The Fallacy of the 'Multiple Strategies' Approach

It has recently come to my attention that many the proponents of the multiple strategies approach to mathematics education are completely hypocritical. Allow me to explain.

Let's take an example to discuss exactly what is meant by the multiple strategies approach. Consider the question 24 x 3. A student approaching this question might use base-ten blocks to model the question as an area, might try to use a 'doubling-halving' approach to alter the question to a basic fact that they already know 24 x 3 = 12 x 6 = 72, or might try finding three groups of 20 and three groups of 4 using the distributive property 24 x 3 = 20 x 3 + 4 x 3 = 60 + 12 = 72. All of these strategies to tackle this problem are taken as equally valid solutions to the posed question.

The strategies above seem fine; however, there is one glaring problem. I haven't included the use of the standard algorithm as an acceptable way of solving the posed question above! What I see in practice is that proponents of multiple strategies dismiss the standard algorithms under the guise that students do not "understand" the mathematics if the algorithm is used. They instead favour visual and tactile models, and mental calculations - all of which are cognitively heavy on our students, and only work in contrived scenarios. Try to use the doubling-halving method on the multiplication 15 x 15. It won't work because one of the numbers must be even to use this "trick"! I have written about this particular "trick" before, stating that it would be more beneficial for students to learn prime factorization (at least prime factorization comes up often in practice and works in more situations than doubling-halving does). Even better than prime factorization is to teach the standard algorithm! When performed correctly, it is much more reliable than mental calculations, and much more efficient than visual or tactile models. The amount of time spent on the inefficient multiple strategies listed above could be more efficiently used to teach the understanding of the mathematics behind the algorithm. Problem solved!

Despite what I have mentioned above, proponents of the multiple strategies approach continue to word assessment questions in such a way so that students are forced not to utilize the standard algorithms. Or if the student does utilize the standard algorithm where appropriate, they are severely punished. So much for being able to use "multiple" strategies to solve a question!

To give some illustrative examples, I have taken some screen shots of an assessment for Grade 7 teachers.

The first figures show the probing question "Write everything you know about fractions." (Aside: What an AWFUL question. How can you determine what is a good answer and what is not a good answer? There are no guidelines for the student to follow!)

The first student shows that he/she knows some basic fractions and that he/she knows what these fractions look like as a part-whole diagram. Yet this student is "not meeting" the intended grade level objective. How can the student know that by looking at the question? The student answered the questions just perfectly! He/she put down knowledge of fractions.

The second student, who can model fractions using diagrams and algebra, and who even knows the ideas of mixed/improper fractions doesn't even meet the standard! The student must connect fractions to decimals to get a "meeting" grade level objective. But was that clear in the posed question?

The third student here demonstrates that he/she can clearly represent fractions in about 2000 ways, so he/she must be meeting the standard. This is exactly what the proponents of multiple strategies want: a student doesn't "understand" fractions unless he/she can model them in many different ways. However, how can our students be expected to recall ALL of this information with such a poorly designed open question? Would it not be better to have them recall specific ideas, rather than overload them cognitively with such a daunting probing question?

Continuing on, let's see another goodie. Here students are asked to order the following fractions, decimals and percents:

But students might use the standard algorithm to convert fractions to decimals. So, to make sure students don't use it, we will insist that students have to use multiple strategies to get to the answer! Again, a multiple strategies approach was not asked for in the question, so now they are simply punishing students for utilizing a perfectly good strategy.

And, finally, here is my favourite clip from that document. In this question, students are asked to calculate 36% of 25.

This student calculated the correct answer using the standard algorithm for multiplication. Very nicely done! But I am sorry to say that the algorithm is not allowed, so you are "not meeting" expectations. What we actually wanted from you is a contrived mental math strategy given below:

 

So, I only hope that this gives you some insight into why I am very wary of proponents of multiple strategies. These individuals are secretly removing the standard algorithms from our curricula under the guise that their "methods" (pictures, mental strategies, etc.) promote "understanding." When, in actuality, their methods are archaic, inefficient and non-transferable to many situations. 

I leave you with two links. One is to the Heinemann blog on the Standards for Mathematical Practice (SMP) for the US Common Core. In particular you should read standard #5, in which John SanGiovanni clearly states that "In many situations, pencil and paper are inefficient and using them is not strategic." He is promoting use of multiple tools to solve problems, yet you can infer that he would not be happy about students using pencil and paper to solve the question using the algorithm for subtraction (he doesn't even mention the standard algorithm as a viable strategy). 

The other link I wish to share is a piece written by Michael Zwaagstra inThe Chronicle Herald, a Nova Scotia newspaper. In this piece, Zwaagstra comments on how Marian Small, a prominent figure for the multiple strategies approach, hypocritically states that different strategies to solve questions are good, but to leave out the standard algorithms from the curriculum.

 

Do Educational Organizations Have Any Clue?

Recently a post by the Canadian Education Association (CEA) came across my desk. It was titled "The Facts on Education: Is Inquiry-Based Learning Effective?" so of course I was curious to see what the CEA had to say. The full post can be found here. And the sources referenced can be found here.

What I want to focus on is how the educational organizations have no clue about how to properly teach children. Let's analyse this passage:

So they begin by stating that inquiry-based activities can boost "learning" yet they have decided not to define what they mean by "learning." I admit that I did take the time to search Sharon Friesen's literature review on inquiry-based learning since she was listed as a reference (She submitted this review to the Alberta government by the way, completely ignoring any research that didn't support her viewpoint surprise surprise.).  This literature review had 254 instances of the word "learning," yet Dr. Friesen failed to define "learning" once. How can one discuss inquiry-based learning if one doesn't know what "learning" even means? Very peculiar. Seems like the CEA is doing this in the above paragraph as well. How can we trust the CEA that inquiry-based activities can boost "learning" if they have failed to define it?

The CEA continues down the rabbit hole by stating that unguided or minimally-guided inquiry may not work with students who have less previous knowledge. They even mention that "learning" may be blocked, or misconceptions may arise, if inquiry-based methods are used. In these couple sentences they are definitely referring to the referenced work of Kirschner, Sweller and Clark. This seems very odd to me - why go out of your way to claim that inquiry methods are excellent, then reference a paper that argues against what you are trying to state? Kirschner, Sweller and Clark give a very good argument as to why student lessons should be scaffolded with many worked examples - which tends to be the opposite to inquiry-based methods. In fact there is a large body of knowledge (not just from cognitive psychology) that has found explicit instruction to be superior to non-explicit instruction (like inquiry-based learning). See this post by @greg_ashman for a bit more to this picture. So it seems to me as if the CEA really doesn't understand the research properly. It is even more apparent that they are clueless when they make the following absurd claim:

According to the CEA About Us page, their mission is to "transform public education by ... disseminating research that can impact practice." Seems to me as if they are preaching the usual "fuzzy" ideals we see from many educationalists. Do we really need "professional" organizations like this distributing biased research to our teachers and government?