Prior Skills Needed: Students must know: there are 60 minutes in an hour, their ten combinations, and how to count by tens and hundreds.

Further they will need to know or be taught how to use a number line to solve difference (subtraction) problems which I will explain here.

To solve the problem 13-7, a student draws a line and writes 13 on the far right and 7 on the far left. From 7, the student draws line jumps to make 10, so the first jump is +3 to get to 10. The next jump the student wants to get to 13 so the final jump would be +3. Adding the two jumps, 3+3 is 6, which is the answer.

Here’s another 63-28. Draw a line with 63 on the far right and 28 on the far left. Starting at 28, make a 10 by adding 2. So, the first jump is +2 which brings you to 30. Second, count by tens. You are at 30 so you jump to 60 which is +30. Finally, the last jump is +3 ones to get to 63. Add the jumps together, 2+30+3, so the answer to the problem is 35.

Lastly, let’s add 100′s into the mix. Try your hand at 230-119. Draw your line, 230 on the right, 119 on the left. Start on the left, jump to the next 10 (+1 to 120). Jump to the next 100 you want (+100 to 220). Jump to the next 10 you want (+10 to 230). If there were ones, you might have to do a last jump. Add the jumps (+1+100+10=111). Your answer is 111.

Now, try elapsed time using the same method. Let’s solve the following story problem together:

Luìs got on the bus for a classroom astronomy field trip to the Observatory at 1:35p.m. After studying several charts of galaxies and nebulae inside and viewing Saturn’s rings through the telescopes outdoors, Luìs and his class got back on the bus and headed back to school. They arrived 2 minutes late at 5:02p.m. and got to have a pizza party before going home. How much time in-between when Luìs got on the bus at 1:35p.m. and when he arrived back at school at 5:02p.m.?

Draw a line. Write 5:02 on the far right side and 1:35 on the far left side. Start from 1:35 and jump to the next 10. (+5mins. to 1:40). Next, jump to the next hour, just remember that you’re not making 100 but 60 since there are 60 minutes in an hour, so instead of adding 40 and 60 to make 100, you want to add 40 and 20 to make 60 (+20mins. to 2:00). Next, jump by 1′s to the hour you want, be sure to label it in hours (+3hrs. to 5:00). Finally jump the remaining minutes (+2mins. to 5:02). Add up all your jumps (5mins + 20mins. + 3hrs. + 2mins.) Add the minutes and hours separately. You should get 3hrs. and 27mins. as your answer. Notice that you can challenge students to combine the first two steps in to one step and jump straight from 1:35 to the next 60 combination (35+25=60) which can be slightly more challenging. You may also present a problem where the minutes add up to larger than 60 and students will need to convert minutes to hours adding yet another level of challenge. Try several more on your own!

Did you notice how much detail I wrote into this story problem? Besides learning a new skill, the story part of the math problem is developed to engage the student’s sense of curiosity (using the astronomy vocabulary), their sense of meaning (class trip), and appeal to their sense of fun (pizza party). The student is likely excited and engaged and may want to discuss their recent trip or how much they love pizza or what they know about astronomy. STOP! Don’t tell them to be quiet and focus! (This will just make them focus on how they didn’t get to share their story). Let them talk, just give them some parameters. This is great discussion to warm up students’ brains and get them intrinsically motivated and excited to solve this problem. Tell them you will give them 2 minutes to talk to a friend or neighbor about how they can each relate to this problem. They may want to try more and more elapsed time problems and even make up their own stories!

Welcome to my math tutoring website. Here you can find information on my hourly rates, current availability, and my teaching approach as well as a variety of blogs and resources related to math teaching and learning.

I am available to tutor children from Kindergarten through 8th grade to help develop good number sense, mental math abilities, knowing multiple story problem solving strategies, fraction number sense, fraction/decimal/percent computation, build square and triangle numbers, write and compute with exponents and square roots, algebraic relationships, solving for an unknown, balancing equations, building 3 dimensional shapes and finding their volumes, thoroughly understand and convert the metric and U.S. Customary Units during measurement practices, solving logic problems, and reading biographies of famous mathematicians. All this helps to develop your child’s confidence in her or his own self and the ability to problem solve and think logically and mathematically

I can also can help you brush up on the areas of math that you never quite got down in school as a kid.  Do fractions frustrate you? I can help you understand them inside and out in just an hour or two. Does 300 kilometers, 14 degrees celsius, 27 millimeters and 22 kilograms just get weird sometimes? I can help you because I’ve been there myself and believe me, you will feel more confident in your own life.

Email me at: tiatia@gmail.com

Several of my friends recently have been asking me if I know anything about Kumon, an international math and reading center. Parents have legitimate cause for concern as math scores in the U.S. rank considerably lower than other leading nations as well as the Baltic States and the Russian Federation. A New York Times article from last Fall observes that U.S. math scores have plateaued since the Federal Government and No Child Left Behind laws have trumped state standards in education. I did not know much about Kumon so I looked into it.

The environment was clean, comfortable and quiet music was playing in the background. Taek Soo, whose educational background is in engineering, and who has run the Portland Kumon Centers for 16 years, was extremely generous with his time and willing to answer all the questions I had. He even let me take the placement test. Given the time limit of 25 minutes, I feel ashamed to say that I wasn’t able to finish the test within these parameters. I found my computation skills to be sluggish and forgot various short cuts I once knew. Taek Soo did however compliment me on my fraction skills. I found myself so intrigued by Kumon that I decided to sign up for the month of June to try it out for myself. The monthly tuition is $100 plus a one-time $50 sign-up fee. The price is cheaper than a professional development class, though I won’t get any credits for it. The curriculum at Kumon goes all the way through calculus as well as offering additional courses in statistics and engineering math. I am fascinated to see how it will go and look forward to sharing my experience with my readers.

Mathematics curriculum in elementary grades must include much more than teaching the four operations and isolated units on fractions, decimals, percentages, and geometry. The greatest challenge is that many teachers of K-5 do not themselves have a strong foundation in math because their teachers in turn did not. It is time to break that cycle and seek out new understandings in mathematics and learn how intricately connected are these isolated math subjects.

What I read:
http://blogs.edweek.org/edweek/campaign-k-12/2010/03/xx_states_are_named_race_to_th.html

Fifteen states plus Washington DC, out of 25 state applicants have made the cut and will now send a team to Washington the week of March 15th, to make presentations. The finalists are:

Colorado, Delaware, the District of Columbia, Florida, Georgia, Illinois, Kentucky, Louisiana, Massachusetts, New York, North Carolina, Ohio, Pennsylvania, Rhode Island, South Carolina, and Tennessee.

Winners will be announced in April.

What I read
From Science Daily:
“Friends’ School Achievement Influences High School Girls’ Interest in Math”
“Few Gender Differences in Math Abilities, Worldwide Study Finds”
“Believing Stereotype Undermines Girls’ Math Performance: Elementary School Women Teachers Transfer Their Fear of Doing Math to Girls, Study Finds”

From the MathForum:
“Girls’ Attitudes, Self-Expectations and Performances in Math”

From the News Bureau in Illinois:
“Girls’ Confidence in Math Dampened by Parents’ Gender Stereotypes”

From Prufrock Press Inc:
“Helping Teachers to Encourage Talented Girls in Mathematics”

There is a widespread belief in our country that only certain people have the talent to pursue mathematics-related careers and to take advanced math classes. This belief tends to assume that males and Asians are more talented in math than are females and non-Asians. While innate talent is a widely disputed subject, the fact is that if there are some people born with an innate genius in math, those are very few. What matters most is effort, persistence and confidence. It’s time to start letting students in on this “secret” of math.

There are many articles and studies trying to understand when the majority of girls get turned off to mathematics (and science, computers and technology) and how to turn that around. The most apparent issue that came out several times when reading the above articles, is that girls tend to have less confidence in their abilities to do math. When I attempt to track this backwards, it seems that the low confidence is directly connected to math anxiety which stems from two barriers: societal influences and the competitiveness of the math environment.

Let’s talk about the first barrier, societal influences. Girls tend to tune in to their environment and the people in their environment to obtain information about how to act and think. If parents and/or teachers think that boys have a higher aptitude for math, that information is communicated through body language, verbal cues, facial expression, assumptions of girls’ abilities and persistence, and unsolicited help. When adults try to “help” female students with their math, they are often really hurting girls’ confidence in their own abilities. So, note to parents and teachers, lay off the helping and encourage all your students, especially girls to persist in finding their own answers without your help. Another major societal influence for girls are their peers. Girls like to do things together in pairs or groups. If all their friends are going to take an art class instead of an advanced algebra class, which class do you suppose she will choose? The adults in girls’ lives must encourage them to pursue mathematics despite what their peers are choosing. Part of this encouragement is believing in their abilities and having high expectations that they can and will succeed.

The second barrier to girls’ positive attitudes towards mathematics is the competitive environment of the math class. Math classes tend to held in a competitive and fast-paced environment. There is little think time and little opportunity for small group or cooperative learning. While some girls thrive in this competitive math environment, the majority of girls get turned off by needing to come up with answers and solutions to problems as quickly as possible and not having the opportunity to check in with their peers before they decide on a solution. Girls “need classrooms in which they will be heard and understood and where they can discuss ideas before coming to conclusions” (Gavin & Reis-Prufrock Press).

Teachers should do their best to create an environment in the math class that

1. Offers choice between competitive and cooperative work
2. Allows think-time (using think-pair-share or journaling) and disallows shout-outs
3. Encourages risk-taking
4. Includes cross-curricular activities such as math and writing, dance or music
5. Uses assessments such as math portfolios and projects in addition or instead of traditional tests
6. Uses open-ended questions on tests instead of multiple choice
7. Doesn’t give unsolicited help, instead encourage students to persist in finding their own solutions
8. Provides some single-sex learning opportunities
9. Uses co-ed small groups but single sex pairing (since girls tend to defer to boys in partner math situations)
10. Holds Family Math Nights planned and carried out by girls (boys should get this opportunity as well)
11. Gives girls the opportunity to tutor younger children and organize math clubs
12. Ensures math topics are of interest to girls as well as boys (not just sports and construction but also paper folding, spread of diseases, endangered species, scale drawings and patterns)
13. Uses a challenging curriculum that introduces different number systems, Fibonacci numbers, non-euclidean geometry, fractals, chaos theory and other such topics in mathematics
14. Introduces knowledge of female mathematicians such as Hypatia, Marie Agnesi, Sophie Germain, Evelyn Boyd Granville, Sonya Kovalevskaya, and Mary Somerville.

Teacher Resource
Math Publication Best Sellers from Prufrock Press

What I read
Chapter 10 of Elementary and Middle School Mathematics

Part 2 in developing meanings for the operations focuses on multiplication and division, what strategies children use, what types of modeling a teacher should use and how the teacher’s understanding of approaches can enrich students’ understanding.

Multiplication and division can both be learned as early as 1st grade. Sound ridiculous? Not if you understand the relationships between all the properties. For example, when teaching multiplication to children, one of the most used approaches is to show how multiplication is merely repeated addition. For example: Eduardo grew 8 tomato plants. Each tomato plant produced exactly 5 tomatoes each. How many tomatoes will Eduardo harvest? The problem boils down to 8 x 5. There are 8 equal groups of 5 sets. The first factor (number) counts how many parts or sets. The second factor tells the size of each set or part and is also known as the multiplicand. In order to solve this problem using repeated addition, you create 8 groups of objects with 5 objects in each group. The resulting equation: 5+5+5+5+5+5+5+5 = 40 should be written down by the student as well as 8 x 5 = 40 next to it. Students will begin to learn the relationship. Equal group problems may also be considered as rate problems.

The major conceptual hurdle to developing meaning for multiplication and division is understanding groups of things as single entities and also that each group has individual objects. Once this mental obstacle is conquered, children of just about any age can begin working with these operations.

Van De Walle talks about four classes of “multiplicative structure,” and that there are two used most often in teaching multiplication. The first is equal groups, which I have written about already. The second class is multiplicative comparison. There are some fantastic example problems of both types that Van De Walle includes in his book. Comparison problems usually compare two people who have two different amounts of something. For example, Trudy built 5 models during art class. Her friend Clarissa built 3 times as many models as Trudy. How many models did Clarissa build?

Consider the two multiplication examples I have given here. Figuring out how to solve each one requires a slightly different way of thinking and certainly a different strategy. Compare this type of thinking to solving a page of textbook multiplication problems. You will be developing meanings for multiplication, not just how to get the right answer. Understanding these meanings are a foundation for algebra as well as other types of advanced mathematics. Van De Walle suggests each math lesson should include 2-3 problems that allow students to use any technique and any math tool they like. For each problem they solve, they must explain, preferably in writing and then later in a discussion, how they solved each problem. Van De Walle also includes a great activity for finding factors in this book, assigning numbers to students and having them use counters, arrays, etc. and then write two equations, an addition and a multiplication for each factor.

Remember how multiplication and addition have a special relationship via repeated addition? A similar relationship occurs between division and subtraction. Division is repeated subtraction. For example, the problem 24/6 = 24-6-6-6-6. A hundreds chart is a useful tool for solving these problems and helping your students skip count backwards as well as forwards will help them use this strategy more efficiently. Other helpful models for practice are counters, number lines and arrays.

When teaching division, teach it at the same time as multiplication or very soon after so students may practice both types of problems and begin to see their relationships. Avoid using the term “goes into,” which carries very little meaning, and instead use “divided by.” Additionally, don’t be afraid of large numbers. Since the strategies and understanding of products and quotients are the same for small vs. large quantities, solve  2 x 4, 13 x 11, and 102 x 15, 6/3, 24/8, 200/5 and even much larger numbers should be introduced at once. To help students see the relationship between multiplication and division, think about how you will word each problem. For example, asking “6 times what is 24?” is more effective than asking “24 divided by 6 is what?”

A note on remainders. There are two methods for remainders in division problems. Method 1 is to leave the remainder as a quantity left over (i.e. 16 R1). Method 2 is to partition the remainder into fractions (i.e. 16 1/16). Van De Walle prefers method 2. In real contexts, you sometimes need a whole number answer so you must round up or down. In a problem where a certain number of cars are needed to transport students for example, using a fraction or remainder answer is useless. There are three ways to decide whether to round the number up of down. If you discard the remaining amount (nobody gets the leftover jellybean) you round the number down. A forced amount (you need an entire car for the one extra student) you must round up. A rounded number will go up or down depending on whether the remaining amount is greater or less than five (We will need 9.6, or about 10 hours to build a compost container).

Finally, when you write or re-write problems for your classroom, make up problems with different contexts. Instead of just using objects such as brownies or pencils. Be sure to use problems that use length, time and volume.

Teacher Resource
Book: Knowing and Teaching Elementary Mathematics: Teacher’s Understanding of Fundamental Mathematics in China and the United States
This book by Liping Ma was highly recommended to me. I have not read it but thought I would pass it on to you. Let me know by posting a comment, if you have read this book and what you think of it. Ma defends the belief that children need to know both arithmetic as well as theory for solid mathematical thinking. Read the editorial review on Amazon by clicking on the book link above or click here.

What I read
Chapter 10 of Elementary and Middle School Mathematics

Part 1 is a synopsis of Chapter 10 of John A. Van de Walle’s book, how to help children develop an understanding for the meanings of addition & subtraction and how they relate. Part 2 will discuss multiplication & division.

There are two main methods for teaching addition and subtraction. The first method is to use contextual problems. The second method is to use multiple models such as counters, drawings and number lines. Using these two methods will help children construct a rich understanding of the addition and subtraction operations.

The problems presented should be structured like Cognitively Guided Instruction (CGI) if you are familiar with that. In short, instead of “adding” and “subtracting,” you “join,” “separate,” use “part-part-whole,” and “compare.” Any of the quantities you work with (initial/starting amount, change amount, resulting amount) can be the unknown. For example, here is a part-part-whole problem with the initial amount unknown:

Jayden had some jelly beans. She ate 13 of them on Monday morning. One Monday afternoon she ate the remaining 12. How many jelly beans did Jayden begin with?

Contextual problems must be somehow connected to the children’s lives. You could write your own problems easily by reading some samples presented in the book. Problems could be about a recent experience the class had together such as a field trip, a discussion in another content area, or your read-aloud (or another book you’ve all read).

A typical development for students to make in early math is as follows:

1. Uses counters and counts everything seen
2. Counts on from the first set given
3. Counts on from the largest set
4. Uses facts retrieved from memory and relies less on counters

Part of what will help children progress to the next stage of mathematical development is to teach with models, showing what to do every step of the way, being sure to use a variety of kinds [Designs, picture "stories," unifix cubes, 2-colored counters (or other object), or part-part-whole mat] in order to give students different ways to see the problem and to model how to use each kind of model.

When teaching these first two operations, teach addition and subtraction simultaneously. If you ask what 7 – 3 is, you must also ask what 5+3 is  so students will learn the pattern. Subtraction must be taught as “Think-Addition,” instead of “Count What’s Left.” So instead of subtracting 6 counters from a student’s pile of 9 counters and asking them for the answer by counting what remains or asking what is 9 – 6 , help your students ask the question 6 and what make nine? As kids practice with each other in pairs, they say and write the equations that match with their problems. This is actually the beginnings of algebra.

Lastly, something we often don’t think of because it seems so obvious to us as adults. When teaching addition and subtraction, help children understand the “Order Property” (usually known as the commutative property) and the “Zero Property.” Using understanding of the order property, you learn that when adding two numbers together, the order of the addends don’t affect the result. Using the zero property, you learn that any number plus or minus zero is that number. Van de Walle here says that some students really do get confused because they connect the idea of addition with a number getting larger and subtraction with a number getting smaller, so to some, 12 – 0 should be less than 12, so some students may write an answer of 11.

Teacher Resource
http://www.aaamath.com/
This is a nice little site that offers tutorials and definitions of 25 different math topics for K-8 grades. You can sort by grade or subject. After each tutorial there is an interactive practice application that produces problems and automatically checks answers. I advise you to check out the practice prior to assigning it to a student because I tried the fraction subtraction practice (that’s a mouthful!) and it wouldn’t accept reduced answers. For example, it wanted the answer 2/2 and not “1.” This is reasonable, just something the student should know.

NEXT: PART 2: MULTIPLICATION & DIVISION