To help young kids who struggle with math, well-intentioned teachers often turn to non-traditional teaching methods. They use music and movement to involve the whole body. They use hands-on materials such as popsicle sticks to help the students understand tens and hundreds. Or they encourage students to come up with different strategies for solving 7 + 8. One complicated way could be starting with 10 + 10 and then taking 3 away (because 7 is 3 less than 10) and then taking 2 away (because 8 is 2 less than the other 10). After many steps, the right answer emerges. And the students came up with it themselves. Good teaching, right?

Wrong.

A new study concludes that those first-graders who are behind their peers would have learned more if their teachers had just taught them to add and subtract the old-fashioned way. And then practiced it a lot.

The study, Which Instructional Practices Most Help First-Grade Students With and Without Mathematics Difficulties?, was published June 26, 2014 in *Educational Evaluation and Policy Analysis*, a peer-reviewed journal of the American Educational Research Association.

Average and above-average students learn about as much with either the innovative or traditional approaches. It doesn’t much matter. But any random classroom is likely to have some strugglers in it; for them, the researcher conclude, traditional, teacher-directed instruction generally yields better results.

The researchers, led by Paul L. Morgan at Pennsylvania State University, analyzed U.S. Department of Education data from about 14,000 students across the United States who entered kindergarten in 1998. They first looked at how the students performed on math tests in kindergarten. The data included teacher surveys, allowing the researchers to track the methods that the kids’ subsequent first-grade teachers said they used. And finally, they had the students’ first-grade math scores.

The researchers found that the higher the number of struggling students, who scored in the bottom 15 percent in kindergarten, in a first-grade teacher’s classroom, the more likely the teachers were to use manipulatives (hands-on materials), calculators, music and movement (See Table 3 on page 12 in the study). The fewer the struggling students, the more likely that teachers stuck with traditional methods, such as showing the whole class how to solve something one way from the chalkboard and then having students practice the method using worksheets.

Yet, at the end of first grade, the researchers found that struggling students who were given traditional instruction posted significantly higher math score gains than the struggling students who had been taught by the progressive methods. Gains are measured by how much students math scores rose between kindergarten and the end of first grade. (See Table 5 on page 15 in the study.)

“Routine practice is the strongest educational practice that teachers can use in their classroom to promote achievement gains,” Morgan said.

Although many educators dismiss rote learning as both boring and bad, Morgan believes it has its place. “Given my interest in children at risk, it’s a troubling observation that teachers are mismatching their instruction to what children with learning difficulties might benefit from,” Morgan said. “These kids with low math achievement in kindergarten are likely to struggle throughout elementary school and beyond. These kids are really at risk.”

Understanding why the more innovative methods didn’t work spectacularly is a matter of conjecture. Morgan theorizes that, just as children need to practice reading a lot and become fluent readers before they can analyze texts, math students need to become fluent with basic operations before they can talk about multiple methods for solving problems or arrive at deep conceptual understandings. “Maybe children with learning difficulties need more practice,” he said.

Innovative methods can also be more challenging to implement properly, and it could be that many teachers aren’t doing them right. It’s not easy to facilitate a math discussion. Six-year-olds are prone to goof around and stick popsicle sticks in their ears, taking away from precious teaching time. Instructional time can be lost while a teacher is setting up a musical lesson.

Does this mean we should all be drilling our first graders with Kumon worksheets? Morgan says not. “I don’t want kids going to school and doing worksheets all day. We all want kids to view mathematics as something that’s interesting and engaging and useful,” he said. “At the same time, we don’t want to be providing instruction to kids that doesn’t have empirical evidence that it’s effective.”

Then what’s a teacher to do? I’ll leave it to others to figure out how to make routine methods engaging.

## Comments & Trackbacks (34) | Post a Comment

at 1:21 pm

As a math teacher for the last 1o years, I see this discussion of hands-on learning vs. rote instruction to miss the point. Hands-on learning has to be a major part of the picture, but it must be thoughtfully organized. Rote instruction, however, is an extremely inefficient and lazy go-to for teaching such things as math facts. A better approach would include context and pattern searches. The more context and pattern identification present in the important practice side of the learning process, the more complexity there will be to the neural pathways that form for students. This provides more ways for the math facts to “stick” because there are more way to access the new learning.

Please avoid such false-dichotomies as this rote vs. hands-on.

Best,

Rodney Ward

at 2:33 pm

It is an oxymoron to talk about struggling first graders. Children who are in first grade are just beginning the process of learning. I will see you research study and raise you my 40 years plus of teaching to say I seriously doubt that the success you see has anything to do with learning mathematics. All young children can be taught to be “little performing puppies”. You drill them enough on something they can repeat it, but do they really understand what they are doing and can they apply it to new situations. When you take away the tools you are condemning the children to never being more than repeaters of rote learning. You are not teaching them how to think. Children learn best when they interact with their environment and use their vast pattern interpreting skills to make sense of what they see and notice. I thoroughly shake my finger at you. In the math brief this morning there was an article that just followed yours…a study by Stanford school of Education that proves my point at the higher grades…”Test scores improve for students who take more math classes, but the gains are temporary, according to research from Stanford University’s Center for Education Policy Analysis. Researchers found sixth-graders who took an extra math class in a Florida school district performed no better than those who hadn’t by the time they reached high school. ” Drill and Kill does only that.

at 2:41 pm

I find this article concerning. I read the original research published by Dr. Morgan and his findings were sound and important; however it did not advocate the polarization of drill and practice verses conceptual instruction that you allude to in your review of his work. Please read the limitations he listed and pay attention to the data he analyzed (this was collected in 1998- 16 years ago). Your interpretation of his report makes me wonder if your purpose in using it is to arm anti-common core and back to basics advocacy groups with misinterpreted information that supports their cause. An interesting article was published by Fuchs et al (2014) that investigated working memory and conceptual instruction verses drill and fluency work with fractions. This may be of interest in order to have a further look at the importance of conceptual learning. Furthermore, as advocates of education in the United States, all of us must be cautious about interpreting information based on statistical analysis as full truth and avoid political cesspools. We are dealing with children and their futures, and it is best to stay focused on actual truth and avoid potential bias. I receive email updates from this organization and truthfully, this article causes me to question the assumptions being made by your organization. As a teacher of 24 years and a doctoral student in mathematics education, I am interested in information that aids the education of our children; not the machinations of policy makers that have most likely not spent time trying to help students learn mathematics. Thank you for considering a different perspective.

Respectfully,

Claudia Bertolone-Smith

at 9:55 am

@Claudia. I appreciate your thoughts. But I hope my story would not give ammunition to either side of the Common Core debate. I actually asked Prof. Morgan if he thought his findings had implications for the first grade Common Core standards, and we both agreed that Common Core specifies only WHAT students should know, not HOW you should teach it. I would think a teacher could teach a Common Core aligned class using any of the methods analyzed in this study: teacher-directed (what I call traditional in my piece), student-directed, manipulatives/calculators and movement/music.

at 12:46 pm

Hear! Hear!

at 2:48 pm

The Morgan study will be helpful to disseminate the message that practice is important for all learners. Education is the ONLY human endeavor in which practice is denigrated by terms like “drill and kill”. Even your next to last paragraph takes a not-so-subtle swipe at practice. Appropriate practice with feedback is the only way that humans can learn. Maybe you can help to spread the word.

MM

at 2:50 pm

But they are only measuring the children’s so-called math ability by test scores? This seems highly unreliable especially because they are 5 and 6 years old. Children at this age have a great deal of trouble understanding abstract mathematics. It would make more sense to assess what kids understand about mathematics later (and w/o standardized tests). Most schools have always and still rely on giving young children abstract math instruction and it hasn’t created a U.S. population of people who understand math and science.

at 9:49 am

It would be fascinating to track these children’s math scores through high school. These 1998 kindergarteners are now in high school. I wonder if the direct-instructed students are still doing as well.

at 3:19 pm

I had the opportunity to work with first graders this past school year and what research says is true. Too much traditional in math is not good, because first graders need lots of teacher-directed instruction. I saw teachers integrating manipulatives and small groups, but the students were not staying on task. When children are on the rug in close proximity of the teacher, and the teacher is giving direct instruction, students can soak up like a sponge. They can ask and answer questions about the skill; then go back to their seats and demonstrate what has been taught. What is the teacher doing? He/she is monitoring, facilitating, kid-watching, and seeing who needs extra help for a possible small group or remediation session. I could go on and on. I will share this article with the first grade teachers. Great information!!!

at 4:35 pm

As and educator for more than 25 years, I enjoyed reading this article and will be sharing it with my 1st grade teachers.

First grade is one of the most important grade levels in elementary school. As teachers we try everything we can to get our students to learn and yes we may use other curriculum like music to help students learn mathematics skills, but you are right that we should not just rely on those strategies alone.

The most important thing to remember is that students need to have a relevance to what they are learning or as adults say, “the why or reason for learning the skills.” Once the students understand the importance and relevance, they will place more attention and focus on mastering the skill. Student no longer will accept the, “because I’m the teacher and you need to learn this.”

The students of today, learn so much different than in the past 20 years. Any new strategies or approaches should be welcomed and shared with everyone. Thank you for sharing.

at 4:49 pm

This isn’t new. The US Department of Education has been studying this in “Project Follow Through” since the 1960’s. Good luck trying to find the results, though. The conclusion that students do best with direct instruction, rather than any form of discovery method isn’t what those “on high” wanted to hear.

at 9:45 am

Thank you for reminding me of this long-history of the debate over direct instruction. Here’s a link to a 1995 journal, Effective Practices, devoted to the aftermath of Project Follow Through.

at 5:01 pm

As a first grade teacher I use Drills to Thrill or Drill Donuts. This method designed by Kim Sutton can be used for multiplication as well. Most importantly, children of all ages love Drill Donuts and can’t wait to do them. It is totally rare to see 24 1st graders all engaged in their work. Kim Sutton is the best.

at 9:31 am

Thank you for sharing this idea for making routine practice fun.

at 10:15 pm

The first three paragraphs greatly misrepresent the findings of this research. Also, using the terms “innovative” approaches, “traditional” approaches, “rote learning,” and “progressive” methods were not used in the article. The author(s) use the terms “teacher-directed” and “student-directed.”

In addition your seventh paragraph (starting “The researchers found that the higher the number of struggling students …”) misrepresents the findings. The researchers did not find the percentage of students with Mathematical Difficulties in a classroom to be correlated with the “teacher-directed” or “student-directed” factors overall (see the Table 3 you reference, page 10, and the Discussion on page 16). Two specific factor groupings were more frequent with high percentages of MD students.

However, the statement,

“The fewer the struggling students, the more likely that teachers stuck with traditional methods, such as showing the whole class how to solve something one way from the chalkboard and then having students practice the method using worksheets.”

is not based on and misrepresents the findings of the article.

I find the original article and the results very interesting, but I would advise caution in making generalizations about the findings from research that relies heavily on survey data and lots of regression analysis.

at 1:01 pm

Really?! Yes, simply reciting the words “two plus two equals four” is obviously not be enough, but that’s not the point. When second graders are being taught methods for solving trivial arithmetic (2+2, 5+9) that the take more time and paper than a Differential Equation student would use when solving a “Wronskian”, something is seriously flawed.

We have idiots administering education systems that force methods on teachers to feed a system whose sole purpose is to spend as much money as possible making as many C students as possible.

———

LHWalker

July 22, 2014

at 11:15 pm

A study that classifies a calculator as a manipulative along side of tens sticks is seriously flawed. Obviously repetition helps students learn, but simply reciting the words “two plus two equals four” is not synonymous with learning math and attaining a sense of how numbers work together.

at 11:15 pm

A study that classifies a calculator as a manipulative along side of tens sticks is seriously flawed. Obviously repetition helps students learn, but simply reciting the words “two plus two equals four” is not synonymous with learning math and attaining a sense of how numbers work together.

at 9:30 am

Thank you for your comment. I had the same initial reaction, why would you lump hands-on materials together with calculators? Perhaps manipulatives are effective, but calculators aren’t and the calculator negative effect is drowning out the positive manipulative effect? When I posed this question to Prof. Morgan, he said that teachers who reported using manipulatives also tended to say that they were using calculators. The manipulative-calculator data lumped together on its own. Morgan used a statistical technique called factor analysis. My recollection of how this works (from my own grad school stats classes) is that you put all the teachers’ methods in a big metaphorical salad spinner, you start turning and see how the data clumps together. If a high percentage of teachers who say they teach one explicit method from the chalkboard also say that they use worksheets to reinforce the technique, that’s another data clump. Still, I am surprised that so many first grade teachers are using calculators.

at 11:35 pm

I can’t imagine moving to rote instruction if a child does not have an understanding of a concept. In my experience of teaching all of the primary grades, some students need more time at the conceptual level of mathematical topics. When they don’t get it, their mathematical wall has a crumbly base and is weak from that time forward. For most students who are struggling, it is found that they are lacking in number sense which is developed through many experiences with numbers. (It’s hard to believe that providing students with manipulatives is still considered non-traditional methods.) Once a child understands a concept, then practice is in order. If more time with number sense is skipped and substituted with rote learning in order for struggling students to “know” their facts, I fear that their lack of number sense will affect future math work. Too often I have been told that one of my struggling 3rd grade math students knew their facts at the end of first grade. Sadly, without number sense or knowledge of strategies (i.e., 7+4 can be thought of as 7+3 +1 more for the student who has the concept of “tenness”). students have a huge anchor weighing them down as they progress on. I truly believe that our students who struggle early in math need more time with a concept than we are able to give. Giving them that time is not easy, but it is an investment in their mathematical journey.

at 9:21 am

Thank you for your comment. I am fascinated with this chicken-egg debate about whether conceptual understanding needs to precede repetition/practice or whether conceptual understanding comes after the blind memorization of certain facts. I suspect we have all had learning experiences where sometimes we understood the concept first, but other times we didn’t understand the concept well until we could easily perform the procedural operations. Back to first graders and number sense. I wonder if developing a rich number sense involves a certain amount of rote-like repetition, e.g. rolling die many times until a student sees that seven can be 1+6, 2+5 or 3+4 instantly. Of course, that would be using a manipulative, which the study also debunks!

at 7:44 pm

As the editor who worked with Jill Barshay on this column, I’m fascinated to read these enlightening responses. I’d agree with those who say this need not be an “either-or” situation in math instruction, and that Dr. Morgan, in his study, didn’t depict drills vs conceptual methods as polar opposites.

Rodney Ward’s point about forming neural pathways (the first comment) is a very good one, and reinforces the old-fashioned idea of “practice makes perfect,” which is also expressed in other comments. And, of course, Lena Green, Theresa Doerfler and others note, correctly that the teacher’s role and skills are of paramount importance. Thanks, all, for your comments!

at 1:49 pm

By coincidence, What Works Clearinghouse, a division of the US Department of Education, sent out a “newsflash” on July 23, highlighting its evidence-based methods for teaching math to K-5 students. ( http://ies.ed.gov/ncee/wwc/mathhome.aspx). When you click through to “assisting struggling students”, it also advises explicit, direct instruction and practice. http://ies.ed.gov/ncee/wwc/pdf/practice_guides/rti_math_pg_042109.pdf#page=27

at 12:29 pm

As an early childhood educator, I also find this article extremely disturbing. Basically what it is telling us is that if a child cannot meet the standards that are too high for most children of that age, then we should “drill and grill” them until they do. This type of teaching turns children off to learning and sets the stage for a life long distaste and struggle with future school experiences. The research is already there! There is no need to reinvent the wheel.

Young children learn BEST through hands-on, exploitative learning experiences! Period! Please don’t try to change what we already know about child development to meet the needs of those that designed developmentally INappropriate learning standards in the first place and corporations that are making a mint off of tests and tutoring to support their lies. This is a tragedy and the emotional and creative lives of children are suffering because of it. Just take a peek inside most kindergarten and/or 1st grade classes today and count how many children are lost, anxious and acting out because of extreme frustration. Stop the lies that only support corporation’s pockets, and get back to what we know works best for young CHILDREN!

at 12:04 pm

I’m not sure when it needs to happen, but some rote learning is so necessary. I teach middle school and do not need a study to tell me that the kids who have not mastered basic math facts to the point of automaticity hate math and struggle mightily. Calculators do not help. Instead of dealing with higher level thought process to solve problems, their brains are busy working out a simple math fact that they should have just known.

Another thing I’ve noticed is a lack of an overall number sense–the realization that an answer can’t be correct because it’s not logical. I’m not sure whether rote or inquiry would help with that, but it needs to be addressed.

And finally, no one should be allowed to teach elementary who claims to “not be a math brain,” or confesses, “I’ve never been good at math” as if some people have math abilities and others don’t. The fundamental skills and understandings as well as attitudes about math are developed in elementary and it’s unconscionable for a teacher to negatively impact math development at this critical time in a child’s life.

at 11:54 am

I think we confuse arithmetic (number facility) with mathematical reasoning to every learner’s detriment. Both should be taught in creative ways – but we teach arithmetic with the belief that it leads to mathematical reasoning. That is bewildering to me.

What we consider to be good math students are those that are good at procedure not mathematical thinking – with a few exceptions we call “genius.” Its left to chance, ironically, on who really gets it…

Also, we neglect offering a meaningful connection to numbers and counting and math by not sharing the rich history of their invention.

at 10:19 am

I have been at training sessions with Math Educators from Penn State. While there is a beneficial promotion of the C-R-A approach (which is a blend of “progressive” and “traditional”), there is an emphasis on number fluency that becomes merely rote. The idea that students have limited working memory is sound. But the idea that they therefore must have EVERY fact memorized to automaticity is ludicrous. Having an understanding of how numbers work and fluid strategies for “digging up” a few “elusive” number facts does not detract from the learning at hand.

Additionally, we cannot limit “basic skills” to first grade. What is a basic skill? We could argue that they continue through calculus and beyond.

at 2:43 pm

It is so great to see all the arguments against this articles assumptions about teaching learning and struggling students! Here are my two-cents: I find this argument to mirror the pushback against using hands on and the use of manipulatives in math. “Innovative” methods that ask children to analyze problems too early are not innovative. They come from teachers’ misunderstanding of the innovative methods they have been advised to teach. Also there is a false dichotomy between direct teaching and innovative. Teachers who are good teachers ALL use methods that help kids think about the foundational concepts involved in the mathematics being taught, and they provide intellectually engaging support to the children no matter where they are conceptually. If a child “struggles” the effective teacher knows how to modify instruction to better meet the child where he or she is conceptually, not just fall back on memorization. Unfortunately it is our children most in need of opportunities to think and make decisions that get targeted for these misguided and misinformed teaching methods that abandon all thinking for rote instruction.

at 4:37 am

I love the comment made about helping (young minds) learn about math being more than numbers but more about problem solving. Involving minds in the invention of math may help. To me I view numbers as a different set of letters used as part of a language to describe things in a way where categories can be made sense of using our senses such as eyes, ears, mouth …. Therefore, manipulates such as digits of the hands and feet are naturally perfect to start with (who hasn’t had their fingers counted by a parent when growing up) , as are sounds made on a piano scale of 7 tones and 12 semitones, degrees of a circle 360, geometrical shapes using triangles as the basic unit of observation, colours (3 primary) , plus shades and tones of infinite qualities, tastes(I’m sorry but this realm of our senses has yet to be quantified in a standard way).

My point is this. Math is about understanding patterns. If we can engage the student about the process of discovering and inventing their own patterns. Show them the richness of their environment using numbers and words. Then introducing some well known patterns that have become the building blocks of Math, then giving them exercises to reinforce and memorize larger and more complex concepts, they will quite willingly want to move forward to explain and predict parts of the environment in which they live in.

Math and language development ought to progress at the same time in order for the (mind) to have mobility in it’s understanding and expression of this understanding by communicating this understanding to others both in colourful language or complex math.

I totally agree that a reasonable amount of fluency with numbers should help students learn higher concepts more easily, and too much emphasis on precision of facts will quickly lead to boredom and negative feelings.

How boring it would be to have to read a story like the three pigs , if you had to read each word perfectly before going to the next page.

at 12:51 pm

Using websites such is http://www.silvermath.com definitely can make this job more interesting and more easier.

at 1:28 pm

I just discovered this website. I’m a retired woman volunteering with 2nd graders and looking for ways to teach basic math facts. I’d like to propose a possible explanation for why some kids seem to be learning the math facts from non-traditional methods.

They aren’t, they’re learning it at home by conventional methods. And that’s the difference between those kids and the other kids.

Maybe?

at 9:45 pm

My son attends a new charter school in town. He is struggling in math, which is partially my fault and the school he attended for second grade. In second grade I did not know he needed to be learning his multiplication tables, so it set him back. He got caught up. The charter school uses khan academy for math, which is fine with me. The math teacher picks different math subjects to work on. Then the children get on and randomly select what they want to work on from the chosen list. My problem is that he doesn’t teach the kids first. He let’s them jump on and if they have problems he will try to explain. Third grade math on KA isn’t easy. The wore problems are not simple at all. I Dont mind my son being challenged, but the teaching style is not working for him. I have expressed my opinions before but the teacher wants to pull him out of Spanish to work with him one on one. I disagree with that. I think the teacher should explain the subjects on a whole, then let the class roam free on KA. I teach him at home but its difficult. I have no problem doing that but the teacher sends me text and let’s me know what my son needs to work on. He is not failing math but he lives the subject and I think he is getting discouraged because he feels like he is not advancing in the subject. What should I do and should I keep pushing the teacher about his teaching style? Not all students can just jump into the subject and do good. I’m getting angry and I Dont want to be “that parent.” Oh and they Dont give homework because they are in school so long.

at 2:52 pm

I don’t think that Dr. Morgan is implying that drilling and rote memorization are better than hands on concepts- but for struggling kids rote memorization may help increase their capacity for learning, whereas, some kids may not need that extra jog?! By timing things we are constantly measuring and it may not actually mean that children aren’t learning what they are being taught. It’s very important to make this distinction: just because someone can’t grasp certain concepts by a certain age and respond quickly doesn’t mean they are doomed to a life of servitude! And homework, as much as parents complain about it, isn’t gonna kill anyone and if it’s not in excess of 2 hours per night you’ll all be fine. That’s the way I see it. I have a struggling student who is very slow to learn via common core, probably, I believe, because she needs to really get some good memory practice and it will enhance her ability to grasp the concepts they are trying to teach. And I always remind myself how much time children are at school versus at home- many people forget this and may fill up their kids schedules with activities and social events and not recognize that the role we have as parents is crucial! We can’t expect schools or one method of teaching to work out perfectly. Common Core is good and it’s bad. But there are things we can to do preempt the crisis that it has created- we can find the time to enhance our kids education by attending museums, watching educational programming, go to zoos, seeing science exhibits or visiting planetariums. These are all knowledge enhance our creativity and curiosity. Don’t leave it to the schools!

at 3:15 pm

As I read this article, I found myself wondering if the writer has ever been in a first grade classroom. I’ve taught first graders for 20 years, and have swung back and forth with the pendulum. The experts tell us that we need more hands-on, so we teach hands on. Then the experts tell us that we need more rote memorization. So we teach rote memorization. It’s been my experience that every child learns differently. We need to show them different strategies for solving problems, and let them decide what works best for them, with guidance from us, of course. I think the problem that has been seen with this type of teaching is that many teachers put it out there and expect the kids to learn it on their own. That’s not going to work. But if you work with the child and see what strategies they pick up on, and build on that, the kids are more likely to understand what they’re doing. Rote memorization is great for fluency and basic facts, but kids who just memorize often have problems understanding why the answer is correct. Then when they get into higher grades, they have problems because they can’t figure out harder problems. I teach math to first graders using different hands-on strategies, while expecting rote memorization to make them more fluent. It’s time to stop teaching methods and start teaching students.

at 11:54 am

i am a grand-mom. i was in gifted all my life. my daughter struggled in school because she was taught lessons geared toward audio-visual learning & what i call the 2d method=chalkboard, the public school systems norm. she however was more of a tactile-kinetic 3d learner i discovered by allowing her to teach me. now along comes my grand-daughter she likewise is a tactile-kinetic 3d learner. what i discovered was when i was growing up i was first taught the names, definitions and functions of math equations, followed by demonstrations or math in practice, later manipulative, rote drills and etc. the problem with so-called math experts is they know math like their names & assumes everyone else just gets it. whereas the majority of people are literal/language brained v/s math brained. teachers jump in and start doing math w/o explanation. numbers are arbitrary symbols like letters. meaning el gato means cat in spanish but in english the cat. show either to the speaker of that language only and they wouldn’t have a clue show the picture of a cat and they both will understand. likewise i was taught the equation and it’s parts name, function/purpose and the relationship between the parts. so, irrespective of the number i understood what i was suppose to do. i also understood there where only basically two kinds of math functions adding and subtracting. division and multiplication was simply fast subtraction and fast adding. there were only two valuations/qualities to numbers positive or negative and 0. there where two kinds of numbers whole or parts therefore fractions/percents etc, i knew what and addend was, subtrahend, plus sign etc, etc first. then came all the other stuff. simple plain explanations of math and its’ basic components before all this place value, number sequencing and common core stuff. we are so busy teaching how many ways i can build a clock until the children can’t build the damn clock. my grand-daughter could do math when i taught her in kindergarten the above mentioned method, now she can’t remember & gets so worked up because she needs to solve 25 +6= 31 by first taking 5 from six add to 25 then add 1. To prove that she knows how to reason,. which somehow shows she’d know how to solve any math problem. she no longer can due math, is being timed and is frustrated. for some the old fashioned/traditional way is better. or at least a meeting in the middle. but most of all we are not cookie cutter beings and there should be allowances made/ rather consideration for the different style learners, instead of labeling them as disabled learners they are different styled learners. as i say anyone off the street can teach the so-called smart kid, but a real teacher is one who can teach the so-called slow kid. lastly, we should make sure the children understand the terminology of math. decompose and recompose. i remember a lot of children did not understand the term variable instead of teachers explaining the number is not actually a variable. we call the number we are looking for the variable because we don’t know what number it is until we solve or work the problem. but we are trying to discover/figure out what is the second addend for instance if it is an addition problem. example 6+_= 8. when i simplified the explanation and in turn told the children to use a word or explanation of what a variable was to them they could do the problem more readily. math terms and symbols need to be taught, studied and drilled first before kids should be required to do math. we can’t just assume kids just get it because it’s simple to us. it wouldn’t hurt to give spelling like test on symbols and math terms to insure the children get the basics before diving into math concepts and rationales first.