Karen, a middle school math teacher shared her reservations concerning the
viability of using differentiation and accommodation in teaching math. She
asked MiddleWeb readers to explain ways they met individual differences
in their math programs.
I was reading the postings about differentiation and accommodating students
in English and social studies classes, but how do you do it in math? I can't
just not teach fractions or whatever we are learning. Mathematics builds
on a foundation, without fractions, for example, students can't explore
probability, or use ratios or understand percents or solve similarity problems
or investigate trig, etc! I must teach fractions (or whatever) to all of
my students.
I teach at a private school so IEPs are not an issue (we don't have them),
but I have been approached about accommodating my curriculum. Do I just
assign fewer problems? (I only assign about 10 big problems a week!) Do
I grade only problems attempted? (I personally have a problem with this).
I will not delete a topic, unless someone can convince me that it is unnecessary
(not foundation or application) and unimportant. (If it's unimportant then
I shouldn't be teaching it to anyone!)
What do other math teachers do?
-Karen
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Chris, a middle school principal, has had the opportunity to watch a
number of math teachers in action. In this posting he described an exemplary
lesson that he had the opportunity to observe. He explained how this lesson
addressed the many ability levels found in any given math classroom.
I'm not a math teacher but I watch them at work. I've seen math teachers
differentiate instruction by providing a range of examples when they are
teaching new skills or concepts. Without going into details, here's one
example. The teacher was teaching how to change decimals into fractions.
He walked through the basics. After he was sure every student had the basic
format he explained that he would give them 8 examples to try at home, but
that he had a 9th and 10th example that were optional, and extra challenging.
He then he put up a repeating decimal with just one digit repeating, like
.4444.... and another with a pair of repeating decimals. I had no clue myself
how to do these, but he did a good job of expanding the basic algorithm
(is that the right word?) to show how to change repeating decimals to fractions.
He introduced some basic algebra and eventually a bit of what would become
calculus. I was reminded that .9999 is actually equal to one. Then I remembered
that that concept was the same idea as the concept of limits in calculus.
(Wow...I finally used it!) So, in a span of about 15 minutes this teacher
had exposed all his students to a basic skill, moved it up through algebra,
and snuck in some foreshadowing of calculus. I'm guessing most of the kids
got the basics, maybe half of the kids got the single digit repeating decimals,
and the one or two G/T kids may have caught a piece of the calculus -- maybe.
As he had the kids start on the 8-plus-2 problems he mentioned that these
were the kinds of problems that would be on the practice quiz next week,
and that he would be watching to see who was ready for the two challenging
problems.
-Chris
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Mary Anne, another observer of math teachers, told readers of ways she
has noticed math teachers meeting individual needs in the classroom.
Like Chris, I am not a math teacher, but I have learned a lot just by
watching them work. One of the best strategies I have seen is called "most
difficult first." The teacher determines which 5 of the problems are
the most difficult. If a student decides to do the most difficult five problems
first, and gets them correct, he "buys" time to work on a math
project of his choosing. If he doesn't get them correct, he gets an extra
5 points just for trying, then has to practice the simpler ones and try
again!
This particular teacher does a lot with simple things. The kids keep journals
explaining new concepts in words. They do projects--one they are working
on now is calculating the various proportions needed to build the statue
of liberty. They are working on proportions! How big is your nose in relation
to your face!? Interesting--and it keeps them thinking.
-Mary Anne
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Marsha believes that differentiation is just as important in math as
in other subjects.
Differentiation is the backbone of instruction as far as I'm concerned in
all subjects. I would think every teacher would strive to learn what their
kids knew and didn't know, put them in groupings (at least mentally), and
plan instruction around the things they didn't know.
Math is the perfect place to do that. Whenever I start a math unit, I spend
lots of time previewing the unit. I have students map out what they think
the unit will be about from reading through the table of contents, then
filling in the bold words (ie glossary words), creating a word wall for
that unit, and then making predictions on what they'll learn. It's pretty
easy from that kiind of pre-unit activity (usually takes about 2-45 minute
class periods) to tell where everyone is.
So many IEP kiddos do understand the concepts, but they have different ways
of communicating their understanding. I'm working with one little guy right
now that could write down what he knows to save him. But he can talk you
through the steps most of the time. And if I could ever teach him to use
a calculator instead of trying to solve everything in his head, he'd be
pretty accurate. So because he resists the calculator and/or multiplication
table, I'm working with him on mental math techniques. His homework is modified
to include more mental math practice stuff than what we are doing in class
--- he just covers the basics. That's because those mental math skills will
help him in the long run.
IEP kiddos sometimes just need you to re-teach in a medium that they understand.
I'm working with an 8th grader who can understand linear equations when
you phrase them in terms of skateboarding lingo. That required him to teach
me his vocab and I turn around and translate the problems for him. It's
beyond me why he can solve y=4x by itself, but he can tell you how wheels
you'd have if there were 15 boards. (I even know what a truck is now!!!!!!!!!!!!!)
I'll wean him off, once I build his confidence up.
The other part of this involves the testing. You can easily extend worktimes,
move them to a quieter(?is that a word?), have someone read the problems
aloud, etc without much effort. I've even taped me reading the problems
and the student listened to the tape on headphones over and over. Last year
for the first time, I tried highlighting things for them not to miss on
the test. If I'm working on process skills, I highlight the important info
and they have to plug it into the correct place. If I was working on identifying
the important info to solving the problem, I'd have them circle the important
stuff and skip solving the problem. It would depend on my objective.
I also created an answer sheet that had blanks for each Step. And yes, I
put the proper number of steps. That gives struggling math students a boost
of confidence that they can do it as well as some clue on how many things
they need to write down.
I hope this helps give some ideas. I love math and found that differentiation
built up weaknesses and played to strengths. IEP students are only one piece.
There are as many modifications for the adept students.
Marsha
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Jan has many students with learning problems in her math classes.
Most students who have severe discrepancies in math come to ME for math
class. I use the Saxon math program which uses lot of repetition and presents
the concepts in many different ways using mutidisciplinary word problems.
The new concepts only have about 5 problems out of the problem set and the
rest is reviewing of previous concepts. If you do not have any special
ed services, you could have the student do evens (or odds) only. You could
also have them do a portion of their testing (over concepts) orally when
applicable.
I would think, though, that if a student requires special education services
and your school cannot provide them, there could be a problem with liability.
I'm not sure how that works with private schools, though, since the student
is there by choice.
One thought is that the student may not be able to go through the book as
quickly as everyone else so he could work on his own individual assignments
while the class is doing their work. In my (LD) math classes, each student
is working at his own pace on his/her own assignment. They read the lesson,
do the practice, I check the practice to make sure they have the concept,
and then they go ahead and do the Problem Set.
Jan Jewell
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J Scott asked Jan to elaborate on the math series used in her classroom.
Jan, Are you using Saxon math with your LD kids? Whatever you're using,
are they able to read and follow the directions independently? How do you
teach them new concepts? It sounds great to say you do it one-on-one or
even in small groups, but that is very time consuming. How do you handle
it all?
-JScott3335
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Jan responded.
Yes, I am using Saxon math. I ask them to read over the lesson and then
do a Practice Set. Then I check the Practice Set. If they don't "get
it", I go back over the lesson with them right at their desk. Then
they go ahead and do the Problem Set. Saxon has these new student workbooks
from the special ed adaptions they offer. In the front of the workbooks
are all the student reference pages they offer. I encourage the students
to use the reference pages.
Also, I DO have them "recycle" if they have trouble on two successive
lessons in a row, or they score poorly on a test. It seems to be working
real well.
-Jan
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Karen Cole provided Middleweb readers with some very practical ways that
she allows for individual differences in her math program.
I agree that differentiation is important and I have done it in a variety
of ways:
First, my students can approach a math problem using many paths. I (almost)
never teach a prescribed method to find a solution. My students develop
a strategy (with help from their peers and/or guidance from me) that makes
sense to them. Some are sophisticated, others are relatively simple, but
all will eventually get the correct answer. My goal is to help those with
simpler, less efficient, strategies find a more direct route to the solution.
But only when they are ready!
For example, when we explore fraction computation we start by using pattern
blocks, naming each block as a specific fraction of the yellow hexagon (the
whole), and using the blocks in our computation. Most students quickly develop
strategies for adding/subtracting fractions with the same denominators (add/subtract
the numerators) and gradually develop methods for adding/subtracting those
with different denominators (use equivalent fractions with the same denominators
then add/subtract). My students can use the blocks as long as they need
them! Most students are very excited when they no longer need their manipulatives.
Secondly, classes are ability grouped by 7th grade. I have mixed feelings
about groupings. The advanced class definitely benefits. I can cover more
content in more depth, (without losing anyone) than if I had heterogeneous
classes. But, I have always been concerned about the "regular"
class. Do they give up because they are the "dummies"? Both groups
get the same units in the same order, but the advanced class goes at a faster
pace with some more advanced content thrown in.
I like the idea of previewing a unit, but my situation (only 3 hours of
math per class per week and having the same kids for 3 years in a row) makes
it difficult and unnecessary (especially by 8th grade!).
Thirdly, the math curriculum that I use (CMP) is very real-world based so
students that are still developing abstract-thinking skills can tackle problems
effectively. The initial algebra unit (7th grade) painlessly introduces
students to algebraic ideas (graphs, tables, variables and equations) by
using a theme, a bike tour! In this way algebra has meaning for students,
(like the skateboarding 8th grader), it is not simply symbolic gibberish.
Fourthly, some students work quickly, others more slowly. I don't believe
that you have to do something fast in order to do it well. So, students
may complete an unfinished quiz/test at lunch (the only time available).
Students who wish more privacy may take the test with our LD teacher. I
allow all 6th graders to use any notes, old quizzes/homework etc. when taking
tests/quizzes. Hopefully this will encourage them to actually take notes!
7th graders may use a 2 page "cheat sheet" on unit tests and notes
on weekly quizzes. 8th graders may initially use a "cheat sheet"
but by the last trimester they may only use it as a study tool. I have never
actually modified a test for a student, but CMP does provide modified unit
tests in their teacher guides, I should take a look!
I have a few questions for math teachers:
1) How much math time per class per week do you have?
2) Do you modify your curriculum to meet the needs of IEP or LD students?
If so, how?
3) Do you allow notes to be used on quizzes? If so, in what way?
4) Do you collect homework daily, weekly, not at all? Do you go over it
in class? Do you grade every problem?
5) Do your students work together? If so, for what percentage of the time?
6) How many students do you see? How many per class? How many classes? How
many preps do you have?
7) What would you call your teaching style? Is it effective? Would you change
anything? Have you changed your style during your teaching career? Thanks.
-Karen
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Robbyn answered Karen's questions.
I'm not teaching math now, but I will try to provide some info based
on my previous experiences:
1. I only taught math, so I had 4 classes a day for approximately 75 minutes
everyday.
2. I would modify as required by the student's IEP. Most of my students
with IEP's listed use of a calculator as a required accommodation, so, I
always let them use calculators. I would also modify assignments or tests
as necessary. Perhaps by giving fewer problems or allowing notes. Most of
the time I would give all students the same assignment but would make adjustments
if necessary when I graded them.
3. I did not allow notes on quizzes.
4. I collected work daily and we would always go over it in class before
I collected it. I gave full credit if students attempted to work all of
their problems. Homework was worth 10 points per day. It was an easy 10
points, if they did it. On homework or class-work, (homework was just an
extension of what we did not get finished in class), my main goal was for
students to use this as a means to practice the skill we learned in class,
and to check for understanding while we went over the problems the following
day.
5. My class and I work together about 50% of the time.
6. I see 120-124 students per day. (30-31 per period over 4 periods)
I had as many as 3 preps- advanced, honors and regular program.
7. I am not sure what I would call my teaching style. The way I measure
my effectiveness is by the students- if they feel I have made a difference
or not. I taught 7th grade and every year the 8th grade teachers assigned
a Mathography (tell about your previous mathematical experiences) to the
students on the first day of school. I would always hear from the teachers,
that students had very positive things to say about the previous year. I
would also survey students at the end of each year to find out what they
felt was successful and what they didn't. I changed my style every year,
every day, and every period, and sometimes my style would change from one
student to the next. I had to find a style that worked well for them not
just for myself.
An additional comment:
I too have mixed feelings about tracking. But, one positive I have noted
is some students become leaders in the regular class that would not if the
classes were heterogeneously grouped.
-Robbyn
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Christine elaborated on a number of ways she adapts her math program
for students that require support.
I am a Seventh Grade math teacher. I teach at a DoDDS school in England.
I am fortunate to be in a co-teach situation with the Special Ed. teacher
for one of my math classes. She is wonderful and I learn many strategies
just by watching her. About 6 students in that particular class are on IEP's.
So, having the extra help in class is one of the accommodations, but there
are many students in other classes that need accommodations as well.
Usually, I assign and expect the same assignments for my IEP students, unless
I can see that it is too much. On tests, my IEP students take the regular
test, but can be given extra time in their Special Ed. class to finish.
One of my students last year had a problem with writing and so rather than
requiring complete sentence answers, the expectation was that he answer
as completely as he could. For assignments requiring a lot of writing, he
could use a word processor in his learning strategies class ( a Special
Ed. program).
Other accommodations include preferential seating, reduced assignments,
modified tests and if necessary modified curriculum. However, I have not
had to modify the curriculum up to this point. I will say that the Special
Ed. teacher has a lot to do with that since she really follows up and keeps
the kids on track.
-Christine
PS. This is my first try at responding. I have been reading for weeks and
really have picked up a lot of wonderful ideas. I am working on my Masters
in Cooperative Learning and have appreciated all of the information that
has been posted up to this point. If anyone has any other sites like http://home.att.net/~teaching/filecab.htm,
I would appreciate you sending them my way. I thought this one was a gold
mine!
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Marsha addressed Karen Cole's questions as well.
I don't agree with your view of preview. Time invested at the beginning
is well worth it throughout the unit. Preview helps them to find common
threads to tie everything together. A little like scaffolding......
But.....What a brilliant fading out strategy. I too believe that not everyone
works at a speed that I could anticipate as their teacher. So I love your
modification ideas here.
1) How much math time per class per week do you have?
10 hours
2) Do you modify your curriculum to meet the needs of IEP or LD students?
If so, how?
Already answered.
4) Do you collect homework daily, weekly, not at all? Do you go over
it in class? Do you grade every problem?
Yes we grade every problem. But no I don't always "take grades".
On the overhead as they come into class, I have the "answer" listed
or as I worked it. They check their own paper. If they find a mistake, they
must cross it out and write down the correct answer and then solve it using
their own understanding. If they can't rework the problem, then they put
that problem # on the overhead and I work through the problems students
can rework on their own. We've had lots of discussion about "cheating"
and that they are the losers because I won't know they didn't understand.
By about the beginning of November, they've bought into the whole concept
and settle into the learning mode.
5) Do your students work together? If so, for what percentage of the
time?
50%
6) How many students do you see? How many students per class? How many
classes do you see? How many preps do you get?
It depends on the year. Some years I taught 4 sections, some years 2
sections. Usually we have 25-32 per class depending on the school finance
formula that year. I've had as few as 2 preps and as many as 4.
7) What would you call your teaching style? Is it effective? Would you
change anything? Have you changed your style during your teaching career?
I would call my approach "Constructivist". I hope so, but
who really knows? I have dramatically changed my style over my career- less
regiment for the sake of order and more demanding of student's intellectual
effort.
- Marsha
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Melba finished this productive string with some practical advice.
Special needs students whether they are officially identified as LEP,
ESL, Spec. Ed., or any other label MUST be afforded the best instruction
that will result in learning acquisition. I have taught math for 18 years.
I also have had Special Ed students with IEPs, which required modifications.
Each student has specific modifications that are prescribed to meet his/her
needs according to the student's learning disability. The law states that
the regular education teacher MUST follow those modifications. We legally
do not have a choice. We can always do more, but never less than expected.
The best person to implement and design those modifications is the teacher
who provides the primary instruction.
You should know that student better than the resource teacher who may only
see him/her a couple of hours each week. All students must learn the entire
curriculum for each math course they take. Teach a step by step process.
The kids will get it, just give them time, practice, patience, and understanding.
"If they can't learn the way you teach, then teach the way they learn."
I don't know who said or wrote that, but I heard it at an ASCD conference
in Florida and again on this Listserv. It's very good advice.
-Melba
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