The two previous posts dealt with two of three myths that exist in math education: (1) that the current curriculum is a “mile wide and an inch deep” and (2) that the current curriculum is built around discovery learning. This post will address a third: that current advocated methods are radically different from the ones in vogue decades ago.
Before going ahead let’s try to make this clear: the current curriculum advocates a balance between memorization and understanding, Consider, as evidence, the snippet, just below, from the grade 4 mathematics curriculum guide. Read from the second column the two bullet points. Notice that the first one talks about strategies going as far as 9×9 and that the second one states clearly that by the end of grade 4 the students need to have committed up to 7×7 to memory.
So that’s what it is for today, but what about times past, the times when people claim to remember a math curriculum that emphasized rote learning of math facts?
You may recall that in the previous post a comparison was made between the topics covered in the current student support materials and those that were in “Investigating School Mathematics” the textbook series used for most of the 1970s. You may also recall that it was found that the topics were rather similar. In a similar way, the language used in the teacher guides was also examined and, once again, striking similarities were found between that language and that in use today. Let’s look at some examples, all from grade 4.
We will begin with the general introduction to the book. Skim through the quote below, taken from the front matter of the teacher’s guide:
“It is intended that each day’s lesson in which the child is presented with a new concept be divided into four parts: Preparation, Investigation, Discussion, and Using the Exercises. The preparation usually should be kept fairly short, and care should be taken to see that this work does not preempt either the Investigation or the Discussion. Generally, the Preparation should do nothing more than provide the children with that readiness which they need before they begin the Investigation. The Investigation presents the rudiment of the concept treated in the lesson and should be the “main event” in terms of pupil activity and involvement in the unfolding of the concept.
In general, it is expected that the Investigation be done by the children either independently or in small groups. Think of the Investigations student-centred activity. It is fully anticipated that the students will grope, question, search, and explore. Investigations are designed to provide for individual differences; that is, the child is frequently asked to perform a certain task as many ways as he can, or to find how many ways he can do a certain thing. By presenting the child with this type of challenge, at least some degree of success is assured. That is, your slowest student will find that he can do something more than one way, while your more able children will find many ways to do a given task. Thus, as you guide the children through an investigation, it is important for you to recognize that they will achieve in widely differing ways, and that you should give recognition for all levels of achievement. Perhaps the most important thing to remember in working with the children during the Investigation is to encourage them to do the thinking arid exploring.”
Critics of modern-day math education, who vividly “recall” spending significant time memorizing math facts and performing endless drills may find it hard to come to terms with the fact that the above quote came from the text series in use in the province in the 1970s. In particular the “Investigations” may come as a surprise. Yes, even four decades ago, there was an acknowledgement among the teaching veterans that students need adequate time to explore new situations and to try and get them to fit with their preexisting cognitive structures.
It does not end there. The whole idea of students talking about and discussing their math work is not new, as evidenced by this quote from the same book:
“Following the Investigation, the children are given an opportunity in the Discussion section to talk about what they did and to summarize the mathematical ideas in the lesson in preparation for working independently in the Using the Investigate section. Generally, the beginning discussion exercises are designed to stimulate the children to talk about what they did in the Investigation. You should encourage them to discuss the various methods that they used to investigate and explore the concepts. Also, you should follow your teacher’s guide carefully to ensure that whatever mathematics ideas are to be developed in the section are actually summarized an understood by the children.”
And, finally, the current ideas we term “Differentiated Instruction” are also not new. Consider this quote from the front matter of the teacher’s guide:
“Minimum, average, and maximum assignments are provided for each lesson other than review lessons. These assignments are given to assist you in providing for the individual needs of the children. It is not intended that you give the minimum assignment to the slower children, the average assignment to the average children, and the maximum assignment to the more able children. Rather, these designations are given to assist you in making individual assignments according to needs, abilities, and time available for each individual child. For example, if time is short and you need to move rapidly through a particular lesson, you may choose to use the minimum assignment for all children. The minimum assignment will, in general, provide the children with sufficient practice and mastery of skills to move ahead to the next lessons. On the other hand, you may sometimes choose to use the maximum assignment with slower children over a period of two or three days. Also, it is highly likely that you will not want to assign the maximum assignment to the more able children, since quite often they need less practice than some average and below average children. For example, when your more able children demonstrate the ability to perform a particular skill with great efficiency, they should not be made to drill excessively in that skill. In some cases, an asterisk is placed beside an assignment to indicate that the lesson could be omitted without loss of continuity. “
But, enough with the general talk. What follows are images of separate pages from the grade teacher’s guides. Each image contains a page from the student text as well as teacher’s notes. Notice that in each case, while there is ample opportunity for drill and practice, strategies are also presented that emphasize concept development and connections.
Notice in the above case that the students are actively encouraged to relate multiplication with an area model.
Here’s another example.
Notice that, once again, the focus is on strategies. In particular the investigation notes encourage teachers to help students to use already-known facts to help find the ones they need to learn.
Here’s an example from grade 3. Notice, once again, there’s a balance between concept development / exploration and drill.
Notice, from the investigation notes, that a degree of insight and creativity is encouraged.
Now this is not to try and pretend that things like drill and practice do not / did not have a valid place in the curriculum. It is, rather, to point out that things today are not as radically different from the past as some would have us all believe. There is ample evidence of, for example, opportunities to practice with multiplication facts. See the image below–taken from grade 4–for example. The page from the student book clearly is intended as basic drill and practice.
But look a little closer. In particular look at the teacher notes at the right of the student page. Notice that, rather than just assigning the exercise the teacher was presented as a valid option for making a game out if it.
See also the example below from grade 3. Straight up “drill and kill,” right?
But drilling is not everything. Look at the teacher note to the right of the student page above. It’s indicated by the circled “1.” It’s acknowledged that the students don’t need to recall all the facts with speed, and furthermore, that it’s also important that students know strategies that can be used to find facts they do not know rather than having to look them up. So, too, for other grades. From the grade 4 book covering that same ideas comes the page represented below. There are several things to notice.
First, note the paragraph indicated with the circles “1.” From reading it there’s no doubt regarding the importance that’s been set on the students accurately stating the multiplication facts. Notice that quickness is now an emphasis–it wasn’t in grade 3.
But there’s more. Now look at the paragraph that follows, indicated by the circled “2,” and see that the students are expected to develop strategies for determining facts they do not know rather than just learning them by rote. Clearly there’s a balance afoot. Now, finally read the discussion indicated by the circled “3.” It becomes pretty obvious that the strategies emphasized by today’s curriculum were just as important four decades ago when “Investigating School Mathematics” was implemented.
So there it is. There are many, many more examples that could be presented but these will suffice. The long and short of these past three posts? 1. The curriculum today is similar in depth and scope to that of the past 4 decades. 2. the present-day math classes do not rely on the type of “discovery learning” that they may have been led to believe exists and, 3. the methods that were recommended decades ago are not so very different from the ones in vogue today.
Of course that’s nowhere near the end of the story. This post has only discussed what’s recommended, not what’s actually implemented in each classroom. Today, and in times past each school and each individual teacher placed their own interpretation on the recommended curriculum. In the 1970’s, no doubt, some classes used the methods advocated by the textbooks and others the drill-and-kill approach. The same is true today, and perhaps instead of wasting time longing for times that probably never existed we should be more focused on what actually works.