Geometric Containers can be filled with water, rice etc.
Ropes and Ruler for tying knots activity
"Math
Manipulatives" This is common vocabulary for present
day math education and at the same time it should not be unfamiliar to teachers
and students in older generations. In planning a math lesson, today's teachers are
encouraged to be thinking "manipulatives, manipulatives,
manipulatives!". Instead of having that box of tiles or cubes in the back
corner on a shelf, they are up front and centre, used in almost every lesson or
at least it is explained to the students how manipulatives could be used in the
lesson. As a student I remember having manipulatives in the classroom however
they were mostly used as an aid for learning instead of a tool to build
learning concepts. What I mean by this is teachers would not immediately
incorporate manipulatives in their lesson and when students would struggle with
concepts the teacher would refer them to the manipulatives station. This could
have been just from my own personal experience but there seemed to be some type
of stigma surrounding students who required manipulatives in math. It was
almost as if the students who understood the concepts and got the answers right
away without the use of any further visual were somehow stronger than the
others. This rationale of course is not true; manipulatives are used far beyond
the use of a visual aid. The first and most obvious use for manipulatives in
math is to engage students in their learning. There are so many forms of
manipulatives such as apps, games, and the traditional tiles or household
objects all of which make math fun and learning worthwhile for the students. In
our "Teaching Mathematics" class this week our instructor had brought
in sample activities that make use of a variety of manipulatives. One activity was
used for teacher linear relations. It involved ropes which the students had to
tie knots in order to determine the type of relationship between the length of
the rope and the number of knots. No matter what the manipulative being used
is, the teacher has to ask themselves is it useful, is it fun, is the activity
memorable enough for future student reference.
Algebra Tiles
Manipulatives are also used to
develop a student's critical thinking skills similar to the skyscraper problem
in my last post you can have students at all different levels try a problem
with manipulatives and see where it takes them. For all students the
manipulatives should help to visualize their problem and where they need to go
from here. For some of the stronger math students, having the manipulative
might allow them to come to a conclusion alternate to their original thoughts
and they are able to learn there are multiple ways to facing a problem. In
class we experienced something similar when our teacher handed us each a card
with a number or word and in a group we were to create one sentence from our
cards. This activity was used as more of an icebreaker however you could repeat
this process by having different variables on the cards and asking students to
create a function or have descriptive words such as max/min and have students
describe a modeled situation.
Word Cards
Another use for manipulatives in the classroom comes as an advantage
for both teacher and student. In my class we had discussed an article on the
difference between Relational vs. Instrumental understanding. In a mathematical
context instrumental understanding is following rules and applying formulas
without fully understanding the reason why and relational is the latter. In
order for students to have more of a relational understanding teachers can use
manipulatives to make that connection for students between the "what to do"
and "why are we doing it".
Below
I have attached a link to an article titled "Why Teach Mathematics with Manipulatives?". I believe the
article does an excellent job arguing why manipulatives are critical in the
classroom and it clearly explains that through the use of manipulatives
students facilitate their learning by making the strange become familiar.
Retrieved from https://pbs.twimg.com/profile_images/602027362247516160/mc_7Jfjv.jpg
This week in my "Teaching in Mathematics" class
we focused on a student's ability and process in problem solving. Problem
solving
questions are most often the hardest questions for student to complete
and/or understand. The most common phrases from students during problem solving
are "I don't get it!", "What are we doing!", "What's
the answer!" and the worst one "I give up!". My teacher
presented us with "Skyscraper" math problems and let me tell you, the
atmosphere in our classroom full of adult Teacher Candidates was not much
different from a class full of students. To give a brief summary of the
Skyscraper problem, the students are given a handout with a square grid (can be
2x2 and up) and on the sides of the grid each square is labeled with a number
as pictured here in this 4x4 example:
Retrieved from https://solvemymaths.files.wordpress.com/2015/01/ss.png
Retrieved from https://solvemymaths.files.wordpress.com/2015/01/ss1.png
We were given linking cubes by our teacher and were
instructed to find the number of cubes needed on each square in order to
represent the numbers on the side of the square. Huh? Exactly my thoughts! Well
our only clue was that this problem is similar to Sudoku puzzles where a single
number can only be represented once in each row and column. Our group started
to build our towers for each square and right away I noticed our different approaches
to solving this problem. I had taken the blocks and built the required number
of buildings to fill the grid. Based on the Sudoku clue I knew that for a 4x4
grid we needed 4 buildings of each height (4 cubes tall, 3 cubes, 2 cubes, 1
cube). I needed to see what we were working with before I made decisions about where
to place these buildings where as my group members were placing cubes right on
the grid linking new heights and taking cubes away as they filled the grid. Once
our group had reached the point of frustration, we asked our teacher for help
and she explained that we should be looking to count the number of "pop
out" cubes from the top. We attempted from this perspective for a while
with no progress. In the end we did solve the problem by taking on a different
view and counting the visible faces of the cubes. Once we knew the key to this
puzzle we jumped from the 3x3 grid straight to the 5x5 and by applying our accrued
knowledge completed it with ease. Now that we fully understood the problem we
also realized that we could add one block to each building and have the same
correct answer.
The same
process and results can be seen in any problem solving situation, the goal for
us teachers to is change the negative phrases I mentioned earlier to more
positive and hopeful questions such as "What can I do next?",
"What would happen if..?". The issue we should be focusing on is not
that the student is struggling; it is how they react and overcome the struggle.
My university math professor would say to us "struggle is good" and
we would all laugh and possibly cry at the thought of working through some of
those calculus problems but in the end I know that he was right. Whenever I had
a question I found impossible, it challenged me to think creatively, try new
things and not give up. Also for the people reading this saying "I
hate math I would have given up eventually" rest assured I did not
solve all of those impossible questions and I did get the answer from my
friends or the professor but it was because of my initial struggle and failed
attempts that I understood the
problem and the solution. Know that this can be applied to any subject or any
problem in general. Before students give up or are told the answer they
continue to learn as long as they make an attempt, success or fail they are
gaining understanding through experience. This message I am sending is deeply
related to the Constructivist Learning Theory in which "the core idea is
that learners actively construct their own understandings, rather than absorb
what they are told or copy what someone shows them" (The Problem-Solving Cycle: Professional Development for
Middle School Mathematics Teachers The Facilitator’s Guide, 2003).
This week I leave you with this video that explains
the answers are never easy, we have to grow in our experiences and whether it
is in something you hate or something you love it is possible for everyone
learn.
References:
The Problem-Solving Cycle: Professional Development for
Middle School Mathematics Teachers The Facilitator’s Guide. (2003) (1st ed.).
Retrieved from
http://www.colorado.edu/education/staar/documents/FacilitatorsGuide.pdf
Khan Academy,. (2014). You Can Learn Anything. Retrieved from https://www.youtube.com/watch?v=JC82Il2cjqA
My name is Stefanie
DiSimoni I am a Teacher Candidate at Brock University and I am a "Mathemaddict"! I truly
love math and hope to spread the love to students, teachers and parents through
this portal. In this blog I will be posting everything related to math that I
find exciting and/or awe-inspiring. I
also will be posting my opinion on teaching certain topics in math and I will
share my experiences with math in the classroom.
