While looking for some information this morning for a teacher regarding our state test, I found something that just might turn out to be useful. I tend to regard high-stakes tests as a necessary evil, but sometimes good things do come out of them. For example, what I found in my massive files this morning was a document called “10 Practical Instructional Strategies for Grades K-8.” It is centered around preparing students to take the CMT (Connecticut Mastery Test), but the strategies could be applied to any mathematics classroom.
10 PRACTICAL INSTRUCTIONAL STRATEGIES FOR GRADES K-8
Preparing students for the Connecticut Mastery Test (CMT) should be an ongoing
process. The process requires sufficient instructional time and appropriate instructional
strategies. While it is certainly appropriate to conduct some form of review, “cramming”
is far less effective than an ongoing set of instructional practices that naturally and
continually prepare students for the test specifically and for higher levels of
understanding generally. A sound K-8 mathematics program embeds these strategies into
all instructional planning.
Strategy 1: Asking “Why?”
Probably the best way to implement a “thinking curriculum” – a curriculum that is
language-rich, focuses on meaning and values alternative approaches – is by regularly
asking students “Why?” A simple, “How do you know that?” or “Can you explain how
you got your answer?” or the basic, “Can you explain to the class why you think that?”
forms the basis of a mathematics curriculum that goes beyond merely correct answers. A
student who can explain his or her answers often has a stronger understanding of
mathematics and can help other students develop understanding. Questions like, “How
did you get 17?” or “Why did you add?” give students powerful opportunities to
communicate their understandings and give teachers powerful tools to assess the degree
of understanding. Classrooms where students are regularly explaining how and why, both
orally and in writing, are classrooms that effectively prepare students for many of the
open-ended items on the CMT.
Strategy 2: Embed In Context, Present As A Problem
Consider the vast difference between “Find the quotient of 20 ÷ 1.79” on the one hand,
and “How many hamburgers, each costing $1.79, can be purchased if you have a $20
bill?” Both problems expect that students can divide. However, the former directs
students to a single long division algorithm with a three-digit divisor that isn’t even tested on the CMT. The latter places the mathematics in a context and expects students to
understand that division is an appropriate operation to use to solve a practical problem. In addition, the latter encourages estimation and raises the issue of sales tax, all of which is assessed on the CMT. Most importantly, the contextualized problem shows students that mathematics is a useful tool.
Strategy 3: Ongoing Cumulative Review
One of the most effective strategies for fostering mastery and retention of critical
mathematical skills is daily, cumulative review at the beginning of every lesson. Rarely
does one master something new after one or two lessons and one or two homework
assignments. Many teachers call this “warm-ups” or daily-math. Five to eight quick
problems to keep skills sharp can be delivered orally or via visual methods. Every day
teachers should present:
- a fact of the day (e.g., 7 x 6);
- an estimate of the day (e.g., What is a rough estimate of the cost of 55 items at
- $4.79 each?);
- a measure of the day (e.g., About how many meters wide is our classroom?);
- a place value problem of the day (e.g., What number is 100 more than
- a word problem of the day; and
- any other exercise or problem that reinforces weaker, newer or problematic
- skills and concepts.
This form of review, often patterned after the types of items and item formats used on the CMT, embeds review for the test in what is recognized as sound instructional practice.
Strategy 4: Ensure A Language-Rich Classroom
Like all languages, mathematics must be encountered orally and in writing. Like all
vocabulary, mathematical terms must be used again and again in context until they
become internalized. Just as young children confuse left and right until they develop
strategies and connections to distinguish between the two, older children confuse area
and perimeter until they link area to covering and perimeter to border. A language-rich
classroom, in which mathematical terminology is regularly used in discussions, solving
problems and in writing, can make a big difference in how effectively children learn
mathematics. Posting vocabulary in the room, perhaps on a word wall, is one way to
ensure that mathematical terminology is used on a daily basis. While not exhaustive, the
vocabulary word list found in each grade-level section of this handbook should be used to
ensure that the language used and expected on the CMT is never new to students. 12
Strategy 5: Use Every Number As A Chance To Build Number Sense
The development of number sense is one of the overarching goals of mathematics at the
elementary level. Number sense is a comfort with numbers that includes estimation,
mental math, numerical equivalents, a sense of order and magnitude, and a welldeveloped understanding of place value. The development of number sense must be an ongoing feature of all instruction. A review of CMT 4 reveals how much of the test
focuses on these critical number sense understandings. A simple strategy for
incorporating number sense development into all instruction is to pause regularly and,
regardless of the specific mathematics being taught, ask questions such as the following:
- Which is most or greatest? How do you know?
- Which is least or smallest? How do you know?
- What else can you tell me about those numbers? For example, “they are both odd,” “all are mixed numbers,” “their product is about 18 because you can round.”
- How else can we express .2 (2/10, 1/5, 20%, .20)?
Incorporating this strategy into daily instruction creates a mind-set that the numbers in
every problem posed and in every chart or graph used can strengthen and reinforce
number sense. For example, in a simple word problem that asks students to find the sum
of 57 and 67, teachers can first “pluck” the numbers from the problem and ask students to
list four things they can say about the two numbers. Consider how much mathematics is
reviewed when students suggest findings such as the following:
- I see two two-digit numbers.
- Both numbers are odd.
- There is a difference of 10 between the numbers.
- The 67 is greater than the 57.
- The ones digit is the same and the tens digit is one apart.
- One number is prime and the other is composite.
- I see 124. 13
Strategy 6: Draw A Picture (Mental or Real)
We say casually that “a picture is worth a thousand words” but we seldom connect
mathematical concepts to their pictorial representations. A significant proportion of the
CMT asks for pictorial equivalents of mathematics ideas. A powerful way to help
students visualize the mathematics they are learning, or to reinforce understanding, is
with mental images or pictures that students actually draw or create. Consider how
infrequently we ask students to, “Show me with your hands about eight inches” or, “Use
your fingers to show me an area of about 10 square inches.” Consider how important it is
that students can draw pictures of fractions and mixed numbers like ¾ or 2 ½ and of
decimals like .3 and 1.2. Consider how powerful a class discussion about the different
pictures for “three-quarters” can be when students show three quarters (25-cent pieces), a shaded pizza slice, a window pane, three stars out of four shapes, a ruler, a measuring cup and simply ¾! Consistently embedding, “Can you draw a picture of…?” and “Can you
show me what that would look like?” into instruction can pay rich benefits in both
student understanding and in CMT scores.
Strategy 7: Build From Graphs, Charts and Tables
Many real-world applications of mathematics arise from the data presented in graphs,
charts and tables. This is why so many of the CMT items are based on data and include
graphs, charts and tables. To best prepare students for these contexts, as well as develop
the essential skills of making sense of data and drawing conclusions from data that is
presented in graphs, charts and tables, teachers are encouraged to make far greater use of
these forms of data presentation. Given a graph or table, students can be asked (similar to Strategy 5) to identify five things they see in the graph or table. In addition, students can be asked to draw two appropriate conclusions from the data and justify those conclusions. So consider “milking” the graphs and charts found in your textbook or data that students find during “data scavenger hunts” by copying the graph, chart or table for students and asking them to create five questions that could be answered by the information in the graph or table. Ask students to share their questions and generate a list of the best questions for future use.
Strategy 8: How Big? How Much? How Far?
No strand of mathematics assessed on the Connecticut Mastery Test produces student
scores as consistently weak as the measurement strand. Rather than leave all
measurement to a single chapter that is often skipped entirely, teachers are encouraged to make measurement an ongoing part of daily instruction. First, questions like, “How
big?,” “How much?,” “How far?,” “How heavy?” all help to develop measurement
understanding. Second, measurements of things such as arm span, book weight, area of
circles, or breath-holding times all provide great sets of data and, therefore, use
measurement to gather data that is analyzed and generalized – integrating many important aspects of mathematics. Finally, more involved projects like determining the number of students that can fit in a classroom or the number of hours students have been alive are wonderful opportunities to keep measurement on the front burner of daily instruction.
Strategy 9: Omit What Is No Longer Important
A significant amount of time and energy is expended by teachers and students on skills
considered less important by national and state standards and not even assessed on the
CMT, the Connecticut Academic Performance Test (CAPT) or the SAT. District
mathematics curriculums must become more focused on what is truly valued and teachers must give themselves and each other permission to skip textbook pages that no longer serve useful purposes. In fact, the proverbial “mile-wide, inch-deep” curriculum that results in far more coverage of topics than mastery of key concepts undermines many efforts to raise student achievement. In addition, time that is no longer spent on
increasingly irrelevant skills – particularly those done most often with a calculator – frees up valuable minutes and hours for increasingly important skills like estimation, algebraic reasoning and problem solving. So carefully review what is NOT assessed on the CMT – particularly complex, multidigit computation – and redirect what is taught to focus on those skills and concepts that have lasting value and that ARE assessed.
Strategy 10: Focus On Sense-Making As Well As Correct Answers
One of the most powerful test-taking skills for multiple-choice items is the artful
elimination of obviously absurd answers. However, identifying such “obviously absurd
answers” – for example, a sales tax of $129 dollars instead of $1.29 on a $20 item –
requires a mind-set that mathematics makes sense. This “minds-on” approach to
instruction is in sharp contrast to the rote regurgitation of rules and procedures to get
correct answers to exercises that all too often comprises mathematics instruction. For
example, when teaching how to convert mixed numbers to improper fractions and vice
versa, it is imperative to teach why these forms are equivalent. Students who only know
how to multiply and add and not why 3 ¾ is equivalent to 15/4 are at a disadvantage in
life and on the CMT. Focusing on the why – that is, focusing on understanding and sensemaking – emerges from consistent use of many of the preceding strategies, particularly 1 and 6.Teachers can improve instruction significantly by adopting the mind-set that good mathematics instruction begins with an answer. That is, when a student responds (for example) “17,” the next question should be something like, “How did you get that?” When the student responds “I added” or “I rounded” or “I took about half,” the next question should be something like, “Why did you do that?”