Mathematical problem solving is critical in developing logical thinkers and lifelong problem solvers. In order for students to work with authentic situations and approach in problem solving, they need a variety of strategies and tools at their disposal. Therefore, an important element of the math class is to expose students to various mathematical operations and guide them to think critically through the medium of math. With a low ratio of six students to one teacher, math is a highly individualized class that allows frequent opportunities of feedback, independent paper and pencil practice, and a personalized online learning experience. This motivates the student to set his or her own goals and progress at a pace that he or she is most comfortable with. The individualized goals and pacing are so important because they help build confidence in students, a critical element in helping students develop into great math problem solvers.
Our goal is to help students move through at least a year’s worth of math content each year.
By the end of the elementary program, our goal is that students will build the skills and fluency of all four mathematical operations with an emphasis on application of these skills in real world scenarios.
The elementary math program covers a series of courses. If students are with us for three full years, we aim to complete at least the 3rd, 4th, and 5th grade levels of the Aleks program. Each course is expected to take a student one year to complete, but it can be completed in under a year if a student is ready or stretched out over a longer period of time if necessary. Students who are not ready for these courses may begin with an alternate program, based on the Dreambox Learning curriculum. Students who are ready for challenge beyond these levels move into the middle school math curriculum whenever they are ready.
Topics Covered: The 3rd to 5th grade math programs focus on mastery of all four operations with whole numbers, decimals, and fractions, plus a basic understanding of percentages, proportions, and basic geometry. As students enter into the elementary level, they continue to extend their understanding of place value, build fluency with addition and subtraction involving multiple steps, and develop understanding of multiplication and division and strategies for multiplication and division within 100. Other critical areas that students focus on during this program are understanding of fractions, describing and analyzing shapes and figures, and understanding and applying of customary measurement units. During the elementary level, students also develop fluency of multiplication and division of fractions, understanding of the relationship between fractions and proportions as well as decimals and percents, and understanding of geometric measurements. Finally, students cover topics in basic algebra and graphs as a stepping stone to learning more advanced algebraic concepts in middle school.
Students spend a majority of their class time focused on three main activities: looping, online practice, and pencil and paper work.
Looping: In this unique activity, the teacher sits with one to two students and creates customized sets of problems based on a list of skills that the student is reviewing and learning. Looping is a highly interactive process that allows each student to work on skills that are uniquely challenging. During looping in class, the teacher is focused on introducing new skills and concepts while also creating review appropriate to the student. No two students will receive the same set of problems in looping – it is all customized in the moment to the individual.
Online Practice: We believe that online practice is not a substitute for math instruction but can be an excellent companion to a high quality math curriculum. Through online practice, students can practice skills that the teacher has introduced in looping. Our middle school students use the Aleks program for this purpose:
- Aleks is a truly adaptive program that responds to students errors and successes in the moment as well as over an extended period of time.
- The scope of problems in Aleks is not limited to the mechanics of how to solve a problem; understanding math conceptually and mathematical applications are built into the program.
- Aleks is an accredited math program in all states that also aligns to the common core. In fact, it usually goes far beyond the standards.
- Aleks allows students to pass through material they already know quickly. By allowing students to move through material they have already mastered and focusing on new, challenging material, students are motivated to keep pushing forward in their work.
- The numeric nature of Aleks makes the program easy to understand for students. The number of topics mastered is clearly listed as a student works through their ALEKS pie. May students find adding to the list of mastered skills to be very motivating.
- Aleks does not assume mastery of material easily. Students are continually re-assessed on their learning and given the opportunity to forget and re-learn concepts. We believe this is a critical aspect of any math instruction.
Paper and Pencil Practice: There is no substitute for paper and pencil practice, so we incorporate traditional paper and pencil work into one-third of the assignments our students do. We use a wide variety of texts to provide material for paper and pencil work. When we are looking for paper and pencil material, we are looking for practice that provides the student with significant opportunity to work on a skill in a focused manner. If a student needs to concentrate practice on a skill 20 times, we have the resources to be able to do this. The teacher creates options of paper practice for the student to choose from based on the work they have completed together in looping and from the information in Aleks.
Putting it together: Focus Packet Looping, Aleks, and paper and pencil work are all tracked in one integrated focus packet. Each packet is set up to show one set of focused skills from a portion of their Aleks program. As students enter a new focus packet, the teacher will use their Aleks report to determine and highlight which skills the student has already mastered. The teacher then use the packet to track new skills that need to be introduced in looping. After students are comfortable with a new skill, the teacher marks that the student should work on that skill in Aleks. After a student finishes the topic independently in Aleks, the student can then finish work with the skill with paper and pencil work. As work is completed, students continue to highlight and check items off in the focus packet.
Other Activities—Fact Practice
Over time, students are challenged to master math facts in addition, multiplication, and even simplifying fractions. The main goal of fact practice is not to make students fast at facts but rather to make facts automatic for students. Making facts more automatic helps clears room in the students working memory so that he or she can focus on the skill at hand rather than on additional computation. We let students work on facts, especially multiplication facts, for as long as it takes to reach mastery.
Students practice facts during looping using dice or flashcards with their looping partner or the teacher three times a week. When students are working on facts during this time, they are focusing in on the facts that are most challenging to them. As they show mastery on a certain set of facts, the teacher will move them up to the next most challenging level.
Other Activities—Open Ended Word Problems: Another activity that is incorporated into our math class is open-ended word problems. This is an opportunity for students to practice communicating their thought processes as they solve a problem. Solving a problem is common for students; however, being able to explain how they arrive at their solutions can be challenging. With this activity, students can practice explaining their strategies and reasoning behind them.
Writing about math starts with patterned word problem in looping where the teacher asks the student to verbally explain processes and applications. Next, Aleks will reinforce these skills in application problems. Additionally, on a few dedicated days throughout the term, students have a whole class dedicated to working on more formal writing about three or more complex word problems. Students complete their work on detailed, leveled worksheets:
At first, students are expected to write the facts of the problem, write the question being asked, circle the operations needed to solve them, and solve the problem with organized work. This begins with simple, one step problems and progresses to more complicated problems with more steps. Later, students are expected to write the facts of the problem that are necessary to solve the problem (omitting any unnecessary facts), write the question being asked in their own words, solve the problem with organized work, and write at least one sentence explaining the work that they used to solve the problem. This is done as problems continue to increase in mathematical complexity.
Other Activities—Multiple Choice Word Problems: Multiple choice word problems provide another level of mathematical thinking: they ensure students can spontaneously return to old problems. As a side benefit, multiple choice word problem practice provides the student with experience for the types of problems that will appear on standardized tests. Students have a personal goal for the number and level of multiple choice word problems they are to solve. Students work on these multiple choice word problems when they have spare time (such as when they complete their paper and pencil work in fewer than 25 minutes) or when they need a break from other activities (such as when feeling frustrated with not being able to complete topics in Aleks).
Finding a Starting Point: Our goal is to take students from wherever they are to where they need to be. We start at a pace and level that is customized to each student. In order to help us determine the correct level for a given student, we use an Aleks pre-test, which assesses students against level and grade appropriate skill lists. Initially, we will have a student start working at a level that is relatively easy – we want them to start at a level at which they are already showing 30-50% mastery of the overall material. If a student scores below this, we will back a student down a level. From there, the momentum of success tends to continue.
Differentiation: Differentiation takes place every day in the classroom as each student is working on a different set of skills, a different concept, and at a different pace. During a looping session, the teacher is creating a set of problems that is completely customized for the student. In a typical term, a student can encounter hundreds of problems of various concepts that is completely different from another student. This customization is highly effective because it directly targets the student’s learning needs.
Aleks has a built in assessment program that makes sure students have true mastery of the material they complete. These assessments occur after a student completes about five hours of work. On these assessments, the Aleks program will draw from the most recent work the student has done, review work the student completed much earlier in the program, and introduce some future topics to see if the student has already mastered any of the upcoming concepts.
In addition to the assessment that Aleks provides, students take a cumulative, written assessment at the completion of each set of focus skills in their course as well as at the completion of each course. Depending on the results of the written assessment, the teacher will determine whether the student should be moved on to the next course or whether additional supplementary work should be completed.
Every six days of the term, students also complete a mad minute assessment focused on fact mastery. When completing the mad minute, students receive their grade from the previous week and set a new goal for themselves based on their previous performance. If a student is falling behind towards their goals from above, the teacher may intervene and help set goals for the student.
- 50% of the grade comes from achieving the Aleks goal: grade is determined by percentage of goal that student achieves each term, which is based on the student’s overall year goal
- 30% of the grade comes from completing and following the notebook expectations for paper and pencil work.
- 15% of the grade comes from word problem work. Each problem is graded on a four point scale with expectations determined by the level of word problem worksheet the student is working on
- 5% of the grade comes from math facts. Mad minutes are graded for accuracy with each student setting a goal to achieve each week based on past performance