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.

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Part-Whole Bar Model – Division

In the last post, we looked at the use of part-whole bar models for multiplication word problems. Here's a similar application of the part-whole bar model method – this time for a division word problem.

As with previous bar modelling videos, this video was produced using Matholia's 'Bar Model Maker' – just one of more than 50 primary maths tools available on the Matholia portal.  Email us now to start a free 3-month school trial now.

Part-Whole Bar Model – Multiplication

The use of bar models was first implemented into the Singapore maths syllabus approximately 30 years ago as a problem solving tool to assist pupils in visualising word problems. The pedagogy of bar modelling is now an integral part of the teaching and learning of primary mathematics in Singapore.

In this video, we look at how a part-whole bar model can be used to represent a multiplication word problem.




This video is part of the Matholia professional development video series. To see all of the videos in this series, follow this blog, or visit the Video Library on www.matholia.com.

Comparison Bar Model for Subtraction

As promised, here's a video demonstration using a comparison bar model to represent a subtraction word problem.




In the next videos, we'll look at how bar models can be used to represent word problems involving multiplication and division.

This series of training videos is produced using Matholia Tools. Tools are available on Matholia and can also be downloaded for free on  the iTunes App store.

The tools are just a small part of the comprehensive primary mathematics resources available on Matholia. Get your free trial account today!

Comparison Bar Models for Addition

A comparison bar model is useful for visualising word problems where a known quantity is compared to an unknown quantity. This video provides an examples of a comparison bar model for addition.






The above video was made using the Matholia Bar Model Maker. For loads more tools and comprehensive primary maths practice, head to the Matholia portal. In the next post, we'll look at how comparison bar models can be used to represent a subtraction word problem.

Matholia Review and Assessment

The Matholia online portal includes comprehensive review modules ideal for placement tests, continual assessment and end-of-year evaluation.

Key features include:
• pre-test from K to Primary 6.
• review modules that follow the spiral approach of the Singapore syllabus.
• pre-test, term, mid-year, full-year and topical reviews
• teacher-assigned reviews.
• instant scores and progress tracking.
• suitable for classroom assessment of competencies across the syllabus.
• allows pupils to gain confidence by re-testing at any time.

Review modules can be accessed on the Matholia website or via the ‘Matholia Essentials’ apps. All reviews on mobile devices are logged to the students’ main Matholia account.

Exploratory Maths Tools and Virtual Manipulatives

Matholia Tools consists of a comprehensive library of virtual manipulatives that are ideal for engaging teaching in the classroom or encouraging exploratory learning at home.



Used in combination with the resources and videos in the Matholia learn environment, the tools can also facilitate autonomous learning in a flipped classroom situation.

Matholia Tools can be accessed with any active membership. An app containing all tools is also available for free download on the iTunes app store and Google Play.

Used in combination with the resources and videos in the Matholia learn environment, the tools can also facilitate autonomous learning in a flipped classroom situation.

Matholia Tools can be accessed with any active membership. An app containing all tools is also available for free download on the iTunes app store and Google Play.

Matholia is Singapore's most comprehensive primary mathematics portal. Create a free trial account on Matholia now, or email support for more information.

Endless, Dynamic Fun Practice

The Matholia portal brings together a diverse range of learning resources based on the pedagogical principles of the Singapore MOE syllabus. A core component of Matholia is the practice environment, consisting of over 1600 dynamic practice modules from K to Primary 6.

Dynamic Practice
Each Matholia practice module uses dynamic question generation with extensive variables and question types. This minimises question repetition and ensures students have a solid understanding of the mathematical concept they are practising on completion of the module. Like many of Matholia’s resources, practice modules are sequenced to follow the Concrete-Pictorial-Abstract (CPA) approach.










Instant feedback and rewards allow students to self-assess their knowledge and skills as they progress through the curriculum. All progress and achievements are recorded in the Matholia progress tracking and reporting environment.

Create a free trial account now at www.matholia.com or email school inquiries to the Matholia support team.

What is Matholia?

Matholia is Singapore’s most comprehensive online primary maths portal.
The portal provides students, teachers and parents with dedicated content based on the underlying principles of the Singapore Ministry of Education’s mathematical framework. Matholia’s diverse resources are developed in close consultation with teachers, experienced MOE-approved authors and higher-education researchers. Key components on the Matholia portal include:

• dynamic practice across the entire primary mathematics syllabus.
• comprehensive collection of instructional videos and learning sheets.
• pre-test, term, mid-year, full-year and topical reviews.
• progress tracking and reporting for parents and teachers.
• maths tools and virtual manipulatives.
• full range of mobile apps for iPad and Android devices.
• digital textbooks.
• printable resources.
• speed skills mental maths activities for years 1 to 3.

Matholia is a true cross-platform resource. It can be accessed on all operating systems and web browsers. Free apps are also available for use on iPad and Android tablet devices. To ensure Internet safety, all of Matholia’s video content is hosted on Wistia.

For more information on Matholia school adoptions, visit www.mathoila.com or contact support@bre.com.sg.

Why Singapore Mathematics?

Singapore's Achievement in Primary Maths

For over two decades, comparative studies on mathematics achievement have consistently ranked Singapore’s primary students as one of the top achievers. In the most comprehensive study, the Trends in International Mathematics and Science Study (TIMSS), Singapore consistently ranks in the top position internationally.

Source: Trends in International Mathematics and Science Study (TIMSS)

The Singapore MOE Mathematics Syllabus

Singapore’s consistently high achievement in mathematics is attributed to its research-based pedagogy and the underlying principles of the Singapore primary mathematics syllabus that is infused into all Ministry of Education approved resources.

The syllabus sees problem solving as central to the learning of mathematics. It emphasises the importance of conceptual understanding, skill proficiency and the development of thinking skills in building problem-solving ability. The importance of differentiated instruction and the integration of technology in learning and assessment is also emphasised.

Through its comprehensive Matholia learning portal, Blue Ring Education is dedicated to delivering innovative Singapore mathematics resources to schools internationally.

For more information on Matholia school adoptions, visit www.mathoila.com or contact support@bre.com.sg.

Virtual Manipulatives and Tools

Virtual Tools in the Elementary Classroom

When I think about it, words can sometimes get in the way of my math teaching. There are moments in my elementary math classes where I am striving mightily to put something into words that would be so much better understood without words. I wish, sometimes too late, that I had prepared a physical object, a picture, or a moving picture to show students what is happening when, say, one is regrouping in addition. There have been plenty of these moments in my teaching, where in the back of my mind I am wishing I had prepared a visual instead of trying so hard to put a math concept or process into words. I can picture myself, like a native speaker trying to explain driving directions to a tourist who doesn’t speak English.

Computer technology is helping me with this challenge in teaching elementary math. Over the years, I have become more and more dependent on visual representations developed by others that I can show to my students instead of telling them about it. One of my all-time favorites is this simple animation used to help children understand the relationships between circumference, diameter and pi:

Source: Wikipedia (http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif

            

Perhaps you will see what I mean when I talk about how this picture is worth a thousand math teacher words.

Of course, planning hands-on experiences with manipulative materials is a foundation of elementary math teaching around the world and is often the best way to get away from words. When I do not have a physical object for children to explore, or when circumstances in the classroom do not allow for the time necessary for managing the stuff for each student, it is this growing trove of helpful visuals that is becoming a well-worn tool of my math teaching toolbox.

For years I have relied on the simple but effective visual teaching tools provided in the National Library of Virtual Manipulatives (NLVM). The tools are also collected for easy access so I do not have to spend a lot of time searching the internet for the right demonstration. Consider the “fractions - adding” activity as an example of the power of these tools.

Matholia (www.matholia.com.sg) also offers an updated collection of virtual tools for active demonstration to students. I’ll give you an example of how I used the collection. I was doing some remediation work with a second grader who was using the standard algorithm for addition relatively well, but his errors showed that he only understood the procedure, but had weak understanding of the meaning behind the procedure. I pulled up Matholia’s place value tool, and we added numbers together, watching what happened on the screen, and connecting the action to what was happening in his pencil-paper work. I was proud that I could restrain myself from too many words as I had him articulate for himself the process of regrouping on the screen, and then the representation of regrouping on his paper.

The Matholia Place Value Tool (http://www.matholia.com.sg/apps/tools/mt_dd92c_r0dn0?cid=601)

I know that I am relying on these virtual tools more and more. Because they are so powerful, I want to access them easily and quickly. I want the whole toolbox in one place so that when I find myself in that moment when words are actually getting in the way, the visual representations are nearby and easy to use. I don’t think we are there quite yet, but we are close. I wonder, would every elementary math teacher’s toolbox be the same? Would one size fit all?

Matholia's growing library of over 100 virtual tools and manipulatives is included in all Matholia subscriptions. If you are already an adopting school and would like an offline version for the tools kit, contact support@bre.com and reference this blog.

by Dr Kevin Mahoney

Dr. Kevin Mahoney is an academic researcher and Math Curriculum Coordinator for an independent school outside of Boston, MA. With over 20 years of elementary teaching experience Dr. Mahoney also works as a consultant and trainer to schools and teachers implementing Singapore Math Curricula.

Bar Models

From the common core standards for mathematical practice:

They (mathematically proficient students) are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

What is the point of bar models?

The Singaporean teaching strategy of bar modelling has spread worldwide. What I want to address here is the common misconception I run into with teachers across the United States: the idea that bar models are just for problem solving.

Now, don’t get me wrong, Singapore’s Model Method is a very effective problem solving strategy. My own academic research attests to that fact. But bar models can help teachers meet many of their other goals essential for good elementary math teaching.

Bar Models are useful for teaching concepts

I use bar models to help children understand addition, subtraction, multiplication, division, fractions, ratio and percentage. This simple representation, used in conjunction with others, provides a flexible and clear model of all of these mathematical relationships. I don’t need to present a bar model as part of a word problem in order to use it to show students instead of telling them.

Bar Models are good for developing reasoning and communication

When a child draws a model of a word problem, she provides evidence of her thinking without words. Bar models can take the place of paragraphs of writing required when a student must explain how they arrived at a solution. I have found over and over again that when a child and I can point to and touch a model that they have drawn, we are able to communicate with each other more efficiently and precisely. 

Bar Models develop algebraic reasoning

Bar Models were designed, in part, as aids to algebraic reasoning. When the child can see from her bar model that 3 boxes hold 45, she can see that division can be used to determine that one box holds 15. Once the value of one box is discovered, she can use that information to determine several different relationships, differences and totals. This is concrete algebraic reasoning presented at an early age to elementary children. For the students we work with at our school, such reasoning will be natural to them by the time they leave us for middle school.

So consider the lowly bar model. It may be more powerful and useful than you once thought.

Note:  Bar models are used throughout the Matholia practice modules, instructional videos and e-books. Bar model tools can also be found in the Matholia Tools environment.

by Dr Kevin Mahoney

Dr. Kevin Mahoney is an academic researcher and Math Curriculum Coordinator for an independent school outside of Boston, MA. With over 20 years of elementary teaching experience Dr. Mahoney also works as a consultant and trainer to schools and teachers implementing Singapore Math Curricula.

10 Things I Have Learned as a Teacher of Singapore’s Math Curriculum

A guide for parents

by Dr Kevin Mahoney

1. Don’t fret! It is still elementary math.

Singapore’s Math curriculum is from half way around the world, but the program is still just elementary mathematics. The fundamental concepts, procedures and habits of mind do not differ radically from the elementary mathematics curriculum currently taught in the United States. Your child will still learn to work fluently with addition, subtraction, multiplication and division – often in ways that you would recognize from your own elementary math education. Measurement, geometry, graphing and statistics, fractions and decimals and ratios and proportions are all addressed through Singapore’s math curriculum.

2. The curriculum relies on helpful and meaningful visuals.

Where Singapore’s math curriculum will probably look different to you is in the strategies used to teach that elementary level math. In Singapore, there are multiple languages spoken at home, but English is the language of instruction. This fact forced Singaporean educators to develop their curriculum with less text and more visuals to accommodate a student body that is learning English at the same time as it learns mathematics. The unique toolbox of visual teaching strategies is where most observers first notice differences between Singaporean and American approaches to teaching math. A great number of the parents that I have introduced these visuals to have remarked that elementary mathematics makes more sense to them when taught using these visuals. They wish that they had been taught math in this way.

3. Some teaching approaches will look different to you.

While the math is still just elementary math, Singapore’s curriculum is based on a fundamental model of teaching mathematics widely known as the Concrete-Pictorial-Abstract (CPA) approach. This approach is different from what you would see in many elementary classrooms. What would typically happen in these classrooms is that the teacher, like the Singaporean teacher, would introduce a new math concept or skill with a concrete experience, usually introducing an object or counter that is a concrete, physical representation of the math idea at hand. Blocks and “manipulatives” help students physically move and experience the relationships between quantities, for example. I am sure you have seen this in classrooms. What usually happens after this concrete experience is that the child is then asked to move to an abstraction of that physical experience. For example, a second grader might be asked to act out an addition problem using special blocks. Then, she would be asked to use what she understands from using the blocks to using the column addition procedure that so many of us learned in elementary school. Do you see the move from the concrete to the abstract? Now, in Singaporean teaching, the approach adds a new pictorial component. After concrete experiences, children are taught to model the concept or procedure with a drawing or diagram. These diagrams act as a bridge between the concrete and the abstract and help the child solidify the concrete experience and connect that experience to the abstraction of the mathematical procedure. Bar models and diagrams for mental arithmetic are particularly powerful demonstrations of this pictorial phase of learning.

4. Problem solving is a core activity.

Majority of American math curricula require children to solve problems. I am quite sure that you solved a plethora of them yourself in your elementary math career. The difference you may notice with Singapore’s math curriculum is that problem solving is a core activity. In many classrooms using Singapore’s curriculum, your child will be solving problems on an almost daily basis. This is particularly interesting because all over the world, problem-solving is seen as quite a challenge for young children and one of the most difficult aspects of teaching elementary mathematics. In Singapore’s curriculum, it is no surprise that children are asked to draw diagrams (C-P-A approach in action!) that help them model the mathematical relationships in a problem. Proficiency with this technique enables children to tackle far more complex and challenging problems than typically presented to American children. I know this first hand through my own original academic research. These diagrams, referred to as bar models or model drawings, are one teaching tool that will seem very different to you. Most American teachers have to be trained in using these pictorial techniques before they can teach Singapore’s curriculum. While the diagrams may seem complex and confusing to you at first, take the time to talk with your child and your child’s teacher about the technique before you judge their effectiveness.

5. Mental math is used to build number sense.

Your child is going to be working with mental arithmetic in a direct way. Let me define my terms here. Mental arithmetic is the application of strategies for breaking down numbers into easier parts to make mental computation easier. Most of us use some collection of strategies for working with numbers when we have to make quick mental calculations. Some U.S. programs emphasize this kind of work, such as TERC Investigations or Everyday Math, but many other programs do not. I used to think that mental math was a good thing for children to learn because it is so useful for all of us in everyday life. I have since learned from experience that Singapore’s emphasis on mental math strategies is not just for grocery store estimations. In fact, through teaching with Singapore’s curriculum I have come to see that mental math work with children is how number sense is built. Number sense is not some innate trait genetically passed down to only a few children. Number sense is an absolutely key ingredient for success in elementary mathematics, and this facility with numbers is built through work with mental arithmetic. This is why there is a greater emphasis on mental math strategies than with typical U.S. math curricula.

6. Algebraic reasoning is baked into the program.

In Singapore, most children are prepared to take the equivalent of an Algebra 1 course in grade 7. We typically teach this course in grade 8. Thus, Singapore’s program must develop algebraic reasoning, without formal abstraction in algebra, from the earliest grades. This is achieved in the curriculum through work with problem solving, bar models, mental arithmetic, and the visual representations I have been talking about here. Like mental math work, algebraic reasoning is a topic that is considered important in most elementary math programs worldwide. However, in U.S. textbooks the emphasis placed on algebraic reasoning varies greatly.

7. It is about a year ahead of typical U.S. curricula.

Here is another difference you may notice with Singapore’s curriculum; the math content taught is typically about a year ahead of standard American math programs. Of course, this may not be true of your child’s particular math program or school, but in general, more complex mathematics is being taught at each grade level than in typical U.S. classrooms. How is this possible? I asked the same question myself as I began to investigate Singapore’s curriculum for the first time. These results are achieved through some of the features I have already mentioned here. Visual representations, the C-P-A approach, and consistent and coherent focus on only the most important mathematics that elementary students need to learn. Why would U.S. textbooks appear to be about a year behind? This difference comes, at least in part, from the fact that the U.S. education system is founded on local control. In math education terms, that translates to elementary math textbooks being written to meet 50 different sets of state standards. Topics are not treated with enough depth and focus because there simply are too many topics to deal with in a typical U.S. elementary textbook. What you get is something that will please everybody, or a math curriculum that is often criticized as “a mile wide and an inch deep”.

8. You have to help out with math facts.

You might think that this list of lessons learned is an advertisement for Singapore’s math curriculum. Well, here is where we get into my criticisms of the program; gaps I have discovered through years of research and practical experience teaching the program to children. One thing you will need to know as a parent is that you will have to help. Singapore’s curriculum is quite rigorous and rich, but it was developed in a country where many parents have their children practicing mathematics outside of school. The memorization of math facts, those basic tables of addition, subtraction, multiplication and division that we were all asked to memorize as kids, is essential for success in this curriculum. That task begins in the classroom, but you should know that helping out with this task will be one of the best ways that you can help your child feel competent and comfortable in math class. Start early and practice often, working in partnership with your child to discover the most effective ways to practice and complete this memorization task. I often ask families to prepare good old-fashioned flash cards and stuff them in the pockets of the family car for 5 minutes of practice during a drive to school. Of course, family discussions around mathematics is another essential home-school connection, but that is an entirely different article right there.

9. The program lacks applications to science.

Singapore’s math curriculum is not taught in an integrative way. Sure it is world class, sure it is focused, coherent and powerful. But mathematics is the language of science, and almost no connections are made between science and mathematics in this curriculum. These connections are vitally important to make with children so that they grow up understanding that these two disciplines are interwoven and that mathematics is a way of thinking that we can use to powerful purpose. At my school, we are developing integrated STEM units and projects to illuminate these connections and help children see how mathematics is applied in meaningful ways. I wish that the curriculum had more of this important theme written into it.

10. The program lacks application projects.

One way that science and other disciplines could be integrated into Singapore’s math curriculum would be to develop more open-ended problems to solve or projects to complete that ask children to apply mathematics to real-world situations. As I stated above, problem solving is a core activity, but most of the problems presented have only one answer. Children absolutely need to be exposed to open-ended problems because the real world is a messy place, and there is much joy to be found in discussing and understanding the grey areas of real-world problems and math solution paths of others. Projects also give children a chance to demonstrate their understanding in more diverse ways, and thus engage the creativity of diverse learners.

So, there you have it, a collection of lessons learned from years of teaching Singapore’s elementary math curriculum. It is abridged and condensed for easy digestion, but these are the big ones out there for me. If you child’s school is using the program, or considering adoption, perhaps these lessons will help you see the nature of this world-class curriculum from an American educator’s perspective.

Dr. Kevin Mahoney is an academic researcher and Math Curriculum Coordinator for an independent school outside of Boston, MA. With over 20 years of elementary teaching experience Dr. Mahoney also works as a consultant and trainer to schools and teachers implementing Singapore Math Curricula.