When it comes to improving students’ understanding of math skills and concepts, it is essential for them to feel a sense of efficacy, motivation, and engagement with the material. One way to ensure this is by implementing methods that not only help students solve the problem in front of them but also develop cognitive skills to solve more difficult and complex problems independently.
The Branching Minds RTI/MTSS system-level education platform houses the most robust library of evidence-based learning supports and interventions across academics, behavior, and SEL.. Through our work with thousands of schools that take advantage of this curated and tested library, we are able to glean insights on what has been the most used interventions. The following strategies are ones that were commonly used in 2020 by our Branching Minds district partners and are also techniques that are based in research and best practices for teaching.
It isn’t surprising that manipulatives were the most commonly used Math support on Branching Minds in 2020. These tools can be used across grade levels and math content areas, such as counting, place values, fractions, decimals, area, volume, addition, subtraction, multiplication, and division. The purpose of manipulatives are to help students visualize and develop a deeper understanding of a variety of math concepts. Research has also shown that using manipulatives in math, compared to abstract symbols, results in higher retention, problem-solving, transfer, and justification skills.
It is important that the manipulatives being used are appropriate for the concept or lesson being taught. For example, when using manipulatives to develop early numeracy skills, students should be given objects to help them count or be asked to show a specific number using the counters. Teachers should always observe the student as they use the manipulatives the first time to ensure they are using them correctly. For example, does the student exhibit 1:1 correspondence with the counters and demonstrate organization with their counting (e.g., pulling objects counted away from objects not counted; organizing into a row; organizing into groups)? Finally, the teacher should ask follow-up questions to ensure the student understands the concept. In the counting example, the teacher can ask the student how many objects there are. If the student restates the last number, they understand the concept of cardinality of sets.
This is a strategy in which the teacher provides scaffolding through models or examples for students as they proceed through a set of math problems. It is also known as interleaved solutions and problem solving. The technique can be used with any problem solving content area in which the student is struggling. Proving these types of examples and models helps students understand the process for solving multi-step math problems. A large number of laboratory experiments and a smaller number of classroom studies have demonstrated that students learn more by alternating between studying examples of worked-out problem solutions and solving similar problems on their own than they do when just given problems to solve on their own.
Before the lesson, teachers develop sample solutions to problems. The worked examples are introduced to students in small groups, one-on-one, or by providing examples on the assignment that is given to students. Students alternate problems to be solved with examples that already demonstrate the solutions (i.e., worked problems). Initially every other problem is a worked problem, but as students proceed through the problem set, the ratio of problems to solved and worked problems changes. If it is being used as a one-on-one strategy, the student should explain to the teacher the steps of the worked problem. If done in a small group, students can independently solve problems and then take turns explaining the steps of the worked problem to the group. Gradually, the teacher reduces the number of worked problems per worksheet.
Structured organizers can be used in many ways but are commonly used in Math to help students break down world problems. This process scaffolds students to help them understand the problem and the steps required to solve it. It is especially helpful when working with students who don’t read story problems carefully, sometimes due to challenges with attention and organization. It can be used one-on-one, with small groups or the whole class.
The organizer itself should include different sections for the different components of the word problem. Examples can be found here. Teachers should begin by modeling how to fill out the organizer to solve a problem while students complete their own at the same time. Next, students should try to solve the problem based on what was written in the organizer. Students should then try another problem and fill out the organizer on their own. Corrections should be made as necessary and eventually students will be able to transition to solving problems without the organizer.
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4) Explicit Time Drills for Math Fluency
This commonly used math activity can help students develop a sense of fluency when it comes to specific math skills, such as addition, subtraction, multiplication, and division. When using this strategy, students are timed on math fact worksheets. If used correctly, it can help train the automatic retrieval of math facts, which is a critical skill that helps students as they work on more difficult math problems.
Students should be provided with a math worksheet with simple math facts and be given a specific amount of time that they have to work on the sheet. Set a timer and have students complete as many questions as they can. They should indicate the question they are on when the timer goes off, and then it can be repeated for another few minutes. When teachers collect the worksheets they can assess each student’s math fluency based on the amount of accurate work they were able to complete during each time set. It is important that students are not singled out, as activities like this can be stressful for students who struggle with math. Students should understand the skill they are trying to develop, rather than it being a race or competition.
Using mixed difficulty assignments in Math can improve students’ motivation and engagement by intermixing easier problems and challenging ones. Research shows that students are more motivated to complete computation worksheets when they contain some very easy problems interspersed among more difficult items. In other words, successfully solving easier problems provides students with a boost of confidence needed to take on the more challenging ones. One research study found that sixth grade students completed more math problems correctly when easier problems were interspersed among the more difficult problems at a 1:3 ratio.
When using this strategy, teachers should identify one more “challenging” problem types that are matched to their students' current math-computation abilities (e.g., multiplying a 2-digit number by a 2-digit number with regrouping). Next, identify an "easy" problem-type that the students can complete very quickly (e.g., adding or subtracting two 1-digit numbers). A series of student math computation worksheets can be created with "easy" computation problems interspersed at a fixed rate among the "challenging" problems. If students are expected to complete the worksheet independently as seat work or homework, "challenging" and "easy" problems should be interspersed at a 1:1 ratio (that is, every "challenging" problem in the worksheet is followed by an "easy" problem). If you will be reading the problems aloud having students solve the problems mentally, writing only the answer, the items should appear on the worksheet at a ratio of 3:1 (that is, every third "challenging" problem is followed by an "easy" one).
For students who benefit from visualizing word problems, having them draw a picture can be an effective strategy to effectively solve these problems. This method has been shown to be especially impactful for students with learning disabilities and is similar to the structured organizer approach. Sometimes, just being able to visualize what the word problem or story is describing can help students understand the scenario and what exactly they are being asked to do. In addition to traditional drawings, other visuals can be used, such as venn diagrams or number lines.
Students should first read the problem and paraphrase if necessary. Next, students should be prompted to ask themselves what they know about the problem. If applicable, students should decide the most logical picture form to draw. Be sure drawing is a representation of the problem, not just pictures of the items mentioned in the problem. Students will likely need some help and scaffolding when starting with this method. Teachers can even provide drawings for the first few word problems and eventually transition to students creating them on their own. This can also be an effective partner assignment when working with the entire class or in small groups.
Flashcards are a commonly used strategy for most subject areas, but when used appropriately in math they can help students develop fluency in basic math facts and concepts. This method involves pairing unknown computation items with a steadily increasing collection of known items through concentrated and repeated practice with one fact at a time. The key component of this strategy is incremental rehearsal; this means that both known and unknown concepts are presented together which allows students to keep their momentum and stay engaged. It also gives them an opportunity to practice or rehearse the concepts they have just learned. The use of incremental rehearsal has been shown to improve math skills specifically among children with learning disabilities.
This strategy is most effectively implemented when working with students one-on-one. Teachers start with a set of math fact cards, including ones that the student is expected to know. The first run through the deck is meant for teachers to separate the “known” and “unknown” cards. Then the cards can be selected so that students are presented with “known” cards alongside “unknown” ones. When the concept is learnt, the card can move to the “known” set and these are continually rehearsed as well. Overtime, new concepts/cards can be added to the deck once students master the ones that have already been presented.
This self-monitoring strategy is designed to support students who know basic facts but have difficulty solving multi-step problems. It is also a scaffolded approach where students are provided with a checklist or mnemonic device that they can use independently to monitor their own progress. This type of approach not only develops students’ math skills, but it also fosters skills foundational for all learning, such as self-management, self-awareness, and executive functioning. It also will allow students to figure out more complex problems independently in the future.
When using this strategy, teachers should first be aware of the specific types of problems the student is struggling with. All of the steps required to solve these types of problems should be outlined and turned into a checklist that the student can understand and interpret. The list should be explained to the student one-on-one, or in a small group, and then the teacher can model how to use it. For students needing additional support, the teachers should go through the checklist alongside a few problems with the student so they fully understand how it should be used. As students move towards working through the problems and checklists independently, teachers should still check in and make sure they are using it properly and getting the correct answers. To fade back reliance on the checklist, the student can be given a mnemonic device to help her/him remember the steps.
This targeted approach is ideal for teaching students early math skills, such as addition and subtraction. The student is provided with a number line and taught to start at the larger number and count up by the amount being added or count down by the number being subtracted. Being able to visually see the numbers on a line in combination with counting out loud can help students understand the underlying concepts of addition and subtraction. Although students can practice this strategy independently, it is intended to be implemented by a tutor, either one-on-one or in a small group setting.
Once the counting up and down methods are taught, teachers can present students with the addition and subtraction questions via flashcards. Students should first be able to try and answer the questions based on memory—but that if the student does not know the answer, he or she should use the appropriate count-up/down strategy. The teacher then reviews the flashcards with the student. Whenever the student makes an error, the teacher directs the student to use the correct strategy to solve. The goal is for students to eventually learn all of the math facts and phase out the use of the strategy. One randomized controlled study found that students who engaged in this practice one-on-one with a tutor for 16 weeks (3 sessions a week; 20-30 minutes per session) improved significantly in their number counting fluency and procedural calculation skills, compared to students in comparison groups.
This is a strategy to help students solve applied math problems by going through a series of four steps. Similar to the word problem strategies above, it is helpful for students to break down these problems into concrete steps. Each of the four steps includes approaches mentioned above, which allow the student to become more aware of the problem itself and what it is asking as well as how to go about solving it and checking the results. For many students, it is helpful for them to use these types of organizational strategies to get them used to going through the steps required to solve a problem. This is also a very useful strategy for students who tend to rush through problems and thus make errors and not get the overall correct result.
The four steps include the following:
Like most strategies, this method should be modeled first to students and scaffolded so that students have help going through the steps the first few times. Eventually, students should be able to work through the steps independently. Over time, the process of explicitly writing out the four steps can be phased out or shortened as students are able to think through and develop them automatically.
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At Branching Minds, we pride ourselves for having the most robust library of evidence-based K-12 interventions and accommodations of any online platform available to schools. Our library includes over 2000 evidence-based activities, strategies, tools, apps, and programs collected from the most trusted and respected hubs of evidence-based supports, including the Florida Center for Reading Research, What Works Clearinghouse, Evidence for ESSA, Intervention Central, the IRIS Center from Vanderbilt University, and Sanford Harmony. All of the supports have been reviewed by our team of learning scientists and sorted by ESSA guidelines for determining tiers of evidence. The supports include detailed descriptions to help teachers understand what each support is, why to use it, how to use it in alignment with the evidence, what the evidence is and provide them with any materials necessary to implement the support.
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Essie Sutton is an Applied Developmental Psychologist and the Director of Learning Science at Branching Minds. Her work brings together the fields of Child Development and Education Psychology to improve learning and development for all students. Dr. Sutton is responsible for studying the impacts of the Branching Minds on students’ academic, behavioral, and social-emotional outcomes. She also leverages MTSS research and best practices to develop and improve the Branching Minds platform.