A group of teachers involved in
embedding NRICH tasks into
their everyday practice decided they needed to address the
(im)balance between teacher and student activity in their
classrooms. In this article they share the issues they identified
and what they have been doing to address them.
'We wanted to place a greater
emphasis on lesson planning so that teachers and students could be
more productive during lessons.'
We are using more of our time thinking about and preparing for
lessons, and placing less emphasis on marking the output of the
lessons. Preparation has focussed on how we can support students in
tackling problems for themselves, rather than on demonstrating
strategies for finding answers. The Teachers' Notes that accompany
the NRICH problems have proved very helpful. See, for example, the
Teachers' Notes
for Triangles in Circles
How well are we
doing?
Lessons were observed and evaluated by peers, with planning forming
the main focus of discussion. The status of the planning process
was raised by setting aside time for colleagues to work together on
lesson preparation - this was effective use of teachers' limited
time.
Teachers spent time thinking about how to present tasks that might
maximise opportunities for students to develop their own ideas,
including which key questions would be asked. The Teachers' Notes
to the NRICH problems always list some key questions. See, for
example, the Teachers' Notes for Reaction Timer
http://nrich.maths.org/reactiontimer
Some schools are monitoring students' attitudes to the changing
nature of lessons by surveying opinions.
Peer observation outcomes showed that staff valued an emphasis on
planning.
Teachers are planning collaboratively and sharing ideas.
Teachers have begun to utilise Kenneth Ruthven's (1989) idea of:
Exploration $\rightarrow$ codification $\rightarrow$
consolidation
'We wanted our students to take
more responsibility for their learning.'
In our schools we have worked on establishing classroom communities
of enquiry, where everyone's contributions are valued.
By setting aside a few minutes for students to consider, on their
own, how they might approach new problems, schools are encouraging
all students to become more resourceful and less dependent on
others for guidance.
Students are encouraged to jot down notes as they think around
problems, using mini-whiteboards or separate space set aside in
exercise books for this purpose.
'We wanted the students and the
teacher to work in a more active way.'
We have been encouraging students to present their findings and
compare their results. They are encouraged to address their work
to, and look for feedback from, the rest of the group, rather than
just the teacher.
Schools are using a range of evaluation methods including peer
observation and student feedback to assess their progress towards
this aim.
'We wanted to create lessons which
are less dependent on talk led by the teacher, with more time spent
by students discussing the ideas they are
investigating.'
We have chosen rich tasks from the NRICH website
http://nrich.maths.org which encourage discussion. For
example
Students have compared the efficiency of various strategies
when working on questions that have been solved in a variety of
ways. As well as comparing their own solutions, students also
discuss published solutions from the NRICH website. See, for
example, the range of published solutions to: Ben's Game, Temperature
The problem includes the following question: Is there a
temperature at which Celsius and Fahrenheit readings are the
same?
This is an extract from a
solution sent in by Seb:
There is a temperature at which Celsius and Farenheit are the same.
It is $-40$, because $\frac{9}{5}$ of $-40$ is $-72$ and
$-72+32=-40$. I decided to look at negative numbers because
starting with a positive number and multiplying it by is going to
increase it and so is adding $32$ so you're always going to end up
with a number greater than the number you started with. However, if
you start with a negative number, multiplying it by decreases it,
and adding $32$ increases it, so I realised that with the correct
number, Celsius and Fahrenheit might be the same.
I decided to go down in tens: $\frac{9}{5}$ of $-10=-18$ and
$-18+32=14$, so that doesn't work; $\frac{9}{5}$ of $-20=36$ and
$-36+32=-4$, so that doesn't work; $\frac{9}{5}$ of $-30=-54$ and
$-54+32=-22$, so that doesn't work But $\frac{9}{5}$ of $-40=-72$
and $-72+32=-40$ so it works.
The reason it works is because multiplying by $\frac{9}{5}$ is
equivalent to adding $\frac{4}{5}$ of it, and for $-40$ adding $32$
is equivalent to subtracting $\frac{4}{5}$ of it (because $32$ is
$\frac{4}{5}$ of $40$). Because of this, Farenheit and Celsius are
equivalent ONLY at $-40$.
Alex, Ben, Chris and Paul also
used a trial and improve approach:
First we started going down in tens of Celsius from $0$, and we
found out a pattern: the difference between F and C was getting
closer by eights every time. When we got to $-30$C the difference
was only $8$. So $-30$C is equal to $-22$F. Then we tried $-40$C
and found out that $-40$C was the same as $-40$F. So the answer is
$-40$.
David used a graphical approach: I plotted the lines of the
simultaneous equations against each other and found where they
crossed. In the graphs $y=F$ and $x=C$.
Phoebe offered an algebraic
solution:
To solve it algebraically I can create two simultaneous equations:
$$F=C$$ $$F=1.8C+32$$ Therefore $C=1.8C+32
C=\frac{32}{-0.8}=-40$
Students have been working on justifying that they had a full
set of possible solutions to certain problems. See, for example: Two and Two, Isosceles Triangles, M, M and M
Schools have used the Curriculum Mapping
documents published on the NRICH website to select suitable tasks
which stimulate discussion.
How well are we
doing?
Teachers encouraged students to develop convincing arguments by
challenging them to: 'Convince yourself'' $\rightarrow$ 'Convince a
friend' $\rightarrow$ 'Convince an enemy' as suggested by Mason et
al. in Thinking Mathematically (Mason, Burton, Stacey 1982) One
teacher encouraged discussion of findings by asking students to
write down their solutions anonymously. These solutions were then
used as a focus for review and comment by other students. Some
schools reviewed the way they measured student performance in
lessons. They not only considered how many questions had been
answered correctly but also how students had reached their
solutions and discussed their ideas. Rich tasks from the NRICH
website have been embedded in curriculum documents and shared
informally between colleagues.
'We wanted the students to focus
on the mathematics, leaving the teacher free to focus on the
students.'
Students have been learning to work independently whilst teachers
have been learning to stand back. Teachers and students have begun
to recognise that although it was frustrating when teachers
withheld advice, it was a necessary part of learning to be a
problem-solver.
Teachers have been able to give more of their attention to what
students know, and this offers greater opportunities for
identifying and addressing misconceptions.
'We wanted students to influence
where problems lead.'
During planning, we have tried to anticipate potential areas of
further exploration. We have made use of solutions to problems on
the NRICH site, which often give an insight into multiple routes to
solutions and other areas for exploration. The Teachers' Notes to
the NRICH problems also suggest possible extensions. See, for
example, the Teachers' Notes to Square
Coordinates, Pick's Theorem
How well are we
doing?
Teachers have started to adopt flexible approaches which enable
them to respond to the unexpected, such as students identifying a
novel approach or different 'What if'?' questions.
In the most successful lessons students' questions have influenced
the direction of the lesson.
'We wanted teachers and students
to view the journeys to a solution as a valuable learning
experience.'
We have focussed on listening carefully to students in order to
draw out key ideas that could inform the next steps that students
might choose to take.
We have often acted as if we are solving the problem alongside our
students, sometimes acting as if the problem was new to us and
sometimes discovering new directions on the way.
Our lessons have focussed more on the strategies being used, rather
than just on reaching the answers. See, for example, Twisting and
Turning
How well are we
doing?
Teachers and students show interest in the strategies students
adopt to reach their solutions.
Students and teachers are developing mathematical 'habits of mind'
(Cuoco et al 1996)
References
Ruthven K (1989) An Exploratory Approach to Advanced Mathematics.
Educational Studies in
Mathematics 20: 449-467
Mason J with Burton L and Stacey K (1982) Thinking Mathematically,
Addison Wesley Publishers
Ltd.
Cuoco A , Goldenberg E P, Mark J (1996) Habits of Mind: An
organising Principle for Mathematics Curricula. Journal of Mathematical
Behaviour 15: 375-402
This article is the result of the
collaborative work of:
Susanne Mallett, Steve Wren, Mark
Dawes and colleagues from Comberton Village College Amy Blinco, Brett Haines and
colleagues from Gable Hall School Jenny Everton, Ellen Morgan and
colleagues from Longsands Community College Craig Barton, Debbie Breen,
Geraldine Ellison and colleagues from The Range School Danny Burgess, Jim Stavrou and
colleagues from Sawston Village College Catherine Carre, Fran Watson and
colleagues from Sharnbrook Upper School David Cherry, Chris Hawkins,
Maria Stapenhill-Hunt and colleagues from The Thomas Deacon
Academy Charlie Gilderdale, Alison Kiddle
and Jennifer Piggott from the NRICH Project, Cambridge
For similar articles about teachers using NRICH go
here.