In a kindergarten classroom at Robinson Elementary in Fresno, Juh'Ziyah Atchinson leans in to help Aubrey Blancas, both 5, create a row of dominoes so they can practice counting. Credit: Lillian Mongeau, EdSource

In a kindergarten classroom at Robinson Elementary in Fresno, Juh’Ziyah Atchinson helps Aubrey Blancas, both 5, create a row of dominoes so they can practice counting. Credit: Lillian Mongeau, EdSource

Teaching elementary school math just got trickier, or at least deeper.

The new Common Core State Standards require students to demonstrate a deeper understanding of math concepts, which means teachers will have to change how they teach those concepts too.

The new standards, adopted in California and 44 other states, have ushered in a whole new set of academic standards for math, with significant changes in the early grades – kindergartners must now be able to count to 100 by the end of the year, for instance, rather than 30. Second graders will no longer learn multiplication tables; that’s now a third grade task. And geometry standards are now less about identifying and measuring shapes and more about building and deconstructing them.

Common Core standards for kindergarten through 3rd grade math also require students to demonstrate a greater depth of understanding than needed under the previous California State Standards, established in 1997. While the old standards were often criticized for an excessive reliance on memorizing certain facts or procedures, the new standards routinely call for students to solve problems that require a strong grasp of mathematical concepts and to explain their reasoning.

“Most people think math is computation at the elementary level – drilling them in the skills,” said Jeanie Behrend, an education professor focused on math education at California State University, Fresno. “Math is really about application and problem solving.”

Major challenge

Making this shift in the classroom – from focusing on computation to focusing on problem solving strategies – stands to be one of the biggest challenges teachers face as they work to implement the new standards, Behrend said. Students will take practice tests aligned to the new standards this year in preparation for the first statewide assessment tied to Common Core in spring 2015. While some teachers are ready to tackle the new standards, others said they have received scant training in how to teach math as a hands-on, exploration based activity.

“It’s easy (for teachers) to focus on memorization of facts and memorization of procedures without really identifying the important mathematics” behind them and teaching those concepts to students, Behrend said.

Memorization has never been the best way to learn math, Behrend said, but it was often enough to meet many of the old standards.

For example, California’s 1st grade students have long been required to “commit to memory” all the ways of adding two numbers to come up with sums between two and 20.  While the Common Core standards do require 1st grade students be able to solve addition problems with sums between two and 20, the requirement that they commit the equations to memory has been moved to the 2nd grade. Students can now use multiple strategies – drawing a picture or using building blocks as counting aides, for example – to find their answers and demonstrate proficiency.

Strategies based on the theoretical underpinnings of mathematics are also emphasized in the new standards. First grade students may not have to memorize as many addition equations, but now they must be able to understand and demonstrate the use of the commutative property of addition, that is:  6 + 2 = 2 + 6. Previously, this idea was not introduced until 2nd grade.

(See chart below comparing the old California State Standards to the new Common Core Standards.)

By removing memorization standards and requiring teachers to cover fewer topics over the course of a year, the new standards are also meant to encourage teachers to spend more time on the underlying concepts behind mathematical concepts.

Changing how math is taught in California elementary schools could be critical to student success. California 4th graders now rank 46th in the nation in math, with only 33 percent considered proficient, according to scores on the 2013 National Assessment of Educational Progress.

Teaching the teachers

The shift in what students need to know is requiring teachers to learn new concepts, as well.

In teacher Jenny Aguirre’s 1st grade classroom at Ayer Elementary in Fresno, students were learning about addition during a class at the start of the school year. Aguirre has taught addition many times in her 12 years as an elementary school teacher, but she was going at it a bit differently this year.

Each of Aguirre’s students had a yellow laminated mat and eight poker chips. The number eight was written at the top of the specially designed mat, followed by space for an equation and two equally sized boxes. The goal was to split the chips between the two boxes to find different ways of combining two numbers to make eight.

First grader Eliana Garcia, 6, works on an activity meant to teach her that there are many ways to make eight at Ayer Elementary School in Fresno. Credit: Lillian Mongeau, EdSource

First grader Eliana Garcia, 6, works on an activity teaching her various ways to add to eight at Ayer Elementary School in Fresno. Credit: Lillian Mongeau, EdSource

Seated toward the back of the room, Eliana Garcia, 6, put five chips in one box and placed the three that were left into the second box. Next, she wrote an equation to illustrate her finding: 5 + 3 = 8

The goal of the activity was to allow the 6- and 7-year-old students to discover an important property of arithmetic on their own. They knew the answer was always going to be eight because the number was written on every mat and they had eight chips. But they were learning that there are multiple ways to add up the chips to equal eight. Most students started by splitting the chips evenly into two groups of four, but quickly progressed to trying new combinations and flipping the order of their equations.

Leon Thao, 6, even discovered that 0 + 8 = 8.

Aguirre was so impressed by Leon’s work – a big milestone in the understanding of addition in 1st grade – that she asked him to come up to the front of the room and show his classmates how he’d figured out what happens when you add eight to zero.

In years past, Aguirre said, she would have started with traditional addition facts and left the lesson with the mats, which was not technically required under the old 1st grade standards, to later in the year. She also might not have asked students to work in pairs or spent time letting a student haltingly explain his mathematical reasoning. This year, she’s doing all of those things in an effort to comply with the new Common Core standards.

Aguirre said she’s excited about the new standards, but she’s not completely clear on how they will change what she’s expected to present in the classroom.

“There’s not (been) a lot of training on how to teach (Common Core) math or on what we should expect of the children,” she said in August. “This is just a total shift for us.”

Fresno Unified has since begun their effort to train teachers in Common Core by offering several district-wide trainings as well as more focused trainings for teachers in small groups.

Killing ‘drill and kill’

Third grade teacher Jaime Button enrolled in a three-year training program offered to teachers in her district by the Shasta County Office of Education, in the hopes of improving her math instruction.

Button, who teaches at Alta Mesa School in Redding, said she has felt pressured over the last decade to use a “drill and kill” method to prepare students for state standardized tests – drilling students repeatedly on specific skills without spending much time explaining the underlying concepts. This clashed with what she’d learned when she was training to be a teacher two decades ago, she said, but her school and district leaders wanted teachers to cover all of the 3rd grade material included under the California State Standards and she didn’t feel she had time to explore other methods. 

“The saddest thing is that (my students) could parrot back answers, but they didn’t have a real understanding” of the math, she said.

Button said she has been thrilled to find that the new training is encouraging her to return to a more hands-on way of teaching math, one in which students can puzzle out math problems using objects or drawings or by working in teams, rather than following along in a book or filling out a worksheet. She’s also pleased that she’ll be required to cover fewer topics, allowing her to spend more time on complex ideas.

In another Fresno elementary school, kindergarten teacher Joe Dawson worked with his students on counting in late August. He sat at a small table in his classroom at Robinson Elementary School watching his students try to count sets of 20 plastic bears he’d given each of them. Though some students counted the bears easily, others made all sorts of mistakes: counting some bears twice, skipping others, pointing to each bear but saying the numbers in the wrong order.

Juh’Ziyah Atchinson, 5, was one the students who couldn’t seem to get it right. Dawson, who has been trained in a method of teaching mathematics called cognitively guided instruction, was more interested in understanding why Juh’Ziyah kept counting bears twice than he was in correcting her. The idea behind this method is for teachers to build on a child’s existing knowledge about math to guide them to the correct answer, rather than quickly correcting them. Instead of simply telling Juh’Ziyah to only count each bear once, Dawson started a conversation with the little girl to determine if she even understood that basic counting rule.

Path of discovery

“I’m not an imparter of information,” Dawson said later. “I want my children to discover.”

Getting them to discover often means conversations where Dawson simply says out loud what he’s seen a child do – “I noticed you counted a few (bears) twice.” Observations like this are meant to make a child reflect on what she has just done and hopefully learn something new by being directed to focus on the actions called out by the teacher.

In Juh’Ziyah’s case, Dawson concludes that she does know objects can only be counted once but is having trouble keeping track of which bears she’s already counted. He shows her a tool to help with that: Organize the 20 bears into several short rows, rather than trying to count a messy jumble.

Kindergarten teacher Joe Dawson help Isaiah Aguliar, 5, count to 20 using little plastic bears at Robinson Elementary in Fresno. Credit: Lillian Mongeau, EdSource

Kindergarten teacher Joe Dawson help Isaiah Aguliar, 5, count to 20 using plastic bears at Robinson Elementary in Fresno. Credit: Lillian Mongeau, EdSource

Megan Franke, an education professor and teacher trainer focused on math education at the University of California, Los Angeles, said research shows that teaching math like Dawson does is more successful than teaching methods focused on memorizing right answers. The Common Core elementary school math standards, Franke said, are based on that research.

“The math practices (in Common Core) are asking kids to explain their mathematical thinking, to represent their ideas and to make sense of other people’s ideas,” Franke said. She hopes these principles encourage a shift in how math is taught that will ultimately change what children know about math.

Dawson is confident that teaching math based on these principles will help Juh’Ziyah and his other students meet the Common Core Standards. In previous years his students have exceeded the old state standards, he said.

By the week before Thanksgiving, his confidence proved warranted: Juh’Ziyah was not only counting to 100 by ones (i.e. 1, 2, 3…) and 10s (i.e. 10, 20, 30…), she was solving three-part word problems using subtraction.

“If Juh’Ziyah had 15 cookies and wanted to give three to one friend, three to another friend and two to a third friend, would she have enough?” Dawson said he asked the girl in November. “And if she had enough, how many would she have left?”

Juh’Ziyah, who hadn’t been able to count to 20 three months ago, was undaunted. Using small objects to represent the cookies, she nailed it: Yes, she’d have enough cookies to share, she told Dawson, and she’d have seven left.

Lillian Mongeau covers early childhood education. Contact her or follow her @lrmongeau. Subscribe to EdSource’s Early Learning RSS feed.

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  1. Paul says:

    Manuel, I’d encourage you to visit the Nueva School next time you’re in the Bay Area.

    One could say that this private school has been piloting the Common Core Stabdards for Mathematical Practice since the school’s founding in the late 1960s.

    The approach works. Very young children with widely different levels of math proficiency do just the sort of work that I described, and there is nothing mechanical about it. (A symbolic, pencil-and-paper approach *would* be mechanical.) Even for students who are just beginning to understand addition and subtraction, the use of familiar representations (blocks, kittens, verbal expressions, e.g. “a dozen eggs” or “the fingers on your hand”); the experimentation with grouping, adding, and removing items; and the dialogue with peers and adults; are all valuable. These elements provide a path to mastery, and an effective one at that. (Just as teaching the “subtraction property of equality” to an algebra student is not an effective path to mastery. It reduces math to symbolic manipulation, useful for someone like me, who studied logic as an undergraduate and built formal systems from first principles, but not useful for a 9th-grader.)

    I was fortunate to do a little teaching at Nueva, not just to observe. I brought an article about the polydactyl cats on Hemmingway’s estate, and our class of second-graders spent the afternoon finding various reasonable combinations of fingers and toes that added up to the likely numbers of fingers and toes for polydactyl cats. One thing that amazed me is that these students were truly in the habit of modeling. They used disparate counters (a mix of discs, blocks, beans, and other items) in such a way that they understood that even if two were not of the of the same color, size, or shape, they could still be aggregated. Conversely, when grouping was necessary, the students were able to create arbitrary groups, such as “the red ones”, or “the round ones”. Anyway, within an hour we had several students who were writing binomial expressions like the right-hand side of this equation: = ( +/- x ). (I don’t remember now whether front or back paws have more digits, but it’s not important.)

    To the commentator who talked about middle-grades math, you raise very important points. Are you familiar with the CPM curriculum? That is a long-standing example of Common Core-style principles in action in Grade 6 general math, Grade 7 general math, Algebra I and II, and Geometry. I’d love to hear more about how the integrated math program that you mention will be structured, i.e., about how topics from algebra, geometry, and statistics will be matched up in each year of the sequence. It could be a good or bad initiative, depending on the thought put into the design.

    1. Paul says:

      Oh dear, the parser removed my equation, because of the brackets. Here it is again, without bracketing:

      fingers on one paw = toes on one paw + or – x

    2. Manuel says:

      Thank you for the invite.

      It is possible that I may have come across as a total nay-sayer.

      I’d like to state that my discomfort is on how adults are arguing over the description of the standards and I have concerns on the likely improper implementation. As navigio says, there is plenty of harm if implemented wrong.

      Equally, I don’t doubt that some and maybe even many students will do well under the new system. My concern, I repeat, is that some teachers will see this as a new set of blinders. It will require administrators to keep an eye out for this.

      We should have this conversation again in at least six years and see if the promises have come true.

  2. Mike McMahon says:

    Old standards or new standards we need to teach math differently. In the video Why People Hate Math explains why:

    1. navigio says:

      Thanks for that. I’d argue that that video is not about teaching math differently, rather its about the need for a cultural paradigm shift that prioritizes recognizing the wonder of nature and thought. Our culture does not do much of that, whether in math or philosophy or history or art or, or, or..

  3. H. Woods says:

    It used to be that the methods being suggested in the Comon Core standards was called multi modal learning, allowing each child to learn from all medalities (seeing, hearing,touching etc.). In K and grade 1 I used math manipulatives. When my students understood what they had seen, felt, and heard as they minipulated material of different kinds, they went on to higher grades understanding the concepts. They were able to calculate and solve problems through visualizing or recalling their other early learning from past math experiences. Using such hands on learning in K and 1st grades had excellent results and as an aside, increased self confidence where math was concerned.

  4. Joe says:

    I have read all the comments from these articles and found them very enlightening. Although I don’t consider myself and expert, I have a few thoughts. First, after teaching for many years, not once did I see the memorization of facts lead to deeper understanding in math. Conversely, I have seen multiple times it leading to confusion and misconceptions. I’m not saying that automaticity is not important, it’s just not foundational. Building the understanding of number and how it works is the foundational. The debate on methodology will never cease. However, I can speak from experience as one who has taught using both approaches. I have found that my students who have been allowed to “discover” mathematical principals and apply them in new situations have far exceeded those in previous years that just imparted information. In my kindergarten classroom, I now have students that are not only reaching for automaticity in addition, but are now working on understanding the commutative property. There are even some that are now starting to have the beginning conceptual understanding of multiplication. The students may know an addition fact, but more importantly explain what it mean. However, it is all meaningless unless they can apply it in new situations-which they can do.

    My Second thought is the education is not finite. That means as a K teacher, I am not limited by the standards that are assigned to me. I view my job more as taking a student form where they are to where they are going. Sound nebulous, but I am not bound to keep a student in kindergarten standards if they are ready higher standards. The idea that learning material of a particular grade is confined to that grade is not teaching. I think most teachers, if not all, would agree with that. “Sorry Johnny. I know you are ready for third grade math, but you are in second grade. I can only teach you second grade math.”

    My final thought, I think, There are standards for automaticity in kindergarten. The State of California has never had those type of standards in kindergarten. I remember when I started teaching this grade 28 years ago. The standard was only having the students know and count to 12. We have come along since then. I would venture to guess that if I gave my current students the assessment from 28 years ago they would “ace the test.” The whole point here is that standards that are established are high throughout the grades. Students are learning today in elementary school what we all were learning in middle school. If you look at other nations in the world that we would like to be like (Japan) they are using the very methods that are being advocated in CCSS.

    Finally, the need for divergent thinking in the business place is paramount. CCSS was established in part as a result of CEOs saying that students are coming out without the ability to think and solve problems. They can tell you that 2+3=5, but they couldn’t design or create. As a result businesses are going to countries that have people who can and hiring them. If the education of a child is to ultimately make them career ready, we need to start changing and adapting they way we teach to better prepare students.

    1. Paul Muench says:

      My view of public education’s impact on business is different. Fundamentally business moved to other countries because skills were available at a much lower price. And from the education side that’s because students in other countries see STEM careers as a way to a better life. It’s not clear to me that students in other countries are any more impassioned about STEM, but they are willing to be a lot more disciplined about learnng it. So perhaps the Common Core standards will inspire teachers to teach math in a more enganging way and that will influence more students to pursue STEM careers. Although that won’t necessarily change the cost equation so its hard to predict any systematic change. But at least some students may be relieved from dreadful math instruction,.

      1. navigio says:

        The low cost of labor elsewhere is a function of historical economics and mostly has zero to do with those cultures’ desire to pursue STEM fields. That will have an impact someday, or maybe is even already starting to, but its not the basis for those low costs, and it will also be countered by increasing standard of living.

        I also think the divergent thinking take on common core is mistaken. CC seems to be more analytical, which is generally considered pretty much exactly the opposite of divergent thinking. There may be ways to foster divergent thinking within any educational environment, but I dont think CC inheres that in any way (and on its face appears to even contradict it–this is one of its real dangers if done ‘wrong’).

        Its probably heretical, but i think this fad focus on divergent thinking is hokey. Divergent thinking is great, but its not everything. Furthermore, it arguably ironically works best in conjunction with convergent thinking, and maybe more importantly, an exclusively divergent thinking group is likely to be less productive than an alternative. I dont think any CEOs are interested in less productivity. And to the extent they are interested in creativity, its generally for a small subset of their workforce (yes there are exceptions, but they are just that).

        I’ll go out even further on the limb and claim that everyone is a divergent thinker, but that it is stifled or does not have a chance to fully develop when people’s world views are limited. I think this is one of the side-effects of poverty concentrations and high-stress neighborhoods. Its one reason art and field trips and diversity are so important. Its one reason going outside, participating in community events, being with other people and even playing sports are also important. If its even true that CEOs are saying things like this, then I think CC itself will never address their concerns and in fact, might even make things worse.

        Our saving grace may be that parents and teachers recognize this. Given the opportunities, they will foster these things in children. Not given the opportunities, they will have a much harder time doing that, or will not be able to at all.

        Finally, I think any discussion of common core is misguided unless it includes the distinction between standards and curriculum and frameworks and testing and whatever else gets brought into the mix. Sometimes those distinctions mean everything.

        1. Paul Muench says:

          Skilled people have to be available before they can leverage historical factors to lower the cost of labor. I’m not sure how to assign a precise percentage to education vs. historical factors, but my general sense is that the educational factors are far from zero.

          1. navigio says:

            Historical factors are not leveraged, they pre-exist. To put it another way, skilled people can exist in societies where the cost of living is a mere fraction of what it is here. The fact that those people became skilled is a function of education, however, the fact that the cost of living is so much lower is not a function of education–it pre-exists. It is true that when you have a glut of skilled labor, that will eventually drive down cost, but that is not what caused the historical low cost of living in those areas being exploited now. (though it will begin to have an impact on costs at some point). Not sure that is any clearer. :-)

            1. Paul Muench says:

              It’s not clearer. Seems you are talking about cost-of-living and I am talking about cost-of-labor. Cost-of-labor is a more global measure whereas the cost-of-living is a more local measure. So the overall statement I was making is that cost-of-labor depends upon both the cost-of-living and education. I chose not to delve into the reasons behind cost-of-living. But I do think that STEM education is seen as a much greater opportunity in countries other than the United States. And going by Amanda Ripley’s recent book, students in the United States just don’t want to work hard at something to get a job, they want to do something because they love it. So maybe Common Core can inspire more students to love math and pursue STEM careers. We’ll see what happens.

              1. navigio says:

                Ok, yes, that makes sense. Especially this sentence: “So the overall statement I was making is that cost-of-labor depends upon both the cost-of-living and education.

                My statement was merely to express that I believe cost of living has a much higher impact on global cost of labor than education does. This is because education (or experience) is something that remains more or less constant within a profession, whereas cost of living can vary wildly. I could move to cambodia and demand, say, a tenth of what I’d make here and still live similarly, for example. Alternatively, if I want to do work that doesn’t require a local presence then I have to compete against people with the same skills who happen to live in Cambodia, for example.

                Anyway, enough confusing people, sorry. I also hope that CC brings something positive. But I would point out that it only covers Math from the STEM part. That said, there are other efforts running parallel to CC implementation that I think will have more impact on STEM desire than the standards themselves will.

                And finally, its possible to love working hard. :-)

  5. SDParent says:

    As a STEM graduate and someone who learned early Math using the Golden Beads in Montessori school, I absolutely see value in adopting a deeper understanding of Math, and I’ve tried to foster that with my children. However, there seems to be so much focus at the state and district level on CCSS instruction in elementary school and very little about what to do in middle and high school. And while I don’t doubt that in the end, CCSS instruction will be done effectively and be beneficial for students, there is little discussion about the students stuck in the middle of this major transition, being taught by teachers with limited professional development and without the benefit of CCSS Math texts, which are not expected to be published until 2016.

    I’m particularly concerned about the transition for students who are currently in more advanced Math (such as Algebra, Geometry, Intermediate Algebra). In our district, they are switching from a traditional Alg-Geom-Int. Algebra sequence to Integrated Math 1-2-3 sequence, which teaches Algebra, Geometry, Probability and Statistics together each year. There is ongoing decision-making (and debate) about which class students currently in the traditional pathway should articulate/matriculate to and really no discussion about how to scaffold their learning now and in the future. According to what I’ve read, these students will be taking SAT and ACT exams with computer-adaptive, open-ended questions(as early as 2015) without the benefit of years of (any?) exposure to CCSS instruction.

    It would be really refreshing to see a story about these students who are caught in the middle as the rug is pulled out from underneath them, because as far as I can tell, no one at the state/national level seems to be thinking about them.

    1. el says:

      I agree with your concerns about kids caught in transition. The big policymakers rarely seem to consider them, and I’ve observed here before that no child ever seems to get one curriculum/standard set for K-12… curricula are deemed failures and replaced usually twice over a child’s schooling, even if the child stays in the same school system from K.

      These transitions always have a cost in learning and understanding that will be paid by the students. Hopefully the new standards and curriculum will be worth the pain.

  6. navigio says:

    While the Common Core standards do require 1st grade students be able to solve addition problems with sums between two and 20, the requirement that they commit the equations to memory is gone.

    Well, it is not gone, it’s just been moved to 2nd grade: “Students are expected to know from memory all sums of two one-digit numbers (2.OA.2▲).

    1. Manuel says:

      See? The bar has been lowered.

      It’s the bigotry of soft expectations!!!


      1. Manuel says:

        Oh, wait, it is the “soft bigotry” of “low expectations”…


    2. Lillian Mongeau says:

      Thanks Navigio – “Gone from 1st grade,” is what was meant.

  7. Bill Honig says:

    For those of you who want further assistance in teaching and creating curriculum for the Common Core Math Standards I would suggest looking at the recently adopted California Math Framework or just google California Mathematics Framework 2013. It has excellent suggestions for organizing instruction in the primary grades (and the other grades) based on the standards. Additionally, the State Board of Education has just adopted (as advice only) basic instructional materials for K-5 and middle grades which contain a scope and sequence, curriculum, and instruction based on the new standards.

  8. I know you’re not supposed to harken back to the ancient past, but we have come full circle in putting Common Core multiplication tables back into the third grade basket of tasks. That’s where it was in the 1950′s. There IS a need for automaticity in multiplication, and it doesn’t belong in second grade.

  9. Paul says:

    Manuel, with respect, I’ve found introducing 8 + x = 11 much more effective than *starting* by teaching rules for explicit subtraction (young students) or, worse yet, equation properties and/or a definition of negative numbers plus rules for computing with them (older sudents). [11 + x = 8 is probably the other problem that we both have in mind here.]

    Your error might be an expectation that students solve 8 + x = 11 by pencil and paper arithmetic or algebraic means. The Common Core Standards for Mathematical Practice (and what used to be skill in teaching math) would have students formulate the problem in ways that makes sense initially. A young student might add three red counters to a container of eight blue ones, or draw eleven kitten faces and cross out three. A slightly older student might respond to a verbal representation: “What number, added to eight, produces 11?”

    My experience is that young students can solve such problems, and make sense of them, to boot. Conversely, older students, well-taught under the old standards, will cheerfully use (and possibly even name!) the “Subtraction Property of Equality” to solve 11 + x = 8 but, the very next day, guess at the value and sign of the answer to 11 + (-3) or be completely baffled by -3 + 11. (NB: a leading -x term in an equation is taught as a special case in many algebra books, and in many more classrooms.)

    I had this very debate with a first-grade teacher at a Silicon Valley Math Group workshop a few years ago. Despite the venue, she insisted that kindergarten and first-grade students were not ready to subtract. Having learned from Ruth Parker, Jo Elsner, and David Foster, after a point I just smiled and thoughy to myself, “Glad I’m not a student im your class!”

    Regarding multiplication tables, I’ve never found anything in the content or methodology (“Practices”) parts of the Common Core math standards that argues against the need for automaticity in multiplication.

    1. Paul says:

      That should read, “thoughT to myself”, and I cut off my compliment to Lillian for an extraordinarily thorough exploration of Common Core elementary math!

      1. Lillian Mongeau says:

        Thanks Paul!

    2. Manuel says:

      Paul, all I know is what I hear from certain members of my family as well as the experience I had with my kids as they were navigating through their elementary school years. Clearly, I am not in the trenches as you are.

      I don’t doubt that kinder and 1st grade students can subtract. Just try taking away stuff from them and you’ll find that out.

      What is bothersome to me is that the “standards” are described in such a way that tells me (and I could be wrong) that there’s an expectation of deep understanding that only comes from doing full-fledged Algebra. If this is the requirement, how else is it going to be achieved?

      To mechanically scratch out kittens or take away Lego-like blocks is not deep understanding (to me, at least, but so is counting with your fingers [and toes!]). But if that is all that is demanded of the kids when they take whatever assessment is required, then that’s fine by me. But just don’t call it “deep understanding.”

      As for your anecdote of the older student, well-taught under the old standards, well, all I can say is that this student did not understand the standard. S/he has no idea what “subtraction property of equality” represents, perhaps because her brain can’t manipulate the concept or s/he just simply learned to parrot the line upon given a clue.

      It is fine and good to theorize what a kid should learn at what age. But the proof of the pudding comes from those in the field: they are the ones who will be implementing these new ideas and assessing their success. It is my belief that the Common Core should have been piloted, preferably with open-minded teachers willing to try new methods, before being imposed across the entire state. Just because a particular set of scholars comes up with a new method does not mean that it will work better than the old one. After all, the previous method had been created by an equally qualified group.

      As for my comment on the multiplication tables, they are not required in the third grade and no addition tables are not mentioned at all. Why not? They should be taught as a method to get the kids to solve the problems faster than sitting there crossing out rabbits or what have you. This means that whoever wrote the standards (a committee!!) has no use for tables. A that’s a shame. (That’s almost as bad as banning Taylor expansion in college physics courses.)

  10. navigio says:

    Odd, I was just reading a Fordham institute study yesterday that graded all states’ old standards and found California’s to exceed the quality of common core’s (just barely in math but significantly in ELA). Whether to trust that source notwithstanding, if that’s true, it seems difficult to imagine that the switch to a ‘worse’ standard will somehow contribute to improved student success as claimed. This supposed link is even more curious given California had essentially the best standards in the nation but nowhere near the best test scores.

    1. Manuel says:

      navigio, this is just another case of the road to hell being paved with good intentions. OTOH, “lowering the bar” might be an example of facing reality: kids simply could not process the information that was required of them under the old standards and it had not much to do with methods of teaching. There simply was not enough time for them to learn the concepts because it was a death march, no different than the one 1st year college students are put through in the “weeder” courses.

      I am, however, mystified by “them” vilifying the learning of addition and multiplication tables. What are they expecting kids to do in higher grades? Break out the plastic bears to do addition?

      I am also intrigued by how teachers are going to introduce algebra to 1st graders. How are they going to teach how to solve “8 + x = 11″ without using the other rules of algebra not mentioned (distributive, identity, and inverse)? In order to successfully “solve” and truly understand this, a basic understanding of the concept of “negative” numbers is needed. I don’t see that mentioned at all. Are 1st graders developmentally ready for that? Are 2nd graders, who are being asked to understand it? Oh, and what about 3rd graders understanding rational numbers and, by implication, ratios?

      All this talk about “comprehension” rather than “parroting” has me discombobulated. What is taught to kids in elementary school is arithmetic, not mathematics. For that, you need both comprehension and when that is obtained, you accept that you need to learn those tables or else you’ll never finish the quizzes. And learn them you did.

      Besides, we have never praised comprehension enough at almost any educational level because if we did we would not have timed tests or tests designed such that an average student can’t ever complete them correctly. Anyone who has gone through a standard college track in the sciences knows this.

      1. navigio says:

        Great point about timed tests. Seems like a good example of the test tail wagging the curriculum dog. I seem to remember one of the criteria used in the shorter path algorithm for the computer adaptive tests being length of time required to answer the question. If that’s true, seems obvious that a memorization “strategy” would be pursued as the one of choice.

      2. Steve Ryan says:

        the point of using “plastic bears” is to help students develop an actual understanding of what addition is. later, when you seem to be worried that such approaches will fail, having memorized 2 + 3 = 5 is of little value when they need to add 1/2 + 1/3. knowing that the operation of addition requires common units (bears or not) provides students with insights that memorizing and timed tests fail to develop. reading some research on so called tried-and-true methods, one might actually know how detrimental some of those practices are. a bit of critical reflection on why so many adults and their children declare that they “just aren’t good at math” might help us see that those tried-and-true methods are largely to blame with our poor mathematics achievement as a society.

        1. Manuel says:

          All I know is that a certain 1st grade teacher I know (and all her colleagues) have been using the plastic bears (actually stackable blocks you get at the local education store) for years without it being dictated as the “preferred” method.

          The fact is that memorization of addition tables is only good for solving the quizzes. If the kid has no idea what 5 represents (“five fingers!!”) then there is no hope s/he will ever get what one-half is. As for adding 1/2 plus 1/3, I have to confess I don’t remember at all how I was taught to do it. But I can tell you the handy-dandy way of doing it in a jiffy and I know that it matches any graphical method. And it will be right.

          Please don’t misunderstand my suspicion of Common Core as suspicion that kids can’t learn math. My suspicion is about how adults are going to twist themselves into a pretzel in order to comply with its language.

          True, it seems I had no problem in learning math via the traditional methods practice in antediluvian times. It is equally likely that these new methods won’t work with 100% of the kids either. Why are we still insisting that one-size-fits-all is going to work just because this one is more free-wheeling?

          One more thing: it has been my experience that when one truly comprehends mathematics one has to understand it at a conceptual, not a visceral, level. To me, math is like music: I may understand it, I may even enjoy it, but I don’t have to know how to play an instrument for that. And we all know that some otherwise reasonably “smart” individuals cannot carry a tune.

      3. el says:

        My mom used to teach remedial math at the high school level, and I remember that frequently she found that students that got to her were extremely rigid about the answer being on the right hand side. IE:

        2 + 2 = __ was easily 4 but
        ___ = 2 + 2

        was extremely uncomfortable for them. And this was part of their trouble with algebra.

        I’ve noticed in my daughter’s education that starting in kindergarten, the answer could be found all over the equation, on the left or right side, and also equations like 4 = __ +2 . I’ve always associated these kinds of changes as being an attempt to help kids get more comfortable with some of the mechanics that used to be introduced only with algebra.