You have good materials. Students have unpacked the question and know what they are looking for. Students freeze.

Now what?

Most problem-solving materials will tell you to use the four steps:

- Find out what the problem is about.
- Choose a strategy.
- Solve it.
- Look back.

I suspect the people who ‘invented’ this process were rock star problem solvers in school. Then there are the rest of us…

So what is wrong with the process? How can we help students think like mathematicians and be less intimidated by the process of problem solving?

**Don’t ask students to ‘Choose a Strategy’.**

The ‘Choose a strategy’ step of the problem-solving process implies that there is one, correct strategy – students just need to find it. The ‘you-need-to-choose-the-correct-strategy’ is especially logical to children when you consider that, in other strands of mathematics, students find the *one right way* to get the *one right* answer. In students’ minds ‘picking’ may be something like drawing straws. They hope they pick the right one.

Often, multiple strategies can (and should!) be used. A student might…

- draw a diagram of the initial story,
- put the numbers into the diagram, and
- logically discover the missing piece(s).

In another problem, students might…

- draw a diagram of the initial story,
- label and organize numbers,
- find patterns in the table, and
- logically decide how the pattern in the table would continue.

In each of these scenarios, students are using multiple strategies. No need to pick or name the strategies.

If you want students to name things, have them name the math vocabulary words that match the particular problem. *Larger conceptual understandings form when students connect the problem with the concepts and vocabulary of other mathematical strands*.

**‘Solve it!’ discourages experimentation.**

‘Solve it.’ What do you think when you hear those words? How do those words compare to ‘Play with the numbers” or “Find patterns”?

For me, ‘Solve it’ translates into ‘Just get there already…’

The problem becomes all about the destination.

*The process should be as important as the product. Students understand this better when they are allowed or encouraged to play with numbers and number patterns.*

**The Abercrombie Process for Problem Solving (or APPS):**

- Find out what the problem is about.
- Label the numbers.
- Find, describe, and continue any patterns you see.
- Relate the patterns or ideas to previous mathematics lessons.
- Look back at the exact question to locate the answer.
- State the answer in a complete sentence.

*Find out what the problem is about.* See the second part of this series.

*Label the numbers.*

Part of finding patterns is labeling the numbers. Numbers are *adjectives*. They never stand alone. Replay scenes of The Count from Sesame Street (if you have to) to reinforce this point.

*Find, describe, and continue any patterns you see.*

Mathematicians don’t just compute. They find patterns. They continue patterns. They make predictions based on those patterns.

Mathematicians look at number patterns like artists look at sunsets.

Allow students to play with the numbers. Encourage them to label their numbers on post-its and move them around the desk. Perhaps students want to move labeled numbers around on an Interactive White Board or on Wallwisher.

Help students be specific when they describe the patterns they see. Instead of saying “This column is ‘plus two'”, teach them to identify the numbers in a column as being two more than numbers in another column. What if this pattern continued? Can you predict what would happen if the table continued to…?

This is the place where students begin to connect the problem to other mathematical concepts. Are the numbers in one column all even? square?

*Relate the patterns or ideas to previous mathematics lessons.*

The NCTM process standards now include *Connections*. Student should connect the problem to mathematical ideas they have previously encountered.

While the idea of making connections to past lessons seems obvious to us, the idea is not obvious to most elementary students.

Create a classroom mantra: “Never forget a math lesson.” Let students look back at previous work to find the rules of divisibility or the formula for circumference if the concepts relate to the larger problem.

Do the number patterns remind you of other mathematics lessons? Do the patterns result in square numbers? Do rules of divisibility apply? Area and perimeter? Rectangular arrays?

*Look back at the question to locate the answer.*

How does the actual question relate to the patterns you have found? Previous math lessons? Make it like a treasure map. The answer is somewhere in the pattern…but where?

*State the answer in a complete sentence.*

Not only do students need to locate the answer, they need to see whether or not the answer they have located actually answers the question. They can do that by putting the answer in a complete sentence.

Notice that the complete sentence is the *only** *complete sentence I ask them to write. Do students really need to write, “First I…Then I…Next I…” to effectively communicate their thinking?

The next post will show you why such sentences are unnecessary. It will also include a newly-created problem-solving rubric you can use for assessment.

*To be one of the first to receive this rubric, sign up to receive Expat Educator by email. Also, please consider passing this post on to colleagues or friends.*

photo credit: Old Shoe Woman via photopin cc

Other posts in this series:

- Classroom procedures that help students explore and construct problem-solving strategies.
- Ways to make sure your low readers and second language students are not at a disadvantage.
- Ways to merge mainstream mathematics textbooks with problem solving resources.

Mrs. Moeller-Abercrombie,

Great blog. I’m following you on twitter but would love it you put a RSS subscription button so I can add your posts to my google reader.

You bring up a very good point about strategies. Why must they pick just one? I teach students how to utilize strategies early in the school year, but we shouldn’t teach students to fall in love with just one.

Mathematical reasoning is making headway, and thank god for that. I look forward to learning from you in the future.

Gary Johnston

Thanks, Gary for the comment. I have added the RSS feed button to the sidebar. Let me know how it works.

I like what you say about helping student to not get too attached to any one strategy. Pedagogy associated with mathematical reasoning has been a tough thing for me to learn. Since I didn’t learn reasoning in math when I was a student, it’s been difficult for me to create a classroom environment that includes mathematical reasoning.

Please let me know how you facilitate it in your classroom – I’d like to learn from you too! :)

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