A Bunch of Questions

This is my response to a set of interview question by a parenting magazine.

👀How can I help my child nurture a love for Mathematics?

😜Children who grow up having a love for mathematics are those who have cultivated a productive mindset about what mathematics is. 

You are unlikely to enjoy mathematics if you think that it is all about carrying out rote procedures, memorising and doing tedious computations. 

You are likely to love mathematics if you see it as a field where you figure things out and one where you see patterns.

Children who learn mathematics by manipulating concrete materials and using diagram, children who learn mathematics by interacting with other people, children who learn mathematics without being burdened by jargons right from the start, children who learn mathematics by constructing meaning for themselves, these are children who would love the subject.

👀What materials can I use to facilitate my child's understanding of Heuristics?

😜What does the word 'heuristics' mean to you? A heuristic is simply a way to solve problems. And there are many ways - act it out, draw a diagram, make a guess, write an equations are some examples of heuristics. Heuristics are rule of thumb that one use, often in combinations, to progressively move towards solutions of problems. How do adults facilitate children's understanding of heuristics? By choosing problems that lend themselves to the use of specific heuristics you want the child to learn and by letting them use the heuristics in their own way. Do not place too many constraints on the way they use them. Heuristics are pretty flexible and there is no one way to do, say, guess and check.

👀How often should my child practise math modelling so that he can be proficient in it? 

😜I think you mean drawing bar models. Mathematical modelling is more than just bar models. Children should always be encouraged to represent information using diagrams. They develop visualization when they do so. So, the answer is as often as possible.

👀How can I make Mathematics more relatable to real life situations?

😜That's easy because mathematics was created to describe and extend the world around us. You see multiplication when you see cookies on a tray. You create art by arranging shapes in a tangram set. Dice used in by rad games give rise to all sorts of mathematics problems. Quadratic equations describe the path of a ball thrown in the playground. Tennis tournaments can be described by exponential function. Our everyday language is peppered with mathematical ideas - you often say success is a function of hard work. 

👀How do I handle my child's frustrations in being unable to solve Mathematics problems?

😜By not spoon-feeding them right from the beginning. Children who are independent are resilient and not easily frustrated when faced with a problem they cannot solve straight away. 

Children who are frustrated easily when they cannot solve a problem tend to see mathematics as something that involves routines tasks which can be solved quickly using a formula. 

Find out the reason why they cannot solve a problem - is it the comprehension of the problem, is it the inability in handling multi step situations, is it the calculation? And provide help in that area.

👀How can I help/support my child if he is lagging behind in the classroom? 

😜Focus on basic skills and tasks that are routine.

Explore alternate ways - if long division (say 351 divided by 3) is troubling them, get them to see that 351 = 300 and 30 and 21, all of which are easily divisible by 3.

Use diagrams to help them visualize - it is much easier to see two-thirds of three-fourths using a bar model.

Avoid explaining solutions to children who are struggling to grasp ideas. Scaffold their learning by asking questions. Concrete materials and diagrams are always helpful.

👀Should I engage my child in collaborative learning (eg. learning circles) outside of school?

😜Collaborative learning is always good.

Question about Student Support and Teacher Education

Thanks for your presentation at the Blake School on Monday afternoon.  I was attending with a team of math intervention teachers from our school district in Edina, MN, and we have some questions about differentiation.

1. In Singapore, how do schools provide support for students that are not keeping up with the rest of the class?

2. How do schools support students that are consistently one or two grade levels ahead of their classmates?

Additionally, do most of Singapore’s teachers matriculate from the same university? If so, do you think that is a benefit to math education or a detriment?

I’m looking forward to seeing you again this afternoon in Wayzata!

BanHar responds ...

All Singapore teachers complete their pre-service education from the same university (National Institute of Education NIE). Plus, it ensures consistency but we need to make sure this place provide the best teacher education money can buy. On the down side, the institution faces no competition and unless it has the discipline to improve and innovate despite the lack of competition, it can be the start of a decline. I hope our NIE is of the former. (I used to teach there.)

1. In Grade 1, students who are not on grade level competency receives support in mathematics and English Language - it is a pullout (we call it learning support program in mathematics or LSM). At all other levels, they receive extra support during class (in more difficult cases, there is a learning support teacher in class but that is not the norm). Usually, the teacher will have to take care of these students when the others are doing independent work. In many cases, we will meet with them after school for an hour or so once a week for remedial work. Having said all that, the use of concrete materials is to help every learner, including the struggling ones, learn well.

So we have in-class as well as pullout / separate remedial classes. In some cases we have a learning support teacher ( we call them allied educators, they are not trained teachers but undergo a training program for the job).

2. We provide these students with more challenging problems i.e. the enrichment approach. We do not accelerate them. However, these challenge is also made available to all other students. At upper levels (grades five on wards but sometimes as early as grade three, we track these students i.e. they are put into a separate class and the teachers do more challenging stuff with them, again not accelerating but provide more challenging tasks). For the most advanced they join the so-called gifted education program (GEP) from grade four onwards, where they are challenged with all kinds of things include research and project. From grade seven there are schools that they can go to to enhance their ability / interest e.g. schools of science and mathematics etc. 

Lesson Study Question

I am currently leading my school's (primary) lesson study efforts. 

I would just like to seek your expert advice on the number of cycles of Lessons Study. Right now, my school currently embarks on two lesson study cycles. Some teachers have actually asked if the second cycle itself can be done not by observing just one teacher, but by taking the improved lesson plan back to their classes and conducting it. 

Would such a practice dilute the essence of lesson study given that there will thus be only one lesson during the first cycle where teachers observe teaching and understanding? 

I would greatly appreciate your expertise and take on this subject matter. 

Brenda, Singapore

You can have teachers taking the lesson and then teaching it to their own classes after the lesson refinement stage. In this case, the team has completed one cycle of the lesson. It is fine. Of course, going into the second cycle offers teachers with another opportunity to observe and talk about the lesson. If the team is motivated and can afford the time, doing a second cycle should offer them new things to see and should enrich the professional learning.

Pies and Tarts

I am  a parent of a student in the school you gave a seminar. I found a problem in an assessment book which was confusing. Can you please help me to find out a solution to that problem?

I do not understand the way the answer given in the assessment book.
A Parent in Singapore

The Problem

Students with good number sense will know that the number of tarts and pies have to be in certain multiples given that they do not come in halves and quarters in a shop.

The ratio 3 : 2 tells us that tarts are in multiples of 3 (and pies in multiples of two) but as the tarts are packed in boxes of 4's, it must be in a common multiple of 3 and 4 (12). In the same way, the pies come in common multiple of 2 and 3 (6).

The least possible number for this to happen is 36 tarts and 24 pies - 9 boxes of 4 tarts and 8 boxes of pies. These will be $9 x $5 and 8 x $4 or $45 + $32 = $77.

It is also given that if all the tarts and pies are sold then they will bring in $770.

I think you can finish this up now.

This problem is based on ratio (P5), multiples (P4) and some basic multiplication and division (P3) and addition (P1).

This is a challenging P5 or P6 problem.

Note: P5 = Grade 5

Skeleton of the Solution

About Functions

Can you guide me how to solve Question 7b?

This is from iGCSE past examination paper. I could not even start.

Given that f(x) = 10^x.
(a) Calculate f(0.5).
(b) Write down the value of f^-1(1).

Usman

f^-1(x) is the inverse of function f.
Given that function f(x) = 10^x
So, f(1) = 10^1 = 10
f(2) = 10^2 = (10)(10) = 100 and so on
I think you can figure out 10^(0.5) ... it is the square root of ten, isn't it?

Let's get to 7(b), which is the one you wanted help in.

First you need to know the meaning of inverse of a function.
If a function is 2x (doubling a number) then its inverse is (1/2)x i.e. halving the number.

Given that f(x) = 10^x, the second part is asking you what is the value of x when f(x) = 1.
The answer is x = 0 because 10^0 = 1.

(Note: I use ^ to mean to the power of)

You may want to review the idea of inverse function.

About Range

I am a fourth grade teacher and we are currently working with mean, median, mode and range.  I received a parent email asking to clarify the definition of range.  Her email is as follows:

“ A question came up during homework on the definition of range. Apparently the kids' textbook says that the "range is the difference between the least number and the greatest number".  Range = greatest value - least value. That is not the correct definition. Range is the least number to the greatest. For example, if the test scores in one class range are between 90 to 100, and another class between 60-70. The textbook definition would say the range for scores in both classes is 10, which obvious does not make sense. (I think their textbook is defining what is known as the span of the data but that is rarely used.) A number of parents were puzzled about it but we don't know if this needs to be corrected or at this level the textbook definition should stand. Would you please clarify.”

Am I correct that finding the range does require students to subtract the greatest from the least amount in the data set?  I am not sure how to respond to her question.

Jillian, New York

Definition of Range: 

There are many places one can check on the definition - wikipedia quoted reliable mathematics / statistics textbooks. This is another source for a definition.

You can send the parent some of these links. 

We are often interested about the 'average' in a data set as well how the data distributes itself around the average. Examples of average include mean and median. examples of a measure of this distribution includes range and standard deviation.

Range is the size of the smallest interval that contains all the data and tells us about statistical dispersion.

I tried checking what the parent referred to as "span of the data" but could not find any entry on the internet. Apparently, it is not a conventional term. Range is a more conventional term to describe the idea under discussion. 

Question on Division Bar Model

Trish from Hawaii asked a question about setting up the bar model in a division type word problem.

Question by Trish (Hawaii) from Jimmy Keng

Of course it is not necessary to solve every problem using the bar model.