Friday, February 5, 2016


You can probably remember a teacher who touched your life in a positive way at some point. Perhaps it was a third grade teacher who patiently helped you understand what the commutative property of multiplication was. Or maybe it was a fifth grade teacher who was passionate about literature and always found a way to help you make every story come to life. Or it might even have been a high school history teacher who dressed up in period costume for classes about the civil war.

Have you gone back to your school to thank that teacher?

There is nothing quite like the feeling that comes from being a good teacher; whether it's guiding children to a better understanding, sharing one's passion for a particular topic, compensating for a child's tough home life by caring a little bit more, or helping a student develop a growth mindset when they say they'll never understand fractions. Being part of a classroom full  of children for a year is an amazing experience.

It's not easy though, if you want to do it right. It' not something you can do by reading a book, taking a single course, spending a year in a classroom with a practicing teacher or just doing what comes naturally. Being a good teacher and getting the most fulfillment out of it involves a rigorous, challenging but incredibly rewarding intellectual process of thinking, reflecting, acting and developing the professional dispositions of a teacher. It involves having a good understanding of what you will teach, a practical grasp of how children develop and learn and the accumulation of a toolbox of teaching strategies you can call upon when needed.

A great place to do this is in the undergraduate or graduate programs at St. Michael' College where theory and practice are integrated in a way that enables you to be the best you can be.      


Thursday, February 4, 2016

What is Maths?

For the past several year I have defined maths as the science of pattern and the art of making sense. This definition has probably developed as a response to the still somewhat prevalent view that maths, at least at the elementary school level, is the study of numbers.

I have tried to operationalize this idea by having student look at maths from many different angles especially from the aesthetic and relational points of view. I have included the study of fractals in my courses because I feel they epitomize the idea of numerical patterns and relationships which are so important as we try to get children to remember concepts, ideas and factual information.

I have read much by Jo Boaler and other illuminaries who believe that maths should be taught the way I believe it should be but I have never read anything quite as clear and eloquent as this piece by John Seibert of the Math learning Center, publisher of the wonderful Bridges Math Program.

The analogy of  defining maths as the study of numbers and comparing it with literature as the study of the alphabet is absolutely brilliant in its simplicity and absurdity. Numbers, like letter, are simply some of the things we use to convey meaning and communicate with.  And this sentence, "Perhaps it is through the lens of patterns that math can transcend the procedures that have come to define it" is exactly what we should be helping young children do.  Seibert then concludes with this masterstroke, "After all, when we teach our students their ABCs we expect them to one day write their own papers and poems. In teaching our students their 123s, shouldn’t we support the same creativity?".

Wednesday, January 20, 2016

The End Is Nigh

So, I was planning to retire at the end of this academic year but a bit of a serious heart attack got in the way just before the end of last year and I'm forced to take medical leave this semester. I had planned to really enjoy the last semester of my working life but sometimes things just don't work out the way we want them to; just  like some maths problems I have encountered over the years. No matter what you do, they seem to defy resolution. And so it is with my last semester. I was to have eased into retirement tidying up all the loose ends, making  sure my courses were complete and ready for my replacement to take over and make them hers or his.

The remarkable thing about teaching, something maybe only teachers know and understand, is that the courses one develops and teaches become incredibly, unbelievably, and almost inexplicably personal. The three maths education  courses I have taught in the Education Department at St. Mike's for the last several years are each unique and exclusively my own creations. Yes, of course there are curriculum standards and teaching standards that have to be addressed but my courses are my personal interpretations of those standards. I use the philosophical theories that I have experienced and developed over the past forty something years to operationalize those standards for my students. I use the experiences I have had teaching maths to children to illustrate and exemplify those theories and ideas with inspiring stories.

Each course I teach is characterized by an emotional set regarding the content I teach. I choose to focus more on those things that I truly believe work and cause my students to think. I minimize the time I spend discussing and exploring things that I know create negative feelings or are just "tricks of the trade" so to speak. I believe that to teach maths one must intellectualize the process, base ones practices on well reasoned theory or, as John Dewey said, we must use our "executive means" to get as close as we can to our "inspired vision". One must listen to children speaking mathematically and get to know what they think mathematically. One must know, understand, and love the maths intimately. My students were always actively involved in their learning by doing, talking, thinking and reflecting.

I cannot believe how difficult this is;quitting cold turkey, the sense of being shut out, stopping teaching, not meeting my students each week, not reading papers, not standing in front of the class, not telling stories, not conferencing  with students about their dreams and aspirations, not talking with colleagues, not helping students over their maths anxiety, not lighting up students' eyes about the joys and wonders of learning and teaching mathematics.

I hope there's a way I can continue to be a part of the maths education community and make a positive contribution in some way.

Saturday, January 9, 2016

Teaching Teachers to Teach Math

There's an old saying that goes "Those who can, do,  those who can't, teach, and those who can't teach, teach teachers. This is patently untrue since most teachers can do, and most teacher teachers have reached that position by becoming good practicing teachers and then choosing to get into teacher education because they don't want to get into educational administration. At least that's what I did.

One of the uphill battles I have faced most of my professional career is the oft-spoken sentiment that really anyone can teach elementary school math since it is so elementary and the math cannot possibly be that  difficult. Perhaps this is the primary reason why the US has never truly embraced reform in math education. Perhaps this is a more plausible reason than the oft blamed "new math" of the seventies as to why there is constantly such a strident call for "back to basics" math education. Sometimes my professional life seems like one of failure and futility!

The unbelievably sad thing is that we now know so much about how children and students of all ages learn math much of which is, sadly, not applied in schools or in classrooms where children need it the most. We now know that learning to count, for example, is an extremely complex activity in which children pass through several stages using different forms of number before, if they are lucky, they have developed a secure sense of numeracy by the end of first grade. It takes a skilled, knowledgeable teacher to provide students with activities, guidance and practice with this pedagogical content knowledge. It also takes a skilled, knowledgeable teacher educator to teach teachers the many diverse aspects of teaching elementary school math. For example, do you realize that when you count ,say five objects, with a young child, your voice goes down automatically when you say the number "five" while it goes up with 1,2,3 and 4. If you know this is how we develop the sense of cardinality in children you are a much better parent or teacher than if you do it without realizing what you are doing.

There is so much to know about math education and the way children think mathemtically.

Friday, December 11, 2015

Math Test Answers

Last week I posed 12 questions related to teaching elementary school math. The test was specifically designed for those who believe a traditional, "back to basics" approach to math is superior to the type of math we are trying to teach through the Common Core in 2015.

So here are the test questions and answers with the  first answer, the traditional math answer (TMA) and second answer, the 2015 Common Core (CCSSM) answer.

1. What is counting?
TMA - "one, two, three four five etc" with no sense of what the numbers actually mean.
CCSSM -  "one, two, three, four, five etc" understanding that three is one more than two and four is one more than three and that when you count five objects the word "five" refers to all five objects not just the last one counted. This is the idea of cardinality and is key to understanding number,

2. What is addition?
       TMA:  34
                + 48
       CCSSM The above is only an algorithm, a piece of arithmetic. Knowing just this is useless in
       problem solving. To problem solve you need to know the concepts of joining, separating,
       comparing and part-part-whole if you want to use addition or subtraction in anything useful. .
3. What is multiplication?
       TMA  112
                x    5
       CCSSM Just like addition and subtraction the above is just an algorithm, a piece of procedural   
       knowledge. It doesn't help you decide which operation to use in problem solving. You'll need to
      know the repeated addition, multiplicative comparison concepts as well as a few more to be know
      whether to divide or multiply
4. What is division?
       TMA:  5/115
       CCSSM See above
5. What is Pi?
        TMA: 3.14 etc
        CCSSM Yes, but this doesn't help a whole lot. Pi is a ratio between the diameter and   
        circumference of a circle. The circumference is always just over 3 times the diameter; or, if the
        diameter is 7 the circumference is 22.
6. What does the 0 mean in 308?
       TMA: A place holder
        CSSMIt means there are 0 tens. Place holder is a meaningless term than has no conceptual  
         value. Children need to learn initially that each place in place value has a value.
7. What is 1/2 divided by 1/4?
        TMA:   "Change the sign and invert the second fraction"
        CCSSM  This is virtual nonsense and doesn't help at all. Knowing that the problem is asking   
        how many 1/4s there are in 1/2 is of far more value especially in a world of calculators.
8. Make up a word problem for # 7 above.
        TMA: I can't
        CCSSM: How many 1/4s in the first half of a football game 
9. What do  you get when you reduce 4/8?
        TMA: 1/2  Unfortunately, most children being taught to "reduce" a fraction  think that 1/2
 smaller than 4/8 because the word "reduce" means to make smaller and the two numbers.1
         and 2 are  smaller than 4 and 8. I joke about reducing 4/8 to 4/8    
        CCSSM; You don't use the word  "reduce" fractions because children think 1/2 is smaller than
         4/8. You  rename or regroup 4/8.

10. Why do you  put a comma after every three digits in a large number?
         TMA: Because that's what you do
          CCSSM Because it marks the repetition of  100, 10, 1 in each place value referent 
11. What is area?
          TMA: length times width
          CCSSM this only allows you to measure the area of regular shaped objects. Area is the two
         dimensional concept of coverage. You can find the area of your hand by tracing it onto squared
         paper and counting the squares.
12. What does a degree measure in geometry?
         TMA: It measures angles
         CCSSM It measures the rotation of an angle about a point. This is the only way you can explain
         why an angle is 45 degrees 4 inches from its origin and still 45 degrees 4 feet from it;s origin.

Children must understand what they learn at the developmentally appropriate level; this is why the common core is so good.

Tuesday, December 8, 2015

Maths; The Way It Should Be

Last night marked the last class of probably the best group of graduate students I have ever had the pleasure of working with. The course, Math and Diversity, focused on teaching maths to children with diverse needs including ELL, Special education, poverty, maths disabilities such as dyscalculia, and working with mathematically talented students. The course focused on developing the students' relational understanding of math as well as a growth mindset. Here's a sample journal from Bria, one of the students in the class that is pretty typical of how all the students in the course saw their growth as math teachers this semester.

 "I am so glad that I took this class. My mathematics confidence has skyrocketed as a result of this semester. I used to be told that I was good at math, but I didn’t believe it because I was grouped with people who were notably good at math and did not compare to them. I was one of the weaker links in Mathletes, for example, and I felt that I struggled in my college calculus class. I know that I will never be as strong in math as many others, at least as long as it is not my academic or professional focus. But what I am learning is that I am better at mental calculations and quick consolidation of numerical information than many other people I come across in my everyday life. I was able to comprehend and work with new concepts presented in this class fairly quickly, and enjoyed being able to experiment with my new knowledge. By gaining conceptual understandings of things I had learned procedurally growing up, I am able to approach new math problems that come to me in life more thoughtfully, and I understand those thought processes. My new hobby is doing mental calculations of various operations and then analyzing exactly how I came to my answer. Feeling in control of math is a fairly new feeling, and I love it. It makes me more confident in my mathematics skills, and makes me more comfortable when I find myself out of control of math.
In the past I have not been able to be comfortable with accepting that something was challenging for me; I would admit defeat at the first sign of struggle. I understand that this is a common ailment of the person who has spent their childhood floating through academic requirements and being told that they are smart. But through the education I am receiving in the graduate program at Saint Michael’s, I am finally learning to practice what I preach. I am learning that finding something challenging does not put a blemish on my intelligence, and asking for help does not signify weakness or make the person asked think I am less intelligent than they are. I believe these things wholeheartedly when they come out of my mouth as a teacher, but I continue to struggle with it personally because I always prided myself on being “smart” growing up and worried that people would find me less so if I asked for help. But during the second half of this course, when a concept or problem presented in class was challenging I began to actually feel alright about admitting it and getting help from a neighbor, rather than chastising myself for not understanding something as quickly as my classmates. I think that the confidence I built from taking this class has allowed me to get over any math-related anxiety I used to feel, such that I now know I am “good at math.” I understand that people were not lying when they told me this in the past. With this newfound knowledge, I am comfortable with struggling. Finding math concepts that are difficult for me are not something to avoid, but something to tackle head-on. It’s fun now, and I have this class to thank for it."

Another student in the course felt that, as a child,  she was "a conceptual person trapped in a procedural world". I will miss teaching this course. 

Sunday, December 6, 2015

Maths and gender

So finally there is definitive research that shows the human brain of both the male and female of the species is the same. It comprises "a mosaic" of all kinds of characteristics regardless of gender. There is, therefore,  no biological  reason why boys should be better at math than girls.

This means that it is the context in which maths is learned that creates the inequalities, as Jo Boaler  so eloquently points out. It means that we, as teachers, as parents, as a culture need to adjust our expectations of what girls can do mathematically.We need to make sure that girls do not receive subtle or not-so-subtle messages that they cannot do math  and that the types of activities that occur in math class are not gender stereotyped in any way.