FRACTIONS: MATHEMATICAL “TOWERS OF TERROR” TAKE TWO

FRACTIONS: MATHEMATICAL “TOWERS OF TERROR”

TAKE TWO

In our last episode, we discussed recognizing fractions and the terror created by “stacking numbers” into numerators and denominators.  Most students understand “sharing” items to be shared evenly.   This sharing is simply dividing.  In the progression of mathematical learning, students learn to count, add, subtract, multiply, divide, and finally fractions.  This progression is natural.  Sharing with your friends is an idea in fairness and equity.  The difficulty becomes the actual numbers themselves.  

After teaching our students to look a “part of the whole”, we expect them to begin manipulating those strange numbers and perform our favorite operations upon them.  Students are now expected to add and subtract fractions, along with multiply and divide them.  While in the natural progression of numbers, multiplication and division come after addition and subtraction, these operations are the most difficult operations to perform with fractions.  The reason is three very scary words----“Least Common Denomination.”   Telling students that you can not add halves and quarters, because they don’t have the same denominator sends another wave of terror through the classroom.

  The students need a three-step combination to dispel the terror and crack the LCD code.  The first part of the combination is ensuring that we are clear in our instruction and all parties in the classroom are involved in the learning.  It is important for all parties involved in the classroom to work together to create a key to break the code–the code of adding and subtracting fractions.  The classroom should have a class-created and student-contributed anchor chart, which is in plain sight of all the students.  It is also a great idea for each student to have his or her own smaller version of this anchor chart close by to use as a reference.

The creation of the class-contributed anchor chart will allow the students to assist each other in their respective learning.  It gives the students more practice in supporting and critiquing each other’s work.  Only when the entire class works together, can the true learning begin for all.  Each student needs to know that they are contributing to their own education, as well as each others.  

The second number in the combination is ensuring that each student is knowledgeable in fraction equivalents.  Knowing that ½ = 2/4 = 3/6 = 4/8 = 5/10, along with some equivalents for 1/3, 1/5, 1/4.  These equivalents will give the students some beginnings for the addition and subtraction of fractions with different denominators.

The last number in the combination would be a strong grasp of their multiplication factions.  Factoring the denominators is an important step in computing the LCD, which makes knowing their basic factors a necessity.

Once the students have gathered all of the necessary numbers to enter the combination, they are ready to begin.  However, it is the process to the combination that is the major contributor to their learning.  Students need to see that they can do it, before they will begin they can.  This three numbered combination is key to it.

FRACTIONS: MATH’S “TOWERS OF TERROR”

FRACTIONS: MATH’S “TOWERS OF TERROR”

Standard of Mathematical Practice No. 4 states that students need to be able to model with mathematics.  Modeling is practiced early in a child’s math career.  In kindergarten and first grade, students are taught to “draw” a model of their problems.  For example, Brody and his class went to the apple orchard.  Brody picked 3 apples and his friends, Harry and Monty, each picked 5 apples.  How many apples did they pick all together?  

In order to solve this addition problem, the students will make a model of the problem, by drawing a set of 3 apples and 2 sets of 5 apples.  These beginnings allow students to visualize their problems and how to solve them.  Being able to visualize is the first step to understanding fractions.  It has always amazed me how the creating parts of a whole can confuse otherwise mathematically savvy students and their adult homework help.  I have a friend that is able to calculate relatively well, but just freezes when fractions are involved.  I believe she would benefit from creating a “picture” when working with fractions.

Apple Problem Representation

 Once a new mathematician has his or her model, the calculation to 13 apples becomes more manageable, as the "calculator" can count the number of apples to reach the answer.

Some Practical Manipulative Ideas

Being able to create a model that represents the whole and its equal parts is as necessary to beginning fractions, as it was to begin adding and subtracting.  The models allow a student to see the interactions of the fractions with the whole.  It also allows the students to manipulate the fractions and gain an understanding in a concrete way.  This will also allow their understanding to move from the concrete to abstract.


As the saying goes, “A picture is worth a 1000 words.”  I believe a model representation of fractions would save a lot of heartache.

GROWING MEANS KNOWING WHY!!!!

GROWING MEANS KNOWING WHY!!!!

There are many issues facing mathematics classrooms.  One major issue is to move away from how and help students move toward why.  Why does a certain mathematical formula work?  It is becoming necessary for students to justify their work and not just reach the correct answer.

In their article, Moving Students to “the Why”, written for Mathematics Teaching in the Middle Grades, April, 2015, Volume 20, No. 8, pages 484-491,  Michael Cioe, Sherryl King, Deborah Ostien, Nancy Pansa, and Megan Staples conducted the two year JAGUAR (Justification and Argumentation: Growing Understanding of Algebraic Reasoning) project

As students prepare for more higher mathematics, they need to be able to explain the “why.”  Standard for Mathematical Practice No. 3 require the creation and construction of viable arguments and critique of others.  This standard leads to the idea that each student is to have the ability to support their methodology and critique their peers.  These two practices have a great impact on later mathematics. 

The impact that justifying an answer has on a student’s higher mathematical success is shown as they advance to high school geometry.  Geometry is an entire class dedicated to the justification of why two triangles are congruent.  The logic behind these proofs is in actuality proving “the why of shapes”. 

Secondly, critiquing another student’s thinking allows the students to take charge of their own learning and make connections through their peers’ work, as well as their own.  It allows them to work within their comfort zone, rather than outside of it.  It also allows them to see where they may have erred in their calculation and could also lead to a much more streamlined process later.

These reasoning skills are skills we use everyday.  There is belief that algebraic thinking and “math” will never be needed after we finish school.  However, we use math and algebraic processes everyday.  We need to reason through our daily decisions and determine our budgets, when shopping.  These are skill learned in math class and will be carried through our remaining learning and life.

MULTIPLYING AND DIVIDING: AN ALTERNATE DESTINATION

MULTIPLYING AND DIVIDING:

AN ALTERNATE DESTINATION

In Chapter 2, the authors discuss multiplication and division.  Their main focus is on understanding the operations conceptually.   Their point is that a student needs to understand the concepts behind the operations to gain operation sense.  

Operation sense is gained through the use of word problems, prior to introducing any standard algorithms or requiring rote memorization of the facts.  This conceptual knowledge helps a student’s algebraic reasoning and allows them to use what they already know about numbers to solve the problems.

In multiplication, a student is able to visualize equal amounts of objects into a certain amount of groups to reach an answer.  It also allows students to invent their own strategies to reach the solution to the problems.  The progression to multiplication begins in Kindergarten, by learning to count by 10's, 5's and 2's.  This basis is furthered by learning to add doubles in First Grade.  Finally in Second Grade, the student progresses to repeated addition.  These foundations allow a student to reason through arrays and models of the word problems to discover their own way to solve the equation.

PROBLEM

I have decided to redecorate the classroom.  How many desks will we need to have 6 rows of 5 desks?

EXAMPLE 1

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This dot array allows the students to see the equal number of desks in each group and reach the answer of 30 in a multitude of ways.

1: 5 + 5 + 5 + 5 + 5 + 5 = 30

2: Group the rows into 3 tens and repeat the addition. 10 + 10 + 10 = 30

3: Simply count by 5's to reach the answer.

These arrays can be helpful.  The students are given the opportunity to group the number of objects together to discover their own way to reach the answer, prior to learning and memorizing 5 x 6 = 30.  These methods and their knowledge of place value will also help them in solving multi-digit problems, as well.  Students are apt to use the same strategies for multi-digit multiplication that they used in multi-digit addition to answer the problems.  The ability to decompose the numbers into their respective parts and then solve the problem will strengthen their number and operation sense.

PROBLEM

There are 25 desks in our classroom at the moment.  We need to have 30.  How many more do we need?  Also, how many rows of 6 desks can we make, once you have the required number of desks?

EXAMPLE

The children are able to solve multi-step problems by this stage, so reaching the answer of needed desks being 5 is easily reached.

The final question is to divide the total number of desks by the known number of desks in a row.  Again, the students’ number and emerging operation sense will give multiple methods for solutions.  Creating a model  is just one way to answer the problem.

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Students, as they count to 30 and created the rows of 6 desks, can easily see that there will be 5 rows of desks.  This will allow them to come to the conclusion that 30 / 6 = 5.

Students can truly learn their basic facts, when they are given the chance to invent their own strategies to solve problems.  Further, by sharing their invented strategies with their classmates, they are vested in each others educations.

TEACHERS ARE "SEEDS", TOO!

TEACHERS ARE "SEED", TOO!

Teachers sometimes need to be seeds, just like their students.  It is through this process, we are able to improve our skills and become better at our jobs of tending the seeds in our care.  These times for teachers are sometimes through conversations with other teachers, rather than being planted in a garden.  I recently had the opportunity to participate in a Twitter chat with other teachers not only in Ohio, but around the country. Each Thursday, at 8:30 p.m., the Ohio Council of Teachers of Mathematics (OCTM) hosts a Twitter chat.  On January 30, the chat’s title was “Generational Trauma: Learning our students to foster systematic change”.

There were five questions that asked teachers to discover their roles in their students’ lives.  I was riveted by the multiple view points, along with having some of my opinions matter.  The first question asked us to consider different learning types.  There are some students that learn differently than others.

The second question dealt with ensuring all children feel as though their learning is important, even if they are not able to accomplish a certain task as quickly as others.  I believe it is important to help every student succeed.  Some just need a little more help than others.

I was impressed by the discussions that centered around the ensuring that each student have the ability to “find” their own way through a subject.  Students that are given this opportunity are empowered and more receptive to the subject matter when they have the ability to discover the answer on their own.  I believe this is very important in solving mathematical problems.  I have always viewed math in a number of ways, sometimes not the same way as the teacher.  I enjoyed having the freedom to find my own way to solve an equation.  

I really enjoyed the chat and the opportunity to discuss these issues with others, not as a future teacher, rather as a colleague.  It was a refreshing and enlightening evening.  I am looking forward to my next chat.

PLACE VALUE, ADDITION AND SUBTRACTION NOT YOUR MOTHER’S ARITHMETIC

PLACE VALUE, ADDITION AND SUBTRACTION

NOT YOUR MOTHER’S ARITHMETIC

Understanding Place Value

In Chapter 1, the authors discuss the necessity for a proper understanding of place value to perform the arithmetic operations of addition and subtraction.  The authors state that in order to properly understand the standard algorithm for adding multi-digit numbers, students need to have a precise understanding of each number’s place value.  Place value is an understanding of how numbers are read.  These values are what gives numbers meanings.  To ensure that a student has a complete understanding of place value, teachers need to be aware of their students’ prior instruction.  Kindergarten starts place value with 10 ones and some more.  This understanding is expanded through each year, until a student is aware that 3,425 is Three Thousand, Four Hundred Twenty-Five and that there are 3 thousands, 4 hundreds, 2 tens and 5 ones.  Further, a student needs to break down the numbers to understand that each higher place value represents ten units of the lower values.  In other words, 1 thousand is equal to 10 hundreds, 1 hundred is equal to 10 tens and so on.

Once students have a complete understanding of place value, it is much easier for them to perform the arithmetical operations necessary to reach the solutions to an equation.  In reaching the solutions to an equation, the authors emphasize that the “standard algorithm” (the usual way to solve a multi-digit equation) should be an offered strategy, not the only strategy offered to solve an equation.  This concept would be more difficult to include for a teacher that learned the standard algorithm by rote memorization, rather than exploring other ways to represent the problem.  As one of those students that learned to “carry and borrow the one,” it is somewhat difficult to distinguish alternate strategies from my usual method.  

I can see the benefits of allowing students to reach their answers with invented strategies, rather than insisting on the standard algorithm.  A student who discovers that 325 + 435 = 760 with a his/her own strategy allows the teacher to discover the processes that the student uses.  If the student’s strategy is rationally related to solving the equation, then the student is more likely to be able to grasp more complex strategies.

Each of the strategies offered in the chapter allows students to discover their own path to reaching a needed sum or difference.

Breaking Down Story Problems

The use of story/word problems becomes very important in mathematical instruction.  These types of problems allow a student to connect mathematical ideologies to real world settings.  Teachers need to be cognizant of word usage in these types of problems.  It is important for students to not only look for key words, but to read the problem and use their reading comprehension strategies to determine the operation needed to solve the problem.  The use of key words are helpful, but may not be the operation necessary.  There are many anchor charts and bulletin board sets that contain the key words for addition, subtraction, multiplication and division.  These charts can sometimes cause confusion, if the student believes the words are only for one operation.  As an example of the confusion for “altogether”, consider the following word problem.

John has 17 pencils.  How many pencils does John need to get to have 20 pencils altogether?

In this problem, the student is required to subtract 17 from 20.  However, students may confuse “altogether” to be an addition problem and add 17 to 20.  In order to solve the problem correctly, students need to use all of their strategies, including reading comprehension, to solve the problem.

Choosing Our Words Carefully

The authors emphasize that teachers need to choose their words carefully, when teaching a child to regroup.  Regrouping is what most adults would call carry (addition) and borrow (subtraction).  As many of us learned to add multi-digit numbers using the term carry, it is important to consider the possible confusion of the “carried” number’s place if you simply say, “carry the one.”  We understand, now, that carrying the one is actually regrouping 10 ones to 1 ten.  However in the following problem, these words may confuse the students.

        236

+      147

  However, our students may not grasp that concept.  In order to help students grasp the concept of regrouping, it is important to state, “6 + 7 is 13 ones.  We can separate these numbers into 1 ten and 3 ones.  Now, we can place the 1 ten into the tens column and leave the 3 ones in the ones column.”  This wording maintains the proper place value, rather than stating it as “carrying the one.”