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Big Ideas in Mathematics

At the beginning of this blogging journey I really could not see the need for identifying a “Big Idea” for math.  I thought that deciding upon and focusing on one big idea would limit my creativity and force me down a path of “teaching to my big idea”.  I have since discovered that this is entirely untrue.  Having a big idea in mind, if anything, pushed me to find connections in all of the lessons and activities I chose to share.  Not only did it not limit me, it expanded my thinking process. 

The big idea that I chose was one that I feel encompasses so very much in relation to all math processes. 

"Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value."(2005 Charles)

I actually found it very hard not to relate it to my activities.  It fits in with so many things, pictorial representations, manipulatives, symbols, numerical and algebraic expressions, equations, measurements (whether they are direct or indirect).  It just seems to cross the borders of the entire math curriculum.  I feel like this is a strong methodology when teaching students. 

Teaching from this wholistic viewpoint strongly links to the Aboriginal cultural ways of knowing and it has been found that most students (Aboriginal or otherwise) respond well to this type of teaching.  Knowing that all things can be connected gives students a chance to draw from personal experiences and make meaningful connections to their own lives resulting in knowledge that will remain with them.  This gives our students a strong knowledge base to draw upon in subsequent years.  In effect we are giving them tools to solve “problems” that they will encounter now and later in life. 

Teaching with a big idea in mind also helps to form assessment that is meaningful not only to the content that is taught but to the gathering of knowledge that our students are involved in.  Assessments that are created to show connections within learning are rich in potential for student understanding.  Instead of “teaching to a test” assessment can be developed with the needs of students in mind.  Not every child does well on or is even, in some cases, capable of doing lengthy exams to demonstrate their knowledge.  Creating assessments that can be hands-on or delivered in alternate formats is essential in finding the true level of and potential of our students. 

In my own classroom, I will definitely teach math with a big idea in mind.  I believe in a wholistic approach to learning and assessment for students.  I feel like providing connections is a meaningful and very powerful way of passing knowledge to our students.  Giving students the tools for future success is a critical part of teaching mathematics as it provides for a strong basis from which they can grow and expand their thinking.  When students have strong roots in mathematical concepts they are likely to succeed in not only math but other areas of academics.

I have actually enjoyed doing this blog once I got into it and I may even choose to continue it at a later date or create something similar.  It’s always a great idea to share ideas and knowledge that have been given to us. 

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Happy “Mathing”!!

Reference

Charles, Randall I. (2005)  Big Ideas and Understandings as the Foundation for Elementary and
     Middle School Mathematics.  Journal of Mathematics Education Leadership, vol 7, number 3.

COW Grade 5 | Develop Number Sense

COW Grade 5 | Develop Number Sense: Outcomes 7, 8 (page 18)

Outcome 7 - Demonstrate an understanding of fractions by using concrete, pictorial and symbolic 

representations to: 

• create sets of equivalent fractions 
• compare fractions with like and unlike denominators. 

Outcome 8 - Describe and represent decimals (tenths, hundredths, thousandths), concretely, pictorially and symbolically.

My Big Idea:
"Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value."(2005 Charles)

This COW definitely relates to my big idea!  

Equivalent fractions show that even though numbers are written in different forms, they can still have the same value.  Comparing said fractions is a great way of cementing this concept in the minds of students.  Visual aids in the from of pictures and manipulatives give students a strong grasp on the concept and provides opportunities for real-world comparisons further providing students with meaningful instruction and application.

Students will learn that numbers can be represented pictorially or symbolically and still have the same value as the numerical representation.

Here is an example of a great hands-on math activity that will help when teaching fractions to students taken from Hands-On Fraction Games at http://www.ehow.com/list_7158792_hands_on-fraction-games.html

Fraction Pizza

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  For this activity, you need large construction paper circles (the "pizzas") that are pre-divided into halves, thirds, fourths, sixths and eighths (other fractions may be incorporated, depending on the level of the students). You will also need a set of fraction cards (These games all require a set of fraction cards, which can be made easily from a stack of ordinary index cards. Write one fraction on one side of every card, in pencil or some other medium that cannot be seen from the other side of the card. There should be one card for each fraction with a denominator of 2, 3, 4, 5, 6, 8, 10 and 12 (for more advanced students, 7, 9 and 11 may also be included). Have two cards with fractions equivalent to zero (such as 0/5) and two equivalent to one (such as 6/6)) that correspond to the "pizzas"---one for each fraction with a denominator of 2, 3, 4, and so on---and a different set of counters (ideally construction paper pizza toppings, although beads or coins would also work) for each student.

  Each student draws a fraction card, and then puts their "topping" on the appropriate part of the appropriate "pizza." For example, if a student with a supply of paper mushrooms drew a card that said "5/6," he would put mushrooms on five pieces of the "pizza" divided into sixths. Continue until the class runs out of fraction cards or pizza toppings.

  If feasible, this activity can also be done with real pizzas. Simply divide uncooked cheese pizzas with thin lines of tomato sauce, and entrust a real topping to each student.

Read more: Hands-On Fraction Games | eHow.com http://www.ehow.com/list_7158792_hands_on-fraction-games.html#ixzz2Edmnj4XW

COW: Grade 2 | Use direct and indirect measurement to solve problems: Outcome 4

COW:Grade 2 | Use direct and indirect measurement to solve problems: Outcome 4 (page 37)

At first when I looked at this outcome I immediately thought of using different objects as indirect measurement tools. If you go to http://www.beaconlearningcenter.com/WebLessons/AreWeThereYet/default.htm#page6 they demonstrate what I was thinking at first glance.  It is an American website but the ideas conveyed are quite effective and it would be fairly easy to replicate using metric units of measurement.  ie. one meter=the distance between student's outstretched arms.

 I didn't even think of using terms such as largest, longest, thinnest, shortest, thickest and the like.  All of these terms imply indirect measurement when you are asking a student to visually find the answer.  Objects need to be compared to one another in order to find the answer.  I found a cute game on the pbskids website at http://pbskids.org/clifford/games/measuring_up.html that goes through many of these terms.  While I think it goes over the terms quite well, it is at a much lower level than grade 2 and I feel students would tire of it quite quickly.

How does this outcome relate to my big idea?

Using indirect measurement is another way of showing students that a number can be represented in more than one way.  In fact they can be represented in a number of ways even in such a relatively small activity.

1m=student's armspan=100cm=? footsteps=?hands  and so on...

Students can use visual appearances to predict which objects in a set are largest, widest, longest etc and then use a multitude of things to measure them and finally get actual measurements and compare all of their findings.

I think this would be a great hands-on engaging lesson and as a teacher you can easily incorporate student's culture, interests and ways of knowing.

On a side note...my youngest son uses power poles to determine how far we are from the next town when we are driving.  He knows there are 10 poles in a km so if we have 25km to go he says, "Okay, so 250 power poles.  Right Mom?"  You betcha my boy!  :)

More Problems in Planning...Ugh!!

Common problems in planning math lessons may include:

• teacher not anticipating how students will approach content provided
• The teacher may assume that students will struggle or "fly through" a given lesson due to past experiences with other classes and that assumption may be completely wrong.
• lack of planning 
• The teacher is unprepared to present the lesson in an organized manner.
• lack of knowledge
• The teacher is unfamiliar with the content and therefore unsure of how to present it and certainly unclear on alternate approaches to teaching the concept.
• loss of student interest
• Without engaging lesson planning students may become off-task and problems will arise in classroom management.

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Challenges in the Designing of Math Tasks

When creating a math task one must be very aware of the needs of the students and the levels at which they are performing.  Taking into consideration the great diversity of learning styles and levels in every classroom can pose to be a huge task in itself.
Making lessons hands on and relevant to your students is a great first step toward teaching to all of the students in the room.  While you don't want to make the lesson too easy for students, it is important to provide support or tools for dealing with problems that will inevitably arise in every lesson.  Giving students a strong base of knowledge to work form is an important step in bridging learning gaps.
A lesson that seems straight-forward and easy to comprehend may prove to be very challenging for many students.  As a teacher you must be prepared to re-evaluate how the content was presented and try to present it in a way that is perhaps more easy to understand and if that doesn't work then yet another way must be investigated.
This may also be an opportunity for students to extend their learning by posing questions as to what they understand the content to be and how they feel it would be easier to understand.  Perhaps in group discussions students will discover that they actually do understand but where just a little off track in the direction they were headed.  As a teacher you may also discover alternate ways to reach the same goal by letting students work on the content "in their own way".  That is when meaningful learning occurs!

Hot Cocoa Math

My heart is in the lower elementary grades and this activity is a really good illustration why.  Who wouldn`t love having a job that requires you to taste yummy things and play with awesome kids all day?
I love how this activity is so hands on and while students are having fun they are learning crucial math skills that will help them in subsequent grades.
This activity is also another demonstration of my big idea that numbers can be represented in an infinite amount of ways.  Here we see them represented in the marshmallows, as well as on the charts that are created and on the tens squares.  The visual representation of the numbers is wonderful. There can be a lot of discussion and interaction during this activity which makes it much more meaningful to students. This lesson is hands-on and fun for students and teacher alike.  LOVE IT!

This great activity is taken from Apples and ABC's http://applesandabcs.blogspot.ca/2012/01/hot-cocoa-math.html

Hot Cocoa Math

The kids had a wonderful time using marshmallows as math manipulatives.  We are studying numbers 11-20.  The students made a mug out of construction paper and were allowed to glue down marshmallows to display any number between 11-15.

Step 1:  Cut and paste

Step 2: Glue down marshmallows

Step 3: Count marshmallows and write the number on the line

The next part of our lesson, we used our 5 senses to taste and investigate hot chocolate.  After we tasted our cocoa, we graphed our opinion if we liked the taste or not.

This chapter we are learning to count to 20 using 10 Frames.  I decided to use the previous cocoa mugs, to make a center station practicing showing sets of numbers 11-20.  The students chose a mat, and then placed marshmallows in the 10 frame to show the number on the mug.

**I try to post the objective and standards on each bulletin board, so parents and administration know what 

we are working on in class.  Here is what is posted around this board:

Hot Cocoa Math

Objective:  In math, we are focusing on numbers 11-15.  We are practicing counting and writing these numbers.  The students were able to choose how many (between 11-15) marshmallows that they wanted to glue onto their cup of hot cocoa.  We also used our sense of taste to take a sip of hot cocoa to see if we liked it.  Then we graphed if we did or did not like the taste of hot cocoa.  The following standards were addressed:

Math: Number Sense

1.2  Count, recognize, represent, name and order a number of objects (up to 30)

Statistics, Data Analysis, and Probability

1.1  Pose information questions, collect data, and record using graphs

Science: Investigation and Experimentation

4.0   Observe common objects by using the 5 senses: taste, smell

Taken from Apples and ABC's  http://applesandabcs.blogspot.ca/2012/01/hot-cocoa-math.html

I love how the teacher includes parent and administration in what is going on in the classroom and the finished product of this activity makes a great bulletin board for students to enjoy and take pride in their work.  I really like this activity and I think there are even opportunities for assessment in regards to number recognition and representation.

Data Mode!

The following activity about the "mode" of a data set is a great instructional opportunity in the classroom.  It offers an opportunity to teach students mathematical language and activities can be integrated in the classroom to show this in real life situations.  Students can use day of birth and find the mode day within their class or even in a few classes.  They could also pick random numbers from a bag and arrange themselves so that to show the mode and in subsequent lessons the median and mean.  This lesson will set them up well for future lessons on statistical data.

Problem:	The number of points scored in a series of football games is listed below. Which score occurred most often?
	7,  13,  18,  24,  9,  3,  18
Solution:	Ordering the scores from least to greatest, we get:
	3,  7,  9,  13,  18,  18,  24
Answer:	The score which occurs most often is 18.
This problem really asked us to find the mode of a set of 7 numbers.

Definition:

The mode of a set of data is the value in the set that occurs most often.

In the problem above, 18 is the mode. It is easy to remember the definition of a mode since it has the word most in it. The words mode and most both start with the lettersmo. Let's look at some more examples.

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Example 1:	The following is the number of problems that Ms. Matty assigned for homework on 10 different days. What is the mode?
	8,  11,  9,  14,  9,  15,  18,  6,  9,  10
Solution:	Ordering the data from least to greatest, we get:
	6,  8,  9,  9,  9,  10,  11  14,  15,  18
Answer:	The mode is 9.

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Example 2:	In a crash test, 11 cars were tested to determine what impact speed was required to obtain minimal bumper damage. Find the mode of the speeds given in miles per hour below.
	24,  15,  18,  20,  18,  22,  24,  26,  18,  26,  24
Solution:	Ordering the data from least to greatest, we get:
	15,  18,  18,  18,  20,  22,  24,  24,  24,  26,  26
Answer:	Since both 18 and 24 occur three times, the modes are 18 and 24 miles per hour. This data set is bimodal.

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Example 3:	A marathon race was completed by 5 participants. What is the mode of these times given in hours?
	2.7 hr,  8.3 hr,  3.5 hr,  5.1 hr,  4.9 hr
Solution:	Ordering the data from least to greatest, we get:
	2.7,  3.5,  4.9,  5.1,  8.3
Answer:	Since each value occurs only once in the data set, there is no mode for this set of data.

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Example 4:	On a cold winter day in January, the temperature for 9 North American cities is recorded in Fahrenheit. What is the mode of these temperatures?
	-8,  0,  -3,  4,  12,  0,  5,  -1,  0
Solution:	Ordering the data from least to greatest, we get:
	-8,  -3,  -1,  0,  0,  0,  4,  5,  12
Answer:	The mode of these temperatures is 0.

Let's compare the results of the last two examples. In Example 3, each value occurs only once, so there is no mode. In Example 4, the mode is 0, since 0 occurs most often in the set. Do not confuse a mode of 0 with no mode.

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Summary:

The mode of a set of data is the value in the set that occurs most often. A set of data can be bimodal. It is also possible to have a set of data with no mode.

Lesson from Math Goodies Website http://www.mathgoodies.com/lessons/vol8/mode.html