Infusing Dynamism into Daily Problem Solving
As a sixth grade math teacher, creating dynamic problem-solving sessions was always challenging for me. The strongest students often arrived at answers soon after they finished reading the question, while the weakest students felt defeated by wording complexities and would quit before trying. The other children fell somewhere in the middle, and I always felt like I was only engaging30 - 35% of my class.
When I transitioned from teaching to Math coaching, I quickly found that similar challenges persisted in most classrooms, regardless of the school or grade level. The more I observed teachers and demonstrated lessons, the more I encountered the following:
A few students felt under-challenged by the word problems presented to them.
A few students felt over-challenged by the same word problems that under-challenged some of their classmates.
When approaching a word problem, most students resisted drawing diagrams, opting instead to solve mentally &/or scribbling down equations.
Few students spent significant time studying diagrams and their relationship to word problems.
Students spent very little time discussing their thinking during problem-solving sessions.
Teachers felt that they didn’t have enough class time to deliver daily problem-solving routines.
These observations led me to spend the past decade creating methods and tools to infuse dynamism into daily problem-solving routines, culminating in this book.
Method 1: Standard Word Problems
Solving word problems independently is one of the most common metrics to assess elementary-aged children’s mathematical competency. Teachers, therefore, are understandably inclined to provide daily problem-solving sessions even if the curriculum they’re teaching doesn’t call for it. Routines often begin with students chorally reading a problem that consists of a few sentences and a question, before the teacher directs the class to solve it using the Read, Draw, Write process. To learn more about the RDW process, see The Read, Draw, Write Process in the Additional Reading section.
Each of the 152 word problems in this book consists of two diagrams – one that accurately depicts it and one that does not. A teacher can deliver any or all of them as standard word problems by simply hiding the diagrams and only projecting the words.
The correct answer and diagram can be referenced in individual answer keys located on the opposite page.
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Although word problem practice is vital to producing strong elementary math students, maintaining its dynamism can be challenging.
I’ve found that when daily problem-solving time only consists of standard word problems, the routines become stale and student interest fades. Therefore, for students to remain interested and progress, they need to experience problem solving in different ways.
I recommend that teachers adopt a coaching mentality to problem-solving, treating standard word problems as a once or twice/week event, while providing practice opportunities on the other days. Methods two through six are practice options that prepare students for standard word problems.
Method 2: Question-less Word Problems
Math diagramming is an essential skill for students to practice because it creates a common model for class discussions and helps them solve more challenging problems in later units and/or school years.
Compulsiveness and impatience harm many children’s problem-solving development. Eager to finish, some students skim the several sentences leading up to the question and try to find a quick solution. Others become confused and intimidated by tricky wording and, without even trying, assume they can’t get answers correct. Those who can arrive at accurate answers without drawing, are understandably unmotivated to diagram information that they already understand abstractly, and it
becomes hard to convince them that developing strong diagramming skills will help them in later years.
Question-less word problems are problem-solving activities that place their entire emphasis on representing and analyzing information. This creates space for students to practice diagramming without the distraction of a question they are expected to answer, in the process preparing them to independently solve more complex problems. There is no singular method for delivering question-less word problems, but the following procedure often produces dynamic results.
Students chorally read a few sentences of information that does not include a question.
The teacher directs students to diagram what they read.
As students draw diagrams, the teacher probes the room, prompting individuals to do one or more of the following:
Label all parts of the diagram.
Position a question mark in the diagram.
Share and explain drawings with a classmate.
Fill in parts of the diagram that aren’t given in the information provided.
Draw a second diagram to represent the same information.
Create a list of questions that could be answered with the given information.
After a few minutes, the teacher demonstrates drawing a diagram to match the question-less problem. To stimulate flexible thinking, they might create a model that differs from what most students drew.
Students share questions that they wrote, and the teacher chooses one for all students to solve.
This problem-solving method is inherently differentiated, because the assignment requires drawing and thinking, not calculating and solving. There is no right or wrong way to diagram one’s thinking, and children organically write questions that align with their skill levels and/or mathematical understandings.
By hiding the question and corresponding diagrams, all 152 problems in this book can be used as a question-less word problem.
Method 3: Create a Question to Match a Diagram
It’s hard to convince young children that becoming excellent model drawers will help them become better mathematicians in several years. Connecting current content to future learning feels distant, intangible, and irrelevant to seven-, eight-, and nine-year-olds. Writing a question to match a diagram is an activity aimed to counter this resistance, while focusing students on interpreting models that are integral to their present and future learning.
Articulating relationships within a diagram centers children on a specific type of mathematical understanding that often goes overlooked. Translating a diagram’s meaning into words requires students to study and interpret models that they might otherwise dismiss. Both accurate and inaccurate student responses provide rich learning opportunities because thinking and reasoning about the diagram is essential to proving or disproving a formulated question.
This book consists of 304 diagrams, most of which can be used as a question-writing prompt. Accurate responses can be found in the answer keys.
Method 4: Selecting a Diagram that Accurately Represents a Word Problem
Matching a diagram to a word problem requires careful reading and model analysis. Isolating and then synthesizing these vital problem-solving skills helps students develop a stronger understanding of language/diagram interconnectivity. It can also stimulate ideas for different ways to model problems.
This problem-solving method is another way that students can build diagram comprehension. Like Writing a Question to Match a Diagram, students channel their focus on visual representations without the distraction of calculations. Because this problem-solving method can last anywhere from two to 12 minutes, it works well on days when teachers are short on time and can’t deliver standard word problems.
As mentioned above, each of the 152 problems in this book consists of an accurate and inaccurate model representation. The latter often possesses a “distractor” that will lead students to incorrectly select it if they don’t read carefully. The teacher who is pressed for time might conclude the activity after a choral reading, short discussion, and explanation. The corresponding answer key provides the following prompts to extend the activity:
Solve the given word problem.
Why doesn’t the incorrect diagram accurately represent the word problem?
How could you change the word problem to make the incorrect diagram correct?
The second and third bullets are each followed by a possible student response written in italics.
Method 5: Problem Solving Ladders
Problem Solving Ladders are a confidence-building tool that engages and challenges an entire class, regardless of its achievement gap. The tool is designed to give all students an opportunity to reach their personal best during an eight to 12-minute problem-solving session. They incorporate aspects of each of the preceding problem-solving methods through a simple five-step routine.
Chorally, the class reads a story statement with fill-in-the-blank questions unexposed.
Students spend several minutes diagramming the story statement that they read.
The class rereads the story statement, and the teacher demonstrates a method for diagramming the problem.
The teacher reveals fill-in-the-blank questions and students work for several minutes. Letter A is often below grade level and always simple enough for every child to confidently answer it correctly. Letter D is usually above grade level and requires most students to draw a new diagram to solve it. The middle problems scaffold or ladder in difficulty.
The teacher reviews one or more of the answers.