Category: Math Instruction

Unlock the Number Line

Image by:  Tratong

It’s becoming more evident that the Common Core initiative is running at full throttle. Keeping that in mind, I’m now finding myself exploring different methods to introduce fundamental math concepts and create lessons that emphasize number sense skills. These number sense skills are especially needed at a young age and often influence how students perceive / achieve in math later in their academic career.

Having students explain their mathematical thinking is a big focus next school year.  By asking appropriate questions, teachers are able to analyze and give feedback in real-time to best meet the academic needs of each student.  After researching math strategies and participating in a variety of CCSS workshops this summer, I’ve concluded (at least partially) that the number line can be utilized for many math objectives at the elementary level.

Teachers that model the mathematical problem-solving process find that it’s beneficial. The visualization of a numer line can be useful, whether it involves the concepts of measurement (elapsed time), percents (finding discounts), algebra (inequalities) or fractions (computation).  A number line can be useful in solving problems related to any of the above concepts.  If the idea of utilizing a number line to solve math is introduced via classroom instruction, it may be beneficial to give examples of how it’s used outside school as well.

So.. how did you use a number line today?

Math Shortcut or Conceptual Understanding?

Image by:  Renjith

There are “tricks” or “shortcut” techniques that many teachers have in their tool belt when it comes to teaching mathematics.  These techniques are often memorized by students to be used later on some type of assignment.  I’m not saying that these types of techniques are good or bad, but often students come away with little conceptual mathematical understanding of the concept being taught via the shortcut. Click on the picture below for more information on conceptual understanding.

© 2007 – 2012 Regents of the University of California

Without that understanding, students are not necessarily being prepared to apply the concepts later in middle school/high school.  A very small sample of the short cut techniques I’m referring to are below.

PEMDAS – order of operations

King Henry Died Drinking Chocolate Milk – Metric System

FOIL Method – Algebra

Negative Multiplied by a Negative = Positive

Gallon Guy/Gal = Capacity

Students generally remember the shortcuts and utilize them on assignments/tests.  Is that a bad thing?  I can almost hear math teachers around the world grumble.   If the students truly have a conceptual understanding of the concept then why not use these techniques?  Many of these types of shortcuts are used at the late elementary level. When students understand the technique – such as King Henry .. but don’t understand the concept (differences between the units of measurement and in what context they can be applied) then students/teachers run into problems.  Students are expected to be able to apply the concepts in multiple situations. Middle school teachers are then are held responsible to deepen the mathematical understanding of the concepts behind the techniques that were briefly utilized at the elementary level.  This topic has been on my mind lately, and finally made it’s way into this post based on this post.  Elementary teachers, as most teachers do,  attempt to use innovative and engaging methods to produce excitement related to learning and school. That motivation is often contagious and beneficial.  Whether teachers use these techniques or not (obviously, it’s up to you!) students should understand the concepts before memorizing nifty sayings that don’t really relate to the concept itself.  I’m not blaming teachers for using these techniques to engage students, but ensuring that students have a mastery of the concepts should be near the top of the priority list.

Math Reasoning and Feedback

Image by:   J. Creationz

Having math reasoning skills is important.  Generally, math reasoning skills are taught and incorporated in early elementary school.  In math, a problem is what a student is asked and expected to answer.  If a student is unable to answer why their answer is correct, I believe that the student might not fully grasp the mathematical concept.  The student might not be utilizing math reasoning skills.

For example, a student that measures area in linear feet might not completely have an understanding that area is measured in square units.  The student could have the correct numerical answer, but include the wrong unit (centimeters compared to square centimeters).

How is mathematical reasoning taught?  I’m going to be taking a proactive step next year to give opportunities for my students to utilize math reasoning.  I’m deciding to use higher level questioning to enable students to think of the process in finding the solution.  The learning process is key.  I’ve found that math instruction isn’t always linear, just as mathematical reasoning isn’t rigid.  By asking students why/how they arrived at a solution is vital in understanding their thinking.

As I’m planning for next school year, I’ve decided to ask students to explain their reasoning more frequently.  By hearing their reasoning, I’m in a better position to give direct feedback.  All math questions have some type of reasoning.  I believe that multiple solution / open-ended questions can be used to display mathematical reasoning. Students need to be able to explain why they responded with a specific answer and what methods/connections were utilized to solve the problem.  Based on the math Common Core, students are expected to reason abstractly and quantitatively.  When students describe their mathematical process, teachers are better able to diagnose and assess a student’s current level of understanding.  Math reasoning isn’t always quantifiable, but it can be documented via journaling and other communication methods.  More importantly, teachers will be able to provide specific feedback to help a student understand concepts more clearly.  I also feel that this questioning process develops self-confidence in students and prepares them to become more responsible for their own learning.  See the chart below.

Problem –> Reasoning –> Feedback

Math and Tourist Destinations

 Image by: Janoon028

Over the past few weeks I’ve been researching math activities that integrate multiple disciplines.  After visiting a number of sites on Twitter, I found an interactive Google Map.  This activity took students to a Google Maps page that gave information about various landmarks around the world.  Not only was there a social studies connection, but the majority of student work dealt with higher level math. There were links by the questions that gave students opportunities to learn more about the landmark.  Logistically, I decided to group the students in 2 or 3 and gave one iPad/laptop to each group.  The technology was needed to visit the site and find the information.

Some of the questions were quite challenging for my students.  I overheard one student saying that since they are already on the Internet they could  look up the formula.  They asked me and I told them that was fine.  Part of this activity is exploration and finding the information for application on your own.  

This activity is somewhat like a Webquest, but a bit more guided.  Students were asked to complete all of the problems on the site.  There was an actual answer guide near the end of the page that some students found.  I reviewed the process and answers with the students after approximately 45 minutes.  The class then completed a plus/delta chart on the activity.  Overwhelmingly, the comments were positive.  I will keep this in mind as I begin planning for next school year.  Some of the pictures from this activity are below

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Beneficial Math Homework in Elementary School

Image by: Keatti

The issue of homework has been on my radar this school year.  Depending on where you teach or how involved you are in reading the research on homework effectiveness, the topic can bring out strong opinions.  Is homework truly beneficial at the elementary level?  Is homework given because it’s what the community expects? Where does all the homework go after it’s been graded? Back to the students, to the parents, shoved in a desk, in the garbage (see the picture above) … I hope not. I’m not able to answer these questions concisely.  Since I’m a math teacher, I believe that students need practice.  Generally, (I won’t speak for all teachers) homework is a form of practice, but the homework that is usually (once again, not for all teachers) assigned deals with repeated problems associated with a concept.  The problems are rarely practical and focus on repeated forms of one or two particular concepts.  If a student has a problem with the homework, the adult at home helps, they find help on the Internet, or the student doesn’t complete the homework. Regardless, the student isn’t showing mastery or showing what they have learned.  Should homework be graded?  Many education experts believe that homework isn’t beneficial at the elementary level.  I’ve found that homework is beneficial for some, but not for all.

Instead of having students complete “typical” math homework every night, I’ve decided to look into an innovate approach to homework.  Instead of pages of multiple concrete math problems, the “updated” homework revolves around conversations and self-reflection.  What do I mean by this?  Here’s an example:

This is just an example, but I’m sure an educator could create multiple questions that cover an array of topics that could last for many homework sessions. Change this ideas and implement as you see fit.

So, instead of giving “typical” homework every night, the homework is assigned on a weekly basis.  The homework involves a discussion about math with an adult at home.  The discussion will be documented by the student and a self-reflection piece will be included.  Before giving this type of homework it would be a good idea to discuss this with the parents.  The idea is to connect math concepts taught in class to the practical application outside of the school. I believe that a rubric would be helpful in assessing the “updated” homework.

Math and Multiple Solutions

Image by: Krishnan

For the past few days I’ve been reviewing a math unit and have found that the lessons included have very few problems with multiple solutions.  I have nothing against one correct answer scenarios, although I feel as though students should be exposed to problems with multiple solutions.  There are cases where having one solution in math is mandatory, but there are other cases where multiple solutions are possible.  I believe the project in this blog post isn’t completely “open-ended”, although it does have multiple solutions. The concept of open-ended math is important because I believe that this idea is relevant in and outside of the classroom.  Students often seem more intrinsically motivated to complete open-ended problems, as it’s different than the norm.

Recently, I came across a math activity designed for the upper elementary level, (although it could work at middle school) that offers multiple solutions. Since I didn’t personally create this activity, I’d like to give credit to NRICH Project for the original idea.  Multiple math concepts are found in this project.  The concepts covered in this project include a great amount of number sense concepts: factors, multiples, square numbers, even, odd, prime, composite, and triangular numbers. This assignment covers many concepts and a teacher could informally assess students in the classroom as they facilitate the learning process.

Here’s the process that I used:

1.  Download the Word documents (you can easily edit them to meet your needs).  Here is the Word file.

2.  Review the concepts of multiples, factors, square numbers, even/odd, prime/composite, and triangular numbers.

3.  Pass out the sheets to the students. (I had one of the pages a different color than the other – better for organization)

4.  Students cut out and glue the project together. (my class took approximately 30 – 40 minutes)

5.  Review the project with the students

6.  Have the students journal about their math problem solving experience.  Extension opportunities can be found here.  The reflection and assignment could be used to show growth over time and might even be useful in a student portfolio.

Here are a few possible solutions:

Additional answers may be found here.

If you use this, please let me know how this project works in your classroom.

Building Math Confidence in Elementary School

Image by:  DigitalArt 

 I’ve found that math confidence often starts at a young age and develops over time.  Starting on a positive note can instill in students an appreciation for math.  Encouraging students to perceive and experience math in a positive light is important.  Elementary students typically experience math through a variety of hands-on experiences/manipulatives (base-ten blocks, geometric solids, counters, etc…) and then eventually progresses to the abstract.  The more time spent using engaging manipulatives often builds confidence, enabling the students to transfer their math understanding to abstract problems.  Building a solid mathematical foundation at the elementary level can lead to an enriching and encouraging math experience in the upper grades.  If you teach a form of math at the elementary level this concept shouldn’t be unfamiliar.  Most math in the K-5 curriculum is unveiled in a specific instructional order, as federal/state benchmarks indicate.  Publishers may suggest that math is linear, although many experts in the field disagree.   I assume that most teachers agree with a concrete/manipulative (visual representation)  –>  abstract (print) type of instruction model.

I’d like to recommend an edit in this process.  Not necessarily a change, but an addition.  Before moving straight onto the abstract, teachers should encourage students to reflect on their learning experience using manipulatives to solve math problems.  When given an opportunity to reflect on their learning, students often begin to become more responsible for their own learning.  Utilizing self-reflection math journals also allows students opportunities to connect their effort and achievement.  It may also give the teacher insight in how a particular student understands a specific concept and plan for formative assessments.  I’m not suggesting that the transfer from manipulatives to abstract should occur during the same day.  Giving ample time for math connections to fuse is important and will build a solid mathematical foundation.  I’ve found that the more engaged students are in their own learning the more opportunities that they will have to retain and apply their mathematical knowledge.  I believe the process below assists in building math confidence which will enable students to become more responsible for their own learning.  I have provided two flow charts below that may be helpful in explaining this process.

Building Math Confidence
Building Math Confidence

Math and Sports

            Image by:  Idea G.

To be honest, it was a memorable Superbowl game last night.  Watching the game brought back many memories from past professional sports games, specifically the Chicago Bulls dynasty in the 1990s.  One memory that caught up with me dealt with middle school math.  I started to enjoy math in middle school.  I started to view sports as percentages, ratios, and decimals.  This started my journey in appreciating math. Most of my students watched the Superbowl and I think that this is a great opportunity to delve into statistics as well as data analysis concepts in the classroom.

Let’s take Eli Manning as an example ….

*All of the football images and statistics were found on Wikipedia

Let’s take a look at Eli’s statistics:


Using Eli Manning’s statistics, we can view trend data.  Here are some questions to ask the students:

  • How many yards do you think Eli will pass for next year?
  • How many games do you think Eli will play next year?
  • Predict Eli’s passing rating for the 2012 season?  Explain your prediction.
  • Do you notice any trends?  If so, can you explain why the trends exist?

After asking the questions above, we can take a gander at Eli’s post season statistics.

  • Compare Eli’s regular season and post season performances.
  • What differences do you see in the data?
  • What do you think contributes to a quarterback rating.  Why?
  • Using the data, what factors do you think impact a quarterback rating?
After discussing the questions, you may want to have a conversation how statistics often relate to percentages.

Additional Sports and Math Links:

Dice Game

NFL Probability

Math and Football

Incorporating Sports into Math

Velocity in Sports

Math in Sports PowerPoint

Math and Sports Webquest

Wanted: Math Skills for Discounts

             Image by:  A. Balaraman

I recently came across “Number Crunching Coupon Calculating Champ” on a Facebook page.  The screenshot image is found below.  I plan on incorporating this page into my math instruction next week.  According to the website, if I completed the problem correctly I would receive a certain percentage off of my next purchase.  So … completing a math problem on the Internet = a percentage discount?  I don’t recall ever reading a promotion like this before.  Regardless of the motivation behind the company sponsoring the ad, I thought this type of math application could benefit my classroom.

I didn’t fill out the form, but I thought that this type of problem could possibly supplement my instruction.  There are many ways to teach discounts, but real life examples bring awareness that math is everywhere.

There are many different ways to solve this type of problem.  I may have the students work in collaborative groups to find the answer.   All of the different mathematical operations will be useful to practice.  Overall, I feel as though this type of problem will give students an opportunity to utilize their mathematical knowledge in a practical way.

My students enjoyed working out the problem above, although it is quite intense for an elementary student. The picture below might assist in starting a discussion about percent increases and the processes involved in finding the answer.

What is the percent increase from 7 to 10?

Fundamental Place Value

Image by:  D. Castillo

After analyzing a recent math assessment, I asked a group of middle school teachers in another state what particular math topic they would recommend elementary teachers strengthen.  I thought that the middle school teachers could offer valuable input regarding how the elementary schools are preparing students for middle school. After a quick discussion, each middle school teacher (all five of them) decided that place value is a fundamental concept that precedes many other high-level math concepts. After listening to the discussion, I started to think of when place value is actually taught.  According to the math Common Core, an entire math strand is dedicated to place value / base 10 concepts.  In fact the word place value is found 42 times in the Common Core Standards for Mathematics.  Obviously, place value is important, but how do teachers introduce and expand on the topic?  First of all, teachers can use visuals,such as a place value chart.

Ideas and links regarding teaching place value can be found below.

  • Place value can start before first grade – counting and identifying specific values
  • Students can be given the opportunity to utilize base-ten blocks and explore how the pieces are interchangeable
  • Place value can be viewed as a decimal or fraction
  • Use coins to show another way to view place value
  • Tactual place value activities
  • Online games or activities
  • Use a number line to show how place value plays a role in how large or small numbers become.