How Should Students Show Their Math Work?

In this post, I am pinging off Maria Miller of Math Mammoth. I recommend Math Mammoth for its concept-based lesson development and worksheets.

Many students resist showing their work. They feel they are demonstrating their smartness by not showing their work, as in “See, Ma. No work.” However, when you ask these students how they got the answer, they cannot remember what they did. Sometimes they say they used a calculator. OK, I say, but what numbers did you put into the calculator? They cannot tell me. I explain that since we cannot record thoughts the way we can record voices, students need to make a record of their thoughts when they solve a problem. Dispensing with the work is not actually smart at all.

Now, here is where we see the real difference between strong students and weak students. Strong students respond to my words, and start showing work ever after. Weak students respond (eventually) only to action. I make them do their homework again, and I mark right answers wrong if there is no work.

As Maria says:

The purpose of writing down the work allows someone else to follow the person's thought processes. This is of course important for students to learn no matter what their future occupation: they need to be able to explain to others how they solve a problem, whether a math problem or a problem in some other field of life!

As strong as Chinese math teachers tend to be, they do not encourage students to show their work. Chinese teachers expect “clean” papers, with only answers. Chinese teachers check whether answers are right or wrong. They are completely unconcerned with why the student got a wrong answer, or if the answer is coincidentally right for the wrong reason. Retraining my students has been quite a challenge. Today they appreciate the need to show work, and they work hard to demonstrate that their work flows in a logical manner. Today, they show off their work instead of showing off the lack of work.

Even though Chinese teachers do not want to see work in the final product, they actually have high standards for the format of work. They train students from first grade in this format, and one reason they do not care to see the work in, say, fifth grade is they trust the student followed the format to get the answer the teacher does see, a dubious assumption at best.

Maria says she would ask primary student to verbally explain how they got an answer. Chinese teachers expect students to translate verbal (or written) math problem to mathematical expressions. Students learn to write “number sentences” from the very beginning. Perhaps there is a picture of a tree branch with three birds and two more birds landing. The child translates this picture in the number sentence “3 + 2 = “, and then writes “5 birds”.

I modify this approach a little. I expect children to write “3 birds + 2 birds = The idea of ignoring the units and then plugging them back in at the end leads to all kinds of confusion in later grades. Leaving the units out of the work is a major reason students persistently forget to square the unit when finding area. The math sentence should be 4 cm x 5 cm =

When students first begin studying area and perimeter, I make them write intermediate steps. In the case of area the intermediate step is: (4 x 5) x (cm x cm) = 20 cm2. In the case of perimeter, the intermediate steps might be (2 x 3) cm + (2 x 5) cm = 2(3 +5) cm = (2 x 8) cm = 16 cm. There are many types of problems where keeping track of the unit is vital. An early example is division, especially division with remainders. Often the unit for the quotient is different from the unit for the remainder. Knowing the difference is the key to understanding the solution.

I also require the box. The box makes the number sentence a complete sentence. Later, we will replace the box with a variable, and later still the variable may appear somewhere besides the end. Take this problem for example: There are 5 birds in the tree. After a certain number fly away, there are 2 birds left. How many birds flew away? I expect children to translate this sentence to math as written, without doing any preliminary math in their heads. Thus “5 birds - = 2 birds”.

Most teachers have the children write this math sentence as 5 birds - 2 birds = 3 birds. Doing so requires the students to do some math in their head first. The purpose of the number sentence is to accurately translate the problem to math terms. The number sentence must follow the story. The number sentence for a multi-part story should incorporate all parts into one number sentence. When problems become more difficult, the ability to translate the story to math as written becomes essential. The crucial part of solving a math problem is the number sentence. When the number sentence is correct, absent any silly mistakes in the work, the solution will most certainly be correct.

Finally, I require the students to answer the question with a complete sentence. The purpose of answering the question is to help student differentiate the solution from the answer. For example the solution to the question, how many cars do we need for the field trip might correctly be 5.2 cars, but the answer is 6 cars.

Summary

The work for a word problem needs to have three parts.

1.  A translation of the word problem into a complete math expression that includes the units and follows the story.

2. The arithmetic which tracks the units all the way through to the solution and may include intermediate steps for as long as necessary for mastery.

3. The complete answer to the question.

Sample

Math Expression: 10 x [$10.50 – (2/5 x $10.50)] = n

Work: 10 x {$10.50 – [($10.50 ÷ 5) x 2]} = n

10 x [$10.50 – ($2.10 x 2)] = n

10 x ($10.50 – $4.20) = n

10 x $6.30 = $63.00

Answer: Annie's total bill is $63.00 or Annie paid a total of $63.00 for the shirts.

Well-trained fifth graders have no trouble displaying their work as in the sample. This vertical work format, started in first grade, gives the students excellent preparation for mathematics involved in algebra, chemistry, physics and calculus. In fact, starting in third grade, I often have students format their work in two vertical columns, the second column for the math property used, as in this simple sample:

Problem: There are 5 birds in the tree. After a certain number fly away, there are 2 birds left. How many birds flew away?

Number Sentence: 5 birds – n = 2 birds

Work:

---

Arithmetic	Property
5 birds – n = 2 birds	given
+ n              + n	both sides rule
5 birds + 0 = 2 birds + n	additive inverse (opposites rule)
5 birds = 2 birds + n	additive identity
-2 birds = -2 birds + n	both sides rule
3 birds = 0 birds + n	math fact/additive inverse
3 birds = n	additive identity

Answer: Three birds flew away.

Class Policies for High School

These are the class policies I use for junior high and high school math and science classes. They are quite brief, but effective because students perceive right away that I say what I mean and mean what I say. Being straight-forward and authentic is probably the number one key to classroom management. Educators will debate the validity of grades forever, but as long as colleges pay attention to GPAs, teachers will have to figure out a way to determine grades. The following system has worked well for me.

Duties of Responsible Students:

1. Responsible students come to class on time, with their homework and materials laid out on their desk, pencils sharpened, and ready to begin before the scheduled start of class.

2. Responsible students do everything in their power to make it as easy as possible for their classmates to concentrate and achieve.

3. Responsible students turn in work that is neat, complete and on time.

Components of Grade:

1. Classwork 40% of grade (includes quality of work completed in class and responsible behaviors during class. Giving your work your professional best effort will raise this grade. This grade starts at 100% for all students.

2. Homework 30% of grade. The grade is the number of completed assignments out of possible assignments. Unacceptable assignments will receive an “R” which means “redo within one week.” Otherwise, the grade for that assignment becomes 0.

3. Tests 20% of grade. Test are graded as a straight percentage.

4. Quizzes 10% of grade. These are generally pop quizzes. 51% or better on a pop quiz earns a P for pass. 50% or below is a “no pass.” Announced quizzes are graded as a straight percentage.

Formatting Your Work:

1. All work must be done on standard 3-ring notebook paper, or specified graph paper. Do not fold your work.

2. Pencil is acceptable for certain work done in class and for math. Products like lab reports and essays must be written in cursive using blue or black ink only. You may write your work on a word processor, however printer malfunction is not an acceptable excuse for failing to submit the assignment on time.

3. Remember to use your English skills. Even when the work is not for English class, you are still expected to indent paragraphs, maintain margins, proofread and rewrite your work as necessary to submit your best work.

4. All papers must have a proper heading as previously instructed.

How to Evaluate a Math Textbook

Regardless of Common Core, everybody knows that practically speaking, the textbook IS the curriculum. Therefore, it behooves textbook adoption committees to choose carefully. First, ignore the beautiful graphics. The beauty may truly be only skin deep. Reject books that teach tricks, procedures and shortcuts. Choose books that teach the profound understanding of fundamental mathematics. You do not have to read the entire book. Look especially for how the book handles the following topics:

Place value---Place value is arguably the most essential foundation stone of all future math understanding. Yet most textbooks provide only a rudimentary presentation of place value. Students are expected to do no more than name the place of a given digit or write a certain digit in a given place. The understanding of place value actually begins with counting. Make sure children name what they are counting and start with zero, “0 frogs, 1 frog, 2 frogs, 3 frogs...there are 7 frogs altogether.” Remember, place value depends on fully knowing the name of what is being counted, and not simply as part of a memorized pattern. 203 means you have counted 2 hundreds, 0 tens, 3 ones. 203 can also mean you have counted 20 tens, 3 ones. Which version is more useful depends on the context of the real life math. An early emphasis on place value helps students with later concepts such as fractions (203 thousandths), volume and area (203 cubes vs 203 squares), or the difference between like and unlike terms (3a + 2b). There are many more math concepts that depend first on naming what is being counted and understanding the significance of the name to place value.

The equal sign—An equal sign means everything around the equal sign is equal to everything else. Therefore an expression like 2 + 3 = 5 x 4 = 20 is not allowed because 2 + 3 does not equal 5 x 4. However the separator bar within the vertical format is allowed, because the separator bar does not mean equal; it is a separator bar.

Long Division---Although the idea that division is nothing but repeated subtraction is a bit oversimplified, the long division algorithm exactly depends on repeated subtraction because when you multiply within the algorithm, you are multiplying negative numbers. That is why you subtract the result of the multiplication. Look for a text that presents long division as more than memorizing the steps of the algorithm.

Multiplication and Division of Fractions---½ x 2/3 means one-half of two thirds. This example highlights the value of word problems. Word problems put math where it belongs and from where it arises, that is, math is the solving of real life problems. All math problems have a story. A page of naked problems has simply lost the stories. Suppose I have a ribbon 60 cm long. 2/3 of the ribbon is 40 cm, and half of that is 20 cm. 20 cm is 1/3 of 60 cm. Through examples like this, students can see that ½ x 2/3 = 2/6 = 1/3.

Division works the same way. Say I need to measure ¾ cup sugar and all I have is a 1/8-cup measuring cup. How many times do I need to fill my measuring cup to get ¾ cup sugar? ¾ cup divided by 1/8 cup therefore equals 6 times. (Notice again usefulness of knowing what you are counting. In this example, the answer is counting “times,” not “cups”). Texts should require kids to solve math problems by drawing pictures. When the student can reliably use a diagram to solve a problem, they are ready for the algorithm. Only at the end of the learning process should we teach the shortcuts. Math first, then shortcuts. Pictures are also the first step to proofs.

Absolute Value---Make sure absolute value is presented as distance from zero, NOT as simply a negative number turning into a positive number. A football analogy may help. If the quarterback is sacked, the ball may be 5 yards from the scrimmage line, but from the quarterback's point of view, it is still a negative 5.

Canceling---I loathe this word. Students are not “canceling.” They are simplifying a fraction. Simplifying a fraction means finding “1.” It does NOT mean crossing off numbers. Canceling leads students to lose track of the difference between “0” and “1.”

Multiplying and Dividing Decimals---Multiplying and dividing decimals has nothing to do with moving decimal points. It has everything to do with multiplying or dividing by powers of ten. 12 x 1.4 means 12 times 14 tenths. 14 tenths means 14 divided by ten, so 12 x 1.4 means [(12 times 14) divided by 10], which means 168 divided by 10, which equals 16.8. Students can tell where the decimal point goes, not by counting decimals places but by realizing the answer must be a number close to the product of the whole numbers. 12 x 1 = 12, so the answer must be close to 12. 1.68 is too small. 168 is too big. Therefore the answer is 16.8.

It is easy to confuse students by changing the problem slightly to 12 x 1.40. They will likely say they need to count 2 decimal places so the answer is 1.68. Giving them a new rule about ignoring zeroes does NOT build math understanding. Shortcuts are just that: shortcuts---and should be taught only when the student knows the actual road, not to replace the actual road.

Ignore the glitzy graphics and choose textbooks that handle all these topics well.

Exactly Those Contrary Ideas

Twenty years ago the late comedian Bill Hicks felt obliged to defend the blasphemous content of his stand-up routine. In so doing he said something remarkably insightful.

‘Freedom of speech’ means you support the right of people to say exactly those ideas which you do not agree with.

Hicks is right. However, it is too bad he did not actually mean what he said.

The Founding Fathers promoted Freedom of Speech because they did not want to lose their heads merely for disagreeing with the king. They wanted to be able to say exactly those ideas the king would not like. Therefore, the established the right of people to say exactly those ideas other people, especially people in power, do not agree with.

So far, so good.

The problem is that today, many people toss off the phrase “Freedom of Speech” as if it is a constitutional defense of any expression. Freedom of Speech protects ideas, especially ideas that might threaten the interests of the powerful in exploiting the weak. It was never intended to let people say (or draw, or film) anything they want.

If you cannot express your idea in a non-”blasphemous” way, perhaps your idea is not worth expressing at all.

Over time, people have gradually lost the ability and the social censure to restrain themselves. There are kids in school who seem unable to speak an obscenity-free sentence. Voltaire said, “The man of taste will read only what is good; but the statesman will permit both bad and good.” Our society seems less and less capable of producing children (and finally adults) of taste. As Paul of Tarsus wisely advised, “...fill your minds with those things that are good and that deserve praise: things that are true, noble, right, pure, lovely, and honorable.” A mind full of treasure has no room for trash.

Increased Reporting DOES NOT Increase Achievement

So here I am, in the land of tiger moms, where supposedly parents are highly involved in their child's education, even riding that pendulum to the other extreme. I don't see it. What I see is a lot of complaining, but little to no positive action at home.

In my school, some parents complained that if they do not know about the homework, they cannot insure its completion. So the principal decided to send a picture or description of the homework assignments to the parents' cellphones. It made no difference. Kids who regularly completed homework before the messages continued to do do. Kids who did not do their homework still do not do their homework. In fact, the parents of five of my twenty-one students admitted to doing the homework for their children. People always like to recommend more communication. Sounds good theoretically, but "communication" is not the cure-all everyone supposes. However, as the principal said, one benefit is the parents stopped calling.

Of course, I expected the students to make a note of the assignments everyday. It is not the teacher's job to tell the parents what the homework is. Parents should check their child's assignment book. If their child is not writing down the homework as instructed, parents should deal with the noncompliance at home. Schools need to stop giving already busy teachers more useless duties. Parents need to emphasize that knowing and doing the homework is the student's responsibility.

A trend over the past thirty years has been to hold the children less and less responsible for their schoolwork and put that burden on the teacher. Years ago, as long as the child was behaving not too badly, parents heard virtually nothing from the school except for the four quarterly report cards. Parents of high achieving children were fine. Parents of low achievers began complaining. They said they could do nothing at home to mitigate a failing grade if they do not know before report cards come out that their child is failing, In response to these complaints, schools started issuing mid-quarter “progress report”. It made no difference to final report cards. High achievers continued to achieve highly; low achievers continued to fail. The only discernible outcome was that teachers had double the reporting work.

Eventually, even mid-quarter reports were deemed too few and some schools began mandating weekly progress reports. It still made no difference. Schools began requiring students to purchase expensive “planners” on the dubious assumption that students were not writing down their homework assignments because they had no little notebook to record the assignments. This assumption is beyond silly, and of course, made no difference. Responsible students have always written down their assignments, long before planner became the soup du jour. Then schools began requiring teachers to post the homework online. The only apparent effect is to create more busy work for the teacher, and stop parental complaints.

Some teachers take matters into their own hands and require students to do the homework during lunch. This tactic is at least partially effective because it generally ensures the homework gets done. I do not like using the lunch hour because children need to run around and play before settling down to an afternoon of work. We invite behavior problems when we deny them this energy outlet. Furthermore, research shows that exercise increases thinking ability and concentration. I prefer to keep kids after school. I have found it to be more effective at promoting self-responsibility.

There is one major caveat: the assigned homework needs to be worth doing.

College Return on Investment

The Federal Reserve Bank of San Francisco (FRBSF) recently published their evaluation of the lifetime payoff of a college degree, concluding that college is a great investment.

We show that the value of a college degree remains high, and the average college graduate can recover the costs of attending in less than 20 years. Once the investment is paid for, it continues to pay dividends through the rest of the worker’s life, leaving college graduates with substantially higher lifetime earnings than their peers with a high school degree.

Meanwhile, Barry Ritholz, using similar data, concludes that a college education “shows a fairly poor return on investment. No wonder so many college graduates are unhappy with their student debt.”

What gives?

What is clear is that without a college education, the chance of earning the median wage is extremely low. There is really no choice in the matter. It's college or nothin' for most people. FRBSF recommends “redoubling the efforts to make college more accessible would be time and money well spent.”

We have seen this movie before. Not that long ago, people wondered if high school would payoff handsomely. It did and eventually high school for all became the publicly funded standard of the land. Everyone who graduated high school could count on a good job, decent salary and a secure future, so they said. And it was mostly true---then.

We are presently taking another turn around the same merry-go-round. Given current economic conditions and opportunities, governments might well conclude it is in the public interest to require and fund college for all. Once college for all is implemented, the Lake Wobegon fallacy comes into play. Just as with mandatory high school, reversion to the mean will occur.

Like medicinal tolerance, it takes more and more education to get the same effect. Soon, (and some people believe it is already happening), college will not be enough. It's inevitable. The pervasive assumption that if only everyone graduated from college, they could all land great jobs ignores the reality that there simply are not enough great jobs to go around for the people who think they made the investment in themselves to qualify for those jobs. And so reversion to the mean happens. The nature of average is that new data only establishes a new average for some of us to be above and some of us to be below.

Nevertheless, college for all is probably a necessary eventuality.

Websites for Kids

Parents have been asking me to recommend some websites. Out of the universe of possibilities, I have selected several worthy contenders:

RAZ-Kids: This is a leveled set of readers with quizzes. The child reads a book online and then takes a quiz. The child can go on to the next book after passing the quiz. Some of the books are audio, allowing the child to listen and follow along instead of reading alone. Many schools offer similar programs, but they all suffer from being online. Children need to read on paper, not on screens. Besides being unhealthy, screens fail to train the eyes to move properly across the page. Studies have shown that reading on screen reduces speed and comprehension. I really like the old-fashioned SRA (anybody else remember SRA?)

In fact, all online resources suffer this same shortcoming simply by being online. I strongly believe the current fascination and quick adoption of anything labeled “technology” is misguided. Nevertheless, I have collected some websites to help your child learn various subjects. I recommend limiting the amount of time children spend in front of a computer screen.

California Treasures: A comprehensive literature, grammar, and writing program popular in many schools. The main menu is here. Supporting activities are here and here. Unfortunately, some of the links do not work.

National Geographic for Kids: Lots of interesting articles in easier English.

Practice English Grammar Online:

Practice Math Online: IXL

MathABC

Adapted Mind

Math Playground

More Games

E-Learning For Kids

Some of the lessons offer a “certificate of completion” the child can print out and bring to the teacher for possible extra credit. Please monitor to ensure your child actually earns the certificate.

Science Lessons: 40 animated science lessons. Children younger than nine years old may have difficulty with the animation. Older children find it easier to make the connection between animated events and real-world events without confusion. Students can learn about the human body in the “health” section of this website: http://www.e-learningforkids.org/health/

Math Lessons: 336 Math Lessons. I just discovered these lessons online. From the same publisher as the science lessons, the goal of these lessons appears to be preparing students to maximize scores on tests, rather than maximizing understanding of math concepts. Not only is the English is slow and easy, but students can repeat any page they want as often as they want. I recommend using these lessons AFTER the child has completed the in-class concept-building lessons. In fact, I will soon use some of these lessons in a comprehensive review.

Computer Skills: This section, from the same publisher as the math and science lessons, helps children learn various computer skills on their own, such as creating a Powerpoint presentation.

Kids Health: A comprehensive resource.

The Open Door Website:

The Open Door website can be used as a online science text based on IB (International Baccalaureate) standards. The text requires reading skills. There are no audio files, but there are extensive illustrations and photos. As of 2014, the science information is quite comprehensive, but the math sections are still extremely meager. This page is an index of quizzes on many science topics. Most of the quizzes can be saved as PDF files and printed out.

The Utah Education Network has collected this nice group of interactive science activities for grades 3-6. The reading is minimal, and the instructions are mostly obvious and intuitive.

Crickweb: The nice thing about the Crickweb science pages is the simple design of each activity and the vocabulary building which will help ELLs with their regular reading.

BBC: The BBC has a number of math, English and science activities, but the audio and video require a lot of bandwidth and may require too much time too load or sometimes never finish loading. I like this interactive series about the body from the BBC (http://www.bbc.co.uk/science/humanbody/body/index_interactivebody.shtml).

For students who want to DO science experiments and activities, here is a set of websites with lots to choose from.

Science Experiments:

Science Activities:

Web Ranger

BBC

Weebly

UEN