Great Examples of Mathematical Modeling

Posted on April 23, 2012. Filed under: Common Core, Guided math, Mathematical Proficiency | Tags: , , , , , , |

I travel to a lot of different states.  Everywhere I go, everyone seems to be grappling with this idea of mathematical modeling. The New Math Common Core has placed a particular emphasis on Mathematical Modeling throughout. Whether or not you are aligning your curriculum to the CCSS, its explanation is revelatory. It states that:

Mathematically proficient students can apply the mathematics they know to solve problems                    arising in everyday life, society, and the workplace. …Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation…They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense… 

Given this criteria, are your students mathematically proficient?  Do they have a repertoire of models to make sense of their mathematical thinking.  Here is a great resource with plenty of examples to get you started.  I found this online from the publisher.  These explanations of mathematical thinking are in the front of a great series of books on problem solving using bar diagrams. When I use these books I start with a grade level below the grade level I am teaching.  I would even start at the beginning of the series so the students have a conceptual understanding and procedural fluency so when they get to the more difficult problems they have a strong foundation.  See the resource page below of the different types of models and let me know what you think!

Resource 1

Happy Mathing,

Dr. Nicki

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The importance of scaffolding instruction

Posted on November 15, 2011. Filed under: Classroom environment, Elementary math, Guided math, Mathematical Proficiency | Tags: , , , , , |

Don’t take the ball from the person that has it.

Often times when I am in a classroom I see teachers ask a student a question and then when they don’t know the answer, they take the ball away.  They take the ball out of the hands of the person that has it when they say things like “who can help” or  “who else wants to answer.”   I’m always saying, “Don’t take the ball from the person who has it.”

Imagine if a coach grabbed the ball from the girl who couldn’t kick or the guy who couldn’t make a basket everytime and just said, “Oh, somebody else do it.”  “Who can do it that knows how?”  Coaches don’t do this.  Coaches let the person who has the ball keep it and keep at it.  Teachers should do the same.  We all have to learn how to coach kids better at math.  We do a pretty good job at it in reading.  But we tend to drop the ball when it comes to math!

We let the person keep the ball by scaffolding the questioning.  So for example, if we say how do I write 335 in expanded form and a student writes a 3 goes in the hundreds space – then our next move would be to scaffold it.  We might even ask is it a 3 or a 30 or a 300? ( You have to make some crucial decisions because you don’t want to overscaffold either).  We don’t take the ball and say “Who wants to help Andy?”  We have to help Andy! We’re the teachers.  SCAFFOLD, SCAFFOLD, SCAFFOLD.   GREAT SCAFFOLDS COME IN THE FORM OF QUESTIONS, GRAPHIC ORGANIZERS, CONCRETE MATERIALS AND VERBAL CUES. (TO NAME A FEW).

Happy Mathing,

Dr. Nicki



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More RTI Math Resources

Posted on November 7, 2011. Filed under: Assessment, Differentiated Instruction, During the Guided Math Lesson, Elementary math, Guided math, Mathematical Proficiency, Problem Solving, RTI | Tags: , , , , , , , , , , , |

Here is the IES guide to RTI (federal gov) in Math.  It outlines the interventions very clearly, with a discussion about the research findings of each recommendation.  Well worth using as a study guide in PD sessions.  Everybody that works with students in  math should be aware of the information in the packet!  Here are the research based recommendations:

Recommendation 1. Screen all
students to identify those at risk for
potential mathematics difficulties and
provide interventions to students
identified as at risk.


Recommendation 2. Instructional
materials for students receiving
interventions should focus intensely
on in-depth treatment of whole
numbers in kindergarten through
grade 5 and on rational numbers in
grades 4 through 8. These materials
should be selected by committee.


Recommendation 3. Instruction during
the intervention should be explicit and
systematic. This includes providing
models of proficient problem solving,
verbalization of thought processes,
guided practice, corrective feedback,
and frequent cumulative review.


Recommendation 4. Interventions
should include instruction on solving
word problems that is based on
common underlying structures.


Recommendation 5. Intervention
materials should include opportunities
for students to work with visual
representations of mathematical
ideas and interventionists should
be proficient in the use of visual
representations of mathematical ideas.


Recommendation 6. Interventions at
all grade levels should devote about
10 minutes in each session to building
fluent retrieval of basic arithmetic facts.


Recommendation 7. Monitor the
progress of students receiving
supplemental instruction and other
students who are at risk.


Recommendation 8. Include
motivational strategies in tier 2 and
tier 3 interventions.


Happy Mathing,

Dr. Nicki

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Multiplying Fractions: Guided Math and Center Activities

Posted on October 10, 2011. Filed under: Assessment, Common Core, During the Guided Math Lesson, Elementary math, Guided math, Manipulatives, Math is a Language | Tags: , , , , , , , |

Here are some great examples of multiplying fractions that can help students to illustrate what they are doing.  The CCSS states that students will be able to illustrate and explain what they are doing.  We have often stressed the algorithm with no explanation. You should definitely have students illustrating the multiplication of fractions in small groups and in centers.  They should talk out their understanding to each other as they are doing these activities. Here are some tools to help build conceptual understanding:

Happy Mathing,

Dr. Nicki

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Printable Math Strategy and Algorithm Posters: Great to Discuss in Guided Math Groups

Posted on August 27, 2011. Filed under: Assessment, Classroom environment, Common Core, During the Guided Math Lesson, Elementary math, Guided math | Tags: , , , , , , , , , |

Here are some great posters.  You might hang these up as reference posters in your classroom.  You might also also have them nearby when you are talking about different strategies and algorithms in your small guided math group.  These serve as scaffolds (in the form of cues) so that students can remember what they are learning.

Happy Mathing,

Dr. Nicki

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An Addition Algorithm

Posted on August 23, 2011. Filed under: Assessment, Common Core, During the Guided Math Lesson, Elementary math, Guided math, Manipulatives | Tags: , , , , , , , |

The New Math Common Core states that children will understand how to “add and subtract (depending on the grade level the number range varies)…using strategies and algorithms based in place value, properties of operations and/or the relationship between addition and subtraction.”

We have to really think about what does this mean in action? How does it look in the classroom?  How do we get fluent as a teaching community ourselves, so that we can teach this way.  I think one way to start is to have grade level discussions about how to operationalize this.

I do think that we should use small guided math groups to discuss different strategies because you want to give the students a chance to talk about their thinking.

Here are some examples of teaching partial sums.



What do you think of the way the teacher frames the method?

How does it make it more student friendly by saying we are going to “break-down” the parts. 

What do you think of the way that the teacher uses different colored pencils to highlight the parts of the problem?

In this next video notice how Eli is solving the problem by drawing out the base ten blocks.  What does this tell us about his understanding of place value?

Now look at this video of a teacher using technology to teach the partial sums method.

How does this instructional strategy of representing it differ from using the concrete base ten blocks?  Do you see how it clearly shows the relationship of place value while moving towards just the abstract representation – but using the pictures as the ongoing scaffold.

Look at these next two videos:

Video 1

Video 2  

What do you notice about the way that the teacher is talking about the numbers?  Notice how he says “5 tens or 50 ones.” Also notice the very step by step process that he uses to scaffold learning of the strategy.

Here is a Partial Sums Poster.

I wanted to share these videos because they offer different perspectives and nuances on teaching this algorithm.

Happy Mathing,

Dr. Nicki






Partial Sums Poster

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Scaffolding is the Key! Small Guided Math Groups is the Place!

Posted on August 17, 2011. Filed under: Assessment, Classroom environment, Common Core, Differentiated Instruction, During the Guided Math Lesson, Elementary math, Guided math, Guided Math Introduction | Tags: , , , , , |

Josh Rappaport wrote an excellent piece on scaffolding.  He notes that

“No one would attempt to climb Mount Everest in a day. But when we teach math, we often expect something similar from students. We expect them to learn a complex, multi-step process in one lesson, in one hour. We expect them to go from no awareness of the process, to awareness to competence to mastery. And we don’t take account of the fact that many math process[es] require a long ladder of thought steps. In edu-jargon, this process of taking all of the little steps into account — and teaching each step individually — is called “scaffolding.” ”

This is a brillant metaphor.  See his whole post.

Guided math groups are an essential part of the “scaffolding” process.  In a guided math group you can listen to students talk, you can watch them do the math and give immediate feedback and you can do some direct instruction as well.

I encourage us all to think about “How tall are those ladders we are using in math class?”

Happy Mathing,

Dr. Nicki

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Mathematical Proficiency for Everyone!

Posted on August 16, 2011. Filed under: Classroom environment, Differentiated Instruction, Elementary math, Mathematical Proficiency | Tags: , , , , , |

Is it possible to do something in your math class this year, so that everybody develops proficiency?  David Borenstein raises this question and others in his NY times article about math. What happens if we truly believe that everyone can learn math, and that we have the keys to that?  How does the paradigm for teaching and learning math then switch?  How would things change IF we had no doubt that everyone can, will and must do math proficiently?  And we really believed it was possible with some instructional interventions, masterful scaffolding and appropriate math tools? Think about it, seriously and read the entire article here.

Happy Mathing,

Dr. Nicki

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Scaffolding In Guided Math Groups: Part 2

Posted on December 27, 2010. Filed under: Differentiated Instruction, During the Guided Math Lesson, Graphic Organizers | Tags: , , , , |

There are many different types of scaffolds.  We should be sure to use a variety of scaffolds to accommodate our students (Alibali, 2006).  For example, we use scaffolds to accommodate the readiness levels of learners, novice, apprentice, practitioner and expert.  But, we also use scaffolds to accommodate the progression of knowledge among learners, so the apprentice might start with one type of scaffold and then move on to a more complex one later.  Hartman (2002) states that scaffolds may include models, cues, prompts, hints, partial solutions, think aloud modeling and direct instruction.

A procedural facilitator can be a hint, cue card or partially completed example (cited in Van der Stuyf, 2002). For example, when teaching rounding, the graphic organizer with the hills on it and the numbers tell students what to do (GOOGLE ANTHILL NUMBER ROUNDOFF/ ).  Another example for rounding  is a colored number grid Here is another example of a graphic organizer being used as a scaffold for rounding.  This one uses the number line Also, poems like the rounding poem can be considered a scaffold (a type of cue card) because they give prompts to what should happen. Here is another example of a cue card type of scaffold.  Notice the prompts at the top of the sheet. [you have to copy and insert url into your browser directly to get to  site]


Hartman, H. (2002). Scaffolding & Cooperative Learning. Human Learning and Instruction (pp. 23-69).  New York:  City College of City University of New York.

Google: Scaffolding as a Teaching Strategy (Van der Stuyf,  2002)

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Scaffolding Math Instruction in Guided Math Groups: Part 1

Posted on December 24, 2010. Filed under: Assessment, Differentiated Instruction, During the Guided Math Lesson | Tags: , , |

What is Scaffolded Instruction?

Scaffolding is helping students become successful through a series of guided steps.  Bruner (1975) coined the term, based on the work of Vygotsky. The main ideas is that children can become successful doing things that they can’t do on their own yet, with a little help from both their teacher and friends.  In the beginning a great deal of support is given and then gradually the support is decreased until the student can successfully do it own their own.  Remember when you learned to ride a bike?  Those extra back set of trainer wheels were one level of scaffolding.  Then, when they came off, whoever push started you and followed close behind was another level.  Finally, you were off, down the street, doing it on your own, grinning all the way!

Hogan and Pressley (1997) found 8 essential elements to scaffold instruction:

1. Pre-engagement with the student and the curriculum

2. A shared goal  

3. Ongoing Assessment- Pre/During/End

4. Tailored assistance – This may include cueing or prompting, questioning, modeling, telling, or discussing.

5. Ongoing Goal Setting

6. Specific Feedback

7. Attention to student disposition/mental and emotional engagement

8. Internalization, independence, and generalization to other contexts –

(adapted from citation in Larkin, 2002)

Larkin (2001) found that teachers who scaffold also

  • Meet students where they are/ focus on what they can do
  • Scaffold success quickly so that the “cycle of failure” is broken
  • Help students to “be” like everyone else
  • Know when it is time to stop – “Less is more”
  • Foster Independence

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