Six Methods for Daily Problem Solving

January 2, 2020

Six methods for daily problem solving:

1. Standard Word Problems

2. Question-less Word Problems

3. Writing a Question to Match a Diagram

4. Selecting a diagram that accurately represents a word problem

5. Problem Solving Ladders

6. Robot Problem Solver

Standard word problems are the most common way that teachers engage students in problem solving.  Routines often begin with students chorally reading a problem, and then attempting to solve it using the Read, Draw, Write process.

It is important for teachers to use this method at least once/week, so that children have opportunities to solve problems independently. However, doing so too regularly can damage student morale, leading to undesirable outcomes.

Although sound in theory, delivering daily word problems has a way of deflating student enthusiasm. Oftentimes, the most precocious students know the answer as soon as they’re finished reading the question, while the weakest students feel defeated by wording complexities, and quit before trying. The rest of the class falls somewhere in the middle, and at most 40-50% of the class is meaningfully engaged.

The best math teachers adopt a coaching mentality to developing critical thinking skills, treating standard word problems as a once or twice/week event, while providing practice opportunities on the other days.

Question-less word problems prepare students to independently solve standard word problems.  They emphasize diagramming and critical thinking skills, while building students’ working stamina.

Compulsiveness and impatience harms many children’s problem solving development. Eager to finish problems, some students skim the several sentences leading up to the question and try to find a quick solution. Other students become confused and intimidated by tricky wording and, without even trying, assume they can’t get answers correct.

1. The teacher projects several sentences of information, but does not include a question. Oftentimes, they select a curricular word problem and hide the last sentence/question.

2. After a choral class reading, the teacher directs students to diagram what they read.

3. Once students have drawn diagrams, they share their creations with a partner and/or write down questions that could be answered using the given information. For every question that can be answered in their drawing, they write a corresponding question mark into their diagram.

4. After students have done this, the teacher models an accurate diagram and projects a problem for the whole-class to solve. Oftentimes, students will find that the question is similar or exactly the same as one they created.

Question-less word problems are inherently differentiated, because the assignment requires drawing and thinking, not calculating and solving. Children organically write questions that align with their skill levels and/or mathematical understandings.

This method also allows teachers to expand or truncate the problem-solving portion of their class period. On days in which they’re strapped for time, they might project several sentences of information. Then, instead of having their students draw diagrams and/or write out questions, they ask them to think of a concluding question and share it with a partner. If they feel students could use more practice diagramming, they might direct individuals or the entire class to represent the information in a different way.

Writing a question to match a diagram focuses students on interpreting models that are integral to their present and future learning. Although diagrams are tools that help students understand and represent information, certain models must be explicitly taught and learned.  This is especially true in the Eureka Math and Singapore Math curricula.

The drawing portion of the Read, Draw, Write process is frequently met with student resistance. Some children are either too impatient to craft diagrams or feel that drawing is a sign of mathematical weakness.

Good mathematicians solve problems mentally, or – at the very least - by writing equations, the falsely reason. Drawing is for little kids, not mathematicians.

This is an understandable conclusion for children who can comfortably solve most word problems they encounter without drawing. Still, not mastering models in early grades can severely limit what students can understand several years later when concepts become more advanced.  Children might not need diagrams to solve problems in lower grades, but if they’re not proficient drawers and interpreters of them, they will be at a disadvantage when they later learn more complex concepts.

Articulating relationships within a diagram centers children on a specific type of mathematical understanding that often goes overlooked. The stimulus of translating a diagram’s meaning into words requires students to study and interpret models that they might otherwise dismiss (see procedure and examples below).

1. The teacher projects a picture or diagram and says, “What word problem does this represent?”

2. Students write or verbalize the word problem independently before sharing with a partner.

3. After re-centering the class, the teacher asks individual students to share their word problems.

4. For each response, the teacher might require students to write a number sentence to solve the problem.

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 problem solving method is especially useful when teachers notice diagramming resistance.  It’s hard to convince young children that becoming excellent model drawers will help them become better mathematicians in several years. Even if they acknowledge their teacher’s forecast, one or two years is a long time to six, seven, and eight year-olds.  Connecting current content to future learning, therefore, feels distant and intangible.

Selecting a diagram that accurately represents 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.

This problem solving method is another tool teachers use to build model comprehension. Like Writing a Question to Match a Diagram, students channel their focus on visual representations without the distraction of calculations (see procedure and examples below).

1. Teachers project a standard word problem.

2. After a choral class reading, teachers project two or more diagrams, one of which accurately represents the word problem.

3. Teachers direct students to study the diagrams and decide which one represents the given word problem.

4. Students discuss their decision with a partner.

5. The teacher asks students to share their conclusions and provide a written or verbal explanation of their choice. As an extension, they might direct students to solve the problem and/or write a word problem to match the incorrect diagram.

Because this problem solving method can last anywhere from two to 12 minutes, it works well on days that teachers are short on time and can’t deliver standard word problems.  Regardless of how long the routine lasts, selecting a diagram that accurately represents a word problem is especially useful in the days following a newly introduced model.  When many students have a shaky understanding of a diagram, this type of multiple-choice activity is more appropriate than the aforementioned problem solving methods.

1. Chorally, the class reads a story statement(s) with fill-in-the-blank questions unexposed.

   Problem Solving Ladders always begin with one or more story statements but no question. This is very intentional. Advanced students often resist drawing diagrams to support their work and it’s hard to convince them to draw diagrams when they’re able to solve without doing so. Still, modeling is an essential skill for all students to learn and hone. By providing statements without questions, children can either spend several minutes drawing diagrams or refuse to work.  The latter is a disciplinary problem, not a mathematical one.

2. Students spend several minutes diagramming the story statement(s) that they read.

   During this time, the teacher probes the room prompting struggling students with questions that might help them get started. They also look for different ways that children correctly diagrammed the problems. When students finish early, teachers have them share their diagrams with others who have finished, but recorded the information differently. This often stimulates rich math discussions. Teachers also direct early finishers to write down questions that could be answered using their diagrams. This helps students see models more elastically, i.e. tools that can be used to solve more than one question.

3. The class rereads the story statement(s) and the teacher demonstrates a process for diagramming the problem.

   To become proficient modelers, students must regularly observe experts diagram word problems.   This provides insight into the latter’s thought processes, and helps the former learn different methods for representing mathematical information.  If most or all of the class creates similar drawings, the teacher sketches an alternative diagram. If they notice a unique and effective student model, they have the individual present their work to the class.

4. Reveal fill-in-the=blank questions and have students work for several minutes.

   Letter a is simple enough that every child answers it correctly, and is often below grade level. Confidence is essential to excelling in elementary mathematics and, too often, weaker students consider word problems to be something they “can’t do”. Consistently answering them correctly helps reverse this mentality.

   Just as struggling students need success to gain confidence, advanced students need challenges to stay interested in the subject.  When children feel under-challenged, they often become bored and their morale suffers. The last problem is usually above grade level, and solving it requires most (or all) children to draw a new diagram. The middle problems scaffold or ladder in difficulty.

   Students are not expected to answer questions in complete sentences, because a disproportionate amount of the activity would be spent writing. However, for each problem they solve, children are expected to write an equation or inequality to demonstrate how they arrived at their answer.

5. Review answers.

   The middle problems are usually easy for some students but challenging for others. As a result, few children complete Problem Solving in their entirety. The tool is designed to give all students an opportunity to reach their personal best during an allotted time period (9-12 minutes). Regularly accomplishing this leads to realized potential, the goal of teaching and learning.

For more on Problem Solving Ladders, click on the button below:

Robot Problem Solver (RPS) engages students in a communal activity in which classmates practice articulating their diagramming thought processes and/or demonstrating their model comprehension.

There are several ways that RPS can be delivered but each involves students following a set of directions to create a drawing (see example below).

The student is free to choose whatever story problem context they want, but a correct response must match the diagram - in this case, a part-whole relationship between girls and boys that make up 2,317 students. Either of the following word problems would be correct:

• There are 2,317 students in a school. 789 of them are boys. How many girls are in the school?

• Some girls, but no boys are sitting in the school auditorium. After 789 boys enter, there are 2,317 students in the auditorium altogether. How many girls were sitting in the auditorium at first?

There are three ways that Robot Problem Solver is usually delivered:

1. Teacher provides oral instructions and all students draw.

2. Students work in pairs with different problems that they have accurately diagrammed.  They take turns instructing each other how to draw their diagrams.

3. One or more student volunteer leaves the classroom. The teacher provides the remaining children with a diagram and instructions. The volunteer(s) return to the room and receive instructions from classmates before trying to express the corresponding word problem.  

(1) In Daniel Kahneman’s Thinking Fast and Slow, he cites a phenomenon called Ego Depletion.  When a brain is too cognitively busy, people will give up earlier than normal when faced with a difficult task. Back to Text.

(2) Working independently through each stage can be challenging for young students, especially if they can’t read.  Kindergarten teachers modify the routine by projecting pictures and asking a series questions that gradually grow in complexity.  First grade teachers often read problems slowly and tell their students to make math drawings of what they hear.  They then deliver questions orally and encourage their students to solve using their diagrams. Back to Text.