Problem Solving
COMMON CORE
MATHEMATIC PRACTICES
Michael Fitchett
Make sense of problems and persevere in
solving them.
Just Think Mathematics
Reason abstractly and quantitatively.
Construct viable arguments and critique
the reasoning of others.
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Rationale
“Just Right” Problems
Launch, Explore, Summarize Format
Designing Problems
Differentiation
 Extensions
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in
repeated reasoning.
Rationale
Without the application of mathematics to solve real-world problems, mathematics would be a
useless endeavor. Traditionally, math curriculums have emphasized “ procedural skills” that
key on computation alone. Computational fluency is only one part. Without problem solving,
computation would be without a purpose. The most mathematically powerful part of the
school day is the Problem Solving session.
Most, if not all, important mathematical concepts and procedures can best be taught
through problem solving. That is, tasks or problems can and should be posed that
engage students in thinking about and developing the important mathematics they
need to know.
Van de Walle (2001, p. 40)
The value of teaching mathematics through problem solving include:
 Problem solving places the focus of the students’ attention on sense making.
 Problem solving develops “mathematical power.” Students solving problems will be
engaged in all eight outcomes of the Common Core Math Practices.
 Problem solving provides ongoing assessment data that can be used to make instructional
decisions. This also provides an excellent way of communicating to parents about a
child’s progress.
 It is a lot of fun! Teachers who experience teaching in this manner tend to never return to
a teach-by-telling mode. The excitement of students’ developing understanding through
their own reasoning is worth all the effort.
“Just Right” Problems
It is important that the word problems given do not have a prescribed or memorized rule or
method for solving them. The level of complexity must begin at a student’s instructional level.
Problems should have appropriate ideas to engage and solve and still be challenging and
interesting. Word problems given should require justifications and explanations for the answers
and methods used to find them. The word problems can be view as the “vehicle” for teaching the
math content.
In literacy, children need the “Just Right” book to build their reading power. A book that is not too
easy for them and not too difficult. (Vgotsky +1) For this to happen, you will need to know where
your students are with regards to comprehension and fluency. Much like a “Just Right” book,
having “Just Right” problems are important to meet the students where they are currently.
Therefor, it is important to think and plan for the math content level currently and your plan for
where you want your students to grow.
Launch, Explore, Summarize Format
If we are to give children an opportunity to think and persevere through tasks, it is important to
make sure the majority of problem solving sessions is spent on students solving the problems!
Launch:
Launching the problem is simply providing the word problem and the materials necessary to
work. This may include sticker labels (1 inch by 2 inches , Avery 1561) to be put in a page of their
hard back composition books. Launching the problem may also include reading the problem for
your non-readers or having your struggling readers read the problem to you with your guidance.
Be careful not to show or explain how to do the problem. You must be careful not to take an
engaging, cognitive task and “whittle” it down to the point where the students are not having to
think about how to solve at all.
There will be times when you may need to clarify or redirect the learning after you have launched
the problem. If you notice that the majority of students have totally missed the mark by
misreading the problem or the misunderstood the logic in the problem (adding two numbers in
problem verses multiplying them), then stop the lesson and redirect by asking questions
regarding the problem-not telling them! Have them go back in and seek for understanding in the
problem.
Explore:
The majority of time during problem solving should be the students “grappling”, thinking,
reasoning, organizing, conversing, and explaining. Since “meaning making” is at the core of
problem solving, it will be important for students to solve the problems the way they
understand them. It is starting where the students are and gradually “moving/growing” them
to solve more problems with larger numbers using more sophisticated and efficient strategies.
During the explore, the teacher “walks the room” watching carefully for common methods of
solving at the same time conferring with students who need a little direction. As the teacher
assesses the progress, it will be important to be thinking about what will need to be
summarized during the summarize portion of the problem solving session. Was the problem
too easy, too difficult? Was the logic in the problem too clear, too vague? If the problem was
too easy, it may not be necessary to go over it in a whole group setting. If the problem was too
difficult, it may be necessary to adjust the level the next day rather than to push for
understanding. The explore portion of this session is a great opportunity to use formative
assessment to plan for the next day or days word problems
Summarize
The actual “teaching” occurs during the summarize portion of the problem solving session.
Here, the teacher has had a chance to observe how students solved the problems and now has
to make some decisions as to what to teach. This could be reviewing the problems where
students had difficulty. This also could be taking the students to the next level of sophistication
regarding a particular strategy. Many times the summarize portion may lead to a connection
or, re-launch for tomorrow’s problems. There may be times when a teacher need not go over
the problems because the majority of students were successful solving the problems
independently. Sometimes, going over one of the bonus problems (see extensions) is
appropriate because the operation used (addition, subtraction, multiplication, division) is the
same as the word problem but number size in the bonus problem is larger and thus the teacher
can extend the learning using larger numbers.
A word about Math Strategies
Perhaps the biggest teacher decision is deciding which strategy to emphasize when solving
the math computation in the problem. These are the “building blocks” to developing
mathematical power. Movement through the strategies enables students to solve word
problems with meaning. Students become more efficient in the more complex strategies as
their confidence and computational skills increase. When using the operation of addition,
strategies may include the most simple strategy such as counting all when adding two
numbers, to a more complex strategy such as composing and decomposing two-digit
numbers mentally to solve.
Example: 28 + 57
“20 and 50 is 70. I know 8 plus 7 is 15 because I can make the 8 a 10 by giving it 2 from
the 7. That would then be 10 plus 5 which is 15. So, 70 plus 15 is 95.”)
It is important to understand that math strategies are not grade-level specific. Instead, they
are progressions that move through the grade levels depending on the needs of the learners.
The Common Core Standards do call out some of these strategies to guide grade-level goals.
However, it will be necessary to be responsive to all levels of learners in your class.
Designing Problems
Sarah had 5 cheetos. Jillian had
6 cheetos. How many Cheetos
did they have in all?
(4, 5) (6, 7) (19, 6) (28, 57)
Traditionally, word problems were always found at the bottom of the math book page. Many
times these problems were skipped, or given to those students who finished early. Often, they
were problems that students had no real world connection-just another generic problem to
complete. It is ironic that the one “thinking” component of math work was put last.
Looking at the Common Core Mathematical Practices, it is interesting to notice that in order
to: make sense of problems and persevere in solving them, reason abstractly and
quantitatively, and, construct viable arguments and critique the reasonableness of others,
students must have the kinds of tasks to make this possible!
Creating word problems that “connect” to your class may seem arduous at first but when you
begin to look at the sequence of math standards and the “steps” that children go through to
understand number, it becomes clearer as the year goes along.
When designing word problems, the following factors need to be considered:
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Mathematical Operations
Number Size
Logic In the Problem (Is the action clear or more complex?)
Problem Types (Where is the unknown part of the problem located?)
Level of “Real-World” Connection and Novelty
Mathematical Operation (addition, subtraction, division, multiplication)
Number Size
One of the best way to differentiate for the learners in your classroom is choosing the size of
numbers you will use in your word problems. Looking at the word problem above, the different
colored numbers can serve as “entry points” to the word problem. Your struggling learners will
do the problems that are in their comfort zone. They also may attempt the higher numbers but
may use a more basic method to solve such as “counting on” verses a using a doubles strategy.
Thus, number size may determine which strategy to use when solving.
Logic and Action
Logic in a word problem is perhaps the biggest area of need for young learners. It seems that
understanding the logic in word problems has been reduced by merely looking at the two
numbers in a problem and reading the last sentence looking for a clue as to the operation one
might use. Developing a student’s logical “sense” is more than looking at the “catch phrases”
at the end of a word problem like, “How many altogether?” Or, “How many left over.?”
Students need to move from problems that are concrete to problems that are more abstract.
Most problems that we, as adults solve, are not necessarily problems that involve straight
forward, one calculation. Teachers need to start with problems where the “action” (what is
being asked) is very simple and clear. Over the course of the school, year move to non-routine
problems. Example:
Lucy had 14 balloons. 5 balloons
flew away. How many does she
have now?
(14, 8) (16, 7) (15, 6) (22, 8)
Routine Problem
Lucy had 14 balloons and Ben had 8
balloons. Lucy wanted to share hers so
that they would both have the same. How
many balloons would each person get?
(14, 8) (16, 7) (15, 6) (22, 8)
Non-Routine
(14,
8) (16, 7) (15,Problem
6) (22, 8)
Problem Types
In order for students to know what is being asked in a word problem, it is important to give
students a variety of problems so that they are constantly looking for meaning. Many times,
word problems are taught in a unit, such as subtraction where all the problems of course are
subtraction problems. Instead, students need to be exposed to not only all of the operations
(addition, subtraction, multiplication, division), but also problems where the “unknown” (the
answer) is located in different parts of the problem. For example, look at the following
problem:
Result Unknown
Change Unknown
Lucy had 14 balloons. 5
balloons flew away. How many
does she have now?
Lucy had 14 balloons. Some
flew away. Now she has 5. How
many balloons flew away?
(14, 8) (16, 7) (15, 6) (22,
8)
(14, 8) (16, 7) (15, 6) (22, 8)
Start Unknown
Lucy had some balloons. 5
balloons flew away. Now she
has 9. How many balloons did
she have in the beginning?
(6, 8) (9, 7) (9, 6) (16, 8)
In each one of these problems, the unknown part is in different parts of the problem. This
forces students to read and perhaps model to make sense of what is going on in the problem.
Having the unknown in different parts of the problem also may allow students to understand
the inverse relationship between addition and subtraction.
(Notice that in the second problem, students may see that to find the difference between 14 and 5, one
might “add up” starting at 5 to reach 14 instead of subtract.)
Novelty/ Real World Connection
Most of us remember a version of this word problem from our math books: problem:
If the train left the station at 12:00 travelling at a speed of 45 miles an hour, what time will
the train…
Problems like these, although the math content embedded in them was important, probably
had left many of us with a dislike of word problems. They were mundane and most of us
looked at problem solving as a “must do.” Many of the word problems in textbooks do not
connect to our lives in any way.
Students are motivated by solving problems that are connected directly to their lives at that
particular point in their lives. Providing a direct “real world” connection in a word problem
will motivate more students to want to solve the problem.
Here is an example of an actual true story that was turned into a word problem.
I started a 3rd grade year where I taught in a portable. Skunks had somehow gotten under
foundation of our classroom.
There were 4 skunk families living
under our classroom. If each family
had 4 babies how many baby skunks
would that be in all?
(4, 5) (4, 3) (5, 2) (5, 5)
Other examples of making your problems real world:
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Putting the names of your students in the problem.
Connect to something that has occurred in class or at school
Connect from a read-a-loud that was done
Connect to science or social studies content
Connect to holidays
Put spelling words or vocabulary words in the word problems
Connect to students grade level/age (Example: losing teeth in 1st grade)
Differentiation
Even children in primary grades are smart enough to know how they “stack up” against the
other children in the class. This becomes obvious many times because the teacher gives a
different set of problems to struggling students and a different task for higher students. With
problem solving with the sticker labels, all children can have access to the same problem. This
type of task differentiates for students whose number sense may not be as high. It also
differentiates for those students who need extensions into problems with larger numbers.
Number size in the problem gradually changes to meet the needs of all learners in the
classroom. (see sticker label below) Experience has shown that as number sense grows, most
students will eventually solve all the problems though the strategy may change as the numbers
get higher.
There were 2 dragons. Each
dragon had 2 heads. How many
eyes are there altogether?
(2, 3) (2, 4) (3, 4) (3, 6)
Extensions
As the school year progresses, more and more students will
become proficient in solving the sticker labels, it is important to extend their learning into not
only more challenging problems, but problems that vary in operation, and math content
(fractions, measurement, geometry)
I provide the following extensions daily in my classroom:
Bonus Problem
These problems are humorous, silly and usually concern my dog (He has a different career every
day) The operation is usually different than the sticker label for that day. These problems are
more challenging than the sticker problem, but can be solved different ways using strategies that
are simple, or more complex.
Bonus, Bonus Problem
This problem is connected to the Bonus Problem. It has the same context but may incorporate a
different operation.
“Oh Uh” Problem
As students move through the sticker problems, and both the Bonus Problems, it will be
important to continue to challenge those students with problems that are not in the realm of
straight arithmetic. The name “Uh Oh” problem is a term I came up with my first grade class. It
was meant to motivate those who think may want to take on something a little different.
“Oh No You Didn’t “ Problem
You will eventually have those students finish all sticker problems, bonus problems and even the
‘Uh Oh” problem. These are the students who really want to show what they know. The, “Oh No
You Didn’t” problem incorporates the math content we are currently working plus new concepts
that these students can grasp and grapple with. These types of problems are usually concrete
problems usually in the form of equations that need to be solved, or determined to be true.
Example: (March, 1st grade)
Math concepts students know to solve this problem:
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= sign means “the same amount as”
knows “jump up method” for subtraction
knows to solve math in parenthesis first
New math concepts students are learning:
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Knows that x means “groups of”