Understanding can be assessed by asking
students to write their own mathematical problems in response to
scenarios or prompts from their teachers.
When to use
Problem writing can be used at any time during
a unit of work. It can cover ideas ranging from surface level knowledge through to deeper understandings.
It can be useful for:
•
assessing knowledge
•
assessing understanding
•
accessing existing ideas at
the start of a unit of work
•
uncovering common
misconceptions
•
stimulating discussion when
used as a group task
•
checking learning and
deciding on next steps during a unit of work
•
reviewing learning at the
end of a unit of work
•
peer assessment, either as students evaluate if the
problemfrom another student fits
the criteria laid down for it, or if another student can solve the problem
The theory
Constructivist theories of learning consider that
students' existing understandings should be considered when developing
teaching and learning programmes.
How the strategy works
When a student writes a problem of their own
it helps uncover what they know, understand, and value in the particular
mathematical topic to which their problem relates. This topic could be
addition, subtraction, multiplication, division, estimation, or one of
many other maths concepts.
What to do
1.
Introduce the students to
the specific area in which they will be asked to write a problem.
2.
Specify the precise kind of problem that the student should
write.
Examples
"Write a word problem for the equation 4 += 13" "Write a word problem for the equation 18 × ½" "Write a word problem that uses the concept of what happens to
anumber when it is
multiplied by 10." "Write a problem that uses the averaging method of
estimation."
3.
Give the students time to write their problems. This may also
require time to conference students or to get them to edit their work.
4.
Get the students to discuss and solve each other's problems.
They should solve the problem or answer the question posed and state
whether the problem or question is appropriate, and why or why not.
5.
Give feedback to students or encourage them to give feedback to
each other. Utilise what is uncovered in further teaching and learning.
A complete cycle of the teacher posing an original problem,
students discussing it, through to them writing and solving each
other's problems is discussed in Estimation exposed (Neill,2005).
Assessing students problems
When interpreting the problems students have written, look for:
•
The
appropriateness
of how the question meets the criteria. Inappropriate questions
may well indicate a lack of understanding. Example: If the
question asks for a word problem that represents 18 ÷ ½, then
"What is half of
18 apples?" (which is 18 × ½, and illustrates a confusion between
division and multiplication).
•
The level of sophistication
of the question. Is it a simple knowledge question, or does it display
deeper understanding? If students write relatively low level problems,
ask them to write more challenging problems. Examples: (In ascending order of sophistication). 1. Write a problem involving multiplying a number by 10.
"What is 10 ×
3?" (a simple knowledge-based response).
"10 people each
have 3 sweets. How many sweets are there altogether?"
"What happens to
any number in the forties when it is multiplied by 10?"
"What happens to
decimal numbers multiplied by 10?"
2. Write a problem about what happens when you add 0 on to numbers.
"What is 57 +
0?"
"What is 38 - 38
+ 57?" "What happens to
any number when you add 0 to it?"
•
Has the writer given
all necessary information that the person needs to solve
their problem?For example, if
the task is to write a problem about calculating the circumference of a
circle, is the diameter of the circle given (or deducible)?
•
Can the student supply a
correct model answer? The resource Seed
and cards (NM1224)
indicates that more able students tend to do this.
•
Is the scenario the student creates posed as a
question rather than as a
statement?
Limitations
•
A student's ability with
written language may interfere with the mathematics being assessed.
This difficulty with language is also seen when students need to write
extended answers to mathematics problems. Potential language issues
within mathematics are indicated in Neill (2000). Allow students to
give their problems orally. Another approach is to incorporate the
writing into a language session where students are reviewing, refining,
conferencing, and editing by themselves, with peers or with the
teacher.
•
Some areas are more suitable
than others for student problem writing. An example of this is
Estimating
in Sport (NM1206) a resource specifically about the
averaging method of estimation.
This would be suitable to assess student understanding about the
averagingmethod because the numbers they select for their problem
indicate their level of understanding of the technique.A resource about
front-end estimation (e.g.,
Estimating
sums of money, NM1202) would be less suitable as any numbers
chosen by the student could use front-end
method.
Adapting the strategy
•
Get students to ask specific
knowledge questions. This may lead on to developing higher level
questions or posing more sophisticated problems.
•
Using
halving and doubling (NM1246): students could be asked to write a problem
where doubling and halving will make the problem easier to solve.
•
This strategy can be used in
many other subject areas, including Science and English.For example, students could design questions
about
comprehension or inference (rather than about retrieving information) based on a text they have read.In Science, students could pose a set
of questions that need to be answered after carrying out an experiment
or practical task.
Examples of ARB resources
that use problem writing
Below is a selection of resources that ask
students to write problems (mainly in Maths).
Barlow,
A. T., & Drake, J. M. (2008) Division by a fraction: Assessing
understanding through problem writing.
Mathematics teaching in the middle schools, 13 (6),
326-332.
•
Harrell, C. P., (2003) Writing in
mathematics: A powerful tool to support math learning. Math counts: Issues that matter.
Macmillan McGraw-Hill. Retrieved 17 March 2008, from www.mhschool.com/math/2003/teacher/teachres/mathissues/pdfs/math_writing.pdf.
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