The Bar Model: A Visual Problem-Solving Strategy in Singapore Math that Helps Learners Understand Math Better

The bar model is a visual problem-solving strategy used in Singapore Math that has gained widespread popularity among educators worldwide. It is a powerful tool that helps learners develop a strong conceptual understanding of math concepts by representing problems visually. By using bars or rectangular models to illustrate math problems, learners can break down complex problems into simpler, more manageable parts and make connections between different math topics.

How the Bar Model Works

The bar model is a problem-solving strategy that uses visual models to represent quantities and relationships between them. In a typical bar model, a rectangular bar is used to represent a known or unknown quantity. The bar is then divided into smaller sections, with each section representing a specific part of the quantity. For example, if the bar represents the total number of apples in a basket, then each section could represent the number of apples picked by each person.

The bar model is a versatile tool that can be used to solve a wide range of math problems, from simple addition and subtraction problems to more complex multiplication and division problems. By visualizing the problem, learners can better understand the relationships between different quantities and develop a deeper understanding of the math concepts being taught.

How the Bar Model Helps Learners Understand Math Better

The bar model is an effective tool for helping learners understand math concepts because it enables them to:

Visualize the problem: By representing math problems visually, learners can more easily understand the problem and the relationships between the quantities involved. This can help learners see the problem from different angles and develop a deeper understanding of the math concepts being taught.

Make connections between different math topics: The bar model is a versatile tool that can be used to solve a wide range of math problems. By using the bar model to solve different types of problems, learners can make connections between different math topics and develop a more holistic understanding of math.

Develop critical thinking skills: The bar model encourages learners to break down complex problems into simpler, more manageable parts. This helps learners develop critical thinking skills and improves their ability to solve problems independently.

Build confidence in math: By using the bar model to solve math problems, learners can develop a deeper understanding of math concepts and build confidence in their ability to solve problems.

The bar model is a powerful visual problem-solving strategy that can help learners develop a strong conceptual understanding of math concepts. By representing math problems visually, learners can break down complex problems into simpler, more manageable parts and develop a deeper understanding of the relationships between different quantities. The bar model is a versatile tool that can be used to solve a wide range of math problems, and it encourages learners to develop critical thinking skills and build confidence in math. By incorporating the bar model into math instruction, educators can help learners understand math better and become more effective problem solvers.

The Benefits of Adopting Singapore Math for Homeschooling

For many homeschooling parents, teaching math can be a daunting task. Fortunately, there is a well-regarded math curriculum that has gained popularity around the world: Singapore Math. This curriculum is known for its emphasis on problem-solving, visualization, and building a strong foundation in mathematics. In this article, we'll explore some of the benefits of adopting Singapore Math for homeschooling.

Focus on problem-solving: One of the key strengths of Singapore Math is its focus on problem-solving. Students are taught to think critically and approach math problems from multiple angles. This emphasis on problem-solving not only helps students become more proficient in math, but also equips them with valuable skills that they can apply to other subjects and real-world situations.

Strong foundation in mathematics: Singapore Math is known for its strong foundation in mathematics. The curriculum is carefully structured to build a solid understanding of mathematical concepts, starting with the basics and gradually increasing in complexity. This approach helps students develop a deep understanding of math, rather than just memorizing formulas and procedures.

Emphasis on visualization: Another unique feature of Singapore Math is its emphasis on visualization. Students are encouraged to use visual aids, such as bar models and diagrams, to help them understand mathematical concepts. This approach helps students develop a more intuitive understanding of math, making it easier for them to apply what they've learned to new situations.

Clear and concise explanations: Singapore Math textbooks are known for their clear and concise explanations. The curriculum is designed to be easy for students to understand, with concepts introduced in a logical order and explained in a step-by-step manner. This makes it easier for parents to teach math to their children, even if they don't have a strong math background themselves.

Widely recognized and respected: Singapore Math is widely recognized and respected around the world. Many top-performing countries in math, such as Singapore and South Korea, use this curriculum in their schools. This recognition and respect can be a valuable asset for homeschooling parents, who can be confident that their children are receiving a high-quality math education.

Abundance of resources: Finally, there are many resources available for homeschooling parents who wish to adopt Singapore Math. There are numerous textbooks, workbooks, and other materials available for purchase, as well as online resources. Matholia is an online primary mathematics portal based on the latest syllabus from the Ministry of Education, Singapore.The learning portal brings together a wealth of interactive content to ensure that pupils excel in all areas of primary mathematics whilst allowing teachers and parents the opportunity to closely monitor pupils' performance across the syllabus.

In conclusion, Singapore Math is a highly effective math curriculum that is well-suited for homeschooling. Its focus on problem-solving, strong foundation in mathematics, emphasis on visualization, clear and concise explanations, recognition and respect, and abundance of resources make it an ideal choice for homeschooling parents who want to give their children a solid math education.

Counting on a Number Line (Missing Numbers)

Problem Solving Task - Baking Cookies

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.

The baked 320 cookies altogether.

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

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

Problem solving Task

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