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

Virtual manipulaitves in the maths classroom

The use of manipulatives in mathematics are essential for the teaching and learning of abstract concepts. Virtual manipulatives take this to a new level with  engaging graphics with dynamic generation and manipulation. Virtual manipulatives also have an advantage of allowing for universal access (at school or home). Research now suggests that the use of virtual manipulatives can lead to a deeper understanding of concepts when compare with concrete manipulatives.

Here’s a great article from the Center for Implementing Technology in Education on the use of virtual manipulatives in teaching mathematics (http://www.cited.org/index.aspx?page_id=151).

Check out Matholia’s library of over 50 engaging virtual manipulatives found under the Tools tab on the Matholia home page.

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.

Part-Whole Bar Model - Subtraction

Help students visualise subtraction word problems using a part-whole bar model.

The video was produced using the Matholia Bar Model Maker. You'll find this tool along with more than 50 other interactive maths tools on the matholia portal (www.matholia.com).

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.

Singapore Math Classroom Diary

Singapore Math Classroom Diary

Year level: Grade 3 

Topic: Fractions 1

Today, I began teaching the grade 3 fractions unit to a group of students who require more support than the average third grader at my school. I wanted to document the sequence of lessons I taught, in part to think about how I use technology to teach the concepts in the unit.

With the faculty here, as well as in my training seminars, I try to emphasize to teachers that they need to begin their lesson planning with a foundational question: what is the most important mathematics that I need to teach today? Looking through the material I have planned, I interpret the answer to that question to be: Fractions show us parts of a whole. The parts come together to make a whole. This is what I want my students to understand and apply by the end of my 60-minute lesson.


In the textbook, this idea is taught by showing shaded and unshaded parts of fraction diagrams, using multiple representations such as fruit, circles, rectangles, triangles and other polygons. I use a 5-component lesson plan for my math classes, so I will present the lesson through the sum of its components. You can see how I used technology to help me teach concepts and skills throughout the lesson.

Mental Math

Today was the first math class after vacation. I wanted to begin reviewing multiplication and division fact practice right away to clear out any vacation cobwebs. Each student completed a 5-10 minute session with the ipad app connected an adaptive, web-based math facts practice solution that our school has been piloting this year. I am pleased with the independent and differentiated work that went on during this component of the lesson. Each child was practicing facts at their level. I was able to carefully observe one particular child who has not been making much progress. I noted that his attention wandered, and at the same time, he rushed in entering answers he knew. How can I have this child work at a more even pace. perhaps coupling that even pace with saying facts out loud?

Problem Solving

With this group, I have been remiss in keeping up with measurement skills. All this week, I have planned to use problem solving to work with telling time on an analog clock. Today, I presented a clock on my smartboard. I had each child write down the time shown (we practiced time to the nearest 5 minute) and then write down what time it would be 20 minutes later. 3 clocks were presented, one per screen. I could see children make progress with each example, and the group was independent with the task by the final example. The large and moveable clock I had designed on the smartboard was a clear and easy-to-use visual. I realize now that I could have used Matholia’s Virtual Clock Tool with even greater effect. I would be testing out a different Matholia tool later in the lesson, so this would have made more sense. Ahh, 20/20 hindsight. 

Directed Lesson

I began the directed lesson with the fruit examples right out of the textbook. We looked at fruits that had been cut up into equal parts. I emphasized how, in the example of fruit cut into fifths, the two fifths on one plate could be combined with the 3 fifths on the second plate to make 5 fifths of the fruit, which was the same as the whole fruit. This part/part/whole concept was certainly not new to these students, as we have been using this kind of language and representation with them throughout their math learning here. I made a big deal about how these pieces were cut into five equal parts, not just five parts. Fifths means five equal parts. If the parts aren’t equal, we cannot say that it has been cut into fifths.

I then digressed from the textbook and projected several circle models for fractions. I had students write down the fraction that was shaded and the fraction that was not shaded. I pointed out how the two parts (shaded and unshaded) came together to make one whole. Without really thinking about it ahead of time, I pushed the idea further into abstraction as I wrote fraction addition sentences that described each circle representation, adding the fraction shaded to the fraction unshaded and totalling to one whole. I knew that this was pushing the level of abstraction, for this group, but instinctively, I felt it was important for the kids to see the abstraction of the fraction equation right alongside the pictorial representation of the fractions.

Activity

The directed lesson then evolved into a partner activity, moving along with the text, I represented fractions with bar models. We worked through several examples on the Smartboard, with me releasing more and more responsibility to the students in the class to look at the fraction diagram, then point out the fraction shaded, the fraction unshaded, and the corresponding fraction equation.

Independent Practice

The corresponding workbook page helped the students practice the ideas that we discussed and that they had worked on with their partners. It took the group about 15 minutes to complete and partner-check the three assigned exercises.
For the last 5 minutes of class, I brought up Matholia’s Fraction Disc Tool. There, I pulled up the disc for one whole, and then pulled several unit fractions onto the screen. We were all able to see how ⅓ is actually a large piece, while ⅕ or 1/9 are actually very small pieces. The visuals helped to build foundational knowledge (without a lot of words from me) of the meaning of the denominator. On a whim, I used the rotate feature on the 1/12 piece to show how many 1/12 pieces would fit into ¼. This was a bit clunky. I didn’t want to spend a lot of time fiddling with the visual right in front of the class. Perhaps next time I will prepare a few tabs on my browser before class, that way I could quickly shift between set representations without building diagrams on the fly.

So, there it is. The lesson was a success. After having written it up, I can see how I used technology much more than I realized. Writing also gave me more ideas about how to incorporate the powerful tools in matholia and blend them with the Smartboard presentations I have already built for the lessons.

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. Read more about Dr. Mahoney at singaporemathmentor.com.

Follow the matholia blog for more classroom diaries and the latest Singapore math news.
If you have a Singapore math story, experience or teaching tip, we'd love to hear from you. Contact matt@bre.com.sg for more information on how you can contribute.

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.

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.