Hybrid ASE Math Steve Schmidt

schmidtsj@appstate.edu abspd.appstate.edu

828.262.2262

Notable Quote“I’ve come to a frightening conclusion that I am the decisive element in the classroom. It’s my personal approach that creates the climate. It’s my daily mood that makes the weather. As a teacher, I possess a tremendous power to make a student’s life miserable or joyous. I can be a tool of torture or an instrument of inspiration. I can humiliate or humor, hurt or heal. In all situations, it is my response that decides whether a crisis will be escalated or deescalated and a student humanized or dehumanized.”

- Adapted from Haim Ginott

Please Write on this Packet!You can find everything from this workshop at: abspd.appstate.edu Look under: Teaching Resources and scroll down to Hybrid ASE Math

Agenda8:30 – 10:00 Steve’s Confession

Math Research Says . . .

10:00 – 10:15 Break

10:15 – 11:45 Navigating the Online Course

11:45 – 12:45 Lunch

12:45 – 2:00 Math Modeling

2:00 – 2:15 Break

2:15 – 4:00 Everyone Can Learn Math!

This course is funded by:

Steve’s ConfessionI 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 timeto give students the foundation that would allow their students to actually understand what is beingtaught. They may teach students procedures and tricks, hoping that they will retain those procedureslong 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. There is a better way to teach math as you will learn over the next 10 weeks!

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Steve, How Would You Teach Today?I would use an evidence based teaching approach to math.

Classroom AtmosphereI 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.

TeachingI 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.

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What Skills Does It Take to Solve These Problems?DO NOT SOLVE these problems! Just think about the skills needed to solve them!

1. Evening tickets to a play are $24.50 each. Tickets for the afternoon show are $19 each. Janice wants to buy 6 tickets. Arrange terms from the options below to construct the expression Janice would use to determine how much less she would spend if she chooses an afternoon show instead of an evening show. (You do not need to use all of the terms offered as options. Use any term only once.)

+ ( ) 6 - $19

÷ $24.50 ½Source: Kaplan® GED Test

2. A kayaker spends 2 hours paddling up a stream from point A to point B, quickly turns her kayak around, and immediately heads back downstream. It takes her only 1 hour to float back down the stream from point B to point A. If points A and B are 6 miles apart, what was the kayaker’s average rate of speed in miles per hour?

Source: Kaplan® GED Test

What skills are necessary to solve problems like these?

How can we help our students develop these skills?

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What Does Math Research Say?

1. Calculation vs. Reasoning“Over the years, school mathematics has become more and more disconnected from the mathematics that mathematicians use and the mathematics of life. Students spend thousands of hours in classrooms learning sets of procedures and rules that they will never use, in their lives or in their work. Conrad Wolfram is a director of Wolfram-Alpha, one of the most important mathematical companies in the world. He is also an outspoken critic of traditional mathematics teaching, and he argues strongly that mathematics does not equal calculating. In a TED talk watched by over a million people, Wolfram (2010) proposes that working on mathematics has four stages:

1. Posing a question

2. Going from the real world to a mathematical model

3. Performing a calculation

4. Going from the model back to the real world, to see if the original question was answered

“The first stage involves asking a good question of some data or a situation – the first mathematical act that is needed in the workplace. The fastest-growing job in the United States is that of data analyst – someone who looks at the “big data” that all companies now have and asks important questions of the data. The second stage Wolfram describes is setting up a model to answer the question; the third is performing a calculation, and the fourth is turning the model back to the world to see whether the question is answered. Wolfram points out that 80% of school mathematics is spent on stage 3 – performing a calculation by hand – when that is the one stage that employers do not need workers to be able to do, as it is performed by a calculator or computer. Instead, Wolfram proposes that we have students working on stages 1, 2 and 4 for much more of their time in mathematics classes.”

(Boaler, Mathematical Mindsets, p. 27)

So, how much time is spent in your math class on calculation vs. reasoning?

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2. Fortune 500 Skills/ReasoningThe Fortune 500 are the 500 largest companies in the United States. Here is a list of their most valued skills for new employees in 1970 and 1999:

Fortune 500 Most Valued Skills 1970 Fortune 500 Most Valued Skills 19991. Writing 1. Teamwork2. Computational Skills 2. Problem Solving3. Reading Skills 3. Interpersonal Skills4. Oral Communication 4. Oral Communication5. Listening Skills 5. Listening Skills6. Personal Career Development 6. Personal Career Development7. Creative Thinking 7. Creative Thinking8. Leadership 8. Leadership9. Goal Setting/Motivation 9. Goal Setting/Motivation10. Teamwork 10. Writing11. Organizational Effectiveness 11. Organizational Effectiveness12. Problem Solving 12. Computational Skills13. Interpersonal Skills 13. Reading Skills

“In 1970, computation was second on the list. In 1999 the list had changed . . . Computation has dropped to the second-from-the-last position, and the top places have been taken by teamwork and problem solving.

“Mathematics is a very social subject, as proof comes about when mathematicians can convince other mathematicians of logical connections. A lot of mathematics is produced through collaborations between mathematicians . . . Yet many mathematics classrooms are places where students complete worksheets in silence. Group and whole class discussions are really important. Not only are they the greatest aid to understanding – as students rarely understand ideas without talking through them – and not only do they enliven the subject and engage students, but they teach students to reason and to critique each other’s reasoning, both of which are central in today’s high-tech workplaces. Almost all new jobs in today’s technological world involve working with massive data sets, asking questions of the data and reasoning about pathways. Conrad Wolfram told me that anyone who cannot reason about mathematics is ineffective in today’s workplace. When employees reason and talk about mathematical pathways, other people can develop new ideas based on the pathways as well to see if a mistake has been made. The teamwork that employers value so highly is based upon mathematical reasoning. People who just give answers to calculations are not useful in the workplace; they must be able to reason through them.

“. . . Reasoning through a problem and considering another person’s reasoning is interesting for students . . . Adults are much more engaged when they are given open math problems and allowed to come up with methods and pathways than if they are working on problems that require a calculation and answer.” (Boaler, Mathematical Mindsets, pp. 28-29)

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3. Power of Mistakes and Struggle

“[Researcher] Carol Dweck met with the teachers and said something that amazed them: ‘Every time a student makes a mistake in math, they grow a synapse.’ There was an audible gasp in the room as teachers realized the significance of this statement. One reason it is so significant is that it speaks to the huge power and value of mistakes, although students everywhere think that when they make a mistake it means that they are not a ‘math person’ or worse, that they are not smart. Many good teachers have told students for years that mistakes are useful and they show that we are learning, but the new evidence on the brain and mistakes says something more significant . . .”

“When I have told teachers that mistakes cause your brain to spark and grow, they have said, ‘Surely this happens only if students correct their mistake and go on to solve the problem.’ But this is not the case. In fact, [psychologist] Jason Moser’s study shows us that we don’t even have to be aware we have made a mistake for brain sparks to occur. When teachers ask me how this can be possible, I tell them that the best thinking we have on this now is that the brain sparks and grows when we make a mistake, even if we are not aware of it, because it is a time of struggle; the brain is challenged, and this is the time when the brain grows the most . . .”

“We want students to make mistakes, yet many classrooms are designed to give students work that they will get correct . . . Countries that top the world in math achievement, such as China, deal with mistakes very differently. I recently watched a math lesson in a second-grade classroom in Shanghai, the area of China where students score at the highest levels in the country and the world. The teacher gave the students deep conceptual problems to work on and then called on them for their answers. As the students happily shared their work, the interpreter leaned over and told me that the teacher was choosing students who had made mistakes. The students were proud to share their mistakes, as mistakes were valued by the teacher.”

(Boaler, pp. 11 – 13)

A Brain Synapse Fires7

4. Why Use Manipulatives?Manipulatives are objects students can move around that represent math concepts like the algebra tiles shown below.

Manipulatives Slow Down the Action“Using manipulatives slows down the process of explanation so students have more time for understanding. For example, in showing 24 – 19 using blocks, the time required to exchange 1 of the tens of the 24 for 10 ones is much longer than the time required to say ‘Regroup the 24 into 1 ten and 14 ones’ or ‘Borrow 10 from the 20 and write 14 in the ones column.’ This extra time gives the student a moment to absorb what is happening before you go on to the next step.

The Student Controls the Pace of the Work“Furthermore, if the student is using the manipulatives himself, and not just watching, the student controls the pace of the work. Students who are not sure of themselves move more slowly. If you watch students working with manipulatives, you get a sense of their understanding. You can’t see what’s in their heads, and often they cannot articulate their thoughts, but you can tell by their hands where their understanding is faulty.

Manipulatives Help Students Remember“The use of manipulatives provides memory cues for different kinds of learning. The movement of the arm in operations of addition or subtraction, multiplying or dividing, reinforces the meaning of the operations for kinesthetic learners. In the operation of addition, the arm sweeps two or more groups of blocks together . . . In multiplication, the arm moves repeatedly to add a group a particular number of times. The shapes, colors and sizes of the manipulatives provide cues for visual learners, and they carry those images with them when they move to working mentally or on paper. Finally, since talking seems to go with using manipulatives, auditory learners get to tell themselves stories about what they are doing, and hear others talk about the processes they are demonstrating, and this verbal rehearsal of the process is committed to memory.

Students Get the Right Answer“Most important, the students nearly always get the right answer when they use manipulatives. If you ask a student to add 1/3 + 1/6 using manipulatives, the answer will never come out to 2/9, which is a common error students make when adding fractions on paper. This means that the instructor deals with a student who has the correct answer. Rather than having to deal with an error, you can work on extending understanding, or helping the student articulate the concepts. The benefits in terms of student self-confidence are evident. (Nonesuch, pp. 24 – 26)

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5. Teach From Concrete to Representational to Abstract

Teach math from concrete to representational to abstract (CRA):

Concrete Representational

Volume = length x width x height

Volume = 3 x 3 x 3

Abstract

Starting a math lesson at an abstract level is a common problem. Math expert Mahesh Sharma writes: "In many of the regular classroom teaching situations, the teacher may begin at the abstract form of the concept. As a result, the student may face difficulty in learning the concept or procedure being taught. Even if he has understood the procedure for solving that problem, he may soon forget it.

"Later when the teacher begins a new concept he may assume, incorrectly, that the mastery in the previous concept is still present and therefore may begin the new concept at a higher level, i.e., the abstract level, creating difficulty for the student. This cycle continues and eventually the student begins to lose the teacher's explanations. The student begins to have difficulty in learning mathematics, which then results in failure and that develops a fear of mathematics."

Additionally, remember that many students have failed algebra multiple times before entering our programs. If we try to teach them again using the same abstract methods that did not work before, what makes us think they will work now? The definition of insanity is doing the same thing over and over and expecting different results!

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6. Positive Norms to Encourage in Math Class

1. Everyone Can Learn Math to the Highest Levels

Encourage students to believe in themselves. There is no such thing as a “math person.” Everyone can reach the highest levels they want to, with hard work.

2. Mistakes are Valuable

Mistakes grow your brain! It is good to struggle and make mistakes.

3. Questions are Really Important

Always ask questions; always answer questions. Ask yourself: why does that make sense?

4. Math is about Creativity and Making Sense Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss and critique.

5. Math is about Connections and Communicating

Math is a connected subject, and a form of communication. Represent math in different forms – such as words, a picture, a graph, an equation – and link them. Color code!

6. Depth is Much More Important than Speed

Top mathematicians, such as Laurent Schwartz think slowly and deeply. Laurent Schwartz won the Fields Medal, math’s highest award, in 1950.

7. Math Class is about Learning, nor Performing

Math is a growth subject; it takes time to learn, and it is all about effort.

(Boaler, pp. 172 – 173)

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7. Teacher Resistance to New Methods of Teaching Math“When I began teaching basic math (whole numbers, fractions, decimals and percent) to adults twenty-five years ago, I started teaching as I had been taught, that is, the teacher did the math at the blackboard and the students watched the teacher do math, and listened to her talk about doing it. Then they worked in their own books, and took the tests at the end of the chapter and at the end of term. From the beginning, from the very first term, I knew it wouldn’t work. Students were bored and frustrated by their lack of activity and their lack of understanding. I was bored and frustrated by their lack of engagement and their lack of understanding. I wanted more.

“I began to change my teaching practice in a variety of ways—more emphasis on teaching concepts rather than algorithms; more group work, less lecture time; more emphasis on students discovering patterns; more emphasis on math thinking and problem solving; and more use of real life problems and a greater reliance on manipulatives and models. Indeed, some of the very things that I found in the [research] literature when I went searching recently.

“When I began to change my teaching practice, I met resistance on all fronts—my own resistance, students’ resistance, and resistance from the programs I worked in over the years. Overcoming this resistance to new methods became the first hurdle to changing my practice and feeling comfortable with the changes.

“As I changed my ways of teaching to include more student participation, problem solving, math thinking, group work, and use of manipulatives and models, a voice inside my head kept talking to me, saying:

“What you are doing is not real math . . .

“These new things will not help students pass the exam . . .

“Working in these new ways just causes problems . . .

“I dealt with the voice in various ways: I talked back to it, I found support in other teachers (both at home and away), I talked to my students, I made change slowly so I could see what was going on, and I stuck with it. I still hear that voice from time to time, but it is weaker, and I know how to make it quiet now—I point to the positive results I get when I ignore it!”

(Nonesuch, pp. 9 – 10)

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8. Student Resistance to New Methods for Learning Math“Many students, however, are less open to new strategies for learning math; their responses range from silent withdrawal, to questioning their value, to open refusal to use them. Over the years, I have used different strategies to honor student resistance and work with it rather than against it. I find thatstudents need to be able to express their resistance in order to maintain their sense of self in the class, and that when they can do so with dignity, they are more likely to be able to stay present and attend to the work . . . [Researchers found] the more complex and open [student] resistance to [new methods of] teaching, the more likely they were to continue to come regularly.

“This is not real math.Nearly every student who enrolls in a basic math class has years of (unsuccessful) experience as a math student; it stands to reason that they have a firm idea of what math class should be . . . I deal with that resistance by acknowledging that what I am asking them to do is not what they are used to, and it feels strange. I ask them to tell me all the ways they have tried to learn math in the past.

“Then I ask, ‘Does anyone know a way to learn math that really works?’ Invariably, nobody does because they have all been previously unsuccessful. This conversation with students is part of making my work and theory transparent, and makes them partners in designing their own learning. The discussion about past methods of learning math, an evaluation of what parts were more useful orless useful and the conclusion that something new needs to be tried, means that they are part of the team talking about what form teaching will take.

Evaluate Teaching Strategies with the Class“Before I introduce a teaching strategy that is new to the class, and that I think might meet with resistance, I present it, giving my reasons for thinking it would be valuable. I ask for their reactions, then propose that we try it out for a reasonable length of time, for example, three weeks, and that we evaluate it briefly at the end of the first week, and more thoroughly after the trial period. I make it clear that I will act on the decisions made at this evaluation, and stop using the strategy if most students don’t like it or don’t find it useful.

“Honor student resistance. Welcome it as a sign of students taking responsibility for their learning. Don’t get into a power struggle with it.

Why Deal with Student Resistance?“Student resistance will weigh you down and tire you out. If you don’t deal with their resistance, and get them onside with you to work in new ways, their resistance will win. You may give up entirely and go back to teaching in ways you know are ineffective, because that is what students expect you to do.”

(Nonesuch, pp. 10 – 14)

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9. Expressing Feelings“An important strand of my teaching philosophy is to deal with emotions, my own and the students’, so they don’t get in the way of the learning . . . it will probably take less than a minute to acknowledge the emotions that come up in the moment; that if you leave it for an hour, it might take two minutes to deal with them; if you leave it until the next day it might take half an hour, and if you leave it for longer, who knows how long it will take?

“How does expressing our feelings help? It helps us keep control of our emotions, helps us identify problems, and helps us maintain clarity in our relationships with other people.

“Saying ‘I’m frustrated’ or ‘I’m mad’ or ‘I’m happy,’ releases the hold the emotion has on you a little, so that you can concentrate on other things, and think and act rather than just emote. Maintaining control over emotions is helpful in the classroom where so many people are working in a public space . . . I’d much rather someone say, ‘I’m really frustrated when I keep getting these questions wrong,’ instead of slamming his books down and stamping out, swearing under his breath, or out loud. Furthermore, a student who can say what is bothering him may be able to go on working, or ask for help, or use some strategy he has for dealing with stress or anger.

“Fight, flight, or freeze. I’ve learned to recognize all these responses by math students, and gone on from there to take it less personally when students attack me or run from me or disengage. I know it’s not so much me they are reacting to, but to the situation itself.

“For some years I would go around the class, asking, ‘How are you doing? Do you need any help?’ and students would say, ‘Okay,’ or ‘No.’ Usually they kept their work hidden when they answered this way, but often I would find out later that indeed they did need help—they weren’t doing okay at all. Yet they shut me out by saying, “I’m okay.” Why do they lie? Because they are running away from whatever mini-lesson I might give them if they admitted they needed help.

“Sometimes I would invite people to come to see me outside class time to get some extra help, and the answer might be, ‘No thanks, I’ll work with my tutor (or my father or my girlfriend or…’). But I would hear from the tutor that they didn’t show up for a scheduled tutoring session, and I would see no evidence that the alleged sessions with family members bore any fruit. Why would a student invent math learning at some other time? Because they are running away from my math lesson and from panic.

“For a while I took it personally, all this running away, but eventually I learned some tactics for heading it off. I no longer ask, ‘Do you need any help?’ Instead I say, ‘What question are you working on? What can you tell me about your thinking about that question?’ or ‘You don’t look happy. What’s getting you down?’ The student can still avoid me if he wants to, but I don’t make it easy for him. If the student is not struggling, this technique invites the student to articulate their math thinking.”

(Nonesuch, pp. 10 – 14)

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10. Teach with Focus“One of the biggest challenges we face in adult education math instruction is the amount (and difficulty) of material we feel pressured to teach and that students are expected to learn. All teachers want what is best for their students and sometimes it feels like we should try to cover as much as possible, to give them a taste of everything they might see on their High School Equivalency (GED, HiSET or TASC) exam . . .

“When students receive instruction that is ‘a mile wide and an inch deep,’ they often lack the ability to apply what they know to new situations, or remember what they’ve already ‘learned.’ Many of ushave asked ourselves questions like, ‘My students knew how to do this two weeks ago, what happened?’ or ‘Why don’t my students see that they can use what we learned to solve that problem to solve this one?’ The answer to both questions is the same: Without a deep enough understanding, students won’t transfer what they know to new situations or to retain what they do in class. It may feel strange to spend more time on fewer topics, but if we try to cover everything, we end up havingto re-teach things anyway, often right in the middle of a lesson on a completely different topic.

“We should accept that we cannot teach everything and then make some choices about how we will focus our resources, especially our time. We should not let any assessments work us up into such a panic that we lose sight of some of our greatest strengths—our practice of starting from where students are and our serious respect for their learning processes. As a student once told me, ‘You can’t make a plant grow by pulling on it, you only make it rootless.’

“The math problems students will see on their HSE exams will be far more complex than what we find in workbooks . . . Just knowing formulas will not be enough. Experience with routine problems that can be answered simply by using a memorized set of procedures or steps will not be enough. At the heart of teaching with focus is time. We need to give students time. Time to work on and struggle with complex problems. Time to present different solution methods. Time to discuss, appreciate and analyze each other’s methods. Time to debate and to write in math class. Time to revise their work. Time to reflect on what they are learning, what they understand, how they understand it and what questions they still have. Students feel great urgency to find shortcuts and move quickly to the test. It is our responsibility to slow the classroom down so that people learn the content.”

(Brandt, Leece, Trushkowsky, and Appleton, pp. 7 – 8)

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11. Teach Math Using a Problem Solving Approach “No matter how kindly, clearly, patiently, or slowly teachers explain, they cannot make students understand. Understanding takes place in the students’ minds as they connect new information with previously developed ideas, and teaching through problem solving is a powerful way to promote this kind of thinking.”

(Diana Lambdin, 2003)

High school equivalency tests (GED, TASC, HiSET) focus on math reasoning (not computation) and use real life problem scenarios. So how should we teach our students to prepare them?

Teach students a problem solving method that works for any type of problem (see the UPS check method on pages 16 and 17.

When we tell students every move to make, they develop “learned helplessness.”

“Students learn math by doing math” (Dr. Paul Nolting)

Learning is social. Have students work together and talk about math. The one who does the talking does the learning (Lev Vygotsky)

The best way to learn something is to teach it (Lev Vygotsky)

“None of us is as smart as all of us.” (Ken Blanchard)

Instructors should act as facilitators instead of lecturers. The instructor: works to get students to do as much math as possible

teaches with questions to get students’ thinking

realizes students’ wrong answers are the best opportunity for learning

avoids telling students they are right/wrong and gets them to think as long as possible

Our goal should be to produce students who can, without our help, know when and how to use particular strategies for problem solving

What is our end goal: A student dependent on us or an independent thinker who can thrive on their own?

(Nolting and Vygotsky)

UPS ✔ Problem Solving Method15

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 problemMake a list ExperimentDraw a picture or diagram Act it outLook for a pattern Work backwardsMake a table Change your viewpointUse 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

Understand16

Plan

Solve

Check

12. Encourage “Productive Struggle”17

“In order for students to develop perseverance in mathematical problem solving, they have to learn how to work through struggle. They have to build up some experiences of feeling stuck, sticking with it and having a breakthrough. They will not be able to do this if we step in too soon or too often.

“I like to think of our role as teachers as similar to that of a weight-lifting spotter in a gym. If the spotter keeps their hands on the weight and just lifts, he or she will be the one who gets stronger, not the weight lifter. A good spotter watches the person lifting weights and lets them do the work. When the lifter gets stuck, the spotter offers words of support and encouragement. If the lifter still can’t proceed, the spotter helps, just enough to get them past, sometimes only using a few fingers, and doesthe least amount of lifting they can. The spotter keeps the lifter able to work and develop beyond instances of struggle. It may be helpful to think of our work with students in a similar way. You can’t get stronger or develop perseverance watching someone else lift weights. Students have to learn to work through struggle, not stop and wait for someone else to do the work when they get stuck. As a general rule, we should try to never take the pencil out of a student’s hands.

“This can be one of the hardest things for us as teachers to do. It is very tempting to just show students how to solve a problem as soon as they get stuck. It is often what they want us to do and if we do show them, they will be thankful and happy, which makes us feel great. But when we do that, what are we teaching them about their ability and independence? How are we preparing them to keep going the next time they struggle?

“We should be honest with our students and tell them that we are preparing them for HSE exams and college and life, all of which will give them problems they’ve never seen before. In math class, we need to build our tolerance to uncertainty and struggle. We need to separate ourselves from the notion that math problems are like sitcom problems, solved quickly and neatly to perfect resolution. Our students need to understand that struggle is not a bad thing. Too many adult students interpret struggle as a deficit on their part. As soon as they start to struggle, they put down their pencils and say things like, “I just don’t get it. I’m not good at math.” Reacting to struggle that way makes it more difficult to learn, since working through struggle is a necessary part of the learning process.

“When students are struggling, we should aspire to only ask them questions—and to ask as few questions as it takes to get them moving on their own.

- When students ask a question about one of the conditions that make the problem ‘problematic’, encourage them and reflect question back to them

- Answer most questions with “Good question. What do you think?”

- When students start to shut down, get them talking. Ask them to describe the situation in their own words. Ask them what they’ve tried so far.

- When students are stuck, suggest a strategy—for example, ‘Can you draw a picture?’ or ‘What could the answer be? Is there a way you can check that? What have you tried from our list of problem solving strategies?’ “

(Brandt, Leece, Trushkowsky, and Appleton, pp. 11 – 12)

How You Can be Good at Math, and Other Surprising Facts about Learning18

1. Is there such a thing as a “math brain”?

2. What happens to your brain when you make mistakes in math?

3. What is the growth mindset?

4. What suggestions does Jo Boaler offer for improving math instruction?

5. What else interested you as you watched this video?

Source: Jo Boaler, TED Talk https://www.youtube.com/watch?v=3icoSeGqQtY

Advice from Past Hybrid Learners19

- Plan a time away from distractions

- Stay up to date on lessons

- Start on Monday . . . Don’t wait until the weekend

- Make sure you schedule time to work on the course throughout the week...they will tell you this on Day 1. You should heed the advice.

- Hit the ground running. Don't procrastinate. If you hit a snag, get help quickly. Don't let yourself drown in material trying to catch up.

- Ask for help when you need it, don't panic. Start early in the week.

- Try thinking out of the box. It makes the posts much more interesting!

- Stay engaged and have fun interacting with one another!

- Do not work through the course quickly; keep an open mind about all ideas presented.

- Share, share and share with your peers for learning. Participation helps everyone learn something new.

- Be sure to make arrangements to complete the coursework. Collaborate with your peers when possible. Communicate with the instructor. Be willing to learn. Have a positive attitude. Do not stress!

- Don't be afraid or intimidated by this course. It is a very useful tool.

- It will be time consuming and you will have times when you wonder why you are doing it, but it is well worth it. Also, the day of presentations is actually a good day - lots of support in the room!

- Stick with the course and read/respond to as many posts as possible.

- Be creative, do your research, contextualize your knowledge in your approach to teaching, learn a lot, and have fun.

- Just breathe. You are not there yet, but you will get there.

- Enjoy the class!!

Research Base20

Blanchard, K. (1982). The One Minute Manager. New York: Morrow

Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math,

inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.

Brandt, K., Leece, R., Trushkowsky, M., & Appleton, E. (2015). The CUNY Curriculum Framework.

New York, NY: City University of New York.

Dweck, C. S. (2006). Mindset: The new psychology of success. New York: Random House.

National Institute for Literacy. (2010). Algebraic thinking in adult education. Washington, DC: Author.

Nolting, P. (2008). Winning at math: Your guide to learning mathematics through successful

study skills. Bradenton, Fl: Academic Success Press.

Nonesuch, K. (2006). Changing the way we teach math: a manual for teaching basic math to adults.

Duncan, B.C., Canada: Malaspina University-College.

Polya, G. (2014). How to solve it: a new aspect of mathematical method. Princeton NJ: Princeton

Science Library.

Sharma, M. Learning problems in mathematics: Diagnostic and remedial perspectives.

Retrieved from: https://abspd.appstate.edu/sites/abspd.appstate.edu/.../01%20GPS%20Overview.docx

Vygotsky, L. (2012). Thought and Language. Cambridge, MA: MIT Press.

Math Humor

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