Analyzing Math Problem Solving Strategies

I blogged awhile back about using anecdotal records in math problem solving.  And I’ve had a few questions about what these strategies look like–especially what differences we see in relational strategies.

So, let’s chat math problem solving strategies: What they look like, what they tell us about kid thinking, and what we can do to help kids in each strategy. #longpostalert

Direct Modeling

A kiddo who is a direct modeler is one who literally models the story problem directly.  They do everything the story problem tells them to do, in the order it tells them to do it.  People outside of the CGI circle might call this “drawing a picture.”  For this post, let’s use this basic addition story problem:

Whitney picked 42 carrots and 29 green beans from her garden.  How many vegetables did she pick altogether?

What does it look like?  Our direct modeler will draw out all 42 carrots (that look like carrots or just look like circles or dots…doesn’t matter).  Then, he will draw out all 19 green beans.  And then count them all up starting at 1 and counting by ones all the way to 71.  These are my babies that are take 30 minutes just to finish one story problem!

What does this tell us about kid thinking?  Direct modelers are telling us that they cannot think beyond the confines of the story problem.  They are not able to see groups of tens.  They are not able to compose numbers into tens and ones (at least without being prompted to do so).  They are not able to conserve a number and count on.  Sometimes, they are able to do these things, but are stuck direct modeling because they think it’s easier, what they are comfortable with, or the numbers are too high for them to display those skills.  For example, a kid who counts on in a 13 + 14 problem, may not in this one because the numbers are much higher.

What can we do to help a direct modeler?  A direct modeler who is still drawing pictures just because it’s easier just needs a simple push–“Show me another way you can solve this problem besides drawing a picture of every single veggie.”  I always find out who my direct modelers by choice kiddos are during Math Talks.  During math talks, these kids see groups of tens, they count by tens, they count on…but they are struggling with or unwilling to make the transition to showing these strategies on paper independently.  Another routine I have in place is making my kids label their counting.  That means if they direct modeled and counted from 1 to 71, they MUST write all numbers from 1 to 71.  While that may seem excessive…it works.  First, it helps with number writing.  But it also gets tiring writing 71 numbers.  When I hear complaints, I simply reply, “Find a faster way then!”

Direct modelers who are stuck in this strategy–and not by choice–need lots of experiences with counting, skip counting, finding groups of tens in larger numbers!  I do this through Math Talks (read about that here) where we can model other strategies whole group, Counting Collections (read about that here) where they can work with partners to count large numbers using groups of tens, and through small group interventions.  During our counting collection or fact fluency partner work days, I pull small groups where we do some guided math interventions…I often pull my direct modelers and work on counting on and finding groups of tens.

Another way I help direct modelers is during share time.  If I share Whitney’s direct model strategy with a direct model by tens strategy after it, I can ask, “So did Whitney have groups of tens in her strategy too?  Can anyone come up and find a group of ten in Whitney’s thinking?”  And then we highlight groups of tens.  After one person has found one easily, I then call on my direct modelers to force them to find tens.  This helps push them past direct modeling!

Direct Modeling By Tens

A kid who direct models by tens is able to solve with base 10 blocks (or unifix cubes if you’ve thrown away your base 10 blocks like I did!)  They either use the tools or draw a picture of tens and ones.

What does it look like? A direct modeler by tens will use tools to build 42 and 29 with tens and ones.  Then, she will count by tens and ones to get the answer.  The important difference between this and a relational thinker is a direct modeler by tens HAS to draw the picture or use the tools first and a relational thinker doesn’t need the picture.

What does this tell us about kid thinking?  A direct modeler is telling us that she is dependent on tools and pictures.  She can count by 10s and one by ones.  She can decompose a 2-digit number into tens and ones, but needs picture/tool support–it cannot all be done in her head.

What can we do to help a direct modeler by tens? It’s important not to rush kids out of this strategy.  The logical next step is relational thinking with base 10 understanding.  That’s a super abstract strategy that takes time with concrete tools and pictures to help solidify.  My focus for kids in this strategy is two fold: notation and flexibility.  I work on getting these babies to flexibly move between direct modeling by 10’s to counting as needed.  The more flexible their thinking, the more they will be thinking about other strategies and begin to stretch their thinking.  I work on notation because that is the bridge to relational thinking in my opinion.  I teach these kiddos arrow notation and going beyond just writing 42+29=71.  Eventually, when they are ready, the pictures will drop and they will realize that the notation is enough and will be able to follow the abstract steps in the notation.  So, just don’t rush this one!


A kid using a counting strategy is able to count on from any number…not just starting at one.  They can start at the smaller or larger number and still be considered a counter, even though starting at the largest number is more efficient.

What does it look like?  A counter will start at 42 and count on 29 more.  She may also start at 29 and count 42 more.  When I model this strategy, I circle the number to show that I got that number in my head.  And yes, I make them write out all of the numbers just as they counted.

What does this tell us about kid thinking?  Counters are telling us that they can conserve numbers (hold a number in their head and count on).  If they count on efficiently, they are telling us that they can find a larger 2 digit number.  They cannot skip count on from any number.  They may or may not be able to see groups of tens.  Some kids can see groups of tens, but prefer to count anyway because they think it’s faster. (And sometimes it is, like in 42+5.)

What can we do to help a counter?  Counters only need our help if they are stuck and unable to use base 10 to solve problems.  If they are not able to solve a problem using a base ten strategy, then I will sometimes pull them with my direct modelers in small groups to talk about finding groups of tens.  Counting Collections will also help build “ten-ness” in these kiddos.  One important thing to remember is that counting is a GREAT foundation for incrementing (a relational thinking strategy), so I am always careful not to push base ten on these kiddos.  If they understand base 10, can find groups of tens, they will be just fine!

Relational Thinking

Relational thinkers do not rely on pictures or tools.  They are able to solve problems mentally or with equations or notation only.  They need to be able to explain or show more than just the equation for the story problem–more on that in a bit!  There are no pictures in their thinking….All abstract, no concrete.  There are 3 different kinds of relational thinkers and each of them look a little differently, but the abstractness of these strategies can make it difficult to differentiate.  Let’s take a closer look at each.

Base 10:  These sweeties were most likely direct modelers by 10’s before and they just began dropping the picture.  However, we can pick them out easily because we can still see tens and ones in their notation.  How they notate their thinking will depend on what you model in your classroom, but here are several options that I’ve seen first graders notate with and without help.

Compensators: These kiddos rarely show up in my first grade classroom.  Maybe it’s because I struggle to think in a compensating way, but every now and then I will hear this strategy come up verbally in a math talk.  These kids want to work with friendly numbers, so they will compensate to make the equation easier.  Instead of doing 42+29, they will change the equation to 41+30 to make the equation friendlier.  They must have a fantastic understanding of equality to use this strategy!

Incrementers:  These babies were most likely counters before.  I often see counters transition into incrementing once they understand base 10 and counting on by 10’s and ones from any number.  Unlike base 10 kids who decompose both numbers, incrementers hold the first number and only decompose the second number.  They count on by 10s and ones (or really any increment…this is just the one I push to help with base 10 understanding) to find the answer.  This looks different than the other strategies because one of the numbers does not change or get decomposed.  And the increments are the same (all 10s and then ones, not 5 more then 3 more, then 10 more, then 2 more…)

What can we do to help relational thinkers?  There may be three different kinds of relational thinkers but I help each of them similarly…My focus for these first graders is focusing on flexibility.  Many relational thinkers already move between strategies easily, but if they don’t I focus on this by asking them to show me more than one way or partnering them up with other students with different strategies–like partnering a base 10 kid with an incrementer to share and try out each other’s strategies.  I also focus on sharpening their notation skills…which takes longer than first grade to perfect! *wink wink* 🙂

Which of these strategies do you see in your classroom and how do you help kids keep their strategies moving?

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