(From the book Accessible Mathematics by Steven Leinwand,
Principle Research Analyst at the American Institutes for Research
and former president of the National Council of Supervisors of
Mathematics.) NWSC Cohort Workshops February 1-3, 2011 Slide 2
Factors Impacting Student Achievement Coherent Curriculum Guides
Aligned Assessments Access to Viable Print Materials Access to
Technology Administrative Leadership and Support Supportive
Parents/Guardians Quality of Instruction Slide 3 #1 Incorporate
ongoing cumulative review into every days lesson Todays Mini-Math
1. What is 1/10 of 450? 2. Draw a Quadrilateral and all of its
diagonals. 3. The cost of a substance is directly proportional to
its weight. If 30 grams of the substance costs $45, what is the
cost of 6 grams of the substance? 4. A population P increases by 5%
each year for 2 years. Write an expression for the population in
terms of P after 2 years. 5. What number is 1000 less than 18,294
Slide 4 # 1 Incorporate ongoing cumulative review into every days
lesson Your Task: Make a record of concepts covered during first
semester that need to be consistently reviewed. List by Strand
Slide 5 # 1 Incorporate ongoing cumulative review into every days
lesson Whats seen in an Effective Classroom: Ongoing, cumulative
review of key skills and concepts Students given the opportunity to
clarify their understandings Classes that waste no time and begin
with substantive mathematics at the very start of every class.
Classes that end with summarizing questions and/or exit slips The
use of a brief review and discussion of mini-math questions Slide 6
#2 Adapt what we know works in our reading programs... Jane went to
the store Can you read the sentence aloud? Can you tell me where
Jane went? Can you tell me who went to the store? Can you tell me
why Jane might have gone to the store? Do you think it made sense
for Jane to go to the store? Differentiation Through Questioning
Slide 7 #2 Adapt what we know works in our reading programs... Your
Task: Write three open-ended questions that you will use within the
next two weeks of instruction. Slide 8 #2 Adapt what we know works
in our reading programs... Whats seen in an Effective Classroom:
Consistent parallels between the types of questions that require
inferential and evaluative comprehensions in reading instruction
and the probing for ways in which the answers were found,
alternative approaches, and reasonableness in mathematics
instruction. Open-ended Questions and Parallel Tasks for purposes
of differentiating the mathematics Answers greeted with a request
for justification Reasonable homework assignments with the focus on
explanation and understanding Slide 9 #3 Use multiple
representations of mathematical entities. Draw at least 3
representations of 3 quarters Draw at least 3
representations/models for adding integers Systems of equations X +
Z = Y Y + Z = 9 Y X = 2 X + 1 = 6 Slide 10 #3 Use multiple
representations of mathematical entities. Systems of equations
Number Shapes 2 + = 9 - = 2 + = + 1 = 6 Slide 11 #3 Use multiple
representations of mathematical entities. Systems of equations
using 3-Bean Salads Salad #1 contains 2 Lima beans Twice as many
Red beans as Lima beans 10 beans in all Salad #2 contains 3 times
as many Red beans as Black-eyed Peas One more Lima bean than Red
beans 8 beans in all Slide 12 #3 Use multiple representations of
mathematical entities. Systems of equations using Bars Julie packs
her clothes into a backpack and it weighs 29 kg. Xavier packs his
clothes into an identical backpack and it weights 11 kg. Julies
clothes are three times as heavy as Xaviers. What is the weight of
the Xaviers clothes? What is the weight of the backpack? Xavier:
Julie: Slide 13 #3 Use multiple representations of mathematical
entities. Your Task: Look at the skills and concepts youll be
teaching over the next 3 weeks. Determine where you can provide
multiple representations for at least 3 of these skills/concepts.
Slide 14 #3 Use multiple representations of mathematical entities.
Whats seen in an Effective Classroom: Frequent use of pictorial
representations to help students visualize the mathematics Frequent
use of the number line and bar models to represent numbers and word
problems Frequent opportunities for students to draw or show and
then describe what is drawn or show Slide 15 #4 Create
language-rich routines Tell the person next to you three things you
see and/or know about the numbers 73 and 63. Vocabulary strategies
Wordles (handout in workshop packet) Wordles Math Reflections Slide
16 #4 Create language-rich routines Your Task: 1. List 30 key
vocabulary words that need to be reviewed/used this second semester
2. Create at least one writing assignment where students are asked
to explain their understanding of a concept. Slide 17 #4 Create
language-rich routines Whats seen in an Effective Classroom: An
ongoing emphasis on the use and meaning of mathematical terms,
including their definitions and their connections to real-world
entities and/or pictures Student and teacher explanations that make
frequent and precise use of mathematics terms, vocabulary, and
notation An extensive use of words walls that capture the key terms
and vocabulary with pictures when appropriate Slide 18 #5 Take
every available opportunity to support the development of number
sense Number Sense is a comfort with numbers that includes
estimation, mental math, numerical equivalents, a use of referents
like and 50%, a sense of order and magnitude, and a well-developed
understanding of place value. Number Sense is one of the
overarching goals of mathematics learning. How can we work to
develop number sense? Slide 19 #5 Take every available opportunity
to support the development of number sense By asking questions like
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? How else can we express that number? Is there still
another way? About how much would that be? How did you get that?
And by having students estimate the answer first! Slide 20 #5 Take
every available opportunity to support the development of number
sense Your Task: If you began class one day with the statement
provided below, what are ten number sense questions you could use
to help develop number sense? The statement: As of this morning, my
age is 28,935,285. Slide 21 #5 Take every available opportunity to
support the development of number sense Whats seen in an Effective
Classroom: An unrelenting focus on estimation and justifying
estimates to computations and to the solution of problems An
unrelenting focus on a mature sense of place value Frequent
discussion and modeling about how to use number sense to outsmart
the problem Frequent opportunities to put the calculator aside and
estimate or compute mentally when appropriate Slide 22 #6 Build
from graphs, charts, and tables Ticket Sales for Country Music
Concerts ConcertTickets Sold Kenny Chesney Brad Paisley Martina
McBride Lady Antebellum 385,204 259,593 285,447 327,982 About how
many tickets sold? Which concert was probably least popular? About
how many more tickets were sold for Chesney than for Paisley? Which
concert sold closest to 300,000 tickets? About what percent of the
total tickets did the McBride concert sell? Slide 23 #6 Build from
graphs, charts, and tables Your Task: Develop 5 questions around
the following data from the Federal Highway Administration:
Increasingly Crowded Roads in the United States 1996Growth since
1970 Miles driven Number of vehicles Number of drivers Population
Miles of roads 2.5 billion 206.4 million 179.5 million 265.3
million 3.9 million 123% 90% 61% 30% 7% Slide 24 #6 Build from
graphs, charts, and tables Use the 4 Representations Identify where
in the next month you will have the opportunities to work the data
Slide 25 #6 Build from graphs, charts, and tables Whats seen in an
Effective Classroom An abundance of problems drawn from the data
presented in tables, charts, and graphs Opportunities for students
to make conjectures and draw conclusions from data presented in
tables, charts, and graphs Frequent conversion, with and without
technology, of data in tables and charts into various types of
graphs, with discussions of their advantages, disadvantages, and
appropriateness Slide 26 #7 Tie the math to such questions as How
big? How much? and How far? to increase use of measurement Rocky
Rococo Pizza, with its rectangular pizzas, once had billboards
displaying the following: 20% More! We Dont Cut Corners! Slide 27
#7 Tie the math to such questions as How big? How much? and How
far? to increase use of measurement Using your scope and sequence
for the next couple of months, determine where there is opportunity
to work with measurement before the MCAs Slide 28 #7 Tie the math
to such questions as How big? How much? and How far? to increase
use of measurement Whats seen in an Effective Classroom Lots of
questions are included that ask how big? How far? How much? How
many? Measurement is an ongoing part of daily instruction Students
are frequently asked to find and estimate measures, to use
measuring, and to describe the relative size of measures that arise
during instruction. Frequent reminders that measurement is
referential Slide 29 #8 Minimize what is no longer important, and
teach what is important... If we can answer WHY? (or cant answer
WHY?) we teach a concept then we can determine WHAT to teach and
WHAT not to teach. Why do we teach multi-digit multiplication such
as 2953 multiplied by 12.5? Why do we teach division of fractions?
Slide 30 #8 Minimize what is no longer important, and teach what is
important... Your Task: Read #15 Less Can Be More in Faster Isnt
Smarter Note 2 to 3 key points to discuss at your table Slide 31 #8
Minimize what is no longer important, and teach what is
important... Whats seen in an Effective Classroom A curriculum of
skills, concepts, and applications that are reasonable to expect
all students to master Implementation of a district & state
curriculum that includes essential skills and understandings for a
world of calculators & computers A deliberate questioning of
the appropriateness of the mathematical content, regardless of what
may or may not be on the high-stakes state test, in every grade and
course Slide 32 #9 Embed the mathematics in realistic problems and
real-world contexts Dan Meyers Math Class Needs a Makeover Dan
Meyers Math Class Needs a Makeover "I teach high school math. I
sell a product to a market that doesn't want it but is forced by
law to buy it. Dan Meyer Slide 33 #9 Embed the mathematics in
realistic problems and real-world contexts Your Task: Read #1 Math
for a Flattening World in the Faster Isnt Smarter book Discuss at
your table the 4 th and 5 th Questions in the Reflection and
Discussion for Teachers section on p. 6. Slide 34 #9 Embed the
mathematics in realistic problems and real-world contexts Whats
seen in an Effective Classroom Frequent embedding of the
mathematical skills and concepts in real-world situations and
contexts Frequent use of So. What questions arise from these data
or this situation? Problems that emerge from teachers asking, When
and where do normal human beings encounter the mathematics I need
to teach? Slide 35 #10 Make Why? How do you know? Can you explain?
mantras Questioning Templates Slide 36 #10 Make Why? How do you
know? Can you explain? mantras Your Task: Choose one of the
Questioning Templates and make a commitment to use it throughout
the rest of this school year. Make a poster or a laminated bookmark
of it but use it and make it part of what you do in every class
every day. Slide 37 #10 Make Why? How do you know? Can you explain?
mantras Whats seen in an Effective Classroom Every student answer
is responded to with a request for justification Both teachers and
students consistently and frequently use Why? Can you explain that?
How do you know? or equivalent questions Dismissive responses such
as Not, Wrong, Not quite, and their equivalents are absent from the
classroom Slide 38 The 10 Instructional Shifts: 1. Incorporate
ongoing cumulative review into every days lesson 2. Adapt what we
know works in our reading programs and apply it to mathematics
instruction. 3. Use multiple representations of mathematical
entities. 4. Create language-rich routines. 5. Take every available
opportunity to support the development of number sense. 6. Build
from graphs, charts, and tables. 7. Tie the math to such questions
as: How big? How much? How far? to increase the natural use of
measurement throughout the curriculum. 8. Minimize what is no
longer important. 9. Embed the mathematics in realistic problems
and real-world contexts. 10. Make Why? How do you know? Can you
explain? classroom mantras Slide 39 Punting Is Simply No Longer
Acceptable Implementing these shifts takes time and it takes
planning. We are expected to find ways to make math work for far
more kids. We live in a world of calculators and computes and in a
world that expects, even requires, deeper understanding and far
greater problem-solving skills Planning Templates Slide 40 Wrap Up
Faster Isnt Smarter #40 Seven Steps to Becoming a Better Teacher
Evaluations