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 Reasoning and Problem Solving
10:00 – 10:15 Break
10:15 – 11:45Workplace Math
11:45 – 12:45 Lunch
12:45 – 2:00 Teaching from Concrete to Abstract
2:00 – 2:15 Break
2:15 – 4:00Defeating Math Anxiety
Steve’s Confession, “I Was Wrong”
I hate to admit it, but I was wrong. The way I taught math to my adult students for years was not very helpful for them. I thought I was doing the right thing. Since I did not consider myself a “math person,” I taught math how I saw it modeled by my math teachers through the years. Since the “experts” did it this way, surely this must be the way to do it!
I carefully explained to my students how to do math starting with decimals and then moving through fractions, percents, pre-algebra and algebra skills. I taught them the calculations, rote procedures and the tricks I had up my sleeve. I did most of the work and just asked them just to watch. We would do some problems together and then they would go do numerous problems in a study book that I would later check. Most of the problems assigned were identical to the ones we practiced.
Since my students were in a hurry, we never worked on the skill of problem solving or developed math reasoning skills. We avoided word problems because they were hard and caused students stress. We saved algebra for last because it was very challenging. I avoided teaching them math in real world contexts because it was easier just to teach from the book and who has time to be creative? When a student was having trouble, I wanted to be the hero. I would immediately jump in (many times taking the pencil from their hands) and quickly show them how to do the problem. This made me feel great!
When students would come to me after taking their high school equivalency test, they would say things like, “What you taught me was not on the test.” I would laugh it off and tell them that I knew they passed and they usually did. Only sometimes, late on sleepless nights, would this bother me.
While I have helped hundreds of students pass their high school equivalency exams or receive their adult high school diplomas, what happened to these students? While many of them had the goal of graduating from the community college where I worked, I saw very few walk across the stage with their degrees.
I know now that some of my former student did enroll in college. They took a college placement test that they were not prepared for and placed into developmental math courses. These courses were designed to prepare them for college level math but instead took their financial aid dollars, their time and their college dream. Very few students made it through the developmental math course sequence or what I now call the “developmental math death spiral.”
Math researcher Donna Curry says, “Unfortunately, teachers feel the need to swiftly get students to meet goals and expectations . . . Unfortunately, too many teachers feel like they don’t have the time
to give students the foundation that would allow their students to actually understand what is being
taught. They may teach students procedures and tricks, hoping that they will retain those procedures
long enough to at least pass the test.
“However, without foundational understanding, students rarely remember those procedures. Without conceptual supports and without a strong rote memory, the rules, procedures, and notations they had been taught started to degrade and get buggy over time. The process was exacerbated by an ever-increasing collection of disconnected facts to remember. With time, those facts became less accurately applied and even more disconnected from the problem solving situations in which they might have been used.”
To my former students, I can only say I’m sorry.
Steve, How Would You Teach Today?
I would use an evidence based teaching approach to math.
Classroom Atmosphere
I would create a classroom atmosphere where students are supposed to take risks and make mistakes. I would encourage students to openly express their feelings about math so we could deal with them and move on. I would work with my students as partners in their education and help them take responsibility for their own learning. I would frequently ask for student’s feedback on how they felt about the class and the teaching methods I was using. We would practice skills so they could learn to function better in small groups.
Teaching
I would teach students how to solve problems using a general problem solving method (UPS check method). I would teach them specific problem solving skills such as drawing a picture, doing the guess and check method, and making a table. I would spend most of my class time in problem solving tasks and teach calculation skills only in short mini lessons. We would use the calculator as a tool and spend less time learning how to do math by hand. We would learn how to estimate to see if our answers make sense.
I would divide students into pairs or small groups where they would work together to solve meaningful problems based on real life and workplace situations. I would ask students to suggest problems from their experiences that we could solve. I would support my student’s “productive struggle.” When students run into difficulties, I would ask questions to help them clarify their thinking. I would act as the “guide on the side” instead of the “sage on the stage.”
We would use manipulatives like algebra tiles, cereal boxes, and Skittles® so students would gain a concrete understanding of the math they were doing. I would realize that an abstract approach to teaching math didn’t work with my students when they were in school, so how could I expect it to work now?
Instead of trying to teach everything the standards or the high school equivalency says my students should know, I would cover fewer topics more deeply and show my students connections like how fractions, decimals and percents work together to show the same thing.
For my students who were going to continue their education, I would make sure they understood the consequences and rewards of the college placement test. I would show them the benefits of taking more time reviewing for the test to save thousands of dollars and months of time.
Finally, I would put less pressure on myself to be the perfect teacher and be real in the classroom about my own struggles with math and problem solving.
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?
Source: TEAL Math Works! Guide
Research Based Practices Learned from This Lesson. It’s WIOA Friendly Too!
1. The instructor acted as a ______
2. The instructor taught using questions.
3. The lesson was taught in a ______context
4. The instructor never said, you’re right or you’re ______
5. The instructor taught math from concrete to ______
6. The lesson was primarily visual.
Developing Math Reasoning - UPS ✔Problem Solving Method
- Understand the problem
Will a picture or diagram help you understand the problem?
Can you rewrite the problem in your own words?
- Create a plan
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
- Solve
Be persistent
Try different strategies
- Check
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
UnderstandPlan
Solve
Check
Classroom Questions that Develop Math Reasoning
- 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
Math Humor!
Levels of Math Learning
“Levels” are the order that information presented mathematically is processed and learned. Math researcher Mahesh C. Sharma says, “Almost all mathematics teaching activities … take place at the abstract level.” Students need concrete understanding before moving to abstract concepts.
Levels of Learning / Explanation / ExampleIntuitive / 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 (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.
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.
Contextualized Teaching: 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:
Adult Education and WIOA’s Integrated Education and Training
The finish line has changed for our programs. It is no longer get your High School Equivalency (GED, TASC, HiSET) and see ya later! Now it is: “Completion of high school is not an end in itself but a means to further opportunities and greater economic self-sufficiency. “ (WIOA Key Provisions)
Integrated Education and Training is:
- Literacy Instruction (We have always done this but now with contextualized instruction)
We should contextualize instruction toward:
- Career pathways - “Use occupationally relevant instructional materials”
- Transition to postsecondary education/training
- English literacy/civics and career pathways (ESOL learners)
“The adult education component of the program must be aligned to the State’s content standards”
- 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”
- Occupational (Job) Training
Use partnerships like a community college’s certification and degree programs and NC
Works