The Cornerstones of Mastering Math 2/4: Social Learning
The qualities needed in the future life and jobs are for example being able to find, analyze, and use information in new and unfamiliar situations. We see also lists of those most needed skills and many list include problem-solving and team player skills, detail-oriented and the ability to prioritize.
We teachers want also to prepare our students to believe in themselves and to be curious about the surrounding world. We want them to take charge of their own learning so that they could reach their full potential.
One student has a figure and she/he tells to another what to draw. The drawer is not allowed to show what she/he has done. Students have fun while practicing math vocabulary and trying to understand each other!
Jean Piaget (1896-1980), a French scientist, stated that assimilation and accommodation require an active learner, because problem solving skills cannot be taught — they must be discovered. Learning should be student-centered in the classroom, an accomplishment through active learning.
How to develop and support these skills at school and during math lessons? One way is to work in groups which strongly support many qualities of the social learning. Most of us have experienced that the team can be stronger than its members total.
Math can give students an experience, which strengthens their confidence to solve problems and develop their own critical thinking. Working together they can talk, test and interpret solutions. We can give them a learning environment where they enjoy working on problems. Learning situations where students are working and solving problems in groups facilitate them with the opportunity to discuss math – and adopt a “math language”. Social learning includes activities like action tasks in groups, board games, cards and dice games.
In this activity students are investigating dependency between two variables.
The Cornerstones of Mastering Math 1/4: Learning by Doing
It doesn’t matter what age people are, they enjoy when they are doing something and the joy is even bigger when they realize that at the same time they are also learning! The younger the students are the more important is that they can touch, look and test the properties of the possible object.
How many edges are in a cube? Third grade students need a model of cube in their hands so that they can see and count the edges. Most of the six grade students can tell the number of edges without seeing the model.
How many cubic decimeters are 200 cubic meter? Unit conversations have been taught on the 5th grade but 6 grade students did not have any idea what to answer! Once students built a model of cubic meter with sticks they could give the answer by deciding and understanding the size range of answer.
Examine the Platonic solids and determine how many faces, edges and vertices leach of them has. Students from 7 to 8 grades need to build most of Platon solids to be successful in this research.
You can give more demanding research to the older students. What is the pattern in the number of faces, edges and vertices in Platonic solids? Some of 9 to 10 grade students start their research by drawing the solids, but still some students like to build some models.
We think mostly that learning by doing is building models using paper and scissors. Learning can also happen when students are for example measuring. With paper and scissors student is often investigating alone, but also in many cases you can build groups where students are working together. Then the mathematical talking come to more important role. Students’ talking should be important part of learning situations. They use their own language, test it with their classmates and after practicing they begin to use more precise math words.
Different groups got different tasks. They had to estimate first and then measure their object. Here students are counting the number of raw macaronis. Students don’t even realize that they are learning because they’re so interested in what they’re doing.
When learning by doing is a part of learning environment you can see how students are focused, fully attentive, motivated, engaged, and enjoy their work.
Jean Piaget (1896-1980), a French scientist, taught about discovery learning; the idea that children learn best through doing and actively exploring.
It dawned on me, my way of teaching math was wrong!
I was sitting in Leeds when it happened, I realized ‘I have taught math wrong for over a decade!’
I had joined a group of science, math teachers and professors to visit Leeds University, UK, because it was famous for their research of science teaching. It was explicitly based on a constructive view of learning.
I was placed in small groups, forced to solve problems working together. Talking, testing and finally coming to a consensus within our group. I was amazed, even choked when the Finnish professors in our group did not have the right answer to the problem! I could see that they were facing the same situation as me and many other teachers.
We began to think what learning really is. When are you best at learning, what kind of environment helps you to learn. That was the “spiritual” starting point of Paths to Math.
During the following years we began to develop new materials for math lessons. We tested it in our classrooms across Finland. Our material differed a lot from the common math books, which based more on behavioral way of learning. Once we knew we had it, we found a publishing house to print our material into textbooks.
One of the high points in my career was when I had a chance to tell our math ideas for eLearning material to prof. Howard Garner, upon his visit to Helsinki, Spring 2010. My conversation with him only supported our decision. We then started with Paths to Math in the cloud.
Big publishing houses want to sell books with little or no changes from previous books. The knowledge we have regarding learning changes. We have more information and an ever changing environment around us. We are in balance when schools must prove that they are still the best place for learning and developing of a child’s mind.
We refuse to let children do trivial exercises, alone in rows with textbooks, tablets or PC’s. We want to give them a different path for learning – alone, in pairs, groups or individually with the help of eLearning material in various teaching methods and learning environments.
Please feel free to sing-up for your free Paths to Math account from our web-pages. You can see first chapter in every Module. www.pathstomath.com
Attitudes and stereotypes are mathematics’ worst enemy
Have you ever met a person who would proudly tell you that she/he cannot read or write? Is this a strange question? But you have maybe met a person who has told you that he/she cannot mathematics, because also his/her mother did not understand mathematics? I have heard many times from adults all kind of explanations why mathematics is impossible for them to learn or understand – and it is also clear that they don’t even bother!
How do people remember mathematics at school? Probably they remember them sitting all alone in rows and the teacher was solving an equation on blackboard and then it was their turn to repeat it – over and over again. It was often hard for them to see the connection to real life and they started to lose their motivation.
Where from the negative attitude towards mathematics originates? Family member’s attitudes play very important role and we may easily say something negative without understanding its impact to our children. Jo Boaler (2015) writes that the messages we give students can change their performance in mathematics dramatically. Students ideas about their ability and potential seems to be extremely important, more than previously have been understood.
Sometimes newspapers are presenting mathematics in their cartoons with humor that is not favorable for mathematics learning. Cartoons can also be supporting the importance of understanding mathematics (number sense) with humoristic way.
Artist drawing from the module Pre-Algebra, chapter Estimation from www.pathstomath.com
Children are facing with whole punch of negative attitudes towards mathematics around their life, even from family, friends, newspapers and so on. It is not a wonder that every other students would rather take out the trash than do mathematics? Students often see mathematics boring, irrelevant or even frightening. When teachers get new students in their mathematics class they also meet the attitudes of mathematics learning. We have to consider that the problem is not with mathematics itself but part of it can be on the way it is taught.
What students think about math?
Have you ever met a person who would proudly tell you that she/he cannot read or write? Is this a strange question? Have you ever met a person who would tell that he/she cannot math, because also his/her mother did not understand math? I have heard many times all kind of explanations why math is impossible for the person to learn or understand – and it is also clear that they don’t even bother!
What is the reason that student think that math is only memorizing and practicing endless list of equations?
We have asked in the beginning of 7 th grade students to draw a typical picture of math lesson.
Drawing 1. Student and his/her classmates are sitting alone, in rows and the teacher is teaching.
Students are sitting in rows, most of talking is teacher doing, teachers shows how to solve and there is no connection between students. The emotional feelings from their faces are telling us that they are confused, don’t understand what is going on, bored and even frustrated and frightened.
Drawing 2. The student feels anxiety forward mathematics.
How do you remember mathematics in school? I remember us sitting all alone in rows. The teacher was solving an equation on blackboard and then it was our turn to repeat it – over and over again. It was often hard for them to see the connection to real life and we started to lose motivation
It is not a wonder that every other students would rather take out the trash than do mathematics? Students often see mathematics boring, irrelevant or even frightening. The problem is not with math itself but on the way it is taught.
During developing Paths to Math material and using it with students for three years, we have asked them to draw a picture of typical math lesson. The change is huge. They have memories from different learning situations, they are working together and from their faces you can see the interest and motivation to math.
Drawing 3. The drawing is from the bird sight! Formulas are correct.
Drawing 4. Student remembers different action tasks. All classmates are smiling.
Math is not a list of rules to be remembered. Math is a way to look at the world that really make sense. Students can now understand math’s procedures, properties and proofs for themselves. Math is not anymore a dull misery but challenging, stimulating and even FUN.
As a teacher you can feel that you have prepared your student with more possibilities and better probabilities to choose his/her postgraduate studies. He/she can also tell to her children that math was his/her favorite subject!
Please feel free to sign-up for your free Paths to Math account from our web- pages. You can see first chapter in every Module. www.pathstomath.com
Make the Pi Day of the Century in your School
Many of my students have a difficult time understanding some abstract math concepts. The best way to help them is to find activities involving real-life objects and opportunities, engaging students from the concrete to abstract. The concept of π is familiar to students, but they really don’t understand, what it means.
I need to spend some time collecting materials. I try to find circular items like a magnifying glass, a clock, plastic cylindrical containers like soup cans, or oatmeal tins, measuring tapes, ropes or strong yarn and calculators.
Students work in this activity as a pair-share. First students measure circular objects and collect their findings about the ratio of the circumference to the diameter. This part of the activity goes well and they like to shout that they already understand pi!
The second part needs larger space than your regular classroom, because they have to draw a large circle. A playground outdoors or a spacious corridor will do.
Take this activity outside and allowing them to use their own shoe as a measurement, they will be amazed! Drawing a circle in the dirt or on the snow with the help of rope and stick is not so obvious for them.
During measurements they often change the measurer, the other is measuring the circumference and the other diameter. Can you do this! You get fascinating conversation about it.
This activity touches students deeply when they discover that even when using their own shoe as a measurement the ratio remains the same.
In this method they have enacted the Greeks investigation on this idea and obtained the same values as they did. Most students even at the junior high school level have heard of π but only after doing this type of activity do they really understand the concept.
Date 3.14.15 is a great opportunity to arrange this in the school yard! Make the Pi Day of the century.
Find more fun mathematics exercises such as this on in Paths to Math. This you find from the module Geometry 2 , chapter 5 and 5.3 Going around in circles.
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