Tuesday, December 4, 2018

Problem Solving Task

Problem Solving Task

Alijah is trying to mark the height of an aquarium. She also wants to mark different heights of the aquarium to ensure it holds enough water for her fish. She knows that when it is full, it holds 24 litres of water. She also knows that the base is 30 cm wide and 40 cm long.
(a) Find the height of Alijah’s aquarium.
(b) Using a piece of string, how can Alijah mark the heights when the aquarium is ¼, ½ and ¾ full?

Solution and Discussion:

This is a fairly typical fifth or sixth grade volume decomposition problem. It involves several concepts and skills:
   ● Conversion from cm to cubic cm to litre, and back again. 1 L = 1 000 cubic cm
   ● Area x height = volume, and the relationship of straight, square, and cubic units
   ● Understanding of volume as a part-part-part-whole problem, that is, length x width x height =        ghjkvolume, and any decomposition of that
   ● Measurements and fractions of measurements

(b) Alijah can place a piece of string from the base to the top of the aquarium. She can then fold the string in half to find the ½ full mark. Repeating the process again she can mark the ¼ full mark. Finally, by placing the ¼ length string at the ½ way mark, she can find the ¾ full mark.

Wednesday, November 21, 2018

Conceptual Approaches to Teaching Division

When observing a fifth grade teacher the other day, I noticed that while the students were engaged in many positive ways, they relied upon DMSB -- Daughter, Mother, Sister, Brother -- or some other mnemonic to remember the procedure for long division. They were also ignoring the value of the unit they were dividing. This is very common, but there are better ways, as Singapore Math shows us! 

Why avoid mnemonics like these to help divide? 

          ●  They lead to place value errors, because all digits are treated the same. 

          ●  It’s no easier to remember these than an actual procedure. 

          ●  Students can remember them in the wrong order. 

          ●  They reinforce the idea that math is “magic,” 
following rules that no mortal can understand. 

How can division be taught conceptually?

There are multiple approaches that can lead to the understanding of division. In early grades, students take a number of objects and split it into equal groups. A counter such as the unit cube in Base Ten blocks can be used, and the Matholia tool can help with this.

Work with arrays and place value disks can reinforce the idea of division as repeated subtraction as well. So can using bar models, as illustrated in the article Bar Modelling For Two Types of Division.

Some math approaches undercut conceptual strategies by always representing the problem horizontally or in division’s case, using a division bracket. Starting by representing the problem horizontally allows more flexibility in thinking and does not guide the student into immediately jumping into an algorithm. The problem can always be rewritten in algorithm form if desired.
Once the students reach a solid understanding of what is happening in division, the division can be taught in multiple ways. What is commonly considered  the  division algorithm is just one of several, and not even the most efficient. Two algorithms that can be used to scaffold the understanding of the common long division algorithm are:

Partial Quotient Division -- layout 1:
In this style, students use whatever math facts they know to divide the numbers. They then continue to subtract and perhaps repeat the same fact multiple times in order to reach their goal. Then the partial quotients are added to find the total quotient.

Partial Quotient Division -- layout 2:
This style resembles traditional long division more closely. Eventually they will be able to combine steps to find the total quotient.

With practice, students are able to estimate closer to the product they are trying to reach, and they can eventually practice long division as traditionally taught, but while keeping the value of the unit in mind and not needing mnemonics. Teachers can assess understanding by asking the student to explain his or her work using place value language.

The traditional algorithm is usually considered most efficient. However, for students with a command of multiplication, short division is even more efficient. An example is shown here. Again, teachers can assess understanding by requesting an explanation of the thinking involved.
Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

Tuesday, November 6, 2018

Problem Solving Task

Problem Solving Task

Jennifer, Rory, and Wayne were all born two years apart. Jennifer is the oldest, and Wayne is the youngest. The sum of their ages is 39. How old is Rory?

Solution and Discussion: 

This problem is similar to “consecutive integer” problems commonly presented in pre-algebra. Those who are comfortable with variables would likely solve for a variable. However, with Rory being the middle child, a common mistake would be to forget to add 2 to the youngest age to find Rory’s age.

Here is a sample bar model solution. It supports visualization of the situation and the difference in ages. It also supports algebraic thinking and attention to the variable for which the student is solving in the end. Practice in this type of approach leads to fewer mistakes of the type mentioned earlier. The Matholia Bar Model tool can help students develop these visualizations.

Bar Model Solution:

Algebraic Solution:

Tuesday, October 23, 2018

The Importance of the Unit: Part I

An area of struggle for students is manipulating numbers once they get beyond basic counting numbers. A big reason for this struggle is that when manipulating multiple digits, students get intimidated by the larger numbers. They are also confused by how various digits work in algorithms. This shows up with operations on multiple-digit numbers, fractions, and decimals.

One early source of the challenges may be the way we count. In many Latin-based languages, the teen numbers follow their own naming rules and don’t obey anything resembling logic. For example, what on earth does “eleven” mean? And what about “sixteen” -- why is the “teen” or tens named after the ones?

Counting returns to normal once you get to the twenties, with twenty-one, twenty-two, etc. following place value order and the way we write them. But those teen numbers confuse young brains and can be responsible for many of the errors young children make, such as writing “seventeen” as 71 and not 17. They are being logical, generalizing the rules for writing numbers based on language; it’s our language that is illogical.

Contrast this to languages such as those spoken in China, Japan, and South Korea, in which counting proceeds according to place value. They count 1-10 with individual words, like in English, but subsequent numbers are represented by place value: ten-one, ten-two, ...., ten-nine, two-ten, two-ten-one, two-ten-two, etc.

This language advantage leads to a mathematical advantage. Language researchers believe this is the reason children from those countries can, by the end of preschool, count to 100 consistently, and solve almost three times as many simple arithmetic problems as US children. This gap can be closed by teaching the “say ten” way of counting, mirroring the counting used in these Southeast Asian countries. For example, children would learn to count the usual way, but then to rename the counting sequence using Say Ten counting.

In conjunction with working with manipulatives to show that every number is a sum of ones, and that a ten is a shorter way of naming ten ones (renaming), students can develop a strong sense of place value that makes numbers easier to work with. They learn that digits work similarly, but it’s the place that gives the digit its value. Using the Matholia place value disks tool and place value strips tool can help the student see these dynamics in action.

Students can also use the 10-row Rekenrek to track the tens and ones as they count. This will reinforce the concept of the “make-ten” strategy for addition and subtraction, which then can be generalized to multi-digit numbers. For example, adding 12 to 29, a student can decompose either number to make a ten and then add the remainder. A visual example of the mental math, using number bonds, is shown below. These strategies are practiced extensively in Primary 2.

Then if the student encounters 290 + 120, the same strategies can be used, just renaming it as “29 tens plus 12 tens,” understanding that the sum will be in tens, so 41 tens or 410.
The ways this understanding extends to fractions and decimals will be addressed in future articles.
Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

The Matholia Team

Tuesday, October 16, 2018

Bar Modelling For Two Types of Division

Many people aren’t aware that there are two different types of division. Even if you search the web, only one type mostly comes up. It’s the type we learn first: that when we share one quantity, we split it into equal parts. This is ​partitive​ division, or dividing a quantity into parts. When we divide a whole amount, partitive division tells us how many items there are in each group. Let’s look at a couple of examples.

Jeanine bought 24 buns for a party. She wanted to put an equal amount on each of 6 tables. 
How many buns should she put on each table?

Jeanine should put 4 buns on each table.

Armando earned $56 for a day’s work. 
If he worked 7 hours, how much was his hourly wage?

Armando’s hourly wage was $8.

The other type is ​quotative​ division, and it’s less familiar to most people. In quotative division, we find the number of groups, not how many in each group. It is also known as measurement division, or finding how many of a certain unit it will take to measure something. Let’s look at a couple of examples.

When the party planner was setting up, he found he had 32 place cards. He would like to put four on each table. How many tables would he need to set up?

The party planner would need to set up 8 tables.

For a pizza party, a parent has a maximum budget of 6 pizzas. He plans on each child being allowed three slices. This pizza parlour slices each pizza into 8 slices. How many children can be invited to the party?

16 children can be invited to the party.

The above example is as interesting way to model dividing a whole number by a fraction (an upper primary standard in most countries). Although bar modelling may not be as useful as a tool to calculate quotative division problems, it can be very helpful to help students visualize the problem.

Often when teaching such problems, many teachers resort to multiplying by the reciprocal, or 6 × 8/3 (We need to avoid teaching Keep-Change-Flip though, as this 3 mnemonic can lead to lots to misconceptions.).

Did you know that you can also divide using common denominators? You can
change 6 to 48/8  and solve 48/8 ÷ 3/8 . See if you can figure out why this works.

Leave a comment with your thoughts!

Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

Monday, October 8, 2018

What is more important than the right answer in primary mathematics?

“There is no right or wrong in mathematics:” a quote from a memoir by the daughter of Fischer Black, a famed US mathematician, that she remembers him saying often. What does that mean?

When we learn how numbers work, we can do so in one of several ways. The way many of us have been brought up to learn mathematics is through memorization and learning procedures, such as algorithms. One example is the long division algorithm, which is opaque to most students and is arguably one of the most difficult to learn. Teachers and parents alike can relate to frustration as students confuse which number to put above the little house, which below, and more. 

Another issue with this algorithm is that the way most people learn it. The digits are independent of value. For example, when dividing 125 by 5, first you would see how many times 5 goes into 1, then 12, then… but 1 what? 12 what? If it were truly 1 or 12, wouldn’t the digits be written in different places?

Disconnecting the procedure from the meaning behind it leads to magical thinking about numbers: that it doesn’t matter how or why it works, it just does. 

Another place this shows up is in multiplying or dividing by powers of ten. How many of us think of “moving the decimal” or “adding a zero”? Both of these are not only how mathematics doesn’twork, they lead to significant mathematical errors and lack of understanding in students. 

To promote numeracy, or paying the same amount of attention to all students being as successful and capable in mathematics as they are in reading and writing, the focus needs to shift away from whether or not the answer is correct and more towards the way people think about the mathematics. This involves developing the concept so the student can apply whatever problem solving procedure works for him/her. This also means exposure to a variety of representations of the mathematics, to improve flexibility in thinking. The model of teaching mathematics in Singapore, and the US Common Core standards, both push for that type of learning. This way, students learn how mathematics works, not just how to get the right answer.

For example, ten years ago, when working with parents or teachers, I would ask the question, “What is Pi?” The answers would range from something to do with a circle, to 3.14, to “I don’t know.”

These days, after years of celebrating Pi Day in schools in March, the teachers I meet are much more likely to say, “It’s a ratio,” or “It’s the ratio between the circumference of a circle and its diameter.” This is a much more conceptual definition that allows flexibility in thinking and the ability to recreate the formulas for the different measurements of a circle. 

If teachers or parents are working with students on mathematics, why not try shifting focus from the answer to how the students are thinking? Start by giving feedback, rather than “right” or “wrong,” that focuses a student on their thinking and allows the student to identify if they made a mistake in understanding how the numbers work. This should lead to greater competence and confidence in their own ability to learn.

Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

Related article:
Singapore abolishes school exam rankings, says learning is not competition