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Students are exposed to sets very early in their education, so my first inclination is that this would be the best method to give children in the early primary grades an introduction to probability and statistics.

I struggle with the notion of how to best promote the ideas of chance, etc. to students that wouldn't understand, for example, that a smaller set of two items out of a set of four would represent the chances of the smaller set being drawn are about half.

Are fractions and percents -- I know these early concepts are two sides of the same coin, pun intended, but aren't always taught concurrently -- necessary prerequisites for learning probabilistic and statistical concepts, or are there ways to provide intuition (for example with math based on the analog clock) into these lofty ideas without them?

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    $\begingroup$ In this context: What is an example of a sentence you'd say to a student to discuss either the probability of rolling a 4 on a die or - after rolling a die a bunch of times - the average (mean) number of times that you rolled a 4? In particular, what might you say in either scenario to "students who don't know fractions or percentages yet"? $\endgroup$ – Benjamin Dickman Jun 23 '14 at 19:13
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    $\begingroup$ I think in order to answer this question it is important to specify what grade/age level you have in mind, and what exactly you mean by "without fractions". Children learn the word (and concept) "half" well before they encounter the symbolic representation of it as $1/2$, much less how to perform arithmetic operations on fractions, and likewise (to a lesser degree perhaps) for "quarter" and "third". So a student who has not "learned fractions" may very well understand the phrase "half of the time" or "a third of the time". $\endgroup$ – mweiss Jun 23 '14 at 22:47
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    $\begingroup$ @BenjaminDickman I think that's a good answer (assuming you are asking those questions rhetorically). Otherwise, I'm not really sure. I suppose something like the median would be more intuitive to those groups of students than the mean. $\endgroup$ – jonsca Jun 23 '14 at 22:57
  • $\begingroup$ @mweiss I'm thinking along the lines of preK to 3 or so. I agree with your point about understanding half, but I'm not sure that 1/3 is as intuitive. $\endgroup$ – jonsca Jun 23 '14 at 22:58
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    $\begingroup$ Third grade is 8 year olds, at least in the US. I don't know if I would say that "a third" (note, this is different from $1/3$) is an "intuitive" concept, but it is likely a familiar one. $\endgroup$ – mweiss Jun 23 '14 at 23:09
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I definitely think you can get the intuitions behind probability to elementary age kids, and doing so might even be a way to introduce the important notions of equivalent fractions / related rates / etc.

Take a six-sided die and mark 1 on one side, 2 on two sides, 3 on three sides. Show the kids the die. Ask them, if you roll this die a bunch of times, which number do you think will come up most often? Ask them how much more often they think 2 will show up than 1? Some of them should have the intuition that it will be twice as often. Ask how much more often they think 3 will come up than 1.

Have them do an experiment and create a bar chart with their results. Make them do enough rolls to see the expected trend. Ask them why everyone's bar chart is not exactly the same.

Tell them that since there are six sides on the die, and one of them is marked "1", every time you roll the die, there is one chance out of six that it will land on the 1. Ask them how many chances out of six there are that the die will land on a 2. And how many chances out of six are there that the die will land on a 3. Ask them, if you roll the die 6 times, are you guaranteed to get 1 once, 2 twice and 3 three times? Since they have done the experiment they know this is not so. Ask them why they think it would or would not be guaranteed.

Propose different games to be played with this die, and ask the students if they are fair or not. For example, we roll the die once, and if it comes up 1 you win, and if it comes up 2 I win, and if it comes up 3, we roll again. Ask: Is that fair? What would be fair?

Another thing you could do (to get at equivalences and scaling) is ask the kids, if I had a 12-sided die, how should I mark the sides if I want the chances of getting 1, 2 or 3 to be the same as it was for the 6-sided die that we were just using. In other words, how could I label the die so that if you rolled that die a large number of times, your bar charts would look similar to these ones you all made with the 6-sided die. This will be tricky for them because the chances are now "out of 12" instead of "out of 6". Is 1 chance out of 6 the same as 1 chance out of 12? If not, what would be the same?

You can also try doing basic probability with a spinner. Take a game spinner on a background broken into one half and two quarters. Label the spot that covers half of the spinner area A, and the two quarters B and C. Ask the kids, if they spin the spinner a lot of times, which letter will come up most often. Ask them to compare how often they expect to get A versus B, and how often they expect to get C versus B. If you can come up with decent spinners for them to play with, you can have them do that experiment too. Ask them if they can give you a "chances out of" statement for how often B will come up. If they are stuck, try dividing A's region in half with a dotted line, so they can see 4 equal sized (and thus equally likely regions) and then ask them if they can come up with a "chances out of" statement. If they are still stuck, point out that there are 4 equally likely regions on the spinner, just like there were 6 equally likely sides on the die. So they need to figure out how many chances out of four each letter has of being chosen on the spinner.

If you want to try introducing the notion of equivalence and scaling up with the spinners, you could then divide all the segments in half again, so there are now 8 segments on the spinner. Now B looks like it has 2 chances out of 8, and before it looked like 1 chance out of 4. Are those the same thing? Can they explain why they think so?

If they are getting good at it, then you could try some "or" questions, like on the die, what are the chances of getting a 1 or a 2 on any given roll? (3 chances out of 6)

I don't see as easy a way of doing "and" questions, like what is the chance of getting a 1 the first time you roll the die and then a 2 the second time. Making the chart of all 36 combinations, especially when some look identical, will not go well in the primary grades, IMO. If you do want to get into that, get two different colored dice, normally numbered 1 through 6, and ask what is the chance of getting a 1 on the red die, and a 2 on the blue die. And then help them enumerate all 36 combinations. But honestly, I think at that level, you're better off working with slightly older kids who understand multiplication and fractions.

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    $\begingroup$ I like the concrete examples, I think they are the best way to get the point across. $\endgroup$ – jonsca Jun 23 '14 at 22:59
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    $\begingroup$ (to the students, that is) $\endgroup$ – jonsca Jun 23 '14 at 23:15
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    $\begingroup$ Another thing you could do is ask the kids, if I had a 12-sided die, how should I mark the sides if I want the chances of getting 1, 2 or 3 to be the same as it was for "this" 6-sided die (that you had been using earlier), so that if you rolled that die a large number of times, your bar chart would look similar to these ones you all made with the 6-sided die. $\endgroup$ – PurpleVermont Jun 24 '14 at 0:17
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    $\begingroup$ Without an understanding of fractions, what does someone think about how many chances "2" has of coming up out of 6 possibilities? If a student thinks "3" has 3 chances out of 6 coming up, you essentially have to get them to understand part-whole relationships to see the problem. And at that point, you're at fractions. No? $\endgroup$ – JPBurke Jun 24 '14 at 2:27
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    $\begingroup$ @JPBurke Yes, in many ways you are at the very basics of fractions. What I wouldn't expect most kids to get at this stage is that 3 chances out of 6 is the same as 1 out of 2, although some with good number sense will get that (and also be able to extend to 6 chances out of 12). I didn't read the question as a challenge of teaching probability with no notion of fractions or part-whole relationships, but rather to assume no foreknowledge of fractions. $\endgroup$ – PurpleVermont Jun 24 '14 at 3:07
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I'm going to suggest what you're considering is very difficult. And also, I don't think it's something I'd want to do. (But we'll do it below anyway, for fun!)

Fractions are an early concept that involves a multiplicative relationship between things. In the case of fractions, one of those things is a whole. Students may have some understandings of fraction concepts without understanding the multiplicative relationship part. They can view fraction as a number (because a fraction is also a number). But this is an early opportunity for students to become acquainted with the idea of mathematics involving relationships.

So, the question is, what sort of understanding of relative anything (chances or whatever) is possible without other foundational understanding about relationships between numbers (beyond differences)?

Freeing myself to speculate:

What would be an example goal understanding of probability be, which does not involve any sort of multiplicative comparison?

Is it possible with some knowledge of quantitative comparison to understand how changes in a system do not change the relative chance (where "relative" just means that one is a better chance) of two things?

I would try to build off children's understanding of fairness, I guess. (Fairness, as in equal sharing, underlies fractions, but it is not an understanding of fractions).

For instance: There's ten cracker jack boxes to share. A toy is only in one. We could demonstrate that if I take a lot more boxes, I'm probably going to get the toy and you are not. It doesn't necessarily matter how often you get the toy with fewer (let's say "2") boxes. Kids understand "fair." And this is not fair.

Other uses of fair might be related to probability. Instead of equal sharing of objects, equal sharing of chance is underlying the notion of random sampling. If it is possible to demonstrate that the outcome can be different every time, yet everyone was treated in an equal way (fairness) that random sampling is a type of fairness that does not depend on the outcome.

Obviously I'm focusing on concepts here. It is difficult to do anything numerical without the understandings our exercise constrained us from using. But it is a bit of a fun exercise to think about.

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When I taught second grade, we introduced probability without fractions. We would talk about different events and classify them as:

  1. impossible
  2. unlikely
  3. equally likely and unlikely
  4. likely
  5. certain

The students had no trouble understanding classifying events such as: 1. the sun will rise tomorrow 2. i will pick a blue ball from a bag with a 5 blue and 1 red balls 3. I will pick an orange ball from a bag of blue and green balls. 4. It will rain today (this brought many opinions)

The following year fractions were incorporated into probability. It worked very well.

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    $\begingroup$ In a study from 1984 (referred to in Watson: The probabilistic reasoning of middle school students), 33% of 11-16 year olds interpreted unlikely as impossible. Do you have any comments about that? $\endgroup$ – Dag Oskar Madsen Oct 2 '16 at 11:18
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    $\begingroup$ If the students have no experience/examples using these words then they are going to interpret them according to the own experiences. For example, I had one student who refused to classify anything as impossible saying that God can do anything and nothing is impossible. I had to tell him that mathematically impossible means without intervention by God which seemed to satisfy him! I am wondering if the students were taught to use these words in a mathematical context or if they just interpreted the words themselves. $\endgroup$ – Amy B Oct 4 '16 at 17:55
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You need to reduce the nature of fractions and percentages. Some ideas:

  1. Reduce the nature of ratios, Laplace probabilities and relative frequency to x out of y. Representations of this can be:

    • symbolic: as the sentence „$x$ out of $y$“ or even the fraction symbol $\frac{x}{y}$.
    • iconic: Venn-Diagrams encircling the whole set and the event set.
    • enactive: a bowel representing the whole set and glass inside representing the event set, colored wheels of fortune
  2. Only compare different fractions:

    • enactively: put them as beads onto rubber bands with equal distance and draw the rubber bands until equally long. (Beware, that the last bead has a free distance as well.)
    • iconic/symbolic: if you have a fraction lower than $\frac{1}{2}$ and one larger than $\frac{1}{2}$, you can compare them via $\frac{1}{2}$: Are more results inside the event set or outside the event set?
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    $\begingroup$ I agree that this would be a good "prerequisite" lesson for the probability one, but I think only the very advanced elementary students would catch on. $\endgroup$ – jonsca Jun 23 '14 at 23:01
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    $\begingroup$ @jonsca This might be. Besides, I only answered about the fraction part of probabilities. There is a lot to say about probability theory/stochastics/statistics without using fractions. $\endgroup$ – Toscho Jun 24 '14 at 7:05
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Children have some intuitive notions about chance and probability. Here is an example: When my daughters were about 6 and 7 years old, I tried this experiment. We had two urns (well, shoe boxes, important they were identical), in one 8 black balls, 2 white, in the other numbers switched. Then we put the urns in a sack, mixed them and draw one. From the drawn urn we draw one ball. It was black.

Then I asked: from which urn do you think we got the ball? Thy thought about 1-2 seconds, then answered simultaneously: The urn with most black balls! ---Why do you think so? --- because then it is easier to draw black! they said, in chorus.

So thats it, more probable is the same thing as "easier". No need fro fractions!

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Seeing that our mathematically mature students (after a few rounds of calculus and some discrete math with proofs) struggle with this, and that rather simple problems like Monty and siblings lead to heated discussions here and on MSE, I'd forget about it all.

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    $\begingroup$ No, I don't think so. According to Spiral Approach, one can and should do very basic and easy probability and statistics even in elementary. Monty and siblings are no problem, as you don't get that far. $\endgroup$ – Toscho Jun 23 '14 at 12:55
  • $\begingroup$ @Toscho without fractions, when probabilities are between 0 and 1? Forgive me if I doubt it will work. $\endgroup$ – vonbrand Jun 23 '14 at 17:21
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    $\begingroup$ What is important about probabilities lying between 0 and 1? Nothing prohibiting doing this at all. $\endgroup$ – Toscho Jun 24 '14 at 7:04

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