Blog 4 - Invented Strategies

Intro

     As was stated in the Van de Wall text, students do not spontaneously invent wonderful computational methods while the teacher sits back and watches.  The first step is to create an environment where students feel safe to explore numbers and where it’s okay if a strategy doesn’t work out because you can go back and rethink the steps and troubleshoot and you can confer with your classmates and share ideas on strategies that way.  The text also states that part of the role as a teacher is to help “move unsophisticated ideas to more sophisticated thinking through coaxing, coaching, and guided questioning.  This way you could help a student put their strategy into words and be able to apply the polished strategy to other problems.  Number sense, seeing patterns in number, feeling comfortable with manipulating numbers is a concept essential to allow students to develop strategies. 

     I’d like to focus on two problems from my math interview at Houston Elementary School; one problem without assistance and one with assistance.

Problem 1

The first problem asked the student to find the difference between two numbers:  “Jared has 13 pennies.  He loses 4 of them.  How many pennies does Jared have left? (separate result unknown)”

     At this point, the student had become comfortable in using the Unifix cubes and reached for them immediately; I think this is in part because when I would ask him to show me another way he could have solved the problem, I’d suggest he use the cubes.  The student counted out 13 pennies and connected them end-to-end.  He then removed 4 of them and stated the number 9.  I asked him if he could explain what he did.  He said that he counted 13 pennies and he took 4 away because the boy lost 4 of them.  I asked him if he knew another way to solve the problem.  He said that he could also subtract 4 from 13 in his head and also get 9.

From this problem, I gathered that the student was using the take-away subtraction strategy briefly described on page 221 in the text.  It states that the take-away strategy is actually more difficult than the add-on strategy, but that textbooks emphasize take-away as the meaning of subtraction.  Even though the student seemed to have used the cubes at this point to appease me, he clearly showed that he could count back in his head from 13 to 9.  I was not surprised that he was able to do this, but I found out later that the student had trouble dividing up numbers in anything that was not in solid tens.

     Another way I think the student could’ve arrived at an answer would be to use one ten rod and three single cubes to represent the 13 and 4 single cubes of another color to represent the four.  He could then cancel out the singles, followed by breaking down the ten rod to cancel out the final single cube.

     A second way the student could’ve solved the problem is to count up from 4 to 13 and keep track either in his head or on his fingers of the amount between those two numbers.

Problem 2

     The problem the student and I worked together on was a 2 digit by 1 digit multiplication problem.  I originally presented the student with a problem with slightly higher numbers.  I think he realized that he didn’t have enough fingers to count that high and he seemed unsure of where to start using the cubes.  I gave him a similar problem with slightly smaller numbers.  The student again seemed lost.  I offered to work with the student on solving the problem.  From my observations up to this point, I chose to use repeated addition to show the student a way to solve the problem.  The problem asked:

“There are 4 children going to the water park.  It costs 21 dollars per person.  How much money will it cost for all the children? (multiplication)

     I pointed out that if it costs 21 dollars for each student and there were 4 students going, we could add 21 four times.  I wrote out 21+21 in the standard algorithm and had the student add.  He added 21+21, told me the answer, and I wrote down the answer.  Then I asked him if there was another way of finding out 21 and 21.  He said we could go to the side and add 21 and 21 again.  I wrote out 21+21 again and asked him to tell me the answer.  Again he said 42.  I asked him what we could do next to find out our total.  He said we would add 42 and 42.  I wrote what he told me to write, added it and asked if 84 was correct.  He looked over my work and agreed.  I asked him then how much it would cost for 4 kids to go to the water park if it cost $21 for each student to get in.  He answered 42, then quickly corrected himself and said 84.  I asked him which one he was sure of.  He answered 84.

     Repeated addition is discussed in the text on page 227.  I have to agree that it seems like a very inefficient way of arriving at an answer.  And in this case, the student seemed unsure of what result to look to for a final answer.  I am not at all convinced that he knows why we added 21 four times to answer the question.

     To solve the problem, the student could have separated the tens and ones into 2 tens and 1 ones.  The student could then multiply the 2 tens by 4, resulting in 8 and multiplied the 1 by 4, resulting in 4. 

     The student could’ve also used two base ten rods and a single cube and decided that to find out how much it would cost for each student, he would have to have 4 of the same 2 base 10 rods and 4 of the single cubes to represent the cost for each student.

Stepping Into Teaching

From the time I stepped into the school, I felt welcomed.  The lady in the front office was absolutely friendly and seemed happy to help me track down my cooperating teacher.  I'm surprised my neck didn't break in the first 10 minutes in the school - there were so many items on the wall.  While there were a few pre-printed posters on the walls, the majority of material was made by students.  Teachers had a wide variety of ways that they had students tell a little bit about themselves including photographs of themselves, drawn pictures of themselves, and collages of their interests.  From the start, I felt that the school works hard to make the school about the students and celebrate their voices in a variety of ways.

I'm fortunate to be able to spend two math periods with the same teacher, but different students.  This unique opportunity has allowed me to see the way a general education teacher handles the same content with different sets of students.  The majority of the times that I've been in the classroom during math, it is very rare that the students are at their desks completing worksheets.  The students are often engaged in whole group instruction before being released to small groups or pairs to play a variety of "math games."  I really like how the teacher rotates groups during small group time to check in with students and provide extra feedback.  My cooperating teacher is usually in the room during this time as well.  She plays a variety of roles.  I like how she isn't in the room to only attend to students who have IEPs, but any student who is struggling.  I have adjusted my mindset to emulate her in that way - avoiding the mentally of only serving "my kids."

In working with different students in a variety of contexts, I've noticed that when manipulatives are handed out as supports, students don't always use them.  I observed students whom I know have IEPs that allow them the use of manipulatives and calculators, but choose not to use them.  One student in particular seemed intent on only using what was attached to her - her fingers.  I tried to engage her to use the manipulatives or multiplication chart or the calculator - she said she didn't need them.  Another student I observed kind of pushed around the manipulatives, but I couldn't make heads or tails of what he was doing.  I decided to ask him.  He said "I don't know, but I know I'm supposed to use them."  

Lightbulb moment:  we have to teach kids how to use manipulatives!

Today, as I moved around the room from group to group, I noticed that I immediately went to the steps taught since the beginning of time of what order to multiply numbers in within the same, old algorithm.  While I know that they are expected to know this algorithm and use it, I really want to learn other ways/strategies I can show students to apply to their work.  

My so-called math life

 Peak Experience

Math has generally been a subject that’s come easily to me.  On top of that, I had excellent support at home from my father who enjoyed solving math and logic problems with me.  A time when I think my abilities in math were recognized was when I was a sophomore in high school.  This was the year of geometry.  I nailed the first few quizzes and aced the first test.  At this point is when my teacher asked me to work with a couple of students to help them with the material.  I think I recall smiling at this point.  Me?  Helping to teach my classmates?  Cool!  And this was only the beginning.  The teacher allowed me to have some wiggle room in making modifications for some of my classmates and break down problems in a way I thought would benefit them.  Side note:  later on, the teacher decided to not require me to turn in homework.  Score!

Nadir Experience

I’m not sure that the experience I’m recalling was a negative one, but it did bring out some negative feelings in me.  I remember being very frustrated in 1^st grade and having to complete so many worksheets of math problems.  More and more worksheets.  It never seemed to end!  One day, I cracked.  I remember the sheet in front of me had an outline of a pumpkin and different arithmetic problems inside.  Pumpkin or no pumpkin, this was another one of those worksheets.  I reached in my supply box, pulled out a brown crayon, and wrote across the sheet, “I QUIT!”  I think I was at the point of tears.  I walked the sheet over to the teacher, handed it to her, and turned to go to my chair.  She stopped me and took me out into the hallway.  The tears definitely flowed at this point.  Between the snot and tears, I tried explaining how frustrated I was to be completing worksheets with material I already knew how to do and said that what she gave me to do was “worthless.”  Or pointless.  I’m not sure which of those terms I used but I’m sure it was one of them.  Here’s where there was a turning point.

Turning Point

After my fantastic fit in 1^st grade, I had a chat with the teacher during recess.  She agreed to let me work on more advanced math sheets and gave me a Snickers bar.  That moment changed the way I dealt with math teachers since.  In 6^th or 7^th grade, I took a practice ACT and did fairly well for my age.  I negotiated with my teacher to let me work on different math topics than my classmates and she would send me at least twice a week to the computer lab to work on a program that was supposed to help students practice for ACT and SAT questions.  I previously discussed the opportunity in 10^th grade geometry.  Later, in my senior year, I engaged in a bold conversation with my math teacher in which I informed her that I would do very well in her class, but found the slow pace to accommodate other students frustrating.  I proposed that I would come to class Monday mornings to get the assignments for the week and come in Fridays to drop homework and for tests.  I’m not entirely sure why, but she agreed to this as long as I earned at least a 90 every time.  Oh.  I forgot to mention that this was the first class of the morning.  I’m not much of a morning person. 

I’ve thought about my own frame of mind during these encounters and where I had gotten the chops to draw the boundaries and make the rules for such interactions.  However, I don’t know that I’ve ever thought about why a teacher would negotiate with me and what that says about that teacher. 

Other...

There’s a moment when I was three or four years old that mom reminds me of at least every six months.  She knew that she would have to teach me about money and the work that goes into earning money after a visit to Kmart.  I insisted that she buy me a toy.  She insisted that she didn’t have enough money.  I insisted that she could “charge it.”

During the summer of 09, I spent about 6 weeks in Denmark.  After little research, it turned out that Denmark is pretty expensive.  Because of the housing option I’d chosen, I would be responsible for stocking my tiny fridge and cabinets.  This also meant that I would need to do some mental math over and over in the grocery store when deciding what was reasonable to pay for a jar of Nutella.  It also meant I needed mental math skills when shopping for souvenirs, buying lunch after class, and when paying for overpriced beverages at the Mexican-themed bar down the street.  I’d have to calculate if I should purchase a scarf for my mom at this store or the one across the street; not to mention deciding if the price difference had to do with the authenticity of the item in question. 

"Greatest challenge"

In the course of my long college career, I’ve taken a few math classes.  I opted mainly for the easier ones for any number of reasons – keep the ole GPA up, laziness, avoid bruising of the ego, who knows really.  Then I encountered a couple of courses I could not avoid: M316K and M316L.    If they were meant for future teachers, then these courses would teach me how to teach math, right?  Well.  The first course, M316K, consisted of scribbles of proofs on the chalkboards, playing with numbers in formats other than base 10, and playing nim.  What I got out of this course was that a) I was the queen of nim, b) the loser of nim would become my best friend, and c) some instructors will have no interest in learning your names.    M316L is a little bit of a blur.  The instructor didn’t quite get our names, but he did seem passionate about the subject matter and about teaching in a way to really engage students.  I can’t really say that I remember any specifics from that course other than being introduced to the movie “Flatland.”  So, here I am.  I am one semester away from being a certified teacher and am in a course that’s supposed to help me learn how to teach math to students.  So far, the instructor knows my name :-D

Special Education Teacher

Special education seemed like the area I had to teach in after I worked at a day program for adults with developmental disabilities.  It was quickly apparent which clients had had a positive school experience and which had been warehoused.  During the year that I worked with the clients we participated in the Meals on Wheels program, delivered newspapers, went on tours, had drinks at coffee shops, and helped a few with job coaching at a paper company.  The task at the paper company was quite tedious and repetitive, but it led to money; money earned.

Math is a subject I’d like to teach students with any level of disability.  Math skills mean that you can figure out how to divide the money you have to cover your needs and wants.  Knowing how to make change can keep you from getting short-changed.  Math skills facilitate transportation in that you can calculate what time you need to get on the bus to get to a destination at a certain time.  Math skills mean independence.

Response to readings

1.  How does taking a problem-solving approach to teaching math differ from first teaching children the skills they need to solve problems and then showing children how to use those skills to solve problems?

The problem-solving approach somewhat assumes that all of the students come with the same knowledge base and would be equally receptive to the information/format being given by the teacher.  The approach doesn't really give wiggle room for bringing in previous experiences and other frames of knowledge - there's one right way of solving the problem.  Period!

2.  How do you think your experiences, feelings, and beliefs about math will impact the kind of teacher of math that you will be or the kind of teacher of math that you want to be?

One thing I'm working hard at is to put aside the "it's easy!" frame of mind.  I've been fairly fortunate in math to be able to grasp concepts fairly quickly and had great support at home when I didn't understand.  Through my experiences as an intern teacher, my brain has been expanding to explain things in different ways and I've been learning more ways to bring in students' interests to try to make sense of what I'm asking them to do.  I'd definitely like math to be a fun and interactive part of the day rather than something we "have to do."

3.  Not everyone believes in the constructionist-oriented approach to teaching mathematics.  Some of their reasons include the following:  There is not enough time to let kids discover everything.  Basic facts and ideas are better taught through quality explanations.  Students should not have to "reinvent the wheel."  How would you respond to these arguments?

It's frustrating and unfortunate that teachers are constantly (in some districts even more so than others) under pressure to make sure every student in their class is on the same page and can keep up with the pace to make it to TAKS week.  While some think that kids discovering what works for them may be time consuming, it's what works for them!  It is of far greater use for them to actually understand what is happening than to have them memorize formulas and be able to pass the test.

4.  We sometimes want to jump in and help struggling students by saying things like, "It's easy! Let me help you!"  Is this a good idea?  What is a better way of helping a student who is having difficulty solving a problem?

Not a good idea!  This will only make the student feel worse and be down on themselves;  if it's "so easy" then they must be "so stupid." - No!

Helping a student relate a problem to something they enjoy and are familiar with will serve them to not only work on the current material, but also add more problem-solving solutions to their framework of understanding.

5.  Reflecting on how tasks were defined in the Van de Walle chapters, how did the tasks presented in the Behrand article to Learning-Disabled students help in their mathematical development?  Please give specific examples.

Having the students work together and compare ways of solving the problems seemed to have been a great way for them to either reformulate their own answer or to feel even more strongly about their answer and defend it to the death.  It also provided a check that didn't feel like an authority was shooting down their method/answer.