## Section 1.1 Learning How to Learn Mathematics

Mathematics is a creative endeavor. It is full of â€śahaâ€ť moments. Has that been your experience with mathematics? If so, think of a few â€śahaâ€ť moments you've had. Write them down. If not, you're about to embark on real learning of mathematics, learning that does not involve mind-numbing memorization of rules for combining symbols and solving equations.

Math makes sense. Learning mathematics requires making sense by making connections between familiar ideas you encounter in your daily life. You might find that the mathematics you encounter in this course relates to mathematical concepts from your past experiences. Some of it might seem entirely new. Whatever you think of mathematics as you begin this course, it is important that you keep an open mind, bring your creativity into your learning, and try to connect what you are learning to what you already know outside of mathematics.

There are several tools available to help you learn mathematics. In SectionÂ 1.1, we explore some of them.

### Subsection 1.1.1 How to Learn Math for Students

Stanford University offers free classes online^{â€‰6â€‰}. A useful and short course entitled, How to Learn Math for Students, offers several short videos and brief exercises to help you build your confidence in learning mathematics. How to Learn Math for Students will help you learn how your brain works, how your mindset regarding mathematics helps or hinders your learning, how speed is not related to understanding in mathematics, and other useful information regarding how you learn mathematics. As part of the Beginning Algebra Made Useful course, you will also be completing much of the Stanford course, How to Learn Math for Students. Sign up for the course (it's free!!!).

Throughout the first part of the semester, you will be assigned problems that require you to watch a video in the Stanford course, How to Learn Math for Students, then reflect on what you learned. The videos are short, most of them less than 3 minutes in length. After each video, you will find short quizzes and questions to answer along the way. Answer the quizzes and questions as you go. By providing a certificate of completion for the Stanford course, How to Learn Math for Students, you can earn privileges determined by your professor.

#### How To Learn Math For Students Directions.

Your responses to each of the following problems will be collected on a due date provided in class. Answer each question in writing using complete English sentences. Type responses if your handwriting is difficult to read. Show all of the work you do to answer mathematical questions. Include illustrations.

##### How to Learn Math for Students Exercise 1.1.1.1. Lesson 1: Knocking Down Myths About Math.

###### (a)

Watch the second video in Lesson 1, How do People Feel about Math? (3:38 minutes). Write at least 2 sentences to answer the following question: How do you feel about math?

###### (b)

Watch the third video in Lesson 1, Math Myths and the Brain (3:23 minutes). Answer in writing: Why do you think people often don't like math?

###### (c)

Watch the fourth video in Lesson 1. How does the fourth video, Brain Growth (2:04 minutes), make you feel about your ability to learn mathematics?

###### (d)

Watch the last video in Lesson 1, Smashing Stereotypes (1:57 minutes). What toys and games did you experience on a consistent basis as you were growing up? Would you consider any of them to be mathematical? If so, which ones? In your opinion, what made these toys mathematical?

##### How to Learn Math for Students Exercise 1.1.1.2. Lesson 2: Math and Mindset.

###### (a)

Watch the first Math and Mindset video in Lesson 2: Mindset (3:23 minutes).

###### (b)

In what areas (subjects in school, sports, etc.) do you have a growth mindset?

###### (c)

Do you tend to have a growth mindset or a fixed mindset when it comes to learning mathematics? State your evidence.

###### (d)

What messages do you give yourself that help you develop or keep a growth mindset in mathematics?

###### (e)

Watch the video, Messages about Math (1:44 minutes). Some students were praised for being smart. Which problem did they choose? Why do you think that students who are praised for being smart chose the problem they did?

###### (f)

Watch the video, Messages about You (2:29 minutes).

###### (g)

How do the videos in Lesson 2 make you feel about your ability to learn mathematics?

###### (h)

What three messages from these videos do you want to keep in mind for this and other courses you will take in college?

##### How to Learn Math for Students Exercise 1.1.1.3. Lesson 3: Mistakes and Speed.

Watch the first three videos for Lesson 3: Mistakes: New Evidence (1:42 minutes), Mistakes and Success in Life (1:29 minutes), and Math and Speed (3:49 minutes).

###### (a)

Do you see mistakes in math as a sign of failure or a sign of persistence?

###### (b)

Write a paragraph about what you've learned through the videos in Lesson 3.

###### (c)

Open problems are problems for which there are multiple approaches to find a solution or there are multiple solutions. Most of the problems we've been working on in class are considered open problems. How do they help you learn? What about them, if anything, causes you anxiety?

##### How to Learn Math for Students Exercise 1.1.1.4. Lesson 4: Number Flexibility, Mathematical Reasoning, and Connections..

###### (a)

Watch the first video, Number Flexibility (1:24 minutes), then simplify \(18 \times 5\) using a mental strategy. Write down how you thought about the problem as you solved it.

###### (b)

Watch the second video, \(18 \times 5\) (5:27 minutes). In writing, answer the question posed at the end of the video. Match each illustration with the solution process that fits it. Include drawings in your submission.

###### (c)

Watch the third video, \(12 \times 15\) (1:18 minutes). Solve the problem stated without using the usual algorithm. Try to solve the problem mentally first using friendly numbers. Then try to draw a picture that goes with the way you solved the problem.

###### (d)

Solve the problem in TaskÂ 1.1.1.4.c another way, and again without paper and pencil. Draw a picture that illustrates the way you solved the problem this time.

##### How to Learn Math for Students Exercise 1.1.1.5. Continuing with Lesson 4: Number Flexibility, Mathematical Reasoning, and Connections..

###### (a)

Watch the fourth video, Talking About Math (1:41 minutes). Why did the students who worked on math together do so well in their mathematics classes?

###### (b)

Watch the fifth video, Reasoning (1:32 minutes). What is reasoning? Why is it important?

###### (c)

Watch the sixth video, Mathematical Connections (9:10 minutes; though this video is long, listen all the way through). Several connections were drawn in the video. Choose 3 of the examples and explain how they are connected.

###### (d)

Watch the seventh video, PISA Results (0:41 minutes). Write about some of the connections you have made in mathematics in this class. Why are they important to your learning?

##### How to Learn Math for Students Exercise 1.1.1.6. Lesson 5: Number Patterns and Representations..

###### (a)

Watch the first video in Lesson 5, Making Sense and Intuition (4:04 minutes). Solve the problem posed immediately after this video. Show how you solved the problem. Do not go on to the next video until you find a solution you think works.

###### (b)

Watch the second video in Lesson 5, Drawing and Representing (5:11 minutes). How did the video solution compare with your solution in TaskÂ 1.1.1.6.a? How did the video help you think about the problem differently?

###### (c)

What does this video tell you about What does this video tell you about How to Learn Math??

###### (d)

What important information do you want to remember from these two videos?

##### How to Learn Math for Students Exercise 1.1.1.7. Continuing with Lesson 5: Number Patterns and Representations..

###### (a)

Watch the third video in Lesson 5, Looking for Big Ideas (5:27 minutes). Write down and answer the quiz question after the video. Explain your response. Use what you learned in the video to determine which is bigger, \(\frac{2}{3}\) or \(\frac{5}{8}\text{.}\) Include illustrations in your solution.

###### (b)

Watch the fourth video, Fractions: Same Size Pieces (1:11 minutes). Take the quiz following the fourth video. Write down the problem, show your response, and explain your answer.

###### (c)

What fraction represents the area of the white triangle in FigureÂ 1.1.1.8? Explain your answer.

###### (d)

Watch the fifth video, Fractions: Big Ideas (2:01 minutes), more than once if needed.

###### (e)

From the videos in TaskÂ 1.1.1.4.b and TaskÂ 1.1.1.4.d, what big ideas about fractions will help you make sense of them? List at least two.

###### (f)

Watch the sixth video in Lesson 5, Ideas vs. Memorization (0:53 minutes).

###### (i)

Think about the mathematical content in our class. What big ideas have you learned so far? What makes them big ideas?

###### (ii)

What important strategies have you learned? How might these strategies help you in your future mathematics learning?

###### (g)

What has surprised you about your learning of mathematics?

###### (h)

What additional mathematics would you like to learn?

##### How to Learn Math for Students Exercise 1.1.1.9.

Think about your work with mathematics in and out of class so far this semester.

###### (a)

As you work through activities and problems in and out of class, what messages are you giving yourself? Have you been persistent? Have you told yourself you can do this?

###### (b)

The problems we work on in class and for homework often require persistence. What do you do when you struggle with a problem?

###### (c)

Have any of your messages to self been negative? If so, based on the videos from How to Learn Mathematics for Students, what messages help you grow your brain?

##### How to Learn Math for Students Exercise 1.1.1.10.

Answer the following questions. Be specific. Provide examples.

###### (a)

Why is it important to your mathematical learning to talk about math with others, to make connections, and to use reasoning?

###### (b)

How might you use these ways of learning to help you in other subjects?

###### (c)

How might you use these ways of learning to help you in your daily life?

##### How to Learn Math for Students Exercise 1.1.1.11.

Think about your views of your ability to learn math.

###### (a)

Have any of your views about your ability to learn math changed this semester? Explain.

###### (b)

Watch Video 9 in Lesson 6, Summary and Reflections (2:01 minutes). What ideas from How to Learn Math for Students stand out for you.

###### (c)

Take the survey immediately following Video 9. List the top 5 ideas from the course that stand out for you.

###### (d)

Write a reflection, at least 2 paragraphs long, about what you have learned about your ability to learn mathematics.

##### How to Learn Math for Students Exercise 1.1.1.12.

Earn a certificate for completing the Stanford course, How to Learn Math for Students. You may earn privileges as designated by your professor for completing the course and turning in a printed copy of the certificate of completion.

### Subsection 1.1.2 Good Questions to Ask Yourself

When working on mathematics problems, you are in good company if you find that you are often or occasionally stuck. There is always hope. The questions that follow have been used many times to help students find a way out of a temporary dilemma when solving a problem. First, try to identify what's causing your dilemma.

Do you understand what the problem is asking? If not, you are probably stuck at Stage 1.

Do you understand the problem but don't know what to do now? You are likely stuck at Stage 2.

Have you started to solve the problem and gathered some information but don't know what to do with what you've found? You might be stuck at Stage 3.

Have you finished the problem? Stage 4 questions can help you organize what you've learned through solving the problem into what you already know and help you move new learnings into your long-term memory.

Goldin (1998) suggests a hierarchy of four stages of questioning to guide yourself or a peer to solve a problem:

### Subsection 1.1.3 Four Stages of Questioning

#### Stage 1: Understanding the Problem.

Read the problem and take time to make some initial progress on it. Ask yourself:

What is the problem asking me to do? Write down ideas in your own words.

What do I know?

What am I trying to find?

#### Stage 2: Devising a Plan.

If you understand the problem and have trouble starting to solve it, think about general problem-solving strategies that might help. Ask yourself:

Would drawing a picture help?

Would a table of values help to solve this problem?

Would any materials help to model the problem? (Get the materials and use them.)

Is there a similar problem I've done before that might help me get started?

What do I know must happen in the problem?

What do I know cannot happen in the problem?

#### Stage 3: Carrying Out the Plan.

If you get stuck once you are partway through solving the problem, think about more specific problem-solving strategies. Ask yourself:

What patterns do I see in the table of values or drawings?

What rule can be used to describe the relationship in the table of values I created?

What information does the graph give me?

#### Stage Four: Looking Back.

Once you solve a problem, think back about what you did. Think about your thinking by asking yourself:

How do I know my solution is correct? How can I convince someone else?

How did I think about the problem?

Can I solve the problem in another way?

How does this problem connect with other problems I have solved?

Your goal is to help yourself explain the problem and its solution. This stage helps you mentally file your thoughts about the problem in a category with other similar problems, organizing your new learning into your long-term memory.

Table based on Goldin, Gerald A. â€śObserving Mathematical Problem Solving through Task-Based Interviewsâ€ť. In Anne R. Teppo (Ed.), Qualitative Research Methods in Mathematics Education (pp. 40-62). Reston, VA: National Council of Teachers of Mathematics, 1998.

### Subsection 1.1.4 Other Helpful Practices

To best learn mathematics, you will find that each of the following strategies are helpful if you make them a habit:

#### Read carefully.

It might be necessary to read problems or narration several times in order to understand more fully what each intends. Students who do not read carefully always make mistakes that can be remedied by rereading more carefully.

#### Organize your work.

Use tables and graphs to help you make sense of information you find when solving a problem. Don't underestimate the power of trying a few numbers to see what happens. Always keep track of what you try. Do not erase or throw away attempts that don't work. Save them and compare them to an approach that did finally work. Ask yourself these questions: What was I thinking initially? What do I now know? Was I far off at the beginning? If not, what was I thinking that could have led me to a correct solution? If so, what was I thinking that was helpful? What was getting in the way of my understanding?

#### Check your solutions to see that they make sense.

If a context is given, translate your solutions into the context. Check to make sure any numbers you're getting from equations, tables, or graphs make sense in the context. Make sure graphs, tables, and equations give corresponding information.

#### Never give up!

It's OK to take a break from a problem that's frustrating you. Return to it after you've had a chance to relieve your frustration. You can learn mathematics. You can solve any problem facing you, *as long as you never give up!*

#### Study at least a little *most days*.

After each class, take 15 minutes to write down what you learned. Be specific. For example, don't state, â€śI learned about linear functions.â€ť Instead, state specifically what you learned, â€śI learned that the slope of a linear function when shown in a graph is the value of \(m\) in the equation, \(y = mx + b\text{.}\) This makes sense, because slope is how much \(y\) changes when \(x\) changes by 1 so in a graph, the \(y\)-value of a linear function changes by \(m\) every time \(x\) changes by 1.â€ť

#### Do your assigned homework.

Homework provides you the opportunity to grow your brain outside of class. It also prepares you to think about problems that build on your homework when you are in class. Making mistakes is an important part of learning, so grading your early thoughts about a topic is counterproductive. *Doing* homework, however, grows your brain, allows you to get more out of the work you're doing in class, and helps you prepare for your future with technical information. Not doing homework is detrimental to you, and to your current and future work!

#### Commit to your education.

Participate fully in your group and class and make every effort to complete your homework so that you're prepared to participate with your group.

### Subsection 1.1.5 Mathematical Practices

Several practices have been identified that can help you make sense of problems that arise in your daily work and in technical situations. Which practices do you use to learn mathematics and other content fields? When you work on mathematics, use this page to help you choose strategies that might help you learn. Revisit this page often, especially when you get stuck, for suggestions of other strategies you might try as you're trying to solve problems. Resource.^{â€‰7â€‰}

### Homework 1.1.6 Homework

Complete the following homework before the next class period:

#### 1.

Go back to How To Learn Math For Students Directions. Complete How to Learn Math for Students ExerciseÂ 1.1.1.1. You will hand in your written reflections for problem 1 in class.

#### 2.

Read the sections above, Good Questions to Ask Yourself, Other Helpful Practices, and Mathematical Practices. Remember these sections as you work your way through the activities in this course. Refer to each section as needed. Write a one sentence description of each section to help you remember when to consult each of these sections.

`online.stanford.edu/courses/gse-yeduc115-s-how-learn-math-students`

`corestandards.org/Math/Practice/`