Relevance: Discovery, Skills-based, and Manipulatives

Recently I penned My Last Math Class and shared how I made math learning real for an unique group of students.  Following on its heels, I shared information on the 3 R’s —Rigor, Relevance, and Relationships— coined in 2010 by The International Center for Leadership in Education.

Now I want to expand this theme into another area of math education and discuss what is more meaningful and effective — discovery learning or skills-based learning.  Much material is available to fuel this debate, but I don’t lean towards one or the other.  I am quite comfortable with my seat on the fence, where I have a vantage point.

In 1967, William C. Lowery from the University of Virginia listed in the High School Journal the advantages of learning by discovery. They are: (1) The student understands what he learns and is able to remember and use this in other situations.  (2) He/she learns a way to learn on his own; in mathematics he learns techniques, or strategies, for discovering other new things. (3) He develops an interest in what he is learning. Isn’t this what learning is all about?  Shouldn’t this be our goal?  Don’t we want our math students to understand and apply, learn techniques or strategies for discovering other new things, and to develop an interest in what they are learning?

Then we have skills-based learning. Does anyone think that someone can read how to drive a car and do it effectively and safely without practicing?  A case in point, I am relearning how to knit.  I have had two knitting masters demonstrate how to cast on stitches then knit, and I have watched a YouTube knitting video at least 15 times in order to master the steps, but the only way I have improved is through practicing these skills.

So why the debate?  Shouldn’t students be able to participate simultaneously in discovery learning and skills-based learning?  Wouldn’t learning then be more relevant?

After teaching in a school district, I became a professor of teacher education at a university.  I taught undergraduate and graduate methods classes, and supervised student interns.  I had privileges in the surrounding school districts to test in the classrooms methods, strategies, techniques or activities.

In a 3rd grade class I demonstrated and proved to my graduate students that manipulatives or concrete objects could teach whole number division through the understanding of place value.  This all happened in one day. Here is how I did it.

I first asked the 3rd graders if they would be willing to help me solve a problem.  They said they would.

I told them, “My son has found a sheet of Batman stickers that he wants to share with two boys who are going to spend Friday night with him. My son wants to make sure that all get the same number.

“How many friends?” they asked.


“How many stickers?”

“One hundred two.”

Sure, sure, sure.  They wanted to help.

I did have a couple of requirements: (1) they had to work together in small groups of four, and (2) they  couldn’t use paper and pencil.   However, they could use items or tools to help.  At the large table in the front of the room, I had sets of base ten blocks, string, yarn and rubber bands.   I did not explain what these items could be used for, but only  that they could be checked out if they wanted them in their groups.

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I had two hours to find out how many would work on this problem in small groups. Would they use the materials to help solve the problem? After they were introduced to the algorithm, could they apply the new knowledge they had gained?

At first, they talked in the groups about the problem. A bright, outgoing female class member approached me for some materials.  Other group representatives followed suit.

I handed out baggies containing one hundreds block and two units along with their choice of string, yarn, or rubber bands.

Watching them figure out what to do with the string/yarn and rubber bands was most interesting when they were trying to separate the hundreds block into parts for distribution.

Now remember.  I never used the word ‘division,’ and I never said that there were three boys.  They had asked me how many friends were coming over.  I told them two, so they had to extrapolate that there were three total.

Soon, the same outgoing female came up to me and asked if she could trade in some “stuff.”

“Like what?”  I asked.

“Like this block.”

“What do you think you can trade it in for?”

“Well, can I have those sticks (the ten-rods)?”

“How many would make it an equal trade?”  I asked.

She took the hundreds block, put it on the table, then laid ten ten-rods on top of it.

“I’ll take ten of them.”

Other students were watching this process, and they followed the same steps.

After the ten-rods were doled out, they found they had a leftover ten-rod and the two units or ones? Quickly, they realized another exchange was required, and the leftover ten-rod was exchanged for ten units or ones.  These ten units were added to the two units to make a total of 12 units, which were distributed evenly among the three friends. The result?  Each of the three friends should be given 34 stickers.

Although the students had not received formal instruction on how to perform whole number division, the concept of ‘fair shares or equal shares’ was intuitive.   By 3rd grade, they have learned how to dole out shares fairly.  So, they had enough prerequisite knowledge to solve this problem.  They had used base ten blocks when learning to add, subtract, and multiply, so they were familiar with the materials. The problem posed was a real-life problem for an 8 year-old. It was relevant.  They had worked in groups and learned from each other as they used the process to find an acceptable solution.  Relationships.

The beauty of this learning situation came next. Immediately, I introduced the whole number division algorithm alongside the manipulatives.  At the end of the next 45 minute period, all but five of the 28 students knew with certainty that when a teacher says, “3 won’t go into 1 (100),” that she/he means that the hundreds block cannot be divided into three equal parts, but if the hundreds block is traded or renamed into 10 groups of 10, then those can be divided  equally into 3 groups with one group of ten left over.  The remaining ten rod cannot be distributed evenly into the three groups unless it is regrouped into ten ones or units which when joined with the original two units yields three equal groups of four units.

Then, we proceeded to practice the newly learned skill.  I presented similar problem situations relevant to 3rd graders;  the students found the data needed to solve the problem and then set it up to find a solution.  Even the five students who were not 100% confident after the initial exercise showed progress during the guided practice of the skill.

Had the students taken part in discovery learning (understand and apply; learn techniques or strategies for discovering new things; develop an interest in what is being learned)? Yes.   After their formal instruction on whole number division, was practice needed to hone the newly learned skill?  Yes.  Was this learning relevant to the age group?  Yes.

Which is more meaningful and effective — discovery learning or skills-based learning?  No argument. Both are important, and both should be integrated into the classroom.


4 thoughts on “Relevance: Discovery, Skills-based, and Manipulatives”

  1. Pingback: Whole Number Division with Semi-Concrete Base Ten Blocks | Elevated Math

  2. Pingback: 81st Carnival of Mathematics « Wild About Math!

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