Evidence Based Math Instruction
Notes Packet
Steve Schmidt
[email protected]
abspd.appstate.edu
Today’s Quote
“The only way to learn mathematics is to do mathematics.”
- Paul Halmos
Please Write on the Packet!
You can find everything from this workshop at: abspd.appstate.edu
Look under: Teaching Resources, Evidence Based Instructional Resources, Evidence Based Math.
Agenda
8:30 – 10:00
It’s All About Reasoning . . .
10:00 – 10:15
Break
This course is funded by:
10:15 – 11:45
Workplace Math
11:45 – 12:45
Lunch
12:45 – 2:00
Teaching from Concrete to Abstract
2:00 – 2:15
Break
2:15 – 4:00
Defeating Math Anxiety
Are You Familiar with the Content Standards?
The content we will examine today comes from The North Carolina Community College System
College and Career Readiness Adult Education Content Standards. These standards may be found
at: abspd.appstate.edu Teaching Resources, Evidence Based Instructional Resources, Applying
Content Standards: GPS for Success.
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Math in the Workplace
What makes this a poor example of a workplace contextualized problem?
“In the workplace cafeteria, Sally, the cafeteria manager, wants to make milkalopes. (Milkalopes
have ¾ of a cantaloupe and ¼ cup of milk.) Sally wants to make enough milkalopes to feed 300
third shift workers. How many cantaloupes and how many gallons of milk should she buy?”
In real workplace math:
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All problems are word problems
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All problems are realistic on-the-job situations
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None of the problems has explicit math
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No “find the common denominator”
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No written formulas - you are told to rearrange or solve
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You must interpret the English in terms of math and you must choose the correct math tools to
solve the problem
To develop realistic workplace math questions, have:
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More and more extraneous information to sort through
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More and more rearranging of information required to get to the answer
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More and more chained steps - sequencing is important.
To Summarize: Workplace math is critical thinking applied to math
Source: http://curriculumredesign.org/wp-content/uploads/Paris-Workplace-Math-MMayo.pdf
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Developing Math Reasoning - UPS ✔ Problem Solving Method
1. Understand the problem
What are you asked to do?
Will a picture or diagram help you understand the problem?
Can you rewrite the problem in your own words?
2. Create a plan
Use a problem solving strategy:
Guess and check
Make a list
Draw a picture or diagram
Look for a pattern
Make a table
Use a variable
Solve an easier problem
Experiment
Act it out
Work backwards
Change your viewpoint
3. Solve
Be patient
Be persistent
Try different strategies
4. Check
Does your answer make sense?
Are all the questions answered?
What other ways are there to solve this problem?
What did you learn from solving this problem?
Source: Polya, How to Solve It
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Understand
Plan
Solve
Check
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Research Based Instruction in Action
The instructor has a student, Sam, who works at a small restaurant. Sam has told the class about the
tasks he does on his job, so the instructor used that information to provide an activity for the class to
explore and expand upon patterns and to connect patterns with rules.
The Problem
Sam has to make 50 hamburgers for the lunch run. Each burger should be a quarter pound (lb.). The
ground beef comes in 3.5-lb. packages. He needs to figure out how much ground beef he needs to
take out of the freezer to make 50 burgers.
Instructor:
• What exactly are we trying to figure out in this problem? Do we need to find just one answer or
multiple answers to solve the problem?
• Is this similar to problems we have worked on before? What approach did we use in those other
problems?
• Can you think of ways to represent the information we have in front of us other than using words?
• Can anyone predict what they think a reasonable answer might be? We’ll compare that to the final
solution later.
The students used a visual strategy, developing the visual representation shown below:
Instructor: What does each part of the diagram represent?
Andrea:
• Part 1 shows that each package contains 3½ lbs. of ground beef.
• In Part 2, it shows that we know each burger has to be ¼ of a pound. And so, each pound can be
divided into 4 equal parts that equals ¼ lb. of beef. Here we show the breakdown of each pound.
You can get 4 quarter-pound patties out of each pound.
• Finally, in Part 3, by counting them out on the drawing, you can see that each package will make 14
burgers.
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Instructor:
• Does anyone have ideas about other ways we could represent this information visually?
• Does it make sense that the number of burgers in a package would be higher than the number of
pounds of beef in a package? Why or why not?
• Now that we have this information, do we have the answer to our problem? If not, what do we need
to do next?
Andrea:
Next we need to figure out how many packages are needed to make 50 burgers. Let’s make a chart
to show the ratio of packages to burgers. She designed and populated the chart below:
Instructor:
• Based on the chart, how many packages should Sam get out of the freezer? Why?
• Were you surprised that he would need this number of packages? Why or why not?
• Before you started to figure it out, did you think he would need more or less?
• So, the problem was represented in words first, and then with diagrams. What would it look like in
symbols?
Teaching Math Concretely: Levels of Math Learning
The term “level” refers to the order that information presented mathematically is processed and
learned. Mahesh C. Sharma, in “Learning Problems in Mathematics: Diagnostic and Remedial
Perspectives” states that “almost all mathematics teaching activities … take place at the abstract
level. That is where most textbooks … tests and examinations are.” For students who have not
mastered particular math content, he proposes the following order or “Levels of Math” as effective for
teaching mathematics:
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Levels of
Learning
Intuitive
Concrete/
Experiential
Explanation
Example
At the intuitive level, new material is
connected to already existing
knowledge. (The teacher checks that
the connection is correct.) Introduce
each new fact or concept as an
extension of something the student
already knows.
When a student is given
three-dimensional circles
cut into fractional pieces,
he/she intuitively begin to
arrange them into
complete circles, thus seeing the
wedges as part of a whole.
Manipulatives are used to introduce,
practice and re-enforce rules,
concepts, and ideas.
Present every new fact or
concept through a concrete
model. Encourage students
to continue exploring
through asking other questions.
Using the concrete model (in this case
the wedges) helps the student learn
the fractional names. As the student
names the pieces, the instructor asks
questions such as, “How many pieces
are needed to complete the circle?
Yes, four, so one out of these four is
one fourth of the circle. As students
continue to explore they may see that
two of the quarters equal half the
circle.
A Picture, diagram, or image is used
Pictorial/
to solve a problem or prove a theorem.
Representational Sketch or illustrate a model of the new
math fact. Pictorial models are those
pictures often provided in textbook
worksheets.
Abstract
Applications
Communication
When the
student has
experienced
how some
pieces
actually fit into the whole, present the
relationship in a pictorial model, such
as a worksheet.
The student is able to process
symbols and formulae. Show students
the new fact in symbolic (numerical)
form.
After the student has the concrete and
pictorial models to relate to, he can
understand that 1/4 + 1/4 is not 2/8.
Until this concept has been
developed, the written fraction is
meaningless to the student.
The student is able to apply a
previously learned concept to another
topic. Ask student to apply the concept
to a real-life situation. The student can
now approach fractions with an
understanding that each fraction is a
particular part of a whole. The
instructor can now introduce word
problems without illustrations because
students have images in their heads.
A student who is asked to give a reallife example or situation might respond
with 1/4 cup of flour + 1/4 cup of flour
equals 1/2 cup of flour.
The student is able to convey
knowledge to another student
reflecting an embedded understanding
and the highest level of learning. The
student’s success in this task reflects
an embedded understanding and the
highest level of learning.
Ask students to convey their
knowledge to other students, i.e.,
students must translate their
understanding into their own words to
express what they know.
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“Concrete” Math Teaching Ideas
Math Cards
Materials needed: Cards with decimals, fractions, or percents
Give students a set of cards. Working as individuals, they can put the cards in order from least to
greatest. Then, they can work in pairs and put both sets of cards together in order from least to
greatest. They can then work in groups of four and put all four sets together in order.
Acing Math (One Deck at a Time)
Using an ordinary deck of playing cards, this manual has over 50 games that reinforce math skills
including number recognition, ordering, sequence, positive and negative integers,
addition/subtraction/multiplication/division, order of operations and more. “Games help lift math off
the textbook pages, and they support students’ learning about numbers and operations” (Burns,
2009). Find it by Googling: Acing Math One Deck at a Time
Clothesline Math
Materials needed: Sets of math cards, masking tape, and scotch tape
Put a six foot long piece of masking tape on a whiteboard or on the floor. Put some numbers on the
tape such as -3, -2, -1, 0, 1, 2, 3. Have students work together to put a set of math cards on a
clothesline using scotch tape. The activity can be used with fractions, decimals, and percents or work
with all three. Cards can have both numbers and picture representations of numbers like
Ask
students to write equivalent fractions/decimals/percents on blank cards and then attach them to the
clothesline.
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Beach Ball Math
Materials needed: Beach ball; plastic cup and markers to create circles on the ball
This activity allows participants to practice basic operation and mental math skills. It stops worksheet
boredom by playing with a beach ball.
Preparation
1. Cover a large, inflated beach ball with circles by tracing circles using a 3 - 4 inch diameter pattern
(a plastic cup works well as a pattern) and a permanent marker.
2. Inside each circle write math problems that focus on the skills you want to reinforce.
Conducting the Activity
1. Have participants form a circle. If you have a large group you can form several circles of 6 to 8
participants.
2. Explain the rules of the game:
a. Toss the ball to a player
b. The player who catches the ball must read aloud and answer the problem that is covered by the
right thumb
c. That player then tosses the ball to another player
3. Continue to play ball as time allows.
Paper Plates
This activity uses two paper plates. Either buy or make the plates two different colors. Cut both
plates in a straight line halfway across. Put the two plates together so one is on top of the other. This
is easier to see done than describe, so Google: youtube paper plate angles and forward to the 5:30
mark to see this demonstrated.
Paper plates can be used to represent and teach fractions, percents, pie charts, angles, and more!
2 Color Counters
Two color counters, also available at EAI Education, can be used to teach multiple skills like
discovering patterns and signed numbers. A number of YouTube videos can be accessed that show
how to add, subtract, and multiply signed numbers using the counters. You can make your own
counters using pennies and stickers.
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Fraction Strips
Fraction strips are an easy manipulative to create. One can either cut up colored paper or card stock
into one inch wide strips or use white paper and markers. Create different colored paper strips to
represent different fractions. YouTube also has some videos that demonstrate how to use the strips
to represent equivalent fractions or teach adding fractions with different denominators.
Fraction Circles
Window Panes
Egg Cartons
Grid Paper
Teaching Math in Context
ABSPD developed a contextualized math resource featuring math used in healthcare, culinary, and
horticulture careers. Find it by Googling: teaching math in context abspd
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Classroom Questions that Develop Math Reasoning
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What does this mean?
What are you doing here? (indicating something on student work)
Tell me where you’re getting each of your numbers from here.
Why did you decide to…?
I don’t understand. Could you show me an example of what you mean?
So what are you going to try next?
What are you thinking about?
Is there another idea you might try?
Why did you decide to begin with…?
Do you have any ideas about how you might figure out…?
You just wrote down ____. Tell me how you got that.
What are you doing there with those numbers?
Do you agree with ______’s answer? Why or why not?
Is ______ always true, sometimes true, or never true?
Questions adapted from GED Testing Service
Research Base
Barber, D., Knight, C., & Voss, J. (2004). Numeracy. Boone NC: Author.
Polya, G. (1945). How to solve it. Princeton University Press: Princeton, NJ.
U.S. Department of Education, Office of Career, Technical, and Adult Education. (2014).
TEAL Math Works! Guide. Washington, DC: Author.
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