## Tuesday, March 5, 2019

For a school fundraiser, Nicole baked twice as many cookies as Rodney. Joey baked 5 times as many cookies as Rodney. If Joey baked 200 cookies, how many cookies did they bake altogether?

Notes on the problem:

This is a multi-step word problem that works well with a comparison bar model.

When drawing the bar model, it’s helpful to show learners, “One time as many, two times as many…” while verbalizing, so that they can conceptualize multiplicative relationships rather than think of it as, for example five more bars for Joey. This avoids a common mistake of drawing the wrong number of bars in the model.
Here is a sample bar model. In this one, the smallest unit (or baker with the least number of cookies) is identified first, and the relationships to that one are scaled up from there.

Once the model is successfully drawn, we can see that it can be solved in two or more steps.

One way to do it is to find the unit value by dividing what we know, the number of Joey’s cookies by the number of units he has: 200 ÷ 5 = 40

Once that is found, the unit value can be populated into the model.

Concrete learners might write it into every unit, as shown, and then add up all the 40s or solve line by line and add 40 + 80 + 200 = 320.

More abstract thinkers, who understand that 40 is the value of one part, and the whole consists of 8 equal parts, might jump straight to 8 x 40 = 320.

The second solution is shown below.

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, February 26, 2019

### The Importance of Problem Solving in Math

When we look at math accomplishment around the world, some countries stand above others in their achievements. Singapore’s record is outstanding, for example, often taking the number one spot in international testing in reading, science and, of course, math.

Digging deeper into the reasons for Singapore’s success, a pattern emerges. This young country has invested most of its resources in developing a high-quality approach to education. Part of this includes drawing from the best practices from around the world in all subjects. Professionals are sent around the world to study the research and practices in different countries and bring their findings back to the ecosystem of their teacher training.

Another reason for their success is the emphasis on problem solving. Where historically math has been taught at elementary levels as primarily computation practice, Singapore was a leader in changing the focus to problem solving. Not only that, but they took from the work of Jerome Bruner’s The Process of Education (1960) to develop their approach of introducing concepts first through concrete experiences, then pictorially, and then finally the abstract or procedural level. This enables a robust form of learning that supports long-term learning, not only memorization or working memory.

Then in the 1990s, a teacher in Singapore created an ingenious approach to problem solving called bar modeling, or tape diagrams in US Common Core parlance. This approach allows a systematic way to solve word problems that is far more efficient that previous strategies, which would often include the tedious guess-and-check approach.

Bar modeling also supports algebraic thinking, in which the unknown is identified, usually with a question mark. That model then allows powerful visualization in problem solving, so when students make a bridge to algebra, it is demystified and scaffolded.

One reason students worldwide might not succeed in math is because they are not challenged in the zone of proximal development (Vygotsky). They are either given tasks that are too challenging, creating frustration, or too easy, leading to boredom. Also, when teachers are unable to challenge students at the appropriate levels, the temptation is to give students too much help too quickly, or give them the answers. This undermines students’ ability to struggle and persevere in problem solving.

This is connected to the mindset work of Carol Dweck and its application to math by Jo Boaler and her team. When struggle is seen as negative and the right answer is paramount, students feel like failures when they struggle. When productive struggle is encouraged, though, and the process, rather that the answer, is emphasized, new patterns emerge. Students persevere, feel successful when they work hard, develop more confidence, and are more willing to take risks in learning. They embrace mistakes as learning opportunities and become more supportive of their classmates. These abilities then carry on through their school careers and life. This is why the first Standard for Mathematical Practice adopted in the US is “Make sense of problems and persevere in solving them.”

Therefore, it’s extremely important to include challenging word problems, and the appropriate scaffolds for student success, as part of students’ math experiences.

The Matholia bar modeling tools and practice activities can fulfill these needs, and the math problems released in the Matholia Loop newsletter have their place in the classroom as well.

Taken from the Practice environment at matholia.com

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

Find the value of the shapes.

Solve for each shape value. This puzzle is appropriate for grade 3 and up.

Notes on the problem:

This puzzle reflects the algebraic thinking supported and encouraged throughout the Singapore Math curriculum. By recognizing that three of the same shape require a total quantity to be distributed equally between them, students make a connection to the concept of partitive division. They then transfer the value of one unit to find the value of the others. This allows the development of flexible problem solving in multi-step problems. It is related to the thinking used in bar model strategies for word problems.

### The Importance of the Unit, Part II: Fractions as Units

How many times have we heard, “I hate fractions!” from both children and adults? So many times, because after students spend years learning how numbers work, suddenly they are introduced to a new kind of number that breaks all the rules they learned before.

When we add or subtract fractions, we don’t add them the way we add whole numbers, but we have to remember a whole set of new rules. And when we multiply by a fraction, sometimes the product is bigger, and sometimes it’s smaller.

Dividing fractions makes no sense -- so students learn “keep-change-flip,” which if you ask students (or teachers) to explain why it works, you will get blank stares and stuttering. Not only that, but many students don’t remember which fraction to “flip,” or why to flip the divisor and not the dividend. But this mnemonic is so popular that people create videos and other funny ways to remember it and how to apply it. Still, errors abound.

And then there are “improper” fractions too -- and if you ask some students, that term should apply to all fractions, because they don’t follow proper number rules!

So as soon as decimals appear on the horizon, many students will avoid using fractions at all costs, and will run straight for the calculator and converting to decimals in high school. This makes calculating slope, ratios, and a number of other concepts more challenging, concepts that would be easier with a strong grasp of fractions.

THERE IS A BETTER WAY!

Unlike students who learned in procedural ways, in which operations with fractions appear mysterious and illogical, when students learn them conceptually, fractions often become their favourite numbers in math. How does this work?

In early grades, students are introduced to the basic fractions they are most likely to encounter in everyday life: halves, thirds, and fourths. These are introduced in Singapore Math using manipulatives like fraction tiles and circles, and are then mirrored with pictures of these units. Students are given stories involving fractions that relate to real life as well, such as sharing pizza or other equally divisible items. The Fraction Tiles and Fraction Discs tools on Matholia can be used to model these units.

When students are introduced visually to fractions, they are then given the concept that a fraction is a unit just like any other unit. They have been introduced to counting by ones, tens, apples, and so on; counting by halves or thirds is just counting by another type of unit.

Using a number line, students can then count by units in later grades to find non-unit fractions and mixed numbers. For example: one-fourth, two-fourths, three-fourths, four-fourths (or one whole), five-fourths , etc.

Students practice converting back and forth between different forms, including equivalent fractions and mixed numbers. This makes adding and subtracting more intuitive, because the big idea is -- you have to have the same unit to add or subtract! In other words, how to you add two apples and three oranges? You can’t -- but you can convert both to fruit, and then you can add.

In the same vein, how do you add ⅓ and ⅖? You don’t -- but you find the same value in an equivalent unit, in which you can then add or subtract.

In future articles, visual ways to find equivalent fractions, and the way this approach relates to multiplication and division of fractions, will be explored.

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, 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 3 ÷ 4 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.