Alternative Ways to Teach Mathematics for Common Core – in response to video for TERC

03 Aug

What do you think about this?  This came from a site called


Note this link comes from a site that clearly has a negative opinion about these alternative methods.  Is she justified?  Well, let’s not jump to conclusions and say, “Yes.”  I think the answer is “almost yes.”  I am a firm believer that nothing is ever black and white.  Common Core mathematics encourages students to learn alternative ways of thinking about mathematics.  They encourage students to delve deeper into the meaning behind the computation.  They want students to know the meaning behind an algorithm at a mature level.  This is a very LOFTY goal.  In fact, after doing my dissertation, I learned that many teachers don’t even have this level of understanding.  My daughter was taking 5th grade mathematics this past year using the North Carolina Common Core curriculum.  She was required to solve all her problems three different ways and then she had to write a journal entry that explained the WHY behind what she did.  This really isn’t too far off from what is in these videos.  Of course she was also taught or allowed to use the standard algorithm, but only in addition to other less effective algorithms that she had to learn.  These less effective algorithms were meant to build her conceptual understanding.  Was it effective?  I really don’t think so.  I don’t think you need alternative algorithms to build conceptual knowledge.  I think you can just build conceptual knowledge with good teaching.

For example, I have no idea how TERC might teach adding mixed numbers but when I look at how to teach this I know that some teachers (most) would teach it by just teaching the algorithm:

2 3/4 + 1 3/4

1.  Add 3/4 + 3/4  = 6/4

2.  Convert 6/4 to    1    2/4

3.  Add the whole number pieces:  2 + 1 + the extra 1 from step 2 = 3

4.  Final answer 3    2/4  or reduce to 3  1/2


However, a better approach might be to demonstrate this with a concrete approach rather than starting abstractly with just numbers.

Draw 2 and 3/4 a pies and another 1 3/4 pies.  Show how you can move one piece of the second partial pie to fill up the other partial pie, giving you 3 full pies and  2/4 left over pie (or 1/2 left over pie) – final answer 3 2/4 or 3 1/2.

From here you can now relate the concrete picture to the abstract numbers.


This is what I mean by “good teaching” and is much better than using 3-4 additional ineffective algorithms or “discovery” learning approaches.  Discovery learning was something that I was introduced to in my methods class when getting my degree to become a teacher.  It seemed like such a good idea at the time, instead of you telling the student what the answer was or how the problem worked, you devised a lab situation where they naturally “discovered” the solution all on their own.  Since they created it, it was more active learning and hence would stay with them longer than just being told.  The idea sounds great and active learning DOES produce better retention.  The problem is that you have to actually create that situation you are looking for.  If you just give them labs that the teacher THINKS will lead them to building conceptual knowledge but instead just ends up being a lot of extra steps (see the first video example), the impact is the exact opposite.  Our local high school tried this.  They decided to teach Common Core Math 1 concepts through discovery learning.  Instead of giving lectures on exponential functions and showing relationships, demonstrating examples, and having students work problems, they created these lab experiences for these freshmen (and a few sophomores) to do.  The students that I worked with completed the labs but it was clear that they had no meaning for them.  The intent was that this was supposed to build all the concepts they needed about exponential functions when in fact, the students just found answers to questions without piecing together the big picture or linking one lab to the next.  It was an utter failure.  Discovery learning is very hard to pull off.  Not only does it require really good lab experiences but it requires the right kind of student that can learn that way.   Students have a variety of learning styles and many need to be taught by concrete methods, examples, and linking new information to things they already know.

Another important point made by these videos is that it is a waste of time to teach the student the standard algorithm because they will eventually just rely on the calculator anyway.  Is this true?  Well, it is true if we make it true.  It also NEEDS to be true for some students.  If you read my post about accommodations for students with memory problems, they will NEED to have a calculator accommodation because they lack the same ability that the non-LD student has to memorize rote facts.  For this student, I think they do need a calculator so that they can focus more on the big picture and less on things that will never happen for them.  However, what about the student who can learn their multiplication facts but just doesn’t want to put in the time.  What about the 9th grader who still breaks out the calculator for 8×3?  One might argue that we always do have technology at our disposal.  With cell phones that have calculators on them, one is almost never without the ability to use a calculator these days.  Does that mean it is okay to rely on it?  I am a strong believer in technology and feel that we should not be doing tedious problems that one would not do in the “real” world without a calculator, in class without a calculator.  However, for things that one should just “know,” that demonstrates a basic understanding of basic mathematics, yes, students should be required to do these things without a calculator unless them have a documented learning disability.  In other words, let the real world determine the appropriate use of technology in the classroom.

What about the second book that was discussed in the video?  Was it all bad?  I have had students who have used both those algorithms:  the one that uses the place value and the lattice multiplication.  Here are my thoughts on those.  First, I like the idea of showing and having a lab on how 2 digit multiplication can be done with the list of numbers that show the true place value.  This really helps a student “see” what is going on behind double digit multiplication.  In fact, there are other ways that organize it even better that a teacher can use.  We do want to teach in a way that is more than just procedural mathematics.  Not all students will grasp everything but it will help some.  However, this does not mean that the ultimate goal is to leave out the standard algorithm.  The other algorithm is used to demonstrate conceptual understanding, not to be an algorithm that students continue to use.  My student who did try to use this as their only algorithm, did make many more mistakes than students using the traditional algorithm.  As for the lattice method, there is no reason conceptually to show this method.  However, for some students, the organization (keeping things in boxes) helps them from a visual perspective and that is why a teacher might choose to show this.  The student that used this algorithm used it successfully for a long time, he is a rising 9th grader and I wouldn’t be surprised if he doesn’t still use it (I only tutor him on occasion now when he has a test but it is still his algorithm of choice.)  The point of all of this is two-fold: one, use alternative approaches IF it helps a student grasp the concept more and two, allow them to use an alternative algorithm if they can be as fast, as accurate, and as effective with it.  Give the student the choice, it is frustrating when teachers tell students they “have” to use a specific algorithm.  Once a student finds one that meets the criteria of fast, accurate, and effective, they should not be forced to practice other algorithms except in sense of in class concept building labs.

Written by:  Lynne Gregorio, Ph.D. Mathematics Education



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