Category Archives: Concrete-Representation-Abstract
Tagged axis of symmetry, Maria, Priovolou, reflect, reflection, symmetrical, symmetry, task analysis, Topmarks
The work shown below posted on LinkedIn by Maria Priovolou. I think this is awesome.
The photo below shows a focus on just the vertical axis and the student has to reflect one object at a time. This is a nice task analysis approach. The stamp creates the objects which makes it hands on and a little different from just mathy work.
This hands on work can be followed with work on this website. In the photo at the bottom you see an example problem. This can make reflection more concrete and eventually more intuitive for the student.
Tagged adding fractions, conceptual understanding, concrete, concrete representational abstract, CRA, fractions, Legos
The photo below is courtesy of Robert Yu, Head of Lego Education China, as shared by Jonathan Rochelle, Director for Project Management at Google.
The use of Legos shown here is a classic (and wicked clever) example of manipulatives.
Before writing the actual fractions students can use drawings as shown below. The sequence of manipulatives, drawings then the actual “mathy” stuff constitutes a Concrete-Representational-Abstract (CRA) model. Concrete = manipulative, Representational = picture, abstract = symbolic or the “mathy” stuff.
Tagged authentic setting, authentic situation, batteries, comparison, comparison shopping, gas, gas cans, rates, real, real life, shopping, unit rates
Saw the following price tags, shown in the two photos, at an office supply store. $4 for 4 batteries or $9 for 8 batteries. To compare we can double the smaller pack to see that 2 packs would cost $8 for 8 batteries for a better deal.
Another method is to use unit rates. Rates are a measure of one quantity, with units, per 1 unit of another quantity, e.g. you make $10 per hour. To compute
$4/4 batteries = $1/1 battery vs $9/8 batteries = $1.13/1 battery
Below is an example of instruction for unit rates to help a student conceptually understand (pretend that gas price shown on this pump is $2 per gallon). Say you pumped 3 gallons and it cost $6. Show the 3 1-gallon gas cans together and the 6 $1 bills together. Separate them to you have equal groups to get $ per 1 gallon. You can use actual gas cans (unused) or cutouts from Google Images.
Tagged discount, discounted price, percent discount, sale, sales price, scaffold, scaffolded handout, scaffolding
A pseudo- concrete representation of a sales price problem is shown below. This is what I use as an entry point for teaching these problems.
The entire shape represents the total price of $80. This is 100%, which in student language is “the whole thing.”
The discount rate is 25%. Cut with scissors to lop off the 25% which also lops off $20, which is the actual discount. Explain to the student that this 25% is part of the “whole thing.”
What remains is 75% or $60. This is the “new price” which is called the sales price.
Tagged base ten blocks, concept, concrete, concrete representational abstract, CRA, hands-on, mathy
Here is an easy way to create and implement strategy to unpack place value for students (created by one of my former graduate students). I suggest using this after manipulatives and visual representations (drawing on paper) in a CRA sequence. It is hands on but it includes the symbolic representation (numbers). Hence is another step before jumping into the mathy stuff.
The focus can shift to money as well.
Tagged base ten blocks, CRA, fish, individualized, information processing, interest, pond, starburst
I am consistently surprised by the reliance on canned items for students who struggle. There are different reasons students struggle but we know that there are secondary characteristics and factors that inhibit effective information processing that can be addressed with some Individualization.
In a math intervention graduate course I teach at the University of Saint Joseph, my graduate students are matched up with a K-12 student with special needs. The graduate student implements instructional strategies learned from our course work. Below is the work of one of my grad students. From class work and our collaboration we developed the idea of using the fish and a pond as base 10 blocks for the student my grad student was helping. He likes fish and fish will get his attention. The grad student explained that if he has 10 fish the 10 fish go into a pond. In the photo below the student modeled 16 with a TEN (pond) and 6 ONES (fish).
Similarly, another student likes Starbursts and that student’s respective grad student created Starburst packs to represent TENS and ONES (there are actually 12 pieces in a pack so we fudged a little). The point is that it was intuitive and relevant for the student. The student understood opening a pack to get a Starburst piece.
Tagged alligator method, greater, greater than, inequality, less, less than, more, more than
The alligator eats the bigger number is the common approach for student to use inequality symbols (< > < > ). I find that students remember the sentence but many do not retain the concept or use the symbols correctly, even in high school. The reason, I believe, is that we introduce additional extraneous information: the act of eating, the mouth which is supposed to translate into a symbol, the alligator itself. For a student with processing or working memory challenges this additional information can be counter productive.
I use the dot method. By way of example here is the dot method. I show the symbols and highlight the end points to show one side has 2 dots and the other, 1.
Then I show 2 numbers such as 3 and 5 and ask “which is bigger?” In most cases the student indicates 5. I explain that because 5 is bigger it gets the 2 dots and then the 3 gets the 1 dot.
I then draw the lines to reveal the symbol. This method explicitly highlights the features of the symbol so the symbol can be more effectively interpreted.
That is the presentation of the symbol. To address the concept of more, especially for students more severely impacted by a disability, I use the following approach. I ask the parent for a favorite food item of the student, e.g. chicken nuggets. I then show two choices (pretend the nuggets look exactly the same) and prompt the student to make a selection. This brings in their intuitive understanding of more.
I think use the term “more than” by pointing to the plate with more and explain “this plate has more than this other plate.” I go on to use the quantities.
Finally, I introduce the symbol to represent this situation.
Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!
Tagged dividing sausage, fractions, kids, real life, real life application, sausage
I have 3 kids and was cooking sausage for them.
There were 5 sausage links available (below). How do I give each the same amount? Fractions!
Each child gets a full sausage link.
I then cut the remaining 2 sausage links into 3 parts, 1 for each child. 1/3 of a link.
Each child gets 1/3 and another 1/3 or 2/3. So they get 1 full link and 2/3 of a link or 1 2/3. This is an entry point into mixed numbers (whole number and a fraction).