Don't punt; take skills into the end zone

In 2011, the National Assessment of Educational Progress (otherwise known as ``The Nation’s Report Card’’) asked fourth graders to calculate the following:

The results were 83%, 64% and 52% correct, respectively.  Why was performance on the last task, arguably the easiest of the three, the worst?  Answer: probably because it was the only one in which calculators were not allowed.

Part I.

Without obsessing on the calculator issue, the authors of this blog believe in learning basics, not as rote exercises, but for the purpose of being able to apply them in the furtherance of solving more advanced mathematics problems and equally important, to bring those basics to the abstract level.

We don’t know if the poor results on the third question were due to lack of concept (not knowing it’s multiplication?), weak (multi-digit multiplication) skills, or calculation error.

For the moment, we’ll look at the third factor, because of the other seemingly endless debate (besides calculators): times tables memorization.  We’re going to try to end this silly debate once and for all (ya, right!).  There is sufficient rationale for knowing one’s times tables: even at the highest levels of discourse, there comes a time that two numbers must be multiplied quickly.  But beyond its obvious utility, there is greater depth to the times tables than just memorization.  There are important patterns inherent in the times tables that can lead to a deeper understanding of the nature of numbers and ultimately, lead to more profound problem solving and abstract abilities.  The times tables can play a central role in fostering some fairly advanced mathematical thinking, even in elementary school—but more on that later.

The times tables fit as a subtopic in the more general topic of multiplication, the other steps that lead to multiplication facility being: how the concept is introduced, multi-digit multiplication skills, and how multiplication is used in problem solving.

Once again, we have to malign Common Core starting at the very beginning of its treatment of a topic: how it introduces the concept.  CCSSI begins a piecemeal approach to multiplication in 2.OA.4: ``Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.’’

Without saying as much, this is repeated addition.  Certainly that’s how one begins, but then what?  Following CCSSI, students will spend the entire summer after second grade thinking that 4 + 4 + 4 + 4 + 4 = 20, and so what?  It is both ridiculous and demeaning to leave students hanging with repeated addition as the way to total an array.

Instead, finding the total number of objects in an array by repeated addition should culminate in an introduction to multiplication as a shorthand notation.  There’s no rhyme or reason for separating these two concepts.

In other words, repeated addition should be seen, but quickly recognized as an inadequate solution to a common situation, and therefore, should be rejected soon after its appearance.  Repeated addition should certainly never be tested.

Following Common Core, it is not until some months later that students reach 3.OA.1: ``Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.’’  How many months later are students introduced to multiplication proper?  According to CCSSI, which explicitly avoids mandating the order in which topics are presented, 2.OA.4 could come in the middle of Grade 2 and 3.OA.1 a full year later.  Again, any time gap is counterproductive.

Repeated addition as a concept certainly belongs in Grade 2, but multiplication?

If Grade 1 focuses on addition and subtraction, and the first half of Grade 2 is devoted to carrying and borrowing, then in the latter part of Grade 2, students should arrive at the sum of arrays (repeated addition) and an immediate introduction to multiplication as a concept and its notation.  Calculating the times tables (yes, the times tables should be calculated as an activity, one row at a time) allows students to make a valuable connection to counting by twos, threes, etc., the latter skill an explicit requirement which has been left out of CCSSI for some reason. (See 1.OA.5, 2.OA.3 and 2.NBT.2.)

Teachers can post a large blank times table chart on the wall (seemingly mysterious at first), with 0-9 on the top and side, and fill it in slowly as they are calculated.  We’ve seen numerous useless math visual aids plastered around elementary classrooms, but we think an evolving chart would be both motivational and intriguing.

After the times tables have been filled in manually (which justifies each value), it’s time to memorize them.  3.OA.7 states: ``By the end of Grade 3, know from memory all products of two one-digit numbers.’’  Yes and no.

The key to making good use of the times tables and recognizing their critically important patterns is to memorize them first.  The problem with 3.OA.7 is that too many multiplication things are happening in the Grade 3 curriculum for students to lack fluency in the times tables as a prerequisite.  How are students supposed to become proficient at division (3.OA.4, 3.OA.6, looking for the correct quotient) and find equivalent fractions (3.NF.3) during Grade 3 if they can’t already rapidly multiply number pairs?  We won’t dissect each standard, but 3.OA.2, 3.OA.3, 3.OA.5, 3.OA.7, 3.OA.8, 3.NBT.3, 3.MD.2 and 3.MD.7 all involve the use of multiplication.  Students who can recall the times tables fluently will excel, those who cannot multiply with ease will lag.

Memorizing the times tables over the course of the year is too late; students won't make the necessary connections.

Parents: even if your child’s school is following Common Core, reject CCSSI’s approach.  Buy a set of flash cards and drill the times tables into your child’s head over the summer, before she begins the third grade.

NAEP has already highlighted that American students can barely multiply, even with a calculator.  We need to make it a national priority that all students entering the third grade should understand the concept and know their times tables.

In contrast, by introducing repeated addition in Grade 2 and making knowing the times tables the final goal of Grade 3, CCSSI has dragged out multiplication unnecessarily, spoonfeeding the concept and focusing on basic skills instead of developing higher order thinking.

Consider the following ``new style’’ sample question released by PARCC, one of the two major assessment consortia, that conforms with 3.OA.7, fluency of multiplication and division within 100:

Despite the new format, it is still a true/false test of basic skills.  That's not reform.

To conclude this section, we look at 3.OA.8, which captures the culmination of CCSSI’s low-level aspirations for our students: ``Solve two-step word problems using the four operations.’’  Certainly the newly learned operations of multiplication (and division) can be applied to solve word problems: ``8 children attend a birthday party.  Each child gets 3 cookies at the party and 2 cookies to take home.  How many cookies are needed?’’

And the final word on whole number multiplication skills?  The answer lies in 5.NBT.5 ``Fluently multiply multi-digit whole numbers using the standard algorithm.’’  So the ultimate goal is to be fluent at basic skills and get those NAEP questions correct.


Part II.

What is the next step up from basic multiplication skills, i.e., are there other aspects of multiplication that can be explored in a useful educational experience?

Pattern recognition takes arithmetic in a different, deeper direction.  CCSSI has some notion this should be a goal in 3.OA.9: ``Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.’’

[Certainly patterns are a useful thing to recognize, but (as an aside about addition) why is identifying patterns in the addition tables in Grade 3?  Addition tables were finished (and hopefully memorized) by the end of Grade 1, or by Grade 2 at the latest.  Odd and even numbers are covered in 2.OA.3.  What patterns in the addition tables can’t be covered in Grade 2?  Another CCSSI typo?  Or a blunder?]

As for patterns in the times tables, to what kinds of patterns does 3.OA.9 refer?  It’s not clear because CCSSI is (again) not explicit.

We therefore take this opportunity to list some basic patterns in the times tables that students need to explore and recognize.  This discovery should occur during the course of Grade 3, after the times tables are memorized, and should be a learning unit unto itself.  Not all elementary teachers will know to do it; many need to be trained in the math themselves.  Self-motivated students will discover some or all of these patterns on their own, but everyone should know them.  To make sure every student learns them, they have to be an explicit part of the curriculum.

Times table patterns (some more obvious, some less so)
1. 0 x anything = 0 (in later grades, part of the zero-product property)
2. 1 x number = number (in later grades, known as the multiplicative identity)
3. 5 x anything will end in 0 or 5
4. odd x odd = odd; even x anything = even
5. in the odd number times tables, the products alternate between odd and even (a result following from pattern #4)
6. in the times tables for 1,3,7 and 9, the ten products for each number end in the digits 0-9, although in different orders
7. in the times tables for 2,4,6 and 8, the ten products for each number end in 0,2,4,6 and 8 twice over (a result following from pattern #6)

There are more patterns in multiplication, but how many elementary teachers, much less graduating high school students, know all of these simple ones?

Here's a ``tricky'' example we like, to keep students on their toes: 243 x 56 x 9 x 21 x 0 = ?

CCSSI’s deepest foray into pattern recognition comes in 4.OA.5, which states in part: ``Identify apparent features of the pattern that were not explicit in the rule itself.’’  Here, CCSSI is trying to write a standard which asks students to engage in some thought processes, by not giving away answers and letting students figure something out on their own.

Open-ended standards like 4.OA.5 will be the most difficult to assess, so we’ve been curious to see how it will be manifested.

PARCC has begun to release a trickle of sample assessment questions, but has nothing on 4.OA.5.  SBAC, the other major assessment consortia, won’t be releasing samples until 2013.   illustrativemathematics.org, a web site of sample assessment questions, created as a parallel project by CCSSI’s authors themselves, seems incapable of writing its own questions, but instead pays authors $200 for each question it publishes, yet as of this posting has nothing that goes to this standard.  Ultimately, we have to turn to the New York State Board of Regents, which has stepped up (actually stuck its collective neck out) and published a whole series of sample Common Core questions.  We give a lot of credit to the Board of Regents for taking the lead and struggling to align a question with 4.OA.5.

The following question comes from their draft assessment questions for Grade 4.

This problem clearly exposes the struggle: holding itself out as a 4.OA.5 question, the underlying pattern goes to the earlier 3.OA.9, a times table pattern, illustrating pattern #5 above.

The second problem with this question is that none of the answer choices actually complies with 4.OA.5, describing an implicit feature of the pattern; for instance, the number of circles alternates between odd and even.  Instead, each of the choices is either a true or false statement about a single term, not a pattern.  Regents: good try, though.

At the elementary level, there are not that many number patterns besides alternating odd and even and increasing or decreasing by a set amount (to be taught later as arithmetic sequences).  There’s going to be a dearth of testable questions, and students can easily be prepped.  You can cloak the same patterns in different visuals, but the reality is the problem set for 4.OA.5 is going to be finite and quite limited.  The standard and the Regents’ failed attempt to assess it illustrate the limited utility of arithmetic pattern recognition at this grade level.  (Geometric pattern recognition would be a lot more relevant and visually absorbing at this grade level, but more on that in another blog post.)

As a skill of limited variety at the elementary level, parents should know that focusing on pattern recognition in basic arithmetic problems is not going to foster deep thinking skills in your child.  In fact, pattern recognition of this sort as the end goal is a low level skill.  We’re not suggesting it should be skipped entirely, because pattern recognition will evolve later to be a valuable skill, when number patterns are more complex.

Consider the following 2009 NAEP question:

Although this question was presented to 8th graders, the correct answer is a variation on a Grade 3 skill: #4 in our list of times table patterns.

But the question is kind of abstract, and how well are students being prepared to face abstract word questions?  At 36% correct, the conclusion: not well.

Of course, we are concerned about the myriad root causes.  Did students never learn that odd x odd = odd?  Are students so reliant on calculators that they cannot solve abstract non-calculation questions that involve calculation patterns?

The results of this question certainly bolster the arguments of the anti-calculator camp.  Although this problem involves nothing more complicated than addition and multiplication, it cannot easily (or more importantly, quickly) be solved with only a calculator.  In fact, what student who didn't immediately see the correct answer would reach for a calculator?  This problem addresses patterns of numbers, not numbers themselves.  Most importantly, it requires thinking on a more abstract level because it asks for generalizations; mostly likely, students have never seen this particular problem before or practiced problems like it because it is not a ``drill’ question.

Another contributing cause to the difficulty of this problem was posing it as a Jeopardy-like question.  Are answer-first formats appropriate for non-aptitude tests?  That's another debate.

To conclude, pattern recognition, CCSSI’s loftiest goal for arithmetic, should definitely be taught, but it alone should not be considered a high-level thinking skill.  We need to require students to delve more deeply into arithmetic than that.


Part IIIa.

Are there more worthy (and attainable) goals at the elementary level that stem from basic skills and foster real thinking?  We at ccssimath.blogspot.com think so, or we wouldn’t be writing this blog.

To map out our vision, we repeat that we understand recognizing patterns is a step up from basic skills, but it is applying basic skills and using pattern recognition together to solve more complex problems that form the goal to which we should have our students aspire.

We also repeat that CCSSI only would have students reach the first two goals, basic skills and pattern recognition.  We, as usual, have higher ambitions for our students.

Let’s first look at a simple example of how the two skills (multiplication and pattern recognition) can be used in tandem, by reexamining a problem from above:
Obviously, this arithmetic problem can be answered correctly with simple multiplication skills, but using pattern recognition as an analysis tool is a type of parallel thinking process that we should engender.  Not just recognizing patterns, but using them, can help students, if not necessarily arrive at the correct answer, at least reject obviously wrong answers.

The parallel thinking that is possible in this problem is an immensely valuable skill, and transcends the silly ongoing calculators/no calculators and memorize/don’t memorize debates.

For those readers that aren’t getting our gist, let us illustrate: multiplying 74 (a number ending in 4) by 16 (a number ending in 6) will result in a product which ends in 4 because 4 x 6 = 24, which ends in 4.  That pattern exists without exception.  Proficient students know this and therefore have multiple tools at their disposal; the 28% who chose answer A (90) or answer B (518) never considered it, or worse, were never taught it.  (This idea is more specific than simply a step towards the more ethereal goal of ``number sense''.)  Various types of pattern recognition need to be taught, not just as basic skills, but as a means for advanced problem solving that involves parallel thinking.  CCSSI envisions nothing of this sort.


Part IIIb.

Our basic premise: how can students use patterns, in addition to arithmetic, in the course of problem solving, to deepen thinking skills?  It is this question that recounts a theme that runs throughout this blog: taking basic math skills past the point of mere application and raising them to the abstract level.  Using existing patterns to solve complex problems is a significant step past identifying and explaining a pattern, again, CCSSI’s loftiest goal.

The 4th grade NAEP questions we presented are basic skills questions, not abstraction questions.  CCSSI, for the most part, is perpetuating the focus on these low level skills.

The 8th grade NAEP question we presented is on the cusp of abstract ability, and the poor performance shows that students are not close to attaining it.  Unfortunately, CCSSI is continuing to stress basic skills and simple pattern recognition.  There’s nothing in the standards that is addressing the next level of learning.

If students are having difficulty with just simple skills, they are far from the level of math utility we would really like them to achieve, the problem solving abilities which form the real ``career or college ready’’ skills that rise to a world class level.


The authors allege that CCSSI is a ``substantial answer’’ to the problem of ``a curriculum that is a [kilometer] wide and a [centimeter] deep’’.  We here at ccssimath.blogspot.com think the authors (and education pundits in general) have no idea what it really means to ``go deep’’ and have ``focus, coherence and rigor’’ in a curriculum, so we are going to present one last problem that, if taught well, engenders deep thinking, and in fact, cannot easily be solved in a linear fashion.

We like this problem because although it only uses basic addition, multiplication and arithmetic pattern recognition, it joins those same basics at an abstract level.  It focuses the same skills into meaningful problem solving, it coherently brings those skills together, and it involves rigor.

And the best part is: it can be solved at the elementary school level (with proper guidance, of course).

What do we mean by bringing basics to the abstract level?  Beyond memorization or even fluency, important patterns become evident to those who are conversant at basic operations and perform those operations regularly.  It goes back to learning the basics early and effectively: students who are fluent with their addition and multiplication tables are more likely to be able to answer abstract questions like this because they won’t be mired in the arithmetic.

Here’s the problem:

As many boxes as there are, there may be pathways to solve this problem, but that doesn’t necessitate chaos.

As we said, it helps immeasurably to know one’s arithmetic.  More importantly, it is essential to know patterns such as: the upper-rightmost box must be a 5, or you won’t get a 5 in the product.  Like (harder) Sudoku puzzles and the games of chess and Go, this problem encounters branching points and possibly will reach a dead end where a wrong choice becomes apparent and it is necessary to backtrack.  But that’s the whole purpose of this problem.  Keeping track of choices, knowing where choices have been made and you are branching off to a possible dead end, and backing up to that point and trying another solution, on top of basic skills and knowing arithmetic patterns, involves deep level thinking unlike anything that CCSSI envisions.

Anyone who can add, multiply, and knows arithmetic patterns can solve this problem.

Some readers may counter that there are plenty of fill-in-the-blanks multiplication puzzles out there to solve, some with dozens of dead end pathways to follow and reject as mathematically impossible in the course of finding the correct solution.  While problems like those have their place and are suitable for the motivated person to pursue, that is not our intent here.  One particular advantage of mathematics as a learning tool (over, say, interpreting a Shakespearean sonnet) is that a problem can be precisely crafted to achieve a particular learning goal, and the length and pathways can be carefully controlled.  That careful control is what we are trying to illustrate in this puzzle: at key intersections, the number of choices never exceeds 2.  (Contrast that to chess, etc., which may have numerous branching points and pathways.)  Those 2 choices at each branching point are based on knowing arithmetic and patterns.

[We’re assuming that the readers of this blog most curious about our agenda will be able to solve this problem and will see what we mean by ``2 choices at each branching point’, so for the time being, we won’t provide a solution. But a hint is in times table pattern #7.]

We rarely encounter the careful orchestration of problems to be age-appropriate and specific learning goal oriented based on previously learned basic math concepts, with the intent to engage the student in a particular type and level of thinking.  Even when we examine so-called math enrichment problems at summer math academies, after school courses, and the like, the chosen problems may indeed be challenging, but there’s no rationale for how they fit into a learning path or how they connect to what students have learned before and how they are supposed to apply their basic skills in solving the problem.  A math problem is not useful just because it’s challenging and can be solved with a certain degree of insight; math problems, to be effective learning tools which allow students to tie skills and concepts together, must be crafted more carefully.

This problem has our favorite triumvirate of length (involving concentration for an extended period), connectivity (uses multiple skills), and dimensionality (can’t see the end) that we think are necessary to take simple skills like multiplication and pattern recognition and ``go deep’’ with them.   Going from basic skills to pattern recognition to using those skills in an age-appropriate but complex manner is what it really means to ``go deep’’ and is how to vanquish the ``kilometer wide and centimeter deep’’ problem in American education that will be perpetuated, in spite of CCSSI’s efforts.

Such problems belong in elementary schools and will create an opportunity for the early development of real thinking skills.