This article was published in California Math Council's Communicator magazine in March 08. I always go back to it to refresh ideas on improving student achievement in math. Feel free to share with attribution.

1. Prevention Trumps Intervention
by Renee Hill, Riverside
teamteach@sbcglobal.net
T
he urgency for supporting student ◆ Give immediate, corrective feedback. In the
learning in mathematics often comes December issue of Educational Leadership,
too late. Our need for intervention assessment expert Thomas Guskey re-
would be virtually eliminated if we could minds us that “35 years ago. . . [Benjamin]
systematically provide the opportunity to Bloom and his colleagues stressed that to
learn, prevent misunderstanding, teach diag- improve student learning, . . . progress
nostically, analyze errors, and offer immediate checks must provide feedback. . . and be
corrective feedback. followed up with correctives.”
◆ Provide the opportunity to learn. Schools
must deliver grade level content to each
and every student. If students do not have Our need for intervention would be
the opportunity to learn grade level con- virtually eliminated if we could
tent, they will never demonstrate profi-
ciency in an assessment of the standards. systematically provide the
opportunity to learn, prevent
◆ Prevent misunderstandings. This marks the
difference between a good teacher and an misunderstanding, teach
expert teacher. The expert works with her diagnostically, analyze errors, and
professional learning community (PLC) to offer immediate corrective feedback.
go beyond calendaring lessons to planning
the lesson delivery. The expert relies on
personal and PLC experience to know
where students typically stumble and to Strategies That Work for Mathematics
design lessons to prevent those typical Intervention
misunderstandings. The expert makes sure The first step in supporting student learning is
that the topic is sequenced properly and high quality instruction as a preventive mea-
alerts students to problem areas. sure; then intervention strategies can be ap-
plied. The strategies below are some of the
◆ Teach diagnostically. The Mathematics Frame- strategies that I have found to be effective in
work for California Public Schools states: classrooms are listed in alphabetical order,
Most challenges to learning can be corrected with rather than order of importance. Teachers
good diagnostic teaching that combines repetition should not attempt to try all of them but
of instruction, focus on the key skills and under- should instead incorporate those that they feel
standings, and practice. For some students modifi-
cation of curriculum or instruction (or both) may be
would work best in their individual situa-
required to accommodate differences in communi- tions.
cation modes, physical skills, or learning abilities.
(p. 230) Answer Keys
Students solve problems, then check. Establish
The diagnostic teacher asks deliberate a routine and be adamant that students ad-
questions, evaluates student responses, here to it. If a norm of integrity has been
and moves the students forward during established in the student learning commu-
the interactions. nity, students will not copy the answers. One
◆ Analyze errors. This work extends diagnos- method is that students complete the assign-
tic teaching to the guided and independent ment and take only that page to a correcting
practice times as well as assessment and table where the teacher’s edition or answer
homework sessions. Error analysis has to key and highlighters can be found. Students
be paired with immediate, corrective mark the incorrect answers with a highlighter,
feedback. Continued on page 36 fi fi
March 2008 CMC ComMuniCator Page 35

2. then rework those problems. The highlighting digit number, a student might add the clue
lets the teacher know how the student did on “five and one more.”
the first pass and the rework allows the stu- Crutches. Students learn a coping strategy.
dent to persist in solving correctly. Another For example, whenever my sixth grade stu-
method is “Scrambled Answers” where stu- dent Ronisha got an assignment that required
dents receive assigned problems and a sheet multiplication or division, she had to turn her
with answers scrambled. They first solve, then paper over and make her own multiplication
match their solutions. table for 3 through 8 times 6, 7, 8. Since she
had not committed those facts to memory, I
Ballpark Answers preferred to hold her responsible for the grade
Students use their mental math, known opera- level learning rather than spending a gener-
tions, compatible numbers, visual images, and ous portion of time remediating her fact
part-part-whole knowledge (see van de Walle, knowledge.
Ch 6) to “ballpark” the answers before finding
solutions. I use the term “ballpark” to indicate Danger Zone Alerts
a preference for sense-making over textbook- When delivering instruction on the topic, alert
style estimation or rounding, which is too students to areas that are commonly misun-
formulaic and often not useful. I have stu- derstood. For example, when teaching deci-
dents write down the homework problems, mal place value, alert students that decimal
then we ballpark all the solutions. Students place value and whole number place value are
use the ballpark estimates to judge the reason- very specific. For instance, 0.12 might be
ableness of their calculated solutions. mistaken as greater then 0.3. Demonstrate,
using base ten models, that 0.12 has a single
Build, Draw, Write tenth and two hundredths whereas 0.3 has 3
Students model with concrete objects, record tenths pieces.
their work in a drawing, and then record the
same work with numbers and symbols. Some Double Points
people refer to this as concrete, representa- Students receive one point for the correct
tional, and abstract; but I have found that answer and one point (or more) for the correct
teachers and students more easily remember solution. This places grading emphasis on the
build, draw, write. Build, draw, write is not solving rather than the answering.
necessarily sequential. Draw doesn’t literally
mean draw. It might mean visualize, imagine, Focus Friday
or rehearse, and uses the numbers, symbols, On Fridays, students meet in a group that is
and notation of mathematics. If the writing is focused on their specific learning need. For
prose, it is about the how and why of the example, a team of three teachers who are
mathematics—not a step-by-step regurgitation currently teaching multi-digit division might
of how to carry out a procedure. organize a group to review 6, 7, 8, and 9 facts;
one group for review of procedure; and a
Cues, Clues, and Crutches challenge group. The majority of students in
Cues. Students are prompted to be wary of each teacher’s classroom would complete an
their own areas of vulnerability. My third independent assignment. Each teacher would
grade student, Lewis, dove directly into sub- work with the small focus group made up of
traction problems. I asked what would make students from all three classrooms. Groups
him remember to check for regrouping and he could meet weekly, every other week, or once
drew a small cartoon figure he called Rooster per month. This can also be done across
Man. grades when there are overlapping learning
At first, I reminded Lewis to have Rooster needs.
Man help him remember to regroup. Eventu-
ally, he drew the character on his own and Hot Spot Assignments
together they checked for regrouping. Eventu- Student assignments are specific to areas of
ally, Lewis did not need Rooster Man. difficulty that arise when teachers monitor
Clues. Students learn a rhyme, mnemonic independent practice, analyze homework, and
device, or other learning support that helps ask questions. In this strategy, teachers make
them over a rough spot. For example, at the note of the specific areas of need, and then
bottom of flashcards for adding six to a single generate work that untangles the difficulty.
Page 36 CMC ComMuniCator Volume 32, Number 3

3. For example, simplifying fraction solutions Response Cards or Boards
might be overlooked. I recommend giving a Students individually respond to a question or
page of fractions and asking students to iden- prompt by using a white board, scratch paper,
tify whether they could be simplified. Then, as voting device, or response card. The teacher
fraction operation assignments are completed, scans all responses, providing scaffolding,
ask students to review solutions to verify that cues, or questions where necessary. The
all fractions have been simplified. teacher gives specific feedback and provides
immediate correctives as appropriate. You
Immediate Feedback might also use a preprinted card. For example,
Students get an immediate response as they an index card with the letters A, B, C, or D
work each problem. Feedback could be pro- written on each edge would allow students to
vided by the teacher or a learning partner. respond to multiple choice questions. It is
My Mistake critical for every student to respond and for
Students earn back credit for incorrect home- scaffolding, correctives, and feedback to be
work problems by analyzing and noting their offered.
own mistakes (“I added incorrectly,” or “I Right-sized Problems
multiplied instead of divided”). Reworking Students receive problems selected for their
the problem earns back points or fractions of current level of understanding plus a little
points. push. Struggling students get less complex
problems and gifted students or fast finishers
Nontraditional Computation Methods get more complex problems. For example,
Students learn methods other than the stan- when subtracting fractions, struggling stu-
dard United States computation methods, for dents might get problems with simple de-
example, partial products for multi-digit nominators or easily computed common
multiplication or compensation methods for denominators or fewer problems needing
addition and subtraction. simplified solutions. Gifted students would
Preview get mixed numbers with difficult to compute
All students can benefit from participating in common denominators and solutions requir-
a preview of the lesson. For struggling stu- ing simplifying.
dents, this can take place during a before- Sequenced Problems
school session, in a small group a day ahead Students receive specially designed problems
of the whole-class lesson, or in support that progress from easy to hard in order to
classes. This front-loading allows struggling facilitate error analysis. For example, multi-
students to then participate in the whole-class digit multiplication problems can progress
instruction. from no regrouping needed to multiples of
Repeat, Review, Reteach, Remediate tens to regrouping in one place with easy facts
Students require various levels of intervention to regrouping with more than one place with
support and these typical intervention strate- easy facts to regrouping in one place with
gies. Repeat works best when the student has a hard facts, and so on.
grasp of the topic, but might be unclear on Solve, Match, and Challenge
certain areas. Review suits students who do Students are partnered. They solve two or
not need a full repeat. They do not need a three problems independently and then com-
different approach; they just need a review of pare solutions. If solutions do not match,
portions of the lesson. Reteach is for students students challenge the response of the other
who need a different approach. For example, and rework together until they both know
if the teacher presented the topic abstractly, how to solve for the correct solution.
build and draw might help the student.
Remediate is for students needing prerequisite Strategy Instruction
skills. Their missing skills prevent them from Students receive instruction in strategies that
succeeding in the current topic and there are support computation. John van de Walle’s
no cues, clues, or crutches to provide adequate Elementary and Middle School Mathematics
support. offers a wealth of strategies. Addition ex-
Continued on page 38 fi fi
March 2008 CMC ComMuniCator Page 37

4. amples include making a ten, doubles, and lems to all homework assignments for the
near doubles. Multiplication examples are next two or three weeks.
doubles, squares, and times five plus one
more set or times ten less one set. Warm-Ups
Students solve warm-up problems specifically
Technology Assistance selected to clarify areas of misunderstanding.
There are a multitude of computer software Use this method when you do not have the
programs, applets, virtual courses, and web time or the need to dedicate a full lesson to
sites that claim to support students. My per- clarification. Warm-ups can be used in con-
sonal opinion is that it is rare to find a technol- junction with preview or review to its best
ogy solution that adequately supports a strug- effect.
gling student. I have had more success having
proficient and advanced students utilize Workstations and Games
technology while I work with the struggling Students are provided with memorable expe-
students. I have used Excel and Schoolhouse riences where they explore mathematical
Technologies to generate hot spot assign- ideas, connect to prior learning, and engage in
ments. Multiflyer allows students to work on practice. I have seen Marilyn Burns’ About
their facts-to-learn and occasionally refresh Teaching Mathematics, Kathy Richardson’s
their facts-I-know. This site can be found at Developing Math Concepts, Patsy Kanter’s
gdbdp.com/multiflyer/play_online. html. Partner Games, and Frog Publications’ Learning
The National Council of Teachers of Math- Games all used to improve learning. I often
ematics’ Illuminations web site illuminations. recommend that teachers place repeatable
nctm.org/ActivitySearch.aspx includes many activities in the workstation area. For ex-
useful applets as does the National Library of ample, give students three index cards. Have
Virtual Manipulatives at nlvm.usu.edu/en/ them write the decimal form on one card,
nav/vlibrary.html. I have worked with fraction form on another, and a shaded deci-
schools that use Get Ahead Math, Accelerated mal grid or fraction square on the third card.
Math, and publisher intervention sites, but all Once students ensure that the three represen-
must be monitored in order to get the best tations are correct, the cards can be used at a
effect for student learning. Math Forum has a workstation for a matching or concentration
searchable site called Math Tools at game.
mathforum.org/mathtools/.
References
Textbook Burns, Marilyn. About Teaching Mathematics: A K–8
Once, when providing a model lesson in a Resource. Sausalito, CA: Math Solutions Publications,
2000.
fourth grade classroom, I made a study guide
for the geometry chapter. Not a single student California Department of Education (CDE). Mathematics
in the class knew how to use the textbook to Framework for California Public Schools: Kindergarten
find information or learning supports. Stu- Through Grade Twelve. Sacramento, CA: CDE, 2006.
dents must use textbooks to support their
learning. They need to learn the features of Guskey, Thomas. “The Rest of the Story.” Educational
Leadership 65 (December 2007/January 2008): 28–35.
their text that support learning: the examples,
glossary, index, and review pages. Textbooks Marzano, Robert J., Debra Pickering, and Jane E.
are sent home as learning support tools and Pollock. Classroom Instruction That Works. Alexandria,
parents should familiarize themselves with VA: Association for the Supervision of Curriculum
the text. Many teachers also have alternate Development, 2001.
textbooks on hand for reteaching purposes. Richardson, Kathy. Developing Number Concepts (series).
White Plains, NY: Wesley Longman, 1999.
Tune-Ups
Students receive four to ten problems in van de Walle, John A. Elementary and Middle School
addition to the current homework. The tune- Mathematics: Teaching Developmentally. White Plains,
ups address specific areas of need. During the NY: Longman, 2001.
time of the year I call “division season” I
© 2008 Renee Hill
usually recommend that teachers cease devot-
ing full lesson sessions to practice division
and instead add three to four division prob-
Page 38 CMC ComMuniCator Volume 32, Number 3