Thursday, September 25, 2014

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

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

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 – 6th 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)
 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                  
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.
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.
Hung His Wu (2011) Phoenix Rising in American Educator;                       
Skemp, R. (1972) The Psychology of Learning Mathematics, London UK, Penguin Press
Wright, R. (2012) Developing Number Knowledge London U.K. Sage Press

Wednesday, September 24, 2014

The Archway to The Teaching Gardens

The Archway to the Teaching Gardens at St. Michael's College is complete. The work of stone artist Thea Alvin the archway and "plinth" provide a spectacular entryway to the Teaching Gardens.

I make a point of walking through the arch on my way to my office in the morning and on my way home at night. For the past couple of weeks since it was completed I have said to myself, as I walk through the arch "I will retire this year" or "I won't retire quite yet". If I am confident in my

decision I walk to the right of the plinth, if not,    I walk to the left. I'm waiting for the time I consistently say and do the same thing each day.
I call it the Archway of decision.

Mathematically, the arch and the structure itself are very interesting. Initially I didn't think the arch followed the curve created  by the squares of the numbers in the Fibonacci sequence because I only saw the top part of the opening. Now that it is complete you can see the continuously changing radius of the curve from
                                                                                    the bottom of the arch to the middle of the top.
The same sequence is repeated in the line at the top of the wall where the curve continues to tighten toward the right.

Below is a diagram of the Fibonacci curve which is made by squaring the numbers in the Fibonacci sequence; 1 1 2 3 5 8 13 21 34 etc. The sequence is created by adding the last two numbers to get the next one. This pattern occurs throughout the natural world and is also related to the golden ratio.

 Isn't that neat!!!!!

Wednesday, September 3, 2014

No Stone Unturned

After a summer of tree felling to clear a space, work has begun on the stone arch, the latest addition to the Teaching Gardens at St. Michael's College. 

The work of renowned stonemason Thea Alvin, the arch will form an integral part of the ever-expanding Teaching Gardens. Construction of the arch began on Monday September 1st with a seminar about stonemasonry given by Thea Alvin to a group of enthusiastic students and faculty. Participants were then given the opportunity to add a stone or two to the arch so that, in Thea's words, when they visited the campus in 50 years with the grandchildren they could point out the particular stones they had personally added to the arch when they were students at St. Mike's.    

Additional information about Thea's work can be found in this NY Times article , this 4-minute video from the Oprah Winfrey show, and this  You Tube video of a similar installation at Duke University.

To find out more about the Teaching Gardens or the stone arch installation  contact Valerie Bang-Jensen: or Mark Lubkowitz: