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.

Spiral Curriculum

Good day Dr. Yeap! I attended the seminar last May 21-24 at SM Megall (see photo). Thank you for sharing your expertise and time with us. I just want to ask you about spiral progression. How do you apply it in Math? Can you give an example? Cely

Singapore mathematics curriculum emphasizes the spiral approach based on Jerome Bruner's explanation on spiral curriculum. The idea of the spiral curriculum, according to Bruner - 'A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them.' Most people will not miss the idea of 'repeatedly' but may miss the subtle notion of 'building upon them' and 'until the student has grasped the full formal apparatus' of the target concept. In Singapore curriculum, addition is taught four times in Grade 1 (this is a core idea and they are new to it) - addition with 10, within 20, within 40 and within 100. Students get to revisit the idea of addition repeatedly but each time building on the strategies that they already had. When they add with 10, they count all and count on, perhaps with the use of concrete objects and drawings. Later, in addition within 20, they learn to make ten before adding, effectively acquiring the notion of place value. Later they progress to more formal approaches such as adding ones and adding tens in the formal algorithm. Thus, it is not mere review of materials. It involves extension. In a similar way, multiplication of whole numbers is taught in grades one through four; addition and subtraction of fractions is taught in grades two through five; area of plane figures is taught in grades three through seven; solving equations is taught in grades seven through nine. As a result, in Singapore, Algebra is taught across grade levels in high school (grades seven through twelve). Thus, we do not have the practice of teaching Algebra, Geometry etc separately. They are all under the subject of mathematics. There is geometry in all grade levels.

Question on Retention

Question - How do we help students retain what they have learned.

ATeacher in Hawaii

Answer

A good program arranges the topics in a certain way for various purposes. To help students retain, one of the ways is to arrange topics in such as a way that students have many opportunities to revisit a particular key concept or skill.

Let's use the example of equivalent fractions. After it is learned in Grade 3, the chapters that follow it are adding unlike fractions and subtracting unlike fractions (but limited to cases such as a third plus a sixth or three tenths subtract a fifth where it is necessary to rename only one of the fractions to make both like fractions). The two weeks or so of constant writing fractions as equivalent ones as students add unlike fractions help them consolidate the skill learned. It is critical that a new skill is well consolidated before students leave them behind.

The next grade level when student said such fractions but for cases where the sum exceeds 1, students get to review finding equivalent fractions. In grade five, when students are dividing say a half by three, one of the methods involves renaming one half as three sixths before proceeding to divide the three parts into three. Using this method, three fourths divided by three can be done straight away but to divide three fourths by two requires renaming three fourths as six eighths. The opportunity to review equivalent fraction in a different context enhance the retention of the skill of finding equivalent fractions.

Two principles discussed here - ample consolidation after a new skill is learned and review but not a mere repetition but done in other / more challenging contexts.

Singapore curriculum is arranged for this to happen. Good textbook authors arrange the topics for this to happen.