Strengthen Mathematical Understanding in 4 Easy Steps

Mathematical understanding means that students understand the story problem and follow the problem-solving steps. Understanding story problems can be very challenging because it requires multi-step, higher-level thinking processes. Students are required to process several pieces of information before starting any mathematical operations. If students do not understand the problem, they will solve it incorrectly, even if they have a strong repertoire of strategies.  Strengthen your students’ mathematical understanding with these 4 easy steps:

1. Scaffold instruction: Mathematical understanding includes many steps: identify question; identify key information; get rid of erroneous information; determine the operation; solve using an appropriate, efficient strategy. Since mathematical understanding involves so many steps, teachers should teach each step explicitly and introduce the next step after proficiency is demonstrated. This allows the teacher to isolate individual steps first and then gradually integrate the steps together.

2. Incorporate multi-sensory activities: Allow students to act out the problem and use manipulatives to help students build understanding. In my classroom, students use Upton Understanding Fish to help them complete the problem-solving steps. I purchased a inexpensive, yellow Webkinz fish from Ebay to use as our classroom Upton.

Reading with Upton

During direct instruction, I model each of the problem-solving steps, thinking aloud as I go. I hold Upton right beside me and will often talk to him during my think-alouds. The students think it is funny, but it keeps them engaged. During guided practice, I toss Upton to different students, asking them to help me complete one of the problem-solving steps. Students also take turns using Upton to assist during independent problem-solving time.

Reading with Upton 2

The stuffed Upton fish has become a pivotal piece of my problem-solving instruction because it helps kids feel safe to take risks and discuss problem-solving steps. In fact, a few of my students have even asked their parents for their own Upton as a birthday or Christmas gift.

3. Use discussion questions and prompts:Students are more successful at solving math problems when they monitor and reflect upon their thinking and problem-solving steps as they work through problems. We often assume that students know how to thoroughly discuss their mathematical thinking and problem-solving steps, but like an other concept or skill, this must be taught in depth. Teachers must model the self-questioning process and provide multiple opportunities and support for students to practice it until they can use the questioning strategies independently.

I use Upton’s self-reflection questions and peer discussion prompts to facilitate mathematical discussion. I introduce one prompt or question at a time and add additional prompts in subsequent lessons.

MathProblemSolvingSelfReflectionandPeerDiscussionQuestions.pdf-6 (2)

During lunch (before our problem-solving time), I leave the new prompt or question right by Upton as if he is presenting it to the class. When I pick the kids up for lunch, I tell them that Upton left us something and they get so excited to read his new question or prompt.

4. Provide anchor charts: Post the problem-solving steps in a prominent place in classroom. I add a visual cue for each step to promote understanding. As you teach problem-solving, refer to these posted steps and encourage students to do so when solving independently as well.

Upton anchor chart

Download our Upton Understanding Fish problem-solving unit and accompanying self-reflection and peer discussion questions and prompts to help build mathematical understanding in your classroom.



Meet Max Modeling Mouse

Greetings I’m Max the Modeling Mouse.
I solve problems by using things from my house.
Beans, cubes and counters are useful math gear;
Use them to model the problem so it’s clear.

Max Modeling Mouse is the first strategy animal in our Problem-Solving Pond: A Common Core Math Strategy Unit. The Problem-Solving Pond  was created to help teachers overcome Common Core math challenges and employ problem-solving strategies with confidence and fidelity.  Read more about Max’s strategy below or download the complete unit here.

WHAT is modeling?:  Students use manipulatives, counters or drawings to model, or represent the mathematics of the story problem.

WHY is modeling important?: By making a visual representation, students are able to see the situation presented in the problem.  Modeling is critical to student understanding as it allows students to see, feel and process math in a concrete way.

Jake Mouse

HOW do I teach modeling?:  Select appropriate manipulatives (beans, cubes, coins, place value rods) and make a visual model of story problem.  Write equation below the model to solidify understanding.

Alec Mouse

WHEN should I use modeling?:

This strategy is ideal for presenting a new mathematical operation.  Most teachers use this strategy with K-2 mathematicians, but this is also beneficial for older students as it works especially well with money, fractions, ratios and percentages. Modeling is a great way to double-check solutions because the visual representation increases understanding of the problem.

Helpful hints:

  • Create rules and expectations. Before introducing manipulatives, establish rules and expectations for use. Explain the mathematicial purpose of them as well as how and when to use them. Practice essential routines such as getting out manipulatives, freezing at teacher’s signal (with hands-off manipulatives) and cleaning up quickly and quietly.

Manipulative rules

  • Prepare manipulative bags. Count out specific manipulatives and store in individual Ziploc bags or small Tupperware containers so they are ready to use at any time.  These individual units can be stored in student desks or in a large container.

Manipulative bags

  • Store manipulatives in a common area. Purchase an inexpensive storage container with shelves and house manipulatives here. Label each shelf and include a picture or glue on an example for student reference.

Manipulative storage

  • Allow time for student exploration and play. Before students use the manipulatives as mathematicians, provide time for exploration and play. This will help students stay focused and use them correctly during math time.

Check out our other Problem-Solving Pond strategy animals coming soon:
–Drawing Dragonfly
–Tallying Toad
–Counting Crocodile
–Hopping Hare
–Tabling Turtle
–Break-Apart Badger
–Equating Earthworm
–Fact Fluency Fox

Math Animals with names


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