NCII --National Centre for Intensive Interventions
There are 8 recommendations that are widely recognized as important in designing a math intervention program:
Checklist for carrying out the recommendations Recommendation1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk. As a district or school sets up a screening system, have a team evaluate potential screening measures. The team should select measures that are efficient and reasonably reliable and that demonstrate predictive validity. Screening should occur in the beginning and middle of the year. Select screening measures based on the content they cover, with an emphasis on critical instructional objectives for each grade. In grades 4 through 8, use screening data in combination with state testing results. Use the same screening tool across a district to enable analyzing results across schools. Recommendation 2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee. For students in kindergarten through grade 5, tier 2 and tier 3 interventions should focus almost exclusively on properties of whole numbers and operations. Some older students struggling with whole numbers and operations would also benefit from in-depth coverage of these topics. For tier 2 and tier 3 students in grades 4 through 8, interventions should focus on in-depth coverage of rational numbers as well as advanced topics in whole number arithmetic (such as long division). Districts should appoint committees, including experts in mathematics instruction and mathematicians with knowledge of elementary and middle school mathematics curricula, to ensure that specific criteria are covered in-depth in the curriculum they adopt. Recommendation 3. Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review. Ensure that instructional materials are systematic and explicit. In particular, they should include numerous clear models of easy and difficult problems, with accompanying teacher think-alouds. Provide students with opportunities to solve problems in a group and communicate problem-solving strategies. Ensure that instructional materials include cumulative review in each session. Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures. Teach students about the structure of various problem types, how to categorize problems based on structure, and how to determine appropriate solutions for each problem type. Teach students to recognize the common underlying structure between familiar and unfamiliar problems and to transfer known solution methods from familiar to unfamiliar problems. Checklist for carrying out the recommendations ( 12 ) Recommendation 5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas. Use visual representations such as number lines, arrays, and strip diagrams. If visuals are not sufficient for developing accurate abstract thought and answers, use concrete manipulatives first. Although this can also be done with students in upper elementary and middle school grades, use of manipulatives with older students should be expeditious because the goal is to move toward understanding of—and facility with—visual representations, and finally, to the abstract. Recommendation 6. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts. Provide about 10 minutes per session of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval. For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts. Teach students in grades 2 through 8 how to use their knowledge of properties, such as commutative, associative, and distributive law, to derive facts in their heads. Recommendation 7. Monitor the progress of students receiving supplemental instruction and other students who are at risk. Monitor the progress of tier 2, tier 3, and borderline tier 1 students at least once a month using grade-appropriate general outcome measures. Use curriculum-embedded assessments in interventions to determine whether students are learning from the intervention. These measures can be used as often as every day or as infrequently as once every other week. Use progress monitoring data to regroup students when necessary. Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions. Reinforce or praise students for their effort and for attending to and being engaged in the lesson. Consider rewarding student accomplishments. Allow students to chart their progress and to set goals for improvement.
Ø Interventions should begin with 5 minutes or so of basic fluency, things the student already knows how do but needs to get faster at. You can use questioning, drawing problems (questions) on white board, flash cards, playing cards, dice, games, whatever. Typically this is adding/subtracting or multiplying/dividing.
Ø Work on dry erase boards, preferably vertical surfaces (a classroom whiteboard or dry erase poster board fixed to wall). Sometimes with manipulatives its easier to sit and use an individual dry erase board.
Ø Monitor progress: Keep notes on what was covered and every day review the learning the day before.
Ø When stretching to new skills or working on skills, the most important question we ask is “how did you get that?” Reinforce the need for students to verbalize their reasoning. Help them find words to describe what they did.
“OH, you counted up?” or “You worked from a known fact?” “You used what you know about doubles?” “Did you imagine a number line? Counters? Ten frame?” “ Ten-pairs?” But don’t do the talking for them.
Ø Help students get comfortable with struggling mentally. It’s hard to not offer answers! But hold off. If they’re stuck, ask “Well, what are you thinking?”
We think differently about math when we speak about math. Give guidance when you have to.
Ø Paper-pencil practice is necessary and plays a role, but our struggling learners need to talk and interact with someone. Keep interventions as interactive as possible. Allow movement, dialog, and sense making.
Ø C-R-A continuum is very important. Use manipulatives to reinforce procedures. You may have to help students model the same thing repeatedly, over time.
Ø Avoid “tricks” “rules”, and rote memorization. Developing fluency is not the same as memorizing. For instance, lost of kids can skip count by two or five out loud, but its because it has become a “sing song”, something they can recite, but it may not be attached to number sense at all. Students know to “borrow” or “carry” (regroup) because they’ve been shown and they have practiced, but they may not know what that actually means.
Ø Always link addition to subtraction. If working on addition, pause periodically to have student say or write out the fact family or answer related subtraction facts. Same with multiplication/division.
Ø A basis for fluency is being able to compose and decompose numbers (take them apart in different ways).
Why: Students solving 8+6 or 18 + 26 or 1 248 + 96 are all doing the same thing: Because we work in a base ten system, we are always “making ten”.
When we add 8 + 6, we are thinking 8 + 2 + 4
For this reason, we do a lot of work with ten frames. Students should be able to subitize in ten frames and give the related number to make ten.
For instance, you allow the student to glance at this card for 2 seconds, then cover it:
You ask “How many dots”? So the student hopefully saw it was one less than 5, so they say 4.
Then ask “So then how many are empty”? Student should say six without thinking, counting, using fingers, etc. The student needs to get fluent with knowing the pairs of numbers that make ten (“ten-pairs”) because if they are not, they will always need to count on their fingers, or draw dots, or count up mentally.
We are not done with this exercise till kids know all the pairs that make ten.
Fish for ten is an exercise that supports this. Using this skill never goes away—its embedded in all the math we do. If we can’t add, we can’t subtract, and we can’t multiply. If we can’t multiply, we can’t divide or work with fractions, etc.
Ø Provide LOTS of models: Ten frames, base ten blocks, number lines, Number bond diagrams, part-part-whole boxes, fact families.
Ø Games are great interactive and engaging ways for kids to practice. Kids should LOVE intervention time. But even when it’s not their turn, make sure they’re helping do the math—not just answering, but talking about how they got an answer.
Ø Be curious. Listen. Remember the person doing the talking is doing the learning.
Ø Help your student feel comfortable by reminding them that being fast at math doesn’t mean your better at math. Slower processors often think more deeply about the math. We’re not working on fluency to get faster, necessarily, but to get more efficient. If we don’t get fluent with basic facts then our brain is tied up with computation and can’t move on to the deeper mathematics (like the shape, orientation, graph, trend, data, probability, etc).
Ø Infuse your time with students with growth mindset talk. We can be transparent about why we do things. We can respect kids’ maturity by telling them that we do things because research points to positive outcomes —that we need models to attach our thinking to. That we will learn and grow. That making mistakes creates more neural activity than practicing things we already know how to do. And that struggling in math is like body-building: Even if you don’t get the right answer, or even if you don’t get an answer at all, struggling with mathematical reasoning and engaging in mathematical dialog improves your brain and your thinking.
Ø End your notes with progress you saw and things to keep working on, including maybe a quick plan for next session.
Ø Track progress with the I Can statements provided for Sask Curriculum. A more detailed description of what each one means follows in the binder—a summary of Sask Curr’m outcomes K – 9.
Ø And of course, relationships are everything. Help your student see a sense of achievement. Point out progress. Appreciate focus and effort.
Essential Numeracy Skills and Understanding
Counting, cardinality, and quantity
Representing numbers
Subitizing
Compose and decompose number (Addn/subtr), fact families, number bonds (part-part-whole)
One more, one less, two more, two less
Comparing numbers
Models: Rekenreks, ten frames
Making ten **Important
Using place value, counting ten as a single unit
Models: Number path, number line
Ten more, ten less
Understands hundreds chart and patterns within hundreds chart
Number system and place value
Two-and three-digit addition and subtraction
Strategies for addition and subtraction
Multiply and divide
Compose and decompose number by factors
Strategies for addition, subtraction (larger numbers) multiplication and division
Fractions of a whole, of a set, fractions as a number
Modeling and representing fractions
Equivalent fractions
Comparing fractions
Factors and Multiples
Addition strategies:
Using place value, making friendly numbers, compensating, using bench marks and open number lines, adding in chunks, using doubles, near doubles, decomposing by place value, algorithm.
Subtraction: Constant difference, subtracting in chunks, benchmarks and open number lines, compensating, using friendly numbers, algorithm.
Multiplication strategies:
Doubling, skip counting,
Four (double double)
Eight (double double double) ,
Six (mult by three then double)
multiplication by factors of ten,
Nines, elevens
Fives (mult by ten and take half)
Multiplying using partial product (compose and decompose number), area model multiplication, “Fact families” (mult and div), doubling and halving factors.
Division strategies: Dividing in chunks, halving, working from multiplication, compensation, division by factors of ten, algorithm.
