Thoughts on First Grade Math

Random Thoughts on First Grade Arithmetic

Number lines are seldom found in primary school math materials. One exception (?) is the 11" by 16.5", 645-page 1st grade enVision Math textbook from Pearson 2011, but the number line is hardly used--just one lesson: "You can use a number line to find missing numbers: 2 _ 4 with "before" and "after" as vocabulary. A ten-frame is used extensively to model numbers and figure out single-digit number facts, not a number line. I think this is a mistake. Also, memorization and drill are not part of the enVision reform math program. Why not? 

11" by 16.5" enVision Math 1st-Grade textbook. It is 645 pages.

The reform math idea is that students should calculate the single-digit number facts using a variety of methods, not memorize them, which is a shift from the Old School ideas of memorization and drill to develop skill in arithmetic. Frankly, the ten-frame method and counters are inferior to a number line, which shows basic arithmetic. In my self-contained 1st-grade class (the early 80s), I stopped using counters (manipulatives) after the first couple of weeks of school. The number line is essential mathematics, not the ten-frame, counters, or calculators. The 0-20 number line is an extension of the number line kids come to school with. 

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Number lines show key math concepts. Students learn magnitudes, number relationships, patterns, whole number operations, fractions, decimals, etc. The "one more" idea shows children how numbers are built: add one. 

First Grade Number, Line 0 to 20: Equal Units. 
I started with a 0-20 number line, then added a -10 to 10 number line. 

Equal Units

First-grade students come to school with a built-in logarithmic number line (unequal units), but in arithmetic, the number scale is linear (equal units) and needs to be taught that way. The distance between 4 and 5 is ONE unit or just plain 1. The distance between 10 and 11 is ONE unit, etc. Students can see this on a number line. To get the next number (a'), add 1 to a: a + 1 = a'. (FYI: a' is read a-prime, which is the next number) Thus, 2 + 1 = 3,  4 + 1 = 5, 18 + 1 = 19, 56 + 1 = 57, etc. Linear number lines should be used from day-one in teaching arithmetic to first graders, but I seldom see them in primary math textbooks. Counting starts at 1, but number lines and rulers start at 0.  Magnitude is an important idea, too. Students can see that 4 is greater than 3 (4 > 3) or that 3 is less than 4 (3 < 4). These are inequalities, and the symbol always points to the smaller of the two numbers when they are compared. When comparing numbers m and n, only one of these is true: m = n, m > n, or m < n. 

Okay, how do we get from 4 to 5?

ADD ONE to get the next integer.

The number line shows how (and why). We add 1, which, on the number line, means you move one unit to the right of 4 to get to 5: 4 + 1 = 5, that is, to get 5, add 1 to 4. This is the successor rule for integers: a + 1 = a', in which a is an integer {4} and a' (a-prime) is its successor {5}. The idea that you get the next integer by adding one to the previous integer is very important, but, unfortunately, it is seldom taught this way in 1st grade. The successor rule works for all integers. For example, -6 + 1 = -5 or -1 + 1 = 0. You don't need to call it the "successor" rule. For whole numbers, you can call it the "add 1" rule. Furthermore, what seems obvious to adults, is not always clear to 5 and 6-year-olds. Remember, kids don't think like adults, which is a basic premise of cognitive science. Hence, presentation and explanation are important in teaching young children arithmetic as are memorization, "drill for skill," pattern recognition, place value, magnitude, etc. 

Adding 3 and 4

Once children see the basic relationship of 3, 4 and 7, then they should memorize the fact (3 + 4 = 7) and solve missing addend problems such as 7 = x  + 4. Also, note that subtraction is defined in terms of addition: 7 - 3 = 4 if and only if (iff) 3 + 4 = 7. And, 7 - 4 = 3 iff 4 + 3 = 7. It makes good mathematical sense that addition and subraction should be taught together. 

The concepts in arithmetic are elementary, and the number line makes them accessible to 1st-grade students. Children are novices, not experts or little mathematicians. They need to memorize and practice to learn. Learning something is remembring it from long-term memory.

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Also, the standard algorithm is not found in the enVision 1st-grade text. The textbook is too big to take home, so students tear out the lessons. Moreover, there are numerous calculator activities called Going Digital to get kids on calculators. Using calculators starting in kindergarten was an imprudent guideline introduced by the National Council of Teachers of Mathematics (NCTM 1989). You would think the NCTM would know better. 

W. Stephen Wilson, a mathematics professor at Johns Hopkins University, sets the record straight by pointing out that using calculators is "absolutely unnecessary" in arithmetic and algebra. He writes, "The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them."

enVision Math - 1st Grade

You see, where reform math programs such as enVision Math are headed, which is to do away with memorization of facts for auto recall and the standard algorithms (Old School). Calculators (tech) are substituted for paper-pencil standard arithmetic. It is already happening. The reform idea is that computations can or should be done on calculators, even easy ones. Therefore, the student can focus on critical thinking or problem-solving. But, the math doesn't work that way. You cannot apply something you don't know well in long-term memory. Students should master the fundamentals, but many don't. 

The content in 1st-grade enVision Math is significantly below the benchmarks of some Asian nations. American kids start behind and stay behind their Asian peers. Often, in reform math, kids are asked to draw something to get an answer or as proof. A drawing is not math. Counting is not a property of numbers. Using a calculator is pressing keys; it's not math.

Arithmetic is mostly addition, subtraction, multiplication, division, fractions-decimals-percentages, and ratio/proportion. In the early grades, parts of arithmetic are unexciting. Also, it is not much fun memorizing the addition and multiplication tables or practicing the standard algorithms. Reform math and minimal guidance constructivist methods seem to dominate K-8 classrooms. Consequently, many students do not master traditional arithmetic or standard algorithms. They are ill-prepared for Algebra. Indeed, basic arithmetic has not been taught well for years. The problem starts in 1st grade and jumps up the grades.  

Kids are novices and need to memorize and drill for developing skills. Learning arithmetic means remembering arithmetic from long-term memory. Learning arithmetic is work, but it is essential because it forms the cognitive architecture of mathematics in your long-term memory and helps students develop number sense. But, later, students will likely use calculators for more complex calculations, especially in chemistry and physics. I used a slide rule. Today, unfortunately, calculators often cover up weak arithmetic skills even in elementary school. 

Liberal reformers (the progressives) tossed out the Old School stuff even if it worked well when it was taught well. Tossing out the good things of the past is hardly a reasoned strategy. 

The doing of mathematics and the understanding of mathematics are rooted in the symbolic language of mathematics (i.e., abstraction). But, teachers don’t know symbolics or properties (rules) of numbers. They do not know that equality is reflexive, symmetric, and transitive (Peano axioms), much less the ideas that subtraction is defined in terms of addition and division is defined in terms of multiplication. In algebra, subtractions are changed to additions, and divisions are changed to multiplications. I am not sure that teachers know that the sum of identical addends is called multiplication. Counting is not a property of numbers.   

Note: The two primary operations in the set of real numbers are addition and multiplication, which make a Field. A number line is all that is needed to explain addition, subtraction, multiplication, and division of whole numbers and regular fractions. 

The idea that elementary students must know the why of everything and show proof rather than the "how" is nonsense. Indeed, students need to practice procedures until they are automatic, which is what kids in many other nations do, including the East Asian countries that trounce American students in factual and procedural knowledge and creative problem solving on international tests (TIMSS, PISA). Memorization and repetition are keys to learning because learning is remembering from long-term memory. 

Asian children are taught mechanics first with the explanation later, and it works! We do it backward with understanding first, and it doesn't work. In short, the progressive math reforms have not stressed the mastery of standard arithmetic in long-term memory. 

As I had said many times: There is no substitute for knowledge in long-term memory and the practice that gets it there.

The reality is that the more "rote learners" of the East Asian nations have excelled in factual and procedural knowledge and creative problem-solving (TIMSS, PISA), leaving most American students in the dust.

Symbolic Arithmetic

Standard arithmetic is the basis for higher-level mathematics, such as classical algebra. Algebra is symbolic arithmetic. Newton called elementary algebra the universal arithmetic because the calculation of numbers (arithmetic) and of symbols for unknowns (algebra) were the same. The rules that govern the calculations of arithmetic and algebra are called the field axioms. Simply, elementary algebra obeys the rules of arithmetic. A focus should be on learning of field axioms (rules) that govern arithmetic and elementary algebra calculations. It impacts 1st-grade arithmetic and algebra. 

Should we teach algebra concepts in 1st grade? 

YES! 3 + 7 - x = 5 - 2 

(x = 7 to make a true statement. The inverse idea is often used in solving equations. 

Left side = Right side: 3 = 3)

John Stillwell (Elements of Mathematics From Euclid to Godel) said that the point of doing arithmetic is not to do millions of calculations but to learn the axioms that govern them along with efficient calculation methods using single-digit number facts. Also, He said that the algorithms (step-by-step recipes) to calculate numbers should be fast and efficient. For novices, one algorithm per operation is sufficient to start. The operations can be understood using a number line, and when the numbers are larger, the standard algorithms should be used. This idea goes against the "many methods" of reform math that clutter the math curriculum and create cognitive load. Also, children should learn the mechanics first with an explanation later. Learning the mechanics of an algorithm requires practice-practice-practice and auto recall of single-digit number facts.  

In the first week of school, first-grade students should learn at least two field axioms (rules) via the number line: a + 0 = a and a + b = b + a. Also, students should learn an important "common notion" from Euclid: Things that are equal to the same thing are also equal (i.e., the transitive axiom of equality). 

The meaning of the equal sign is important and often overlooked. If 2 + 3 = 5 and 12 - 7 = 5, then it follows that 2 + 3 = 12 - 7 by the transitive axiom (rule) of equality. The left side of the equal sign {5} is equivalent in value to the right side {5}. Simply, students should "Think Like a Balance." Mathematics is built on true statements.

True/False Statements

3 + 4 = 7 TRUE because both sides are 7 (7 = 7)

3 + 4 = 5 + 6 FALSE because 7 ≠ 11

Here are the nine "field axioms." 

Elementary Algebra (i.e., symbolic arithmetic ) obeys the rules of arithmetic.

I have given an example in arithmetic and the grade level of introduction.

1. a + 0 = a 

5 + 0 = 5 (1st, zero property of addition)

2. a ⋅ 0 = 0

5 x 0 = 0 (2nd, zero property of multiplication)

3. a + b = b + a

2 + 3 = 3 + 2 (1st, commutative property of addition)

4. ab = ba

2 x 3 = 3 x 2 (2nd/3rd, commutiative property of multiplicaiton)

5. a + (b + c) = (a + b ) + c 

3 + (5 + 2) = (3 + 5) + 2 (1st, associative property of addition)

6. a(bc) = (ab)c 

4 x (3 x 2) = (4 x 3) x 2 (2nd/3rd, associative property of multipication)

7. a + -a = 0

2 + -2 = 0 (1st/2nd, the addition of opposites is zero; the inverse property of addition. Note 2 + -2 the same as 2 - 2.)

8. a ⋅ a^-1 = 1 (for a ≠ 0)

3 x 1/3 = 1 (2nd/3rd, the product of reciprocals is one; the inverse property of multiplicaiton)

9. a(b +c) = ab + ac 

3 (4 + 7) = 2 x 4 + 3 x 7 (2nd/3rd, distributive property)

Note: Subtraction can be changed to addition, and division can be changed to multiplication. Thus 5 - 3 = 5 + -3, and 6 ÷ 5 = 6 x 1/5, or 6 times the unit fraction 1/5, which is 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 or 6/5 by repeated addition. Why is this important?  Both Addition and Multiplication are commutative. (Subtraction and Divison are not commutative) You can add or multiply numbers in any order.

Rules in math are essential. 

I often hear negative remarks about rules in math. Math is governed by rules such as the properties of numbers and equality. The rules should be learned in the early grades. Simply, young students must know the rules and be able to apply them.

Last update: 11-19-17, 11-21-17, 11-23-17, 12-25-17

©2017 LT/ThinkAlgebra

Alma & Harmony

It's Harmony Zhu, Now 12

Arie Vardi challenges Harmony Zhu to a 2-minute chess game. Guess who won? At age 12, Harmony played the Piano Concerto 2 of Beethoven with Vardi conducting the Israel Philharmonic. See below (April 2018).

How can a child get this good at the piano? It's a rare synthesis of talent, practice-practice-practice, excellent instruction, and circumstance (i.e., chance). Practice does not create talent or ability. It must already be there. Practice develops innate ability. To advance, talented students need the best teachers. 

Harmony is one of the rare young talents in classical music today. But, how did she get that way? (Hint: She started at age 4.)

Alma

Zeitgeist 2015
The first Alma, not the second Mozart. Alma writes music as if she lived in the Classical era of the 18th Century.

In 2015, Alma Deutscher (UK) was invited to Google's Zeitgeist conference. She was 10-years old. Alma explained how she composed music and that melodies float in her mind all the time. Sometimes, she can't stop them. She is also an accomplished violinist and pianist. Guess who came to her interview and presentation at Zeitgeist? Stephen Hawking! (See him in the background with the colored glasses as she plays Bach from memory with the host.) You don't get good at something unless you practice-practice-practice, then practice some more. Oh, did I say you need to practice?

"Alma Deutscher was playing piano and violin by the time she was 3 years old and wrote her first opera at 10. For her, making music seems as natural as breathing." (CBS)

Alma & Pelley CBS News

In early November 2017, Scott Pelley of CBS (60 Minutes) interviewed Alma at age 12. She is homeschooled, has a math tutor, takes composing lessons on Skype, has piano and violin lessons, and reads incessantly. She speaks the language of Mozart and other composers. She writes music in the style of Classical composers. 

Alma is in her own world. During a break, she was outside dancing around and singing melodies that came to her. Making melodies is the easy part, she explains. The hard part is taking those melodies and putting them together in a coherent piece of music with an orchestra. She writes the orchestra parts, too. Composing an opera (age 10) or a piano concerto, or a violin concerto is difficult and takes a long time.  

Alma tells Pelley, "For me, it's strange to walk around and not to have melodies popping into my head." Alma started to play her melodies on the piano at age 3 or 4. She envisions a world in her mind comprised of imaginary composers. She gave each composer a name and often asked them for guidance. And sometimes, her make-believe friends would come up with intriguing suggestions, she says. Alma has no idea how all this works, neither does science. She was born this way, I think. 

Just as scientists don't know the origin of Dark Energy, which takes up about 68% of the universe, they do not have a clue about how Alma's mind works. Like Harmony (See Below), she's unique and singular. Our best minds cannot figure it out. We don't understand creativity that well. Why is one person more "creative" than another unless they are born that way. In education, I think we underestimate the role of nature and overstate nurture. Often, we believe that if we copy what highly creative people do (x, y, z, etc.), then we will become highly creative too. It is a false premise. You cannot copy what Alma or Harmony does--or even come close. They are singular. Click --> Watch Alma improvise and create (60 Minutes Overtime). Creativity like academic or musical ability varies widely. 

Kevin Ashton writes, "We are not all equally creative." But, that doesn't mean that average people with ordinary thinking cannot create. "Creation is a result--a place thinking may lead us. For most people, before we can know how to create, we must know how to think." Having a bunch of ideas in our head is not the same as creating. Everyone has ideas. Creating is hard work, i.e., producing an extraordinary product. It is hard work to compose music. It's hard work to play music. Ask Alma. Ask Harmony.

Even though Alma adores Mozart, she doesn't want to be known as the second Mozart. She is the first Alma. She says she wants to counter the ugliness of the world by creating beautiful music with memorable melodies. Why would anyone want to listen to music that is not pleasing to the ear? If you want ugliness, don't come to my concerts, just turn on the TV news, she retorts. 

Harmony Zhu
At age 10... 

Ten-year-old Harmony Zhu in a Master Class with Arie Vardi at the Aspen Music Festival 2016
To advance, talented students need the best teachers. 

Harmony at 11: Ballade No. 1

Then, there is Harmony Zhu. She is an accomplished pianist and has played at Carnegie Hall several times, even when she was 7. She, too, has melodies pop into her head and composes music (e.g., The Moment). Her latest composition, Ballade No 1, was posted on YouTube on 12-30-17 and dedicated to her fans: Happy New Year. "This is a piece about peace, hope, love, and harmony." Harmony is 11.

Like Alma, she wants to play music that makes people happy. Both girls have engaging, outgoing personalities. You can't help but fall in love with these extraordinary, happy children. They are unique and exceptional. Harmony loves math, music, art, chess, history, singing, dancing, and much more. Watch her play Chopin at the Aspen Music Festival (2016) at age 10. Harmony lives in NJ and attends regular school, but, on Saturdays, she goes to Juilliard in NYC, where she is a student of Veda Kaplinsky in the Pre-College Division.  

Oh, did I mention that Harmony was the World Youth Chess Champion for her age group (2013)? "You get to do battle with your opponent. It's so fun!"

It's so fun!

Special Performance by Pianist Harmony Zhu @ Simms/Mann 2017 Think Tank. After the performance, Harmony was interviewed by Pat Levitt, Ph.D., Professor, and Director of Neuroscience Graduate Program, USC. 

Neuroscience Professor: I love music, but I don't play the piano or chess.

Pianist Harmony (Age 10): It's never too late to start learning!

Audience: Giggle!

Note1: Harmony is a typical 11-year-old. She was Wonder Woman at Halloween, does paragliding (She's a little crazy; you should see her dance, too), participates in sports at her school (Run, Harmony, Run), and much more. Oh, did I mention that she plays piano at Carnegie Hall and attends Juilliard? And, you don't get to Carnegie Hall unless you practice-practice-practice. 

Note2: Students with high musical ability often excel in mathematics. But, being smart does not cause musical ability or excellence in math.


Updated on 1-2-18, 6-22-18
Send comments to LT: ThinkAlgebra@cox.net 
©2017-2019 LT/ThinkAlgebra 

© 2017 LT/ThinkAlgebra

Progressive Ideas Litter Education

Progressive Ideas Litter the Education Playground
The claim that teaching kids math discriminates against children of color is the latest progressive gobbledygook. Professor Gutierrez's remarks are ludicrous and biased. We need to keep politics out of schools. (See Alert below)

E. D. Hirsch Jr. writes, "Critical thinking does not exist as an independent skill. The domain specificity of skills is one of the most important scientific finds of our era for teachers and parents to know about, but it is not widely known in the school world.”

The idea that learning "standard arithmetic" is not that important stems from the progressive self-esteem crusade and reformers who think calculators are a substitute for learning basic arithmetic. Furthermore, progressive reformers also believe that critical thinking and problem-solving skills should be taught independent of content. They are WRONG! 

Children of color can excel in math when it is taught and practiced for mastery. But, this is often not the case. Also, Mark Manson points out that we have grade inflation, participation awards, and bogus trophies "to make low-achieving kids feel better about their lack of achievement." In my opinion, all teachers should teach standard arithmetic and standard algorithms rather than reform math. 
(See Alert below)

Peter W. Cookson Jr., a sociologist, wrote about the future: "Teachers will need new pedagogies and curricula for their students that emphasize problem-solving, higher-order skills, access to machine intelligence, teamwork, and lifelong learning." It's the same old junk from progressive ideologues that failed in the past. Cookson seems to ignore the critical importance of content knowledge in long-term memory needed for problem-solving, learning, and innovation.

Cookson claims that "inquiry skills" will drive learning; however, inquiry learning and similar minimal guidance methods are inferior to explicit teaching, according to Kirschner, Sweller, and Clark, who equate minimal guidance with minimal learning. Also, Cookson stresses "skills without content," which is not possible. Critical thinking or problem-solving is domain-specific. You have to know some trig to solve trig problems. 

Cookson is in a fantasyland. He talks about preparing kids (the digital natives) for an upcoming innovation era, but we have been in a technological innovation era since the invention of the transistor in 1947. Before that, the V-2 rocket that launched the space age. Then, there was Einstein.

The progressive narratives are everywhere in education. Most reforms have failed to turn education around. (I know it will work; we just need more funding, more resources, more tech, more innovation, etc.) Also, teachers are trained in progressive schools of education, so many misconceptions hang around decade after decade. Old ideas that failed in the past are revived with fresh language such as reform math.  

Cognitive skills are important. "The cognitive skills of the labor force as measured by math and science scores are extremely important in an economic sense." There is a link between the scores in math and science, and economic growth says Eric A. Hanushek. In short, school quality impacts economic growth. Progressive ideologues talk a lot about school quality then downgrade knowledge and individual achievement. Knowledge and skills go together. (Quote from The 4% Solution)

The bottom line is that students need to upgrade their math skills to move forward. But, the progressive narrative from the National Council of Teachers of Mathematics (NCTM 1989) is for students to use calculators early on, including in kindergarten children. We are told by reformers that with calculators, the central role of learning standard arithmetic is no longer that important. WRONG! In contrast, Ian Stewart writes (The Story of Mathematics), "Without internalizing the basic operations of arithmetic, the whole of mathematics will be inaccessible to you ... You won't learn to think sensibly about numbers by relying on a calculator." The fact is that standard arithmetic (e.g., standard algorithms, auto recall of math facts, etc.) is not taught well in our schools. Kids don't learn basic arithmetic well because it is not taught well.

Alert, Alert, Alert...  
An "education" professor claims that teaching kids math discriminates against children of color. How stupid! Comment: "Why would anyone think that minorities would be less able to do math than anyone else?"

Rochelle Gutierrez, an education professor, not a mathematician, claims that teaching kids algebra and geometry discriminates against students of color and perpetuates white "unearned privilege." She is wrong! In contrast, I go into urban 4th-grade classrooms to teach little kids algebra basics once a week. Virtually 90% of the students are children of color; most are Hispanic. I volunteer as a guest algebra teacher to provide opportunities for young students that expand their knowledge of mathematics. Indeed, factual and procedural mathematical knowledge in long-term memory is an asset to any learner regardless of color. 

Gutierrez says her work is "scholarly," but how can that be if her premise is weaker than gravity and the so-called investigators are offering only their biased opinions. It is not the cognitive science of learning. "This white paper represents the impressions and insights of Sanchez, Kastberg, Tyminski, and Lischka, [principle] investigators (PI) ... The opinions included in this paper represent those of the PIs as of November 5, 2015." (Gutierrez used the wrong word for PI. She wrote the "principle investigators" rather than the "principal investigators.")

Students need higher-level math courses to expand their career choices later on. Some students may dislike math, but this attitude should not govern their education priorities. Saying algebra is discriminatory is like saying tests discriminate against those who do not study.

Unfortunately, the typical methods--discovery or inquiry math activities in small groups--hold kids back because the curriculum and instructional practices do not optimize the early mastery of standard arithmetic. The constructivist minimal guidance methods are inefficient compared to the explicit explaining of worked examples in math. Also, there is no substitute for drill-to-develop-skill. 

Unfortunately, the minimal guidance methods are the same practices that progressive reformers, policymakers, and professors in schools of education advocate. Consequently, kids don't learn basic arithmetic well because it is not taught well.

Often, classroom teachers are asked to teach "items on the state test" using inferior methods. The state standards are Common Core rebranded. 

©2017 LT/ThinkAlgebra
October 25, 2017. 10-28-17, 10-30-17