Is Math Linear?

Some Twitter users suggest that math isn’t always linear.  The curriculum that math teachers teach may resemble something linear, although some curricula (example: Chicago Everyday Mathematics) may engage in some type of spiraling format.   Even if a curriculum spirals, it is still somewhat linear. Most teachers would suggest that background knowledge is needed to learn higher level concepts.  This is especially the case at the elementary level. Generally, teachers are expected to teach specific math concepts at certain grade levels. Most of these concepts are assessed by the state for that particular grade level.

This past week I was teaching a math session with a group of upper elementary students.  We were having a conversation regarding triangles and angles.  We covered the topic that the measures of the interior angles should equal 180 degrees.  One of my students then asked how do we find an unknown side of a triangle.  I thought that was a decent question, so we took out our math books and started looking for clues in the geometry section.  The book led us to a dead end. So … I thought of my own learning and remembered something about the Pythagorean Theorem helping with this question.  As a class, we traveled on the internet and Googled Pythagorean Theorem.

We explored the following pages:

http://en.wikipedia.org/wiki/Pythagorean_theorem

http://jwilson.coe.uga.edu/emt669/student.folders/morris.stephanie/emt.669/essay.1/pythagorean.html

http://www.pbs.org/wgbh/nova/proof/puzzle/use.html

http://www.teachscienceandmath.com/2010/03/03/pythagorean-theorem-real-world-application/

After digging up a few resources, we finally found a  group of students that created a short video on the Pythagorean Theorem.

After reviewing the video above we decided to practice a few problems using our new knowledge.  The students seemed to enjoy and were motivated to continue on this Pythagorean adventure.   I asked the students to research the Pythagorean Theorem that evening and practice a few practical problems. I also asked them to bring in any practical problems relating to the theorem to school the next day.  The next day the students came in with papers of practiced problems and examples. Overall, I felt as though this was a great opportunity to expose the students to a higher level skill, that isn’t necessarily linear, but may benefit them in preparing for middle school.

Multiplication Fundamentals

Image by:  David Dominici


Learning the fundamentals of a particular subject area is important.   In the realm of math, the word fundamentals can be misleading.  CNN produced an article on the lack of math fundamentals in this piece.  Every math strand requires some type of fundamental understanding.  According to the Core Standards, mastering the multiplication tables should occur in the early elementary grades.  Having mathematical fluency at a young age is important.  In elementary school, students that haven’t mastered their multiplication tables may fall behind and not be able to access higher level math skills that require a concrete understanding of multiplication.  Beyond raw memorization via flash cards, teachers need to find strategies/methods to introduce the multiplication tables.  Over time, I have found that the following strategies enable students to understand and master the multiplication tables (0-10):

1.)  Math Apps

The following apps help students master and deepen their understanding of multiplication:

– MathBlaster Hyperblast

– Math Ninja HD

– Rocket Math

– Factor Samurai

2.)  Manipulatives

Students often need a visual demonstration to come to their own understand of multiplication.

– Egg Carton Math

– Multiplication Balloon (Could easily be created)

– Base Ten Multiplication

3.)  Games

– Math Smart Game (Students could actually create their own multiplication board game)

– Multiplication Four in a Row

– Multiplication / Factor Bingo

4.)  Student Multiplication Projects

– Multiplication House

– Math Wanted Poster

– Multiplication String Art

5.)  Introducing Practical Multiplication Problems

Finding practical math problems is important and gives students an opportunity to apply their learning. Here are a few resources that may introduce problems or communicate the relevancy of using multiplication outside of the classroom.

Real World Math Problems (PDF, Word, and PPT forms)

Elementary Multiplication Examples

Geography and Multiplication (PDF)

Additional Multiplication Resources

IBM Youtube Commerical (Why is math relevant?)

Make a Math and Art Connection

The Real Number Line – In Practice

Image by:  Samana


A little while back I wrote a blog post about how the typical math number line needs an upgrade.  You can find that post here.  I thought and still think that the general math number line that is introduced at the elementary level needs to be enhanced.

I believe that students should encounter all types of numbers on a number line. Students should find whole numbers, decimals, square roots, fractions, percentages, mixed numbers, etc.  Of course, the concept needs to be age appropriate .  So, in my last post I wrote about how students should understand the real math number line.  In theory it sounded like an idea that could be put into practice.  I decided to find out how the theory looked in practice.  I asked students to create a math number line with multiple components. This activity fit in well with the decimal and fraction unit that I’m currently teaching.  I gave each student learning group a sheet like the one below and a specific number range (like numbers 3 – 6).

Every student worked on this project in a cooperative group. Through this experience, I believe the students had a unique opportunity to learn about the many different ways that numbers can be represented.  See below for examples.

Overall, students were engaged and thoroughly enjoyed the activity.  At the end of the project, I facilitated an informal plus/delta chart and the feedback was generally positive.   While students were in their cooperative groups I overheard them debate the differences and similarities of fractions, square roots, decimals, improper fractions, and mixed numbers on the number line.  It was a great learning experience and definitely a project I’ll put in the plan book for next year.

Disclaimer (unfortunate but necessary) : The thoughts and opinions expressed in these pages are my own, and not necessarily the opinions of my employers.

Shaving Cream and Math

Image by:  Salvatore


I’m always trying to find new ways to make math interesting and relevant. Generally, the more interested the students are in the instruction, the more willing they are to apply their learning.  This past week I used one common household item to teach my elementary math class about number lines.  I’m not the only teacher who has used this strategy in the classroom, but I’ve found encouraging results by doing so, that’s why I’m sharing.  I’ve provided a few pictures for those (like me) who need a visual representation before putting a strategy into practice.

Procedure

1.)  Have all the students clear their desks.  There shouldn’t be anything on the desks, including pencils, water bottles, etc.  During this time students get a little anxious in wondering what’s going to happen next.

2.)  The teacher takes out one or two bottles of shaving cream.   I used Babaso, available at the Dollar Tree.  This works much better than some of the more expensive shaving creams.

3.)  The teacher asks the students to predict how the class will be using the shaving cream to learn about math.  You might get some interesting responses with that question.  This may also gains student interest.

4.)  Go over the ground rules.  Everyone should roll up their sleeves, don’t fling the shaving cream at anyone in the class, don’t touch the shaving cream until directed, no one gets out of their seat, etc.

5.)  Go to each desk and spray a bit of shaving cream (4-5 seconds) in the middle of each desk.

6.)  Tell the students that they will be given a few minutes to “play” with the shaving cream.  Ask the students to make different types of polygons, rays, lines, etc. with the shaving cream.

7.)  The teacher models a few number lines on the whiteboard.  Students are asked to create their own number lines.  Ask the students to create multiple number lines.  Once a student creates a number line, the teacher reviews the work (could be a great opportunity to take a picture), gives the student a bit more shaving cream and then looks for another finished project.

8.)  At the end of this project there are a lot of sticky fingers.  The teacher hands out wet wipes or wet paper towels to the students.  The students clean their own desk and hands.

9.)  Before the students leave class, or sometime in the near future, the teacher asks the students to create three additional number lines (addition, subtraction, multiplication) on paper and turn their work into the teacher.

More Examples:

Shaving Cream and Math Ideas

Greenfield Exempt Schools

Mrs. Clayton’s Class Blog – Using Shaving Cream

Disclaimer (unfortunate but necessary) : The thoughts and opinions expressed in these pages are my own, and not necessarily the opinions of my employers.

The Real Number Line

Image by Winnond

Approximately two weeks have passed since the new school year has started and I’m finding that the traditional number line (that many teachers have become accustomed to) needs an upgrade.  My math students are benefiting from the number line, but true understanding of numbers doesn’t come from a number line alone.  For the past seven years I’ve used a “typical” number line from -10 to 100 in my classroom.

Don’t get me wrong … the number line is helpful in teaching many number sense concepts.  In my opinion, the number line offers students a visual/spatial representation of the number system.  I  believe many numeracy concepts are built from understanding the system of numbers.  What is often missed, or not necessarily taught, while utilizing the number line are numbers that don’t fit the category of being whole.  For example, I generally don’t see pi or irrational numbers being part of a number line.


Recently I found a “Real Number Line” poster.  I was fortunate enough to find this poster and have utilized it to teach elementary students about the number system. I think it’s important to communicate that square roots, fractions, percentages, mixed numbers, etc.  should be included on a number line.

I actually created a practical follow up activity in response to this post here.

Instead of purchasing a poster, you could have the students create their own.  A few examples are found below:

4/25/12

I believe that Wolfram Alpha does an excellent job of emphasize the importance of a number line in the answer it provides.  The answer can be represented on a number line.  See the example below.

8/14/12

I’ve been reading How the Brain Learns Mathematics by David Sousa.

David emphasis the importance of the mental number line.  All humans have number sense.  For example:  studies indicate that the brain can decide that 60 is larger than 12, but it takes the brain a longer time to distinguish that 76 is less than 79.  It seems that when the digits are closer in value the response time of the human increased.  Visualizing many different forms of number lines would be beneficial and assist in developing better number sense skills at a young age.

 I thought this quote was beneficial:

“The increasing compression of numbers on our mental number line makes it more difficult to distinguish the larger of a pair of numbers as their value gets greater.  As a result the speed and accuracy with which we carry out calculations decrease as the numbers get larger”

 – David Sousa

Math and Art

Image by Graur Codrin

 
I have found that students enjoy and often thrive when presented with a challenging real world problem (often outside of the textbook). This can be observed during a problem based learning activity.  When students come to the conclusion that their isn’t one specific right answer, they are more willing to communicate their ideas and opinions to one another. Many practical problems outside of the K-12 education realm have more than one “right” answer.  When students are faced with problems that have multiple solutions the class community asks questions that often spark additional questions.  Active learning often comes to fruition through these activities.  In fact, the #realmath hashtag provides practical resources / images related to math found in the real world.  I have provided one example of using math through art below.  This idea comes for a PD opportunity that I participated in last year.


1.)   Begin by showing a small section of a picture.  Ask students the questions below or add your own (my questions are based on an elementary classroom).  Attempt to stay away from yes or no questions as students offer their own perspectives.  If you ask a yes or no question, follow it up with a why.  You might want to tell the students that some of the images don’t fit like a puzzle, as some pieces were cropped at different zoom levels – this adds to the complexity of the activity. I keep a separate chart on the board to write down math vocabulary that is used during this class activity.  Keep in mind that each picture is displayed one at a time, generally in a presentation format.  You may want to randomly have students answer the questions below.

Picture one questions:

  • What do you see?
  • Does this picture remind you of anything?
  • Describe the polygons in the picture.  Where are they?
  • What type of math vocabulary can you use to describe this picture?
  • Why is one rectangle in the picture lighter?
  • Where/when do you think this picture was taken?

2.)  Now show a small section of another portion of the picture.  Follow the same guidelines as step one.

Picture two questions:

  • Using math vocabulary, what do you see?
  • Using fractions tell me more about this picture.  Where can fractions be found in this picture?
  • Where/when do you think this picture was taken?
  • What similarities can be found between this picture and the first picture

3.)  Now show another section of the picture.  Follow the same guidelines as step one.

Picture three questions:

  • How does this picture similar to the first picture?
  • Why are some of the flags horizontal?
  • What type of information do you think the builders needed to construct this building?
  • How tall do you think this building is?  Why?
  • Do you think the building in the very front of this picture is the tallest?  Why?
  • What direction do you think the sun is shining?  Why?
  • What part of the picture do you think is missing?
  • Where do you think this picture was taken?

4.)  Reveal the full picture

Picture four questions:

  • How accurate were your initial predictions?
  • What are the differences/similarities between pictures one, two, three and four?
  • What additional math terms can you use to describe items in this picture?
  • What materials would you need to construct a building like the one in the picture?
  • Why is there a reflection on the building on the right?
  • How could you estimate the height of the building in the center of this picture?
  • Optional -Reflect on today’s activity in your journal.  Describe your reaction and what you learned during this activity.
  • Optional –  Similar to this example, students could take pictures around the school and create their own presentations on finding math in art.  Students could be given a rubric and work in collaborative groups and present their findings to the class.

Of course feel free to modify or change any of the steps above to meet the needs of your specific students.  My example is only a general template.  I’ve used this in elementary classrooms to introduce specific topics.  You could use a variety of images for this project, or have students create / take pictures on their own.  What about using Escher’s artwork below? As you can see, there are a lot of possibilities.

 

 

update 12/29 – An additional resource – Mathematics Meets Photographs

Making Math Relevant and Engaging

I noticed a theme while observing an educational math chat on Twitter.  Many of the participants spoke of how math and reading don’t necessarily have the same “emotional knee-jerk reaction” in education or at home.  One tweet I remember reading stated that there isn’t a math equivalent to reading a bedtime story, emphasis on reading.  As far as I know, there is no such thing as a math before bedtime.  Reading often takes precedence over math, especially at the elementary level.  Reading / Language Arts often requires or is mandated to take 1 1/2 – 2 times as much time as math. Don’t misinterpret what I’m writing here – reading is essential and absolutely needed.  I’m advocating for the math crowd – the people who despise hearing the words “I hate math” coming from anyone.  Math has received a stigma over time and there are even adults (you may be one of them)  who can’t stand thinking about math.  An interesting perspective comes from Michael Schultz in his recent blog post.  As you can imagine people dislike math for a variety of reasons. Unfortunately, many adults remember math as one of the least favorite subjects in school.  Their math teachers were less than stellar and used (and only used) the text book for all math instruction.  How do educators and administrators decrease the negative stigma associated with math?  I believe removing the stigma starts before and during elementary school.  Educators need to make math relevant and engaging.  How does that happen at the elementary level?

Use Manipulatives

Educators understand the often use manipulatives to increase student engagement, especially when introducing a topic.  Looking back at my own experience,  the times I enjoyed or expressed interest in math were when my teacher used manipulatives in the classroom.  Using manipulatives creates student engagement, which often leads to increased learning.  I still remember using the base-ten blocks and geometric solids to learn math back in the day. A couple specific examples:

Base ten-blocks

Balances – Mathfour video

Use Technology

Students use technology everyday.  But teachers need to appropriately (that’s key) utilize technology to increase student learning.  Most curriculum publishers have a technology component (like math games or instruction slides to show students) already part of their program.  There is a wealth of knowledge and information available online for teachers to use.  Personally, I’ve used Youtube, Power Point, Audacity, Google Docs, Movie Maker, and Flip Cam regularly.  There are many more tools available to use – I just wrote down what what was used last year.  I’ve placed a few links below if you’re looking additional content. Using Google Docs

Multimedia in Mathematics – http://davidwees.com/content/presentations

Use Practical Examples and Show Relevancy

Students are much more motivated to complete problems that are relevant and applicable to their lives.  A student wants to know why they are learning specific math concepts.  If students aren’t sure about where or how to apply what they are learning, what motivation is there to stay engaged?  Finding practical math problems is important and gives students an opportunity to apply their learning.  Even having students create and solve their own problems is a good start. Students need to understand that what they are learning in math class is relevant.    I tend to show my students the following video from IBM.  Also, during the first week of school I generally show the students a macro picture of what they will be learning throughout the year and what skills that they will need to proceed through each unit.

IBM Math Commerical

I also ask the students the following questions:

  • Can I think of a story problem where I could apply this concept?
  • How will learning this help me in the future?

Play Games

Play games?  Are you kidding?  I think at times, educators and administrators downplay the importance of playing mathematical games.  Games give students an opportunity to use learned skills, such as, but not limited to:  numeracy, collaborative teamwork, and critical thinking skills. There are online math games and boardgames that are relevant to what is being taught in the classrooms.  For example, games like Battleship can help teach algebra quadrants and axis. An example: Board Games

Disclaimer (unfortunate but necessary) :  The thoughts and opinions expressed in these pages are my own, and not necessarily the opinions of my employers.

Grades for Homework?

Grades

A little background … I’ve always been an elementary teacher that grades just about everything.  I’ve thought that every assignment should contribute to an overall grade.  When I say grade, I mean a point value, such as 8/9 points.  Mainly, I did this because it worked with my grading system.  Parents and students alike understood my grading policy.  My policy allowed little subjectivity, which in my case provided less of an opportunity for arguments over grades.  Quick disclaimer:  I never allowed graded homework to count for more than 25% of the entire grade.  I thought that if I didn’t grade the homework, what incentive is there for the kids to complete the homework?  I gave the kids the “talk” about how homework is practice and will help in the long term, but always attached some type of grade to the homework.  My view on grading has changed over time.  A few years ago I decided to tweaked my system.

For one of my math classes, I decided to not give an official grade for homework assignments.  Instead, I decided to give a check or minus at the top of the page.  A check meaning that the student understands the concept (generally getting 80% correct) – here’s where the subjectivity lies.  A minus would mean that the student received less than 80% correct.  I still graded formative and summative assessments, but decided not to officially grade the homework (and have it count for their grade) for this particular class.

I was waiting for community members to start contact me about how they were confused and didn’t understand my grading … etc.  So what happened, did my inbox fill up like a helium balloon?  Actually…

No, it didn’t.  I didn’t get one email or phone call asking me to clarify the grading of the homework.  In fact, students became much more aware of how they were doing in class based on the check / minus system.  Students who received a minus actually took the initiative to redo the problems without asking.  Also, I was finding myself grading less homework, which allowed me time to focus on creating engaging lessons that promote student learning.  I also expected a drop in achievement and focus – neither happened.  I’m so glad that I took a leap and decided to grade using this new method.

Next year, I’m going to expand the system to the other grade levels that I teach.

This post was inspired from :

http://mctownsley.blogspot.com

http://russgoerend.com/

photo credit: Bunches and Bits {Karina} via photopin cc

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