Mike Flynn is the director of Mathematics Leadership Programs at Mount Holyoke College and runs the Master of Arts in Mathematics Teaching program. Prior to this work, he taught second grade at the William E. Norris Elementary School in Southampton, Massachusetts, for fourteen years. He is the 2008 Massachusetts Teacher of the Year, a recipient of the 2009 National Education Association Foundation Horace Mann Award for Teaching Excellence, and a 2010 recipient of the Presidential Award for Excellence in Mathematics Teaching.
An important thing to know about the Standards for Mathematical
Practice is that they pertain across all levels of mathematical
activity. These are K--12 standards, but they describe the kinds of
activity that distinguish math as a discipline.When I wrote about
the Standards for Mathematical Practice on this blog, and in the
Dummies book, I took the easy way out. I addressed their spirit
without going into all eight of them in depth.But I'm here today to
tell you that Mike Flynn has taken the high road and done the much
more challenging job of treating each of these standards
right.Flynn structured his book Beyond Answers around the Standards
for Mathematical Practice--one chapter per standard. He illustrates
each with a vignette from his own life demonstrating the utility of
the practice in his own daily life, with vignettes from classrooms,
and with clear writing demonstrating the very real and important
mathematical work of which young children are capable.On its
surface, this is a book about elementary children doing
mathematics. But it's really a book about people doing mathematics.
If you're a secondary or post-secondary teacher, and you read this
book without seeing important connections to the work that you do,
I'll buy you a cup of coffee so we can talk about that.Ultimately,
my own critique of the SMP was about them being too numerous to
remember, and about them overlapping in ways that make it difficult
to communicate their individual importance. I still have those
critiques. Flynn doesn't convince me (nor does he try) that this is
the perfect set of such standards. But he doesn't need to do
that.These are the standards we have. They resonate at all levels
of mathematical activity, and in Beyond Answers, Mike Flynn shows
convincingly that young children's mathematical work is not
fundamentally different from that of older students. Mathematics as
an intellectual discipline is alive and well. Christopher
Danielson's Blog
Most mathematics curriculums give a place to the "process skills"
of the subject. When the first English national curriculum was
unveiled in 1989, its "Attainment Target 1" was "Using and Applying
Mathematics"
"Pupils should use number, algebra and measures in practical tasks,
in real-life problems and to investigate within mathematics
itself."
Successive incarnations of the national curriculum had similar
sections or statements. Problem was, we tended not to turn to this
bit. It was the more particular 'add two two-digit numbers'-type
statements that we turned to on a week-by-week basis. Why? Maybe
the process skills are, when expressed this way, too general, too
all-pervasive; it's easier to turn to the more specific, to the
'know the names for various kinds of triangles' sort of
requirement. But the process skills are where the mathematics
really happens! In the US, the Common Core State Standards for
mathematical practice are: Make sense of problems and persevere in
solving them.Reason abstractly and quantitatively.Construct viable
arguments and critique the reasoning of others.Model with
mathematics.Use appropriate tools strategically.Attend to
precision.Look for and make use of structure.I suspect there will
be the same tendency to pass quickly to the more week-by-week
standards. And yet these skills are the core of what it is to work
mathematically. What to do? How to make 'reasoning abstractly and
quantitatively' less nebulous-sounding?
Enter Mike Flynn's book Beyond Answers. Mike devotes a chapter to
each of the mathematical practices. He begins each with a personal
experience. This is useful in a number of ways. One, it's good to
get to know the author of a book you're reading. It somehow makes
the exchange more on the level, more real. Two, he gives an analogy
for the practice from his own experience. I like analogies; they
work well for me, and I guess they do for other people too. They
ground the practices in our everyday non-mathematical experience.
For instance, in the introduction Mike talks about getting lost in
Boston, a town he wasn't that familiar with, and how his brother
helped him to learn to navigate the city without helping him too
much, by asking powerful guiding questions. I like the metaphor in
lots of ways. For one, mathematics is like a city, with links
criss-crossing it in all sorts of ways. Also, navigating is a
primal kind of function for us, one we share with animals, whether
hunting or fleeing, patrolling or harvesting. I blogged about the
link for me once. The book is about how we help students to
navigate for themselves, developing skills to find their way round,
to become at home in the world of mathematics. Another thing I like
about the book lots is that it gives us a lot of classroom
vignettes that exemplify teachers enabling their young students to
develop the standards. It's an approach that's worked well in a
number of great recent teacher education books, because we learn by
concrete example more than by definition. There can often be just
one adult in the classroom, me, the teacher, and though we learn so
much from the students themselves, how do we get to learn from
other adults? Well, these kind of vignettes let us peek in at key
moments, listen to an actual conversation, catch how adroitly the
teacher puts themselves to one side while she allows ideas to come
from the students rather than her. There's a consistent emphasis on
student voice in the book. 'The benefit of taking extra time to
discuss their strategies is that it allows ideas to come from the
students and not from me. That means they have opportunities to
hear mathematical arguments from their peers and to critique their
reasoning (MP3). It also helps to create ownership of the ideas and
changes the power structure in the classroom by showing that we all
contribute to the learning in math class' (p123).
It's important that we think about the big picture of what we're
doing in mathematics lessons, and how it links ultimately with what
we do in the adult world, and Mike takes time to spell out the
essentials of each mathematical practice. In the chapter on MP 4,
Model with Mathematics, Mike discusses one of the essentials of
modelling, the process of abstraction. Abstraction is an essential
tool way beyond the maths curriculum, as I've mentioned in a post
before, but Mike helps us see how it's there in so many of the
simple tasks we give young children. He gives the example of a
first grade class who are asked to make representations of who sits
at their tables (p84). They show it in lots of ways: The students
are learning to ignore, for this task, the myriad other important
features of the situation, the appearance and personalities, how
they're relating and feeling, what their position is in relation to
each other, and just to focus on number and one category, boy/girl.
This narrow focus, ignoring most of the context, is something young
students learn through practice. You can see how this process works
with adult mathematical modelling, how for instance a traffic
planner ignores all sorts of context in traffic, car colour and
make for instance, to mathematise the situation, focusing on
numerical data such as number of cars and speeds. Once she's
represented the structure of the network of roads and the numbers
she needs, she doesn't for a while need to think about them as
roads or cars as such. After calculating possible solutions, she
can then translate them back into road locations, numbers of lanes,
traffic capacities and suchlike, and ultimately actual road
construction can begin. Young children learn to do essentially the
same thing. Mathematising is there all the way. When we count a
handful of pebbles, for a short while we attend less to colour and
texture, to shape and size. Or, if we sort by shape, we might
ignore all the other features. The three-act task, as Mike says, is
a really useful tool for helping students to mathematise. Graham
Fletcher's great collection is a great introduction to this, and
one I've just been sharing with teachers in Qatar. In Chapter 5
(Use mathematical tools strategically) Mike lists examples of tools
in five categories: supplies, manipulatives, representation tools,
digital tools, and mathematical tools. I was particularly
interested in the discussion of the last, also called 'thinking
tools'. As he writes (p111), 'A large part of our work with MP5
developing and supporting our students' metacognition.' We've all
had students that when asked, 'How did you know that?' answer,
'With my brain'! Helping students to recognise how they're actually
thinking, in detail, is such a wonderful part of what we do, and
Mike gives attention to this, with plenty of great examples and
vignettes. I loved 'Are there any other foxy shapes?' in chapter 6
(Attend to precision). On p133-6 three's a great vignette of how
children play a game sorting shapes according to some property
they've chosen. (This would make a great development from a Which
One Doesn't Belong!) One student's category is 'foxy shapes' and
the Socratic way in which the teacher shifts the children from a
sort of holistic recognition of a fox-faced shape to themselves
giving it a precise definition of essential properties is a
delight. I realised that I'm probably an MP 7 guy (of course all of
them are important!) - someone that spends a lot of time in class
getting students to explore and search for structure. I'd not seen
such a complete list of the sorts of mathematical claims young
students might make; is that kind of thing out there widely,
because it should be? I do less problem-solving (learning to do
more!) but focus on this lots.
I've merely touched on a few features of the book. There's so much
more in there. It would be an excellent introduction to what's
important in mathematics education for any beginning teacher of
five to eight year olds. And an excellent tool for reflection for
an experienced one too, bringing the focus back again and again to
students' reasoning and discussion. As Mike says, 'Enjoy these
moments with your students and cherish the opportunities to learn
alongside them, for this work is as engaging for us as it is for
the students.' Simon Gregg's Follow Learning Blog
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