Prague Session | 16 March 2018

CEESA Conference - Session 1

Fractions - Why Do Students Find Them Difficult?

Yeap Ban Har 

Pathlight School and Anglo Singapore International School

yeapbanhar@gmail.com

Twitter @banhar

www.facebook.com/BanHarMaths

Let's solve two problems to see what is it that makes fractions difficult and what are the main ideas students need in learning fractions.

This discussion add to the learning experiences that we discussed at the pre-conference. The learning experiences are needed for all lessons. Today's 90-minute discussion focuses on things specific to learning fractions. 

A major culprit is the notation used to represent fractions. Do students know that the numerator is but a name? If the know that then everything else they need is from whole number ideas - except equivalent fractions.

Do we focus on the teaching of equivalent fractions.

Through Problem 1, we see the learning of equivalent fractions in a conceptual way that requires visualisation and meaning-making. Students lack of relational and instrumental understanding of fractions is one major obstacle in learning fractions.

While we obsess with additions and multiplication facts for whole number concepts, do we pay the same amount of attention to the basics in fractions which is renaming a fraction in multiple ways? In the learning of finding equivalent fractions, are students merely learning the procedures without any kind of reasonable understanding? Is the learning connected to previous knowledge or is it "multiply the numerator and denominator with the same number"?

In summary, problems with fractions can be fixed by teaching the fraction notation and equivalent fractions well.

That's all. 

As I said, this is one of the easiest problem in the learning of mathematics to fix.

Prague Workshop | 15 March 2018

CEESA Pre-Conference Workshop on Teaching Mathematics: Five Critical Learning Experiences

at International School of Prague

https://www.ceesa2018.com/program/pre-conference/item/35-teaching-mathematics-five-critical-learning-experiences-by-ban-har-yeap

Yeap Ban Har 

Pathlight School and Anglo Singapore International School

yeapbanhar@gmail.com

Twitter @banhar

www.facebook.com/BanHarMaths

1. Some Data on Achievement (e.g. PISA)

Another well-known study is TIMSS

In this workshop, we will examine critical learning experiences that can result in adequate learning even amongst the weakest learners with the average learners performing at a high level while the learning of the most advanced learners are not compromised.

We will also get to observe a class at International School of Prague and hear about one initiative at the school from two colleagues.

2. Case Studies for Workshop

The first part was to experience the learning experiences ourselves. Two ISP colleagues showed us one way to use the physical space - students writing on the wall spaces around the room - to provide a better learning experience 

Some of the apps that I use in this workshop are from this list.

Pieces Basic is another free app that I used in place of the base ten blocks from BrainingCamp.

Case Study 1 - Sharing Art Paper

This journaling task requires evaluation between ideas.

It involves little use of the written words. Students use diagrams or artefacts (paper folding).

Case Study 2 - Flowers

In this journal entry, students are describing one method used to solve the problem. 

Here, students use mostly mathematical notations and a little bit of the written words.

Students must have the experience to explicate mathematical ideas in four forms:

- using things

- using pictures and diagrams

- using written words

- using conventional symbols and notations 

Case Study 3 - Dots

This is the class' homework - to figure out a pattern raise the relationship between the he area of the polygon with a dot inside and the number of dots. Did your investigation yield anything interesting?

Case Study 4 - BOAT

By putting three more points B, A and T create a quadrilateral BOAT.

Let's investigate the angles of this quadrilateral.

What do we notice?

How can we prove what we notice?

Key Questions - What are the critical learning experiences? What are the theoretical underpinnings? How can we provide students with these  experiences? What is your key learning today?

Additional Question - How do learning experiences relate to mathematical practices outlined in common core state standards in the US?

Prague Seminar | 14 March 2018

At the seminar, we discussed conditions for learning mathematics.

Let me tell you what happened in one lesson today and invite you to think about the conditions for learning. What must we do to facilitate learning, deep learning. 

What do you do when you are trying to teach the first day lesson on fractions to a bunch of second graders and a student raises her hand a second time today with the same request "I have a question."

The first question has been handled.

The second question is - "What if there are 5 quarters (fourths) and there are 4 friends sharing them (equally)?

Yeah! 

What if ....?

And that is exactly what happened in class today.

We were figuring out what happens when 2 friends share a piece of square art paper as well as when 4 friends do likewise. They had the square pieces of paper to help them figure out the solutions. They grabbed themselves pencils, rulers and pairs of scissors.

All sorts of things came up.

First they shared some ways of sharing the paper between two friends and I documented the triangle solution. I asked them how they can be sure if the two pieces are the same amount of paper.

Someone said "measure them".

How? I asked.

"Use a ruler."

Before I could respond (I took quite a while to respond, to be honest, about five seconds), someone  said to cut the pieces out. Then put "one on top of the other".

But how can you tell that they are the same amount of paper.

They gave sound methods to check for equality. 

Then they did the same for sharing among four friends. 

Along the way they cautioned about using ruler to draw the lines, also to "fold first before you draw the lines". 

All is well.

Someone used the word "quarter" and I latched in that to show them the symbol for quarter using post-it. 

"I have a question." 

One students asked if we also use "4" when we want a symbol for half.

I showed them the symbol for half.

"I have a question."

Another wanted to know if the symbol is written after the numerator the way fractions are written in words - 1 quarter. 

I showed them where to put the symbols for the words quarter and half.

"I have a question." 

Why did they use 4 and 2 in the symbols?

I asked them if that's a sensible thing to do and got them to explain to each other why they all kinda think that those are sensible decisions.

I even got them to tell me the amounts when I showed them 2 quarters and 3 quarters. A few other things happened.

Then the 5 quarters questions came up.

All I want team to learn today is about equal parts and the naming convention. I was going to wait till grade four to discuss improper fractions and mixed numbers.

"i have a question."

"What if there are 5 quarters and there are 4 friends sharing them (equally)?"

I asked that question and got them to chat while I decide what to do next.

I am not proud of this but I decided to lie.

I pretend I misheard her.

"So our friend is asking what if there are 5 quarters and 5 friends are sharing them equally." Despite the wild waving of her hand, I pretended not to see the girl to posed the question.

The class was able to say that it's easy - 5 quarters, 5 friends ... That's 1 quarter each.

Having settled that - which is division of fractions, by the way - I let the girl protest .... "What if the mystery friend is not there?"

She sure was persistent.

So I packed them off with two options for journal writing.

One, to show in their journal what if 8 friends share a square paper equally. Please use the paper and paste your work in your journal and say how much paper each friend receives.

Two, to figure out the solution to the question our friend just posed. What if four friends share five fourths? Show your thinking and maybe invent a symbol to describe the amount each person gets.

Prague Seminar | 13 March 2018

I did a seminar Edge of Education for parents of students at International School of Prague, discussing how mathematics learning can be used as a tool to develop awesome kids. 

We often focus on skills, problem solving and achievement score in discussing mathematics teaching and learning so I wanted parents to see how education, when done well, is ultimately is developing kids into young adults who are intelligent, articulate, kind - essentially awesome.

How can that happen? 

Find out at the talk in Prague.

What the market place demands of our kids are changing and will continue to change. as they say, the only constant is change. 

Are our kids developing the right kind of skills to put them in a place where opportunities continue to be available to them and they continue to have options?

For nations, developing their intellectual capital is critical. For parents, developing their kids into young adults who are well-adjusted is paramount. How can mathematics learning do that?

A glimpse into what the brain does in relation to mathematical tasks.

Kids learn Maths but they also learn about the nature of Maths and who they are as learners. Mindset, as Dweck tells us, is critical in learning. And mindsets shape one's personality.

How can learning in general and mathematics learning in particular help kids develop skills, cultivate attitudes and increase their capacity to figure things out in critical and hopefully creative ways?

Some examples I used to make my points ...

Prague Seminar | 12 March 2018

Using one example, I spoke about five critical learning experiences that students should have for high achievement in mathematics and provided theoretical justifications for these experiences.

In the time that I had I demonstrated expiring, structured discussion and journaling and mentioned that reflection and independent practice are the other important learning experiences.

As I always do, I prefaced my talk with the fact that different teachers may organise their lessons differently but should include these learning experiences on a regular basis. 

I will expand on this topic at a pre-conference session at CEESA Confererence.

Swansea Course | 5.6 March 2018

Baroody's developmental phases in learning multiplication facts

Honolulu Course | 29.30 January 2018

We started the course by looking at some international data connected to students' ability in problem solving.