Tuesday, February 12, 2019

Building a Craft Project

Jae needs to cut wood for a craft project. The instructions provide the size relationships, and Jae gets to decide the actual size. Her first piece of wood will be 𝒙 centimeters long. The middle part has to be 3 times as long as the first. The final part is 11 cm longer than the middle part.

a) Express the length of the final part in terms of 𝒙.

b) If the first piece of wood is 25 cm long, how long will the final part be?




A problem like this would likely be found in Primary 6 or US sixth grade math. Students are encouraged to make the connection between bar modeling and variables.

A unit in a bar model is pretty much just like a variable; it is another way to represent an unknown. Students who have experienced bar models and part-whole thinking are more likely to succeed when it comes to solving for variables.

If students struggle with representing the situations, they can start by bar modeling the problem. One sample solution could be like this. After the model is created, just place the 𝒙 in the appropriate bars.



For students who need support making a connection, ask questions like these:

  • How many times do you see 𝒙?
  • How do we write it when we have 3 times something?
  • Imagine  is a pencil. How could you write it with pencils instead of 𝒙?

A student who finds abstract thinking easier to access might come up with the following solutions immediately:

    a)  3𝒙 + 11, or 3•𝒙+ 11, or 3(𝒙) + 11
    b)  3(25) + 11 = 86
         The final part is 86 cm long.

Did you or a student of your solve it a different way? Share the strategy with us!

Wednesday, February 6, 2019

Buzzwords Decoded: Number Talks and Mental Math

One challenging thing about teaching is that terminology and popular buzzwords change all the time. For example, “math talks” or “number talks” used to commonly refer to a conversation about math in the classroom. These days, though, “number talks” usually refer to a specific protocol for a structured conversation about mental math, as outlined in the book Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10, by Cathy Humphreys and Ruth Parker (2015). 

Why has this protocol taken over this terminology? One reason is that the protocol is effective at reaching all students and achieving multiple goals in developing mathematical thinking. 

The protocol begins with students being presented a mental math problem. Students place their fist over their heart. When they have solved it using one strategy, they open one finger. Two strategies, another. And so on. The teacher gives a set amount of time, and students find as many strategies as they can. 

After one or a series of problems are solved, a class discussion of strategies ensues. Students share the strategies they used to find the solution. 

Why is this effective?

Wait time. In many traditional math classrooms, teachers respond to raised hands, often among the first ones up. This creates several problems:

  • Boys have been shown to raise their hands even if they aren’t sure of the answer, or even if they don’t have an answer, while girls are more reluctant to do so. This can lead to an imbalance of learning and to girls feeling less confident in class.
  • Some students process more slowly than others. If the fastest students are consistently providing the answers, those are the ones seen as “smart,” while other students may have more creative or innovative solutions. This can lead to slower/deeper students checking out mentally in class.
  • When one student is called on at a time, this leads to one student learning at a time when it’s a simple problem. If it’s a question of sharing strategies or ideas, it works to have students share one at a time, because that’s interesting! But for numerical or simple solutions, other strategies, such as choral response, white boards, or Plickers, can be more effective to collect responses and retain student engagement.

Differentiation. The activity is naturally differentiated. 

  • Students who know the answer by rote can find other ways to find the same solution.
  • Students who are still learning the concept have more time to come up with one strategy.
  • Everyone has a chance to feel successful. Even the teacher can participate and learn something.

Rich learning possibilities. When students share their strategies with each other, they may learn much more than with traditional approaches to teaching mental math.

  • Students who have made errors can correct themselves and understand why they made that error, possibly preventing the same error in the future.
  • Students who did not understand a strategy when it was taught in class might understand it when another student presents it.
  • Spending time on multiple strategies can help create stronger neural networks around the concept, making reconnection to the pattern easier in the future.

The Matholia tools can be useful to present a situation for students to consider, or for students to represent their thinking. For example, students who have learned how to add fractions without common denominators might be asked to find the sum of these two fractions. 
Alternatively, they could be presented with 34 and asked to use these tools to represent this fraction as a sum of two or more fractions. 

One example of how a number talk might look in a classroom can be seen here. Note that the fist and fingers response is not shown, but the solution strategy sharing is.

Have you tried this approach? Share your experiences with us!
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

Sunday, February 3, 2019

Teaching with Anchor Tasks

What are anchor tasks, and what place do they have in the classroom?
To understand this, we need to begin with a little neuroscience. According to Dr. David Sousa, the “primacy” effect of the brain means that in a lesson, the first ten minutes are the most important for learning. The brain is able to retain the most in this first period. This means that if teachers waste this with attendance, handing in homework, reviewing old work, or other administrative tasks, they are losing the most valuable time in their lesson!

How can we leverage this knowledge more effectively? Singapore teachers have done so by creating anchor tasks. These are rich tasks that engage learners in discovering concepts through exploration and application and are accessible to multiple levels at once. 

A mathematical problem or situation is presented to students to explore and question, in which the students work in groups and the teacher facilitates. A rich discussion ensues, in which different learners share strategies and learn from each other, and the teacher can facilitate struggling learners to make sense of the concept.

One example is a third grade teacher preparing a class for a class on multiplication. The students are familiar with multiples of 2, 5, and 10. Today she wants to introduce multiples of 3, so she projects this image and gives a copy of this chart to each table.


She also provides manipulatives, Rekenreks, and white boards to the students to support their explorations.

She then asks, “Can you fill in this the missing blanks on this chart? Do it as a team, and share your thinking about how you were able fill in the blanks.”

This will encourage pattern thinking and extension. She may also encourage early finishers to extend the chart and fill in as many blanks as they can, justifying their work with pictures or demonstrations to avoid relying on rote memorization.


One student might make doubles of each number to find the product on the second row, and then another might recognize the commutative property and how that leads to the related product. For example, the student might demonstrate that 3 twos are 6 by laying out 3 rows of 2. Another student might then point out that this is the same as 2 threes if you switch the rows and columns. This leads to an intuitive understanding of the commutative property.



When an anchor task is used to launch a lesson, students are fully engaged and learning, and all students are involved, rather than just a few vocal ones. The practice in the workbook or online then has meaning and relevance, rather than being dry and disconnected. 

Do you have experience with anchor tasks in your classroom? Share it with us in the comments!
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

Problem Solving Task

Honora gave 7 candies each to 4 of her friends. She had 37 candies left. How many candies did Honora have at first?
This is a mixed-operation problem with multiple steps. There are several ways to approach this, but a common mistake might be to subtract from 37 and then divide or multiply. This is often seen when students focus too much on the numbers and not the meaning. This is one way bar models can be very helpful, as it causes a student to pause and think about the meaning of what is happening.

Since this is a part-whole question, a first model 
might look something like this:


Then the student might identify the candies given to friends, like so:


The student now has both parts, and all that is left is to add them to make the whole and identify the quantity for the question mark: 

28 + 37 = 65, so

Honora had 65 candies at first.

Did you or a student of yours solve it a different way? Share the strategy with us!

Wednesday, January 23, 2019

Mental Addition and Subtraction Strategies

Have you ever met someone who could do amazing computations in their head, faster than you could even pull out a calculator?

Conversely, have you seen older students or adults who still have to add and subtract by counting up or down from the starting number?

One way Singapore Math differs from other curricula is that it takes what people who intuitively figure out how numbers work and can calculate mentally, and makes it explicit so every student can use these same strategies.

The approaches begin in early grades, such as Kindergarten and first grade, where children go beyond knowing the counting sequence and learn the different ways any number within 10 can be decomposed. For example, they practice with:
5 is 4 and 1
5 is 1 and 4
5 is 3 and 2
5 is 2 and 3
These are first introduced with decomposing with objects, and then those are related to number bonds with objects, both in concrete and picture form, then to number bonds with pictures, and finally to equations, first in word form as above, then with symbols.
When number bonds are introduced, part-whole thinking can develop, leading to ease with both composing (adding) and decomposing (subtracting). Both the part-whole models and bar models build upon that.

Fluency with these compositions within ten provides the foundation for all the subsequent computations. For example, if each subsequent decade is understood as 10 or a multiple of 10 plus one of the known compositions, it’s a simple matter to apply the same patterns. No longer is each number a discrete entity; each is made up of the same type of puzzle piece as the ones they have already learned.

So students learn to apply the rules they learned to compose and decompose with larger place values, using different manipulatives to become acquainted with the values of the different digits and how they work together. This leads to more complex strategies starting in second grade, such as addition and subtraction compensation.

Addition Compensation

One form of addition compensation is the “make a ten” strategy. This is where one addend is decomposed to make a ten, or a multiple of ten, to make adding easier. Some examples of strategies are the number bond decomposition and the “arrow way,” or step-by-step approach. For example:


Subtraction Compensation

This depends on the understanding of subtraction as finding the missing part of a whole and as a difference. The strategies are a bit different here. Number bonds cannot be used in the same way; instead, the students learn that to find the difference, the same number can be added to both minuend and subtrahend, and the difference will stay the same. Two examples here are making a ten out of the subtrahend, and step-by-step subtraction.


There are so many other strategies and so many ways to use these strategies. Sharing solution strategies can be a great teaching tool. Both my students and I have gained flexibility and ease with numbers the more we practice with these. We can also mentally check our answers if we use an algorithm or a calculator, always checking, “Does that make sense?”

Do you use a similar or alternate strategy? Try these out or share your own!
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, 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.