
perhaps, even fractals?

At the risk,
for a moment, of being an agent provocateur, a fox in the hen house, a cat
among the pigeons, or an Englishman at the Boston Tea Party let me ask a
question that borders on heresy.

**(21)** Why do we teach 28 in this form when
28 + 16 = 44 fits right in with making sense of algebra? __+16__

44

How much better it would be if students spent their time learning about the
relationships between 28 and 16 rather than learning a hand calculation that
can be done by a calculator in a nanosecond. There are so many, many more
wonderful mathematical relationships that can be studied instead of the
desperately dull and dismal task of hand calculations.

What we must
not do is confuse the procedural and conceptual knowledge required of
algorithmic thinking for the conceptual knowledge identified by Carpenter in the
different forms of basic problem solving. Selecting an appropriate algorithm is
a totally different skill from completing the algorithmic calculation **(22).** **Making sense in maths is a function of knowing **__why__ we are
teaching selected mathematics skills and concepts.

Think about 28
+ 16 = 44. As a number sentence it is poised to introduce children to the whole
world of algebra. In this form it introduces children to perhaps the most
important idea in algebra; that the real meaning of the = sign is equality or
“is the same as” **(23).** **Making sense in maths is a function of understanding
the mathematical symbols we use. **

To simplify
things let’s explore 8 + 4 = 12 a little more.

·
**(24)** How about
two groups of people, one of 8 and one of 4 being joined together into one
group? Maybe combining two families for dinner and needing to know how many
chairs to set out (P).

·
**(25)**How about a class
of fourth grade students, in which you know there are 8 boys and 4 girls and you want to know the total number students in
the class. There are no separate groups; just two parts of one whole. (P)

·
**(26)** How about
two bags of candies? One bag contains 8 candies and the other bag contains 4
more than the first bag? This time we start with two groups and one of the
numbers is not a group at all but the difference. Or one bag contains 4 candies
and the other contains 8 more? (P)

·
**(27)** How about
wanting to know how many free tickets you started with when you have given
8 away and now only have 4 left. You can even use addition to find the answers
to a problem that involves separation. (P)

How many ways can 8 and 4 be related to each
other through the addition, subtraction, multiplication or division operations ?
**(28)** **Making sense in maths is a function of understanding these procedural/conceptual
knowledge relationships.**
Perhaps this is similar to what Max Beberman
really had in mind with his ‘new math’!
I wonder what would have happened to Beberman’s ideas if he’d had the same access to
technology and computers that we have
today? **(29)** **Making
sense in maths is a function of the technology we have at our disposal.**
But wait,
it’s too easy to get carried away by the seductive idea of letting technology
complete the graft for us. In her illuminating argument for the inclusion of
the “standard algorithm” in the Common
Core Math Standards Karen Fuson eloquently points out that completing
algorithms by hand is an important way for children to learn to think
algorithmically and to practice and develop their knowledge and understanding
of place value and the Base 10 system. Again, this is what Bob Wright refers to
as the mathematization of students’ thinking. Tools and “concrete”
manipulatives are critical in the initial development of ideas and concepts but
sooner or later thinking has to become abstract; students need the skill of
mental gymnastics developed through involvement with computational exercises.
This mental number sense is part of what
it means to be mathematically literate **(30).**
**Making sense in maths is a function of
developing one’s abstract thinking skills, of mathematizing oneself****.**
At home, my
son Andrew does his calendar every morning, he uses a credit card and a phone,
he is constantly thinking mathematically about the various things that make up
his life, comparing quantities and numbers on his DS, his Wii games or his iPAD2.
I well remember his excitement when, about
three years ago, on a Sunday morning at 6:30 he woke Lucie and I with
the “good news” that he had beaten his Wii bowling score of 299. The perfect
game. He learned about dates by looking
at the date on the underside of his Hot Wheel cars to see if they were older or
younger than him after he had found one, quite by accident, that had 1992 (the
year he was born) stamped on the underside. As a young man with D.S. he has to be careful
about what he eats so he is constantly aware of the dietary numbers such as the calorie count on food wrappers. He
has developed his comparative sense of measurement through our frequent
weigh-offs which he almost always wins and he has learned how to use nominal
numbers in place of measures when he wants to assess a situation. Temperature
readings, for example, name a certain level
of warmth or lack of it for him rather than giving him a sense of difference on
a scale. He just knows 75 is warmer than 62 but is not aware than the numbers
are related through a temperature scale. Each number names a level of warmth.
Does he really need to learn how to do algorithms **(31)?** **Making sense in maths is a function of a student’s individual cognitive
abilities.**

For the past
ten years I have tried to emerge from my math education silo, a place in which
I lived and felt comfortable for many years. I have tried to learn about the
intricacies of the WIDA Standards, the SIOP model and academic language
proficiency as they relate to mathematics but I have remained a true disciple
of the credo that math is the science of pattern and the art of making sense. I
have made Math and ELL presentations at TESOL conferences in front of many
participants and have made the same presentation at ATMNE conferences to fewer
participants. I have also made presentations on math and Diversity at Special
Education conferences such as the MDSC conference to many participants and have made the same
presentation at STEM conferences to fewer numbers of participants **(32).** **Making sense in maths for all students is a function of our
dispositions.**

And so as we
begin the implementation of another evolution in math education, the Common
Core State Standards for Mathematics content and practices, we owe it to all
students the best that is thought of and said regarding math education, to
paraphrase Matthew Arnold. The students
in our classrooms are more diverse than they have ever been; diverse in their
ethnic backgrounds and experiences and diverse in their abilities and
dispositions. Regardless of this broad band of diversity all students have the
right to make sense of what they are learning, to see the patterns inherent in mathematics
that make the recall of factual information easier and more precise and provide
us with effective problem solving strategies. Patterns help us make sense,
reason, construct viable arguments, model, be precise, bring structure,
regularity and repetition to the quantitative aspects of our lives, to
paraphrase the CCSSM practices; the type of patterns we find in the study of
fractals.

Along with
the CCMS comes the Smarter Balanced Assessment Consortium or SBAC for short. By
Fall 2015 in 27 States, I think it is, most students (some 99%) in grades 3 and
up will be expected to take these computer based assessments. Having just spent
many hours reviewing the sample test questions my hope for the finished SBAC
assessments is that they assess whether students are making sense of the math
they are learning as well as recalling the
factual and procedural knowledge they are learning **(33).** **Making sense in maths
must be included in any assessment activities if they are to be meaningful and
relevant. **

Maths must
make sense to everybody who studies it. Maths must also captivate students if
they are to truly make sense of it. Patterns help us see the coherence between
all aspects of mathematics in the same way Hung-His Wu describes mathematics as
a tapestry in Phoenix Rising, his commentary on the Common Core State Mathematics
Standards. He writes, “Mathematics is coherent; it is a tapestry in which all
the concepts and skills are logically interwoven to form a single piece”. As math teachers we must help
children weave the different threads of mathematics together to form a coherent
whole, a tapestry, that forms a picture that makes sense to each and every one
of them.

As we
implement the CCSMs we need to make sure that our students, whether in
kindergarten or twelfth grade, or even in college courses, see that math has an
aesthetic component as one would see in any tapestry; that there are poetry and
creative writing equivalents of the English language arts in maths so that
students find it exciting, interesting, motivating and captivating. We must not
make the mistake again, however, of proposing these things in lieu of the rigor
of remembering factual information and the ability to estimate and persevere.
We do not give up on spelling and grammar when children write creative stories
or poetry.

And finally,
this past summer, I audited an online Stanford math education course developed by
Jo Boaler, one of the current shining lights in maths education. Boaler, also
an expat Brit continues to use the British term maths (with an s) which I have
always thought a more logical abbreviation of mathematics than the word ‘math’,
which is why I have used it today. As I followed the course I was reminded of how
importance it is for all students to develop and use growth mindsets related to learning mathematics
as opposed to the ubiquitous fixed minds
sets so often found behind the statement “I’m no good at math”. Carol Dweck’s
work on fixed mindsets probably has more relevance to math education than to
anything else. We must present mathematics in a way that helps students of all
ages make sense of what they are learning through the patterns they see and
construct, so that instead of saying “I’m
no good at math”’ students say “ah, now I get it, that makes perfect sense to
me”. **(34)** **Making sense in maths requires
a growth mindset, **something we all should have
whether we are kindergartners or veteran
teachers.
Thank you.

References

Carpenter,
T.P., Fennema, E., Loef Franke, M., Levi, L. and Empson, S. (1999) Children’s

Mathematics;
Cognitively Guided Instruction, Portsmouth N.H. Heinemann

Dweck, C.
Mindset (2006) New York, N.Y. Ballantine books

Hiebert, J. (1986). *Conceptual* *and procedural knowledge:
The case of mathematics** *

Mahwah, NJ: Lawrence Erlbaum Associates.

Skemp, R.
(1972) The Psychology of Learning Mathematics, London UK, Penguin Press

Wright, R.
(2012) Developing Number Knowledge London U.K. Sage Press