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Math Instruction, Learning and the Model

A strategy for Problem Solving,

  • The importance of labeling,

  • predicting a reasonable solution and 

  • freedom to use a calculator!

After a  year of using Judith Gould’s Four-Square Writing strategies, I started to connect the scientific model to the 4 square and the problem solving “method.”

Step 1 Determine what the problem is asking.

Step 2 Gather the facts relevant to that problem.

Step 3 Use the facts to try some strategies.

Step 4 Make sure your solution solves

the problem.


Please note:  If you search "4 square problem solving"  You will see some people use different

versions of the same idea.  You can see that understanding how to teach students to break the

problem-solving process is:

1) improved over the last 15 years

2) Still being invented in classrooms after classroom across the country.  

3) being taught to many, many children without certainty of effectiveness.

Yet,  this is a start, folks.

Some problems with a more linear model:

Linear can feel overwhelming to students with developing executive functions because it appears long.

The ‘problem’ is far away from the solution which does not encourage the connection between what the problem is asking and what the final solution.

The good thing about the 4 square model is that the solution is touching each of the other quadrants.


However, this model has a LOT of holes.  And if we alter the ideas just a little, then we can use this model to sort out so many types of authentic concerns that people encounter from math to friendships.


Initially, I offer these steps as an alternative.

Step 1.  Consider what the problem could be.

Step 2  Predict a reasonable solution.  (even going to far as to write the prediction in that bottom right quadrant.)

Step 3 Gather the facts that are relevant to that problem.

Step 4 Try some strategies and continue to work until the answer makes sense.

Step 5  Compare your solution to the one you predicted.  Reflect. If the solution is different, what did you learn along the way?


I implore focusing more instruction

  • Step 2 (below)

  • Labeling

  • calculator use while teaching problem-solving process.


Alter Step 2

Step 1.  Consider what the problem could be.

Step 2  Predict a reasonable solution.  (even going to far as to write the prediction in that bottom right quadrant.)


Seriously.  Predicting a reasonable solution. .changes the game.  It really requires a person to refer back to the problem. To ensure the context is fully understood.  It also begins to frame the steps one might need to take to solve it. Predicting a reasonable solution, also helps a problem solver filter through the facts to extract those most relevant.


When novice problem solvers gather information, initially,

they bring all of the facts.  Even those that are irrelevant.  

If the novice perceives that the solution will include complex computation, their brains default to either the simplest operation, or the one that results in any number that is not overwhelming.


Additionally, it takes a great deal of number sense to predict a reasonable outcome.  Will the final solution represent a larger quantity?  Or does the context offer a reason for a smaller solution?  What types of events cause items to increase in quantity?  What causes a decrease?  What would a reasonable number of hamburgers a person could eat in a day? What about the number of hamburgers a whole class of students could eat?

You can see the depth of knowledge we are already priming the brain for when we predict a reasonable outcome before we dig into the problem and our background knowledge for relevant facts.


Specific Labeling

Spend much time improving and encouraging specific labeling of numbers.  

It is imperative that the numbers are labeled. 

Labeling requires the ‘language’ part of the brain to participate. The ‘math brain’ is powerful. It just wants to play with numbers. It does not care what the numbers mean.  So, a student can enter numbers into a calculator and use any operation the math brain desires, often leading to inaccuracy and always reduces the chances the student is successful.

Instead, invite the language side of the brain.  Label each number. No number can have the same label.


The specific labeling increases the chances the student will choose the appropriate operation, because the numbers have context.


You can infer what the problem was asking by looking at the example with specific labels.

In fact, this type of labeling suffices for any ‘explain how you got your answer’ prompt.

Many teachers have a misconception that students have to write out a narrative of the process. You can see by the example on the far right, that even letters or pictures could suffice.  Specific labeling is more than adequate AND helps the learner make good decisions. It is proven that we make better decisions when there disfluency in a process. One reason that when you GIVE someone digested data, they to not get as much out of it as the person who initially process the data to share it with others.


That is not to say students WANT to label the numbers.  The math brain really does NOT want to work with the language side.  The math side, like order, binary, black and white. We all know that with language things begin to get more complex. You have to teach the students that it is normal not to want to label.  It takes longer.


Additionally, I do not let them erase or scribble out attempts that did not have the desired result.   They may draw one line through it.  It is important for problem solvers to learn to value their attempts.  And that there is value in their first attempts.  Because important information learned in the first attempts  may provide useful later.  AND as a teacher, I'd like to see the progression of their attempts. so I can guide them toward efficiency and away from misconceptions.


Calculator use

I advocate for teaching novice problem solvers how to use a calculator in order to scaffold their learning.

Problem Solving time is not fluency time or algorithm instruction time.


Calculator use leads to better decision making.   When a student  is learning to manage a great deal of information, calculators reduce the cognitive load.  It ensures greater accuracy which increases the likelihood that the student will have success.


It is not ‘easier’ to use the calculator.  Calculator use is a skill in and of itself.


Additionally, I require the students to write the numbers and operation and  specific labels down first.  Students do not typically want to write the numbers down until AFTER they use the calculator.  That leads to less impressive outcomes.

They do not want to write the numbers down FIRST because, they may not be sure their attempt will be 'right' the first time.  So they feel like it is a waste of time.  They want to see what the calculator shows before they commit. Because they are still building their understanding of what operations actually do to the value of the solution.


To increase the rigor and true mastery of concepts, focus the MOST on Predicting a reasonable outcome.  Make certain the engage language an meaning to the numbers by specific labeling.  And with this, teach them to use a calculator.


{more to come}

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