Singapore Math

Often identified as the key component that separates deep understanding verses surface understanding of mathematics, number sense refers to the ability to think flexibly about numbers. This includes breaking them apart and putting them together to solve problems, evaluating the reasonableness of an answer, seeing connections between operations, and engaging in mental math. Students develop this important skill by solving problems more than one way, engaging in problems with more than one correct answer, and exploring strategies rather than following rote procedures.

We do not rush students through concepts via rote memorization. This strategy often leads to students with no underlying comprehension of math and therefore little ability to strategize solutions or apply concepts in new situations.

Instead, we “go slow to go fast.” Spending the extra time developing strong foundations in number sense in the early years allows students to move more quickly and confidently when they encounter Algebra, Geometry, and Calculus years later.

Humans are highly reliant on visual information. With this in mind, Singapore uses the concrete to pictorial to abstract learning approach. A unit begins with a hands-on activity with manipulatives, in which students can physically demonstrate the answer, followed by solving problems using drawings or models, and finally, employing the abstract algorithm with just words or numbers.

In practice: Bar models are a versatile, visual problem-solving tool used in grade 3 math and beyond. By using these, students can holistically organize the text of a word problem into one visual model. Additionally, they provide practice with a visual representation of the algebraic equations students will learn years later. Younger students visualize beginning addition and subtraction using number bonds. The model intuitively communicates the concept that two numbers join to form a third, and conversely, one number can be broken apart into two.

From kindergarteners drawing in their math journals to third graders recording Flipgrid videos explaining their approach to 2-digit multiplication to fifth grade “math talks” about the relationship between rate and ratio, learning does NOT stop at just finding the correct answer. Students draw, write, and talk about their mathematical thinking in all grades. Collaborating, explaining why, defending a strategy, analyzing an error, and adding on to another idea all help develop greater metacognition and deepen understanding of concepts.

In practice: Anchor Tasks are an example of how Singapore Math incorporates this into the classroom. Sarah Schaefer of Mathodology explains an Anchor Task as “a problem given to students at the beginning of a math lesson that provides an opportunity to activate prior knowledge, requires students to collaborate and ask questions of each other, and promotes an environment for students to productively struggle and persevere in problem-solving.”

Jo Boaler explains these as “low floor, high ceiling tasks.” Low floor means that everyone in the class has some prior knowledge to bring to the conversation, even if they cannot get to the solution or find the most advanced method of getting there. “High ceiling” refers to the fact that the problem is open-ended enough for complex approaches or solutions, ensuring advanced students receive challenges in that task.

Mathodology shares the following first-grade example:

A less-confident math student has an access point because he can start by counting the cookies one by one, or possibly two by two. A more advanced learner can begin to explore the concept of multiplication and possibly explore its relationship to repeated addition.

Source: Mathodology 

I’ll confess: I was much too old when I made the connection that finding 32% of 25 is the same as finding 25% of 32- a much simpler calculation. Yes, I knew about the commutative property, I just had never applied it to percent… I had not made the connection that my old learning of multiplication properties and this new topic, percentages, were related.  

Exploratory learning opportunities that lead to these “ah-ha!” moments are embedded into our math instruction. Ideally, students will realize that math is not a million different things to know, but smaller number of ideas being used again and again, in different combinations and new situations. This helps students bring prior knowledge to new learning, improves confidence, and improves the ability to recognize a reasonable answer verses an unreasonable one.

In practice: This representation of an actual Dimensions Math 4^th-grade hands-on activity shows how students “prove” that the sum of the interior angles in a triangle is 180⁰, or the same as a straight line. This can then be connected to the idea that the sum of angles in a rectangle is 360⁰, or the same as two triangles, which connects to the idea that the formula to find the area of a triangle is (nearly) the same as the formula to find the area of a rectangle, divided by two.

Initial placement for new students: Teachers will give content-knowledge placement assessments for students who are new to CHA, including all Kindergarten students. Additionally, students will be holistically assessed within the first few weeks of school utilizing our five success criteria to determine their ability to work independently, communicate thinking, use problem-solving strategies, and exercise a growth mindset. This will happen every year, so at-level placement in K does not necessarily mean at-level through 5^th grade.

Evaluating Returning Students:

In the spring, teachers reassess student placement for the upcoming year. Based on end-of-year content review, completion of extension curriculum, and current year performance, teachers will complete the Math Success Criteria form and contact parents if a change in placement is appropriate.  

If the student is a strong candidate to move a grade ahead, teachers will reach out to parents to schedule the content evaluation. Content knowledge evaluations will be scheduled outside of regular class time, such as before or after school, to preserve instructional time for all students. 

This change has been made so that the teacher who knows the student best, and has worked with them for the academic year, makes the decision rather than the receiving teacher in the fall. This allows us to make a more holistic evaluation of the student’s ability, including strengths, growth, and areas of opportunity, as observed through the year rather than deciding based only on performance the first days back at school.  

Parents may reach out to teachers in the fall to share information or concerns that may affect math placement. Please see “Changing Placement” for more information. 

While every effort is made to ensure the correct placement, additional information may become available after the first few weeks of school. Teachers will communicate changes with parents and move students as needed to ensure best-fit math placement. Changes in placement are considered based on new evidence related to the five success criteria and academic performance. Students moving to the accelerated level are moved on a one-trimester trial basis. Teachers will be in communication with parents about the adjustment to the new level.

Above-level placement: Students who have excelled in the at-level extension work and demonstrate the need for an additional challenge will be placed in the above-level math group. This is a fast-paced group with fewer opportunities for review and reteaching and more emphasis on challenge work, communicating strategies, and extension materials. Students are moved up on a one-trimester trial basis. Performance on the five success criteria will be monitored and any areas for improvement will be shared with students for further practice and with parents to support at home.

At-level placement: This is not your average math class. Students have the opportunity to work in a challenging extension curriculum and are exposed to above-level standards through regular class instruction. When applying out in middle school, our at-level students have as much success as our year-ahead-level students in receiving offers and many place into advanced math when joining those schools.

How we support students: Larry Martinek has a powerful quote: “Children don’t hate math. What they hate is being confused, intimidated, and embarrassed by math.” When we recognize that students have a gap in their understanding, our goal is to address it quickly and get students back to feeling prepared, capable, and confident. At school, students may receive additional practice or reteaching of a concept by their teacher or an Academic Specialist. Teachers may also suggest review activities to be completed at home, such as flash cards, supplied practice worksheets, or IXL exercises.

When academic challenges persist, teachers will communicate with parents to discuss their concerns and, if appropriate, suggest testing with a medical professional. For example, a hearing exam might explain why a student struggles with following directions, and a neuropsychologist evaluation can identify strategies that will best address the needs of a student with impaired executive functioning.

How we challenge students: Our goal is to provide engaging opportunities that keep all students growing and learning. Our Dimensions curriculum includes opportunities to explore above-grade standards as part of the regular class instruction. Additionally, teachers have an extension curriculum that provides challenge problems for every chapter of the Dimensions textbook and multi-day, applied conceptual projects for each unit. This allows students to explore the concepts more deeply, developing strong number sense, flexible thinking, and communication while learning multiple problem-solving strategies.

Students who demonstrate strong performance in our five success criteria and exceed expectations in the extension curriculum will move to the next grade level. Teachers can provide more information about grade-level specific expectations (e.g., what metacognition looks like in second grade) and placement.

Additional Factors

Outside math instruction: In short, we place students based on in-school performance. We understand that many of our students engage in additional math learning outside the classroom; in fact, we are proud that several students have been ranked nationally and placed in competitions such as Math is Cool and Math Kangaroo. Each program has its own unique approach and emphasizes different aspects of mathematical thinking, making it difficult to clearly align a level of Russian Math to a level of Dimensions Math, for example. Additionally, the structure of the school classroom versus an after-school program or working at home with a parent may affect student performance.

Compressed instruction: Our Dimensions curriculum is designed to take 90 minutes daily, over the entire nine-month school year. This includes lessons and activities outside of the workbook/textbook, including sprints, Anchor Tasks, math talks, and peer-to-peer activities, such as error analysis, math games, projects, and hands-on learning.

For these reasons and more, we do not recommend students attempt to learn a year’s worth of concepts over the summer months. In our experience, this leads to a surface understanding rather than the deep conceptual understanding and fluent application of strategies that we strive to develop. Completing a year’s worth of curriculum over the summer will not automatically place the student in the next year’s curriculum and can cause frustration for students who then repeat the material (albeit in more depth) the following school year.

Q: Why are students working at different grade levels combined in the same classroom? 

A: We combine grade-level learning and utilize the math workshop model in our self-contained Kindergarten and first-grade classrooms. This means that students work in small groups with the teacher while 1-2 other small groups are engaged around the room. As students rotate through the workshop stations, they may play a math fluency game, complete independent work, explain their thinking in a math journal activity, or continue to be challenged via extension activities. Additionally, our team of Instructional Assistants supports mixed-level math blocks. 

Like the Daily 5 model used in ELA, the math workshop model allows students to engage in a wide variety of activities while building independence. Teachers may list “must do” and “can do” assignments to indicate what is required and what is optional, allowing students to choose the remaining rotations. 

Looking at the ITBS results, students' acceptance rates at the local elite collegiate middle schools, continued discussions with these schools' Admissions Directors and our students who took math placement tests and were placed one to three years ahead even when they worked at grade level math in their time at CHA demonstrates these systems work and students were well prepared to demonstrate their learning in a variety of modalities and the rigor of our curriculum. 

Our teachers promote a healthy balance of challenge and instill a joy for math. This is critical in students believing in their ability to be and seeing themselves as successful mathematicians. 

Q: Is math workshop used in classrooms where one level of curriculum is taught? 

A: Yes, it is one of many class structures that teachers may select to best deliver instruction. Teachers often need to meet different student needs within the same level of curriculum- some students are ready to move on, while others need to practice more examples. The workshop model is a useful strategy to ensure students get what they need, even when there are different needs in the same classroom. Here is an example of using the model in a class where some students have mastered the previous day's lesson and others need a review before moving on:

All students start the class with a challenge problem. They work in mixed groups, sharing ideas and strategies. After students share their solutions, the teacher breaks them into small groups. Students needing an extra review of the concept taught yesterday first work with the teacher to review the previous lesson and get additional practice with manipulatives, then demonstrate their learning by completing independent practice at their seats. Students who have already mastered a concept first work at their desks to complete extension work in small groups, then rotate to the teacher’s table to review their answers in a small group.  At the end of class, all students complete an exit ticket to show what they have learned before cleaning up their materials. This exit ticket checks to ensure all students are now ready to move on to the next topic.