2. Statistics is the art & science of gathering,
organizing, analyzing & drawing conclusions
from data.
Retrieved from: http://www.learner.org/courses/againstallodds/unitpages/unit01.html
What is Statistics?
(And without rudimentary knowledge of how it
works, people can't make informed judgments
and evaluations of a wide variety of things
encountered in daily life.)
3. Probability is the mathematics of
chance behavior.
Retrieved from: http://www.learner.org/courses/againstallodds/unitpages/unit18.html
What is Probability?
(And can help predict events such as
the daily weather, or whether an
asteroid will collide with Earth.)
4. What aspects of Statistics & Probability
are taught in the 6th Grade Classroom?
Organize, interpret, analyze, and display data to predict trends.
5a. Construct, interpret, and explain line graphs, double bar graphs, frequency
plots, stem-and-leaf plots, histograms, and box-and-whisker plots. (DOK 2)
5c. Predict trends based on graphical representation. (DOK 3)
5b. Determine how changes in data affect mean, median, mode, and range.
(DOK 2)
Source: Mississippi Department of Education 2007 Mississippi Mathematics Framework Revised Vertical Alignment
Data Analysis & Probability
25. Source:
Mississippi Department of Education 2007 Mississippi Mathematics Framework Revised Vertical Alignment Data Analysis & Probability
Vertical Alignment
27. Statistics & Probability
in the Classrooms:
Expanding Our Horizons
Presenter: Carl Mahlmann
The
End
The
Start
28. References
Kysh, J., Dietiker, L., Sallee, T., Hamada, L. & Hoey, B., (2013). Core Connections
Geometry. Sacramento, CA: CPM Education Program
Mahlmann, C. (2014). Module 1 Assignment: Statistics & Probability in the K-12 Setting,
Unpublished manuscript, University of Texas at Arlington, Arlington, TX.
Meyers, Paul (Senior Editor) (2008). Buckle Down Mississippi Mathematics Level 6 2nd
Edition. Iowa City, IA: Buckle Down Publishing
Mississippi Department of Education, 2007 Mississippi Mathematics Framework Revised,
Vertical Alignment: Data Analysis & Probability, Jackson, MS: Mississippi Department
of Education
29. Slide 10: [Elphaba/The Wicked Witch of the West - Photo].
Retrieved February 14, 2014 from: URL (http://sciencefiction.com/2013/10/28/witches/)
Slide 12: [Street - Photo]. Retrieved February 14, 2014
from: URL
(https://www.cityofeastlansing.com/Home/Departments/PublicWorks/EngineeringDivision/)
Slide 14: [Quintuplets - Photo]. Retrieved February 14, 2014
from: URL (http://mitziscollectibles.typepad.com/mitzis_collectibles_blog/2010/04/dionne-
quintuplets-101.html)
References
Good afternoon, fellow teachers. Thank you for coming.
As you read in your invitation, the purpose of today’s meeting is to initiate a dialog regarding how we might be able integrate the topic of Statistics & Probability both within our different subject areas and across grade levels.
As some of you are aware, for instance, I have already begun integrating other subjects into my classroom in order to expand students’ study and further their understanding of Statistics and Probability concepts:
Literature, for example – I recently had students read mystery stories in order to teach probability concepts;
Science and Business -- I have been showing videos on these subjects as vehicles to teach statistics; and
c) Even Phys Ed is being integrated – not only have we been using sports data in our statistics studies, but we have also used the gym to create our own data to study Measures of Central Tendency; we had students shoot “free-throws” in order to determine things like averages, and so on.
So, I hope to accomplish two things with today’s meeting: -- to expose to you what we are doing to integrate other subjects into my classroom in order to further students understanding of Statistics & Probability concepts, so that:
You might provide me with further ideas to incorporate in my classroom; and
Get you thinking about how you might integrate the study of Statistics & Probability within your own classrooms/subject areas.
So, I guess we should start at the very beginning; by discussing exactly what is meant by “Statistics & Probability.” When you hear the word “Statistics,” what comes to mind? (Write down audience’s answers on board.)
Thank you. As you can see, we have several answers; Statistics encompasses and applies to many different areas. Farmers, for instance, may use statistics formulated from data derived from rainfall, temperature and yield when planting crops. Memphis city planners look at traffic data when they think about building new roads. Statistics are also used to keep track of things like sports scores, weather forecasts, and to show what music is most popular.
Retrieved from: http://www.learner.org/courses/againstallodds/unitpages/unit01.html
Meyers, Paul (Senior Editor) (2008). Buckle Down Mississippi Mathematics Level 6 2nd Edition. Iowa City, IA: Buckle Down Publishing
And now, let’s talk about “Probability.” When you hear that word, what do you think of? (Write down audience’s answers on board.)
Thank you.
And so we see, that, Probability is also used in a wide array of aspects of everyday life – from seeing whether or not today’s weather forecast calls for a “chance” of rain, to deciding whether or not to invest in a stock, based on its likelihood of increasing or decreasing in value.
Retrieved from: http://www.learner.org/courses/againstallodds/unitpages/unit18.html
Within our 6th-Grade classroom, Statistics & Probability means:
Creating and working with graphs to display and interpret data;
Making predictions based on the information displayed in those graphs; and
Calculating Measures of Central Tendency for given data sets, and determining how changes in those data sets affect those measures.
Let’s talk about Graphs first…
Regarding graphs, students learn that, while it is sometimes difficult to make sense of a bunch of #s, if they take those #s & organize them into a graph…
… that graph can tell them a visual story about the #s.
They also learn that there are many different types of graphs; and that each has its own purpose. They learn that, just like you use different types of tools to do certain jobs (i.e., you use hammers to drive nails, and wrenches to work with bolts), you use different types of graphs depending on the type of data being used and what your needs are.
In summary, they learn that:
1) Graphs are used to organize data so that the data can be visualized.
2) Conclusions can be drawn – and predictions can be made -- based on data sets displayed in graphs.
There are many different types of graphs and each has its own purpose.
P.S.: The 6th-grade curriculum does not include all the graphs shown here. However, students will learn about all of them by the time they are in 8th grade. So, I included them here because we’ll also be talking more about vertical alignment in a little while.
Now let’s talk about Mean, Median, Mode & Range – a.k.a., “Measures of Central Tendency.”
The curriculum for 6th grade involves teaching students to determine how changes to a data set affect the Measures of Central Tendency. Before being able to determine that, one must first have an understanding of what “Measures of Central Tendency” are. So, I think it would be a good idea to do right now what I usually do with my 6th grade students before jumping into this objective – which is to review what “Measures of Central Tendency” are, and how to determine them.
First, I’m going to ask you to look at those words – “Central Tendency,” and tell me what comes to mind. (Write down audience’s answers on board.)
Thank you. When I think of “Central Tendency,” what comes to my mind is a central pattern – i.e., what “typically” is happening? In a data set – unlike a dartboard – there are many different ways of finding this “center.”
One way to find this center is by using the Measure of Central Tendency called the “Mean” – typically, what we refer to as the “Average.”
_________________________________________________________________________________________________________________
[Elphaba/The Wicked Witch of the West - Photo].
Retrieved February 14, 2014 from: URL (http://sciencefiction.com/2013/10/28/witches/)
The average is determined by:
Adding up all the numbers in your data set;
Counting the number of numbers in your data set; and then
Dividing your sum by your count.
It’s actually a “balancing” of the numbers in the data set – accomplished by redistributing parts of the higher value data points to the lower value data points, so there’s an equal value in every data point.
In other words, let’s say that you and your 2 friends have 18 cents altogether to buy candy. However, you have 12 of those cents, another friend has 6 cents, and your other friend only has 3. In the name of fairness, so everyone could buy the same amount of candy – and if you wanted to be really nice – you could give your friend some of your money so that everyone has an equal amount.
Using the method I described earlier, you could add all your money together (12 cents + 6 cents + 3 cents = 18 cents), count the number of numbers in your data set (3 stacks of pennies), and divide your sum by that count (18 / 3 = 6) to find what each friend would get – i.e., 6 cents.
So, “6” would be your Mean, in this case.
However, using the Mean – or “Average” – is not always the most useful way of determining the typical pattern (or Central Tendency) of a data set. For instance, suppose, instead of 12 cents – you had 12 dollars. If your friends still had 3 cents and 6 cents, I doubt very much that you would want to give away your hard-earned dollars to them (unless you were really, really nice – and maybe foolish – it just wouldn’t seem “fair,” would it? So, we would need a better way of finding the “Center,” in this case.
In this case, your $12 would be considered an “Outlier” in the data set. (And, although “Outliers” are not part of the 6th grade curriculum, I teach it to my students anyway because it helps them better understand the usefulness of Medians, and also helps prepare them for 7th grade.) If we included your $12 to find the Mean of this data set, it would really skew the data – it really wouldn’t tell us what the typical amount of money is that each friend has.
So, another way of finding this typical value is something called the “Median.”
_________________________________________________________________________________________________________________
[Street - Photo]. Retrieved February 14, 2014
from: URL (https://www.cityofeastlansing.com/Home/Departments/PublicWorks/EngineeringDivision/)
The Median is simply the middle value in the data set, when the values in the set are arranged in numerical order -- from least to greatest (or greatest to least).
In this case, the Median number (9) is probably more representative of the typical values in the data set than would be the Mean (which – having already done the math – happens to be 246). So, you can see here how an Outlier – in this case, the 1200 cents – might skew our interpretation of data.
P.S.: The data set which we just discussed is an odd-numbered set, therefore, it’s easy to find the middle data point – it has an equal number of data points above and below it. In an even-numbered data set, the Median would be determined using the Mean of the two middle data points. (This is a concept with which I’ve found many students struggle…)
Yet another Measure of Central Tendency is something called the “Mode.”
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[Quintuplets - Photo]. Retrieved February 14, 2014
from: URL (http://mitziscollectibles.typepad.com/mitzis_collectibles_blog/2010/04/dionne-quintuplets-101.html)
The Mode, quite simply, is the number which appears most often in a data set. In this case, the number 6 appears 2x, while each of the other numbers in the data set only appear once. So, “6” is the Mode, in this case.
There can be more than 1 mode in a data set – for example, if both the number 6 and the number 3 appeared the same number of times, the mode would be “3 & 6”. When a data set has 2 different modes, it is called “Bimodal.” A data set can also be “Multimodal” – i.e., if it has several different modes.
A data set can also have no mode, if each value in the data set only appears once.
Yet another Measure of Central Tendency is something called the “Range.”
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[Range - Photo]. Retrieved February 14, 2014
from: URL (http://www.fanpop.com/clubs/disney-crossover/images/29241589/title/empty-backdrop-from-home-on-range-screencap)
Range is determined by subtracting the least number in the data set from the greatest.
When I think of Range, I think of the “span” of the data – in this case, the “span” is from 3 to 27, or a span of 24. When I analyze my students test scores, I use all the Measures of Central Tendency, including Range. I find it useful to know, for instance, that the Range of my students test scores may have been from 50 – 100; a Range of 50 points. It it’s a relatively large range, that tells me that students either “got it,” or “didn’t get it.” So, that tells me that some re-teaching using some differentiation may be in order.
There are other Measures of Central Tendency – Standard Deviation, Variance, etc. – but those are not included anywhere in the middle school curriculum, so we will save those for another conversation.
Of all the different mathematics strands, Statistics & Probability especially lends itself to classroom activities for learning.
One thing I have found works very well is for students to create their own data with which to work. They are especially invested in gathering, displaying, analyzing and interpreting data that has personal meaning to them. For example, one of the things we did in our classroom was to measure each student’s height and weight, and then used that data for both graphing and to find Measures of Central Tendency.
(Could this activity somehow be integrated with Science / Biology?)
Students also recently created graphs using data derived from surveys they took regarding their own learning styles and personality preferences.
Additionally, students regularly graph their test scores. By using bar charts for this, they are able to easily see their growth over a period of time. (They also calculate their to-date Measures of Central Tendency from this data.)
(Could this be integrated in Reading – i.e., students charting the number of books and / or pages they read over a certain span of time?)
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[Young Analyst - Photo]. Retrieved February 14, 2014
from: URL (http://www.vertex42.com/edu/charts-and-graphs-in-excel.html
As mentioned earlier, I have begun integrating literature into the classroom to teach math concepts. Shown here are a couple of books that have recently been used to teach concepts regarding probability and properties of circles.
Not only does this use of literature integrate the subjects of Reading and Mathematics, but the books then become a perfect segue into activities the students can do themselves, based on the stories they just read.
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[Sir Cumference - Photo]. Retrieved February 14, 2014
from: URL (http://www.paperbackswap.com/Sir-Cumference-Dragon-Cindy-Neuschwander/book/0439180317/
[Whodunit- Photo]. Retrieved February 14, 2014
from: URL (http://www.amazon.com/Whodunit-Math-Puzzles-Bill-Wise/dp/product-description/0806958960)
By their very nature, games of chance lend themselves to helping students learn Probability concepts.
Recently, we played “Let’s Make a Deal,” “Craps,” “Rock, Paper, Scissors,” and Spinner games, among others. (No real money was involved…)
Suffice it to say, student engagement was very high. (It might have been even higher if real money was involved…)
_________________________________________________________________________________________________________________
Kysh, J., Dietiker, L., Sallee, T., Hamada, L. & Hoey, B., (2013). Core Connections Geometry. Sacramento, CA: CPM Education Program
[Let’s Make a Deal - Photo]. Retrieved February 14, 2014
from: URL (http://www.damninteresting.com/lets-make-a-deal/)
[Craps - Photo]. Retrieved February 14, 2014
from: URL (http://rapgenius.com/Ab-soul-soulo-ho3-lyrics#note-774679)
[Hands - Photo]. Retrieved February 14, 2014
from: URL (http://www.howtocopewithpain.org/blog/5369/awesome-things/)
I mentioned a couple of things earlier regarding review in 6th grade of concepts students learned in prior grades, as well as preview in 6th grade of concepts to be learned in later grades. I’d like to focus on that a bit more, now.
What you see here, and will see continued on the next slide, is the Vertical Alignment of Data Analysis & Probability mathematics strand from 4th thru 8th grades, prepared by the MS Department of Education.
This 1st slide reflects the Vertical Alignment of Graphs (displaying data).
_______________________________________________________________________________________________________________________________________________
Mississippi Department of Education, 2007 Mississippi Mathematics Framework Revised, Vertical Alignment: Data Analysis & Probability, Jackson, MS: Mississippi Department of Education
This next slide reflects the Vertical Alignment of Measures of Central Tendency and Probability.
I know the type here is small – it’s not necessary that it be read, because I’m going to summarize it for you here:
After examination of the mathematical strand of Data Analysis & Probability appearing in the Mississippi state standards framework, the key differences and patterns I’ve noted across grade levels are as follows:
Probability: The most obvious difference appears in the area of Probability – it is not an objective in 4th – 6th grades. Studies of this objective begin in 7th grade.
2) Measures of Central Tendency: The differences appear to be in the levels of cognitive demand from one grade to the next..
In 4th grade, students are essentially learning how to find Mean, Median, Mode & Range of data sets. In 5th grade, students are using these measures to analyze data sets. In 6th grade, students are then presented with the challenge of determining how changes to a data set affect those measures. (So, basically, they have to find the measures of more than one set of data and then compare those measures to see how they differ.) In 7th grade, this idea is expanded further with the introduction of Outliers, and students are asked to determine how these data points affect the Measures of Central tendency of a data set. In 8th grade (Pre-Algebra), this idea is expanded upon even further, requiring students to compare multiple data sets and how changes in data values affect their Measures of Central Tendency, as well as requiring students to determine the most appropriate Measure of Central Tendency for a particular purpose. (Which is, I believe, the key point to this objective – using Measures of Central Tendency to deduce meaning from a mass of data.)
3) Organize/Display Data (Graphs): The differences in this area are in the types of graphs with which students are required to work, as well as in levels of cognitive demand (especially in 7th grade, where students are required to synthesize concepts – i.e., proportions, percentages, estimation, and statistical displays – which previously were taught as distinct objectives).
The objectives for graphs are scaffolded across grade levels. In 4th grade, students work with bar graphs, line graphs and stem-&-leaf plots. In 5th grade, histograms and box-&-whisker plots are added. In 6th grade, double-bar graphs are added. In 7th grade, circle graphs and scatter plots are added.
In summary:
It follows logic that the differences from grade to grade also reflect the growth in student understanding of concepts. As stated above, when we look at the Measures of Central Tendency, the differences across grade levels appear to be in the levels of cognitive demand; from lower order thinking skills such as knowledge and comprehension to higher order thinking skills such as synthesis and evaluation (Bloom’s Taxonomy).
Students in 4th grade are finding mean, median, mode, and range, requiring them to memorize algorithms and “crunch numbers”. But by 8th grade, they are analyzing, synthesizing, comparing multiple data sets, and choosing and evaluating the best measure of central tendency to bring meaning to a data set. This holds true for the progression of skills in creating and interpreting graphs. Each year at least one new graph is introduced while building upon prior knowledge. Finally, probability is not introduced at all until 7th grade.
_______________________________________________________________________________________________________________________________________________
Mahlmann, C. (2014). Module 1 Assignment: Statistics & Probability in the K-12 Setting, Unpublished manuscript, University of Texas at Arlington, Arlington, TX.
Mississippi Department of Education, 2007 Mississippi Mathematics Framework Revised, Vertical Alignment: Data Analysis & Probability, Jackson, MS: Mississippi Department of Education
What this means for us as teachers, is:
Because of the spiraling nature of the curriculum, the most important implication for instruction is that students master the skills at their grade level. The concepts introduced at each level provide the prerequisite background knowledge for concepts and skills to come. When mastery is not achieved, teachers at higher levels spend inordinate amounts of time reviewing and re-teaching, leaving insufficient time for new skills to be taught. The result of non-mastery of prerequisite skills is that students slip farther and farther behind in successive years. The expectation for schools and districts should be that students master the skills of their grade before moving on to the next grade or class.
Another implication that this spiraling curriculum suggests is that teachers give students a “taste” of what is to come in higher grades. For example, since probability is not introduced until 7th grade, it might be helpful for some groundwork to be laid in this area prior to that grade. Suppose a 6th grade teacher is teaching ratios. S/he might have the class calculate the ratio of girls to boys in the room. Following that, s/he might ask, “If I were to randomly pick a name, do you think the odds would be that I would pick a girl or a boy?” Then the teacher could introduce the term probability, and explain to the students that what they are learning about ratios will help them in 7th grade. This would not take a great deal of time, but would help students understand that what they are learning will benefit them in years to come. This might build engagement and motivation for learning some content now.
Finally, teachers at every grade level should present students with problems, inquiry, or tasks which challenge them to grapple with their content in depth. Such tasks that require higher-order thinking skills would promote students’ mastery of current content as well as preparing them for the higher-order thinking challenges to come in later grades.
_________________________________________________________________________________________________________________
Mahlmann, C. (2014). Module 1 Assignment: Statistics & Probability in the K-12 Setting
[Dog - Photo]. Retrieved February 14, 2014
from: URL (http://bowwowblogger.wordpress.com/2012/12/06/turning-people-down/dog-question-mark-curious/)
So, as mentioned at the beginning of my presentation, the purpose of today’s meeting was to initiate a dialog regarding how we might be able integrate the topic of Statistics & Probability both within our different subject areas and across grade levels.
I trust the information provided to you today has given you some “food for thought.” I welcome your suggestions.
Thank you all for coming.
Questions?
