Evidence Based Math Instruction

Steve Schmidt

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

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.

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:

·  All problems are word problems

·  All problems are realistic on-the-job situations

·  None of the problems has explicit math

·  No “find the common denominator”

·  No written formulas - you are told to rearrange or solve

·  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:

·  More and more extraneous information to sort through

·  More and more rearranging of information required to get to the answer

·  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

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 Solve an easier problem
Make a list Experiment
Draw a picture or diagram Act it out
Look for a pattern Work backwards
Make a table Change your viewpoint
Use a variable
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

Understand
Plan
Solve
Check

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.

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?

Research Based Practices Learned from This Lesson. It’s WIOA Friendly Too!

1. The instructor acted as a ______

2. The instructor taught using ______

3. The lesson was taught in a ______context

4. The instructor never said, you’re ______or you’re ______

5. The instructor taught math from ______to ______

6. The lesson was primarily ______

Math Humor!

Teach Math from Concrete to Representational to Abstract: 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:

Levels of Learning / Explanation / Example
Intuitive / 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.
Concrete/ Experiential / 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.
Pictorial/ Representational / A Picture, diagram, or image is used to solve a problem or prove a theorem. Sketch or illustrate a model of the new math fact. Pictorial models are those pictures often provided in textbook worksheets. / When the student has experienced how some pieces actually fit into the whole, present the relationship in a pictorial model, such as a worksheet.
Abstract / 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.
Applications / 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 real-life example or situation might respond with 1/4 cup of flour + 1/4 cup of flour equals 1/2 cup of flour.
Communication / 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.

Adult Education and WIOA’s Integrated Education and Training

The finish line has changed for our programs. It is no longer get your GED and see ya later!

Now it means developing career pathways so students can earn credentials allowing them to get jobs that pay family sustaining wages.

Integrated Education and Training is:

1.  Literacy Instruction (We have always done this but now with contextualized instruction)

2.  Workforce Preparation Activities (Employability Skills)

“Help participants acquire a combination of basic academic, critical thinking,

digital literacy, and self-management skills including: using resources, using

information, working with others, understanding systems, and gain the skills

necessary for successful transition into and completion of postsecondary

education/training/employment”

3.  Occupational (Job) Training

Use partnerships like a community college’s continuing education and curriculum

programs and state employment agencies

Contextualized Instruction

Three main areas of contextualization:

1. Career pathways - “Use occupationally relevant instructional materials”

2. Transition to postsecondary education/training

3. English literacy/civics and career pathways (ESOL learners)

“The adult education component of the program must be aligned to the State’s content standards”

“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, white board, and scotch tape

Put a six foot long line on a whiteboard or on the floor. Put some numbers on the line 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.

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.