@AdrianaLaGrange @DavidStaplesYEG

I am included you because I believe this is "rich feedback" that does not require classes to pilot it.

This is my follow up for the Grade 5 curriculum. It focuses only on one learning outcome: Algebra.

There is an awful lot to unpack here.

2/ Organizing Idea: Algebra
Learning Outcome: Students interpret numerical and algebraic expressions

The Knowledge section emphasizes knowing that expressions involving numbers are called numerical expressions.True, but not something that a curriculum document usually states

3/ The people who need this are teachers or parents who are using Google to find additional work or other creative ways to teach concepts around numerical expressions. Knowing the difference between a numerical and algebraic expression allows the search to be refined easily.

4/ The breakdown of numerical expressions according to “conventional order of operations” (why “conventional”?) outlines what is commonly referred to as BEDMAS fairly well. It is interesting that this document refers to “parentheses”, as opposed to “brackets”.

5/ In Canada, we have generally used the word brackets and the acronym BEDMAS, when teaching this concept; in the US, parentheses is usually used, with the acronym PEMDAS. Largely insignificant in the big picture, but it gives me an idea of where this curriculum is coming from.

6/ Between the second statement in the Knowledge section and the second statement in the Understanding section there are a couple of statements that demonstrate either an oversimplification or a lack of mathematical understanding.

7/ We are told that “[p]arantheses change the order of operations in a numerical expression”. For the purposes of what this draft is doing and at a grade 5 level, this is true enough, but the reasons behind it actually align with the DMAS part of BEDMAS.

8/ For example, in a question such as this:

3 + 2(4 + 7) traditional order of operations tells us to do the brackets first:

3 + 2(11) followed by the multiplication (as represented by the brackets):

3+ 22 and then the addition:

25. But why?

9/ The reality is that the brackets are telling us that both the 4 and the 7 are being multiplied by 2. This is what the distributive property will be used for.

10/ When dealing with numerical expressions, it allows us to potentially solve the question this way:

3 + 8 + 14 and then use addition to obtain 25.

11/ In this way, order of operations has been maintained because still did all of the multiplication before we did any of the addition. Knowing what the brackets actually mean allows us to maintain the order of operations for multiplication, division, addition, and subtraction.

12/ This is important because when students start to deal with algebraic expressions they will NEED to know how to distribute values; there will be occasions where the values inside the brackets will NOT belike terms so this is the ONLY option to correctly solving the expression.

13/ It is also interesting that the entire section on numerical expressions does not make reference to exponents at all. It is my assumption that this is because the numerical expressions being explored will not have exponents in them; fair enough.

14/ As I will show later that is consistent with other curriculums (who teach this concept in grade 4, but I digress). The other curriculums overtly state this omission so that teachers clearly know what is to be taught and where the limits are.

15/ This draft does not do this; as a result, teachers may go much farther than the draft curriculum intends, causing much frustration to their students, who are not mentally ready to incorporate exponents into order of operations.

16/ Side note: a numerical expression does not always represent a quantity of known value, as stated in the Understanding. If so, give the defined and known value of this expression:

(4 + 8)/0

17/ Of course, it also needs to be discussed whether or not these concepts are age appropriate for grade 5. In Ontario, this is taught in grade 6 (and has some of the same muddled explanations that the draft has, with regards to defining order of operations.

18/ But it does use the term brackets. It also overtly states that students should be able to perform order of operations on whole numbers,decimals,and fractions. The draft curriculum does not say which number systems students should be able to evaluate numerical expressions for

19/ I would assume that the draft also expects whole numbers, decimals, and fractions, but it is not overtly stated.

20/ In Australia, it appears as though students are expected to be able to perform order of operations involving multiplying, dividing, adding, and subtracting in grade 6, and also explore the use of brackets.

21/ It appears, although I am not certain, that only whole numbers are used at this point, as students do not learn how to multiply and divide fractions until grade 7. Decimals may be included here, though, as students will have learned all four operations with them by grade 6.

22/ The Common Core is a little messy. In grade 5 students are introduced to order of operations (where the terms parentheses, brackets, and braces are all used) but they are only taught them with whole numbers.

23/ In one part of the document it says that students should be able to evaluate simple expressions; in another, it says they are supposed to interpret them without evaluating them.

24/ In any case, it is very clearly limited to whole numbers (no fractions, or at appears, decimals, and definitely no exponents, as that is explicitly introduced in grade 6).

25/ No matter which curriculum is referenced, the intention of the Alberta draft curriculum goes further than any other does in grade 5 and may go further than many do even in grade 7.

26/ Unfortunately, there is a real lack of clarity across the board, not just in the draft curriculum that makes it difficult for me to make as strong of a comparison as I have in other places.

Sadly, this is only the first part of this outcome.

27/ The draft also wants students in grade 5 to explore algebraic expressions. Based on what I read, there is no expectation that students will evaluate or simplify these expressions, but I am not sure as the Skills and Procedures statements are more vague and garbled than most

28/ The Knowledge section:

We are told that “[a] variable can be interpreted as a specific unknown value and is represented symbolically by a letter”. This is true, a variable can be interpreted this way. Sometimes.

29/ It may actually have more than one specific unknown values, it may have non-permissible values and other restrictions, it may have an infinite number of possible solutions to the expression. This definition is terrible.

30/ We also have an oversimplification of how variables are represented with multiplication and division. In multiplication, it is common that 3 times a variable is represented by 3n, but that is not exclusive.

31/ It is also common to see a dot placed in between the two or a conventional multiplication sign, or brackets, like so 3(n). The draft states as absolute fact that products are done one way. That is just unmitigated garbage.

32/ Similarly, while quotients are generally expressed as fractions and it makes for much better and easier to read math when it is, it does not have to be. The division sign may also be used.

33/ And when students get to high school, it is possible that they will learn to perform long division of algebraic expressions using the standard algorithm, as opposed to writing it as a fraction. Once again, an absolute statement of the type used here is inappropriate.

34/ Lastly, a quick statement about constant terms. Yes, a constant term is represented by a number but students will later learn that it actually does have a variable (that variable has an exponent of 0).

35/ Of all the errors in this section, this is the least offensive to me, as it is not something that students will see until much later; but it is still technically oversimplified.

36/ In terms of age appropriateness, it is tough to compare because the 3 curriculums I am using have their own vagueness.What this draft asks students to do appears to align with the grade 6 curriculum in Ont, although there are elements of this outcome in the grade 5 curriculum

37/ For example, in Ontario, evaluating an algebraic expression is a grade 5 objective - when dealing with whole numbers only. The same objective is used in grade 6 to incorporate decimals as well, with integers incorporated in grade 7.

38/ Nothing is mentioned about fractions, although I assume this would be included at the same time as decimals and little is mentioned about the representation as a product or a quotient (although it does restrict the expressions to those that are linear, so no exponents).

39/ The draft curriculum offered to Albertans has no limits or restrictions, so I honestly don’t know whether this objective is intended to include only whole numbers, or it is to include decimals and fractions as well. That would seem to be an important piece to tell teachers.

40/ More importantly, there is a massive emphasis on pictorial and graphical representations so that students can begin to make connections to patterns that they will use in middle and high school.

41/ This allows students to visualize what a variable truly means and understand why using a letter to represent a value can be important. The connection to linear equations will also be helpful, as this will help students when developing tables of values, linear graphs, etc.

42/ Most of this is done at a grade 6 level in the Ontario curriculum, but it is also introduced, especially concretely at the grade 5 level as well. None of this is even mentioned in the draft curriculum.

43/ It is almost as if the writers believe that algebra should be taught in absence of these connections and patterns, as though it is something abstract, a construct of mathematicians to play with numbers and little more.

44/ This emphasis may allow students to do rote algebra (if they are mature enough to understand abstract patterning) but it does nothing to encourage students to be interested in math itself.

45/ Many, many students will likely be left behind because their brains are not emotionally mature enough to understand these concepts without concrete, pictorial and graphical representations.

46/ In Australia, it appears as though the introduction of variables into algebraic expressions of the type mentioned in the draft does not occur until grade 7 (although variables and substitution are mentioned earlier, dealing with multiple terms seems to happen later.

47/ Again, just by looking at the curriculums only it is tough to say this with 100% certainty). Patterning and connections that model algebraic expressions seems to occur in grades 5 and 6.

48/ In grade 5 and 6 there is an absolute emphasis on concrete patterns such as dots, although the graphical connections appear to be omitted until grade 7, where those connections are made.

49/ Here, too, there is a progression from concrete, to pictorial, to graphical that threads its way through three grades. Maybe the draft has some of this as well but without a scope and sequence or even standard organizing ideas it is incredibly difficult to follow.

50/ Plus, I don’t if the grade 7 curriculum will have at all. The draft includes all of its talk of linear functions in the pattern organizing idea the connections between algebraic expressions with two terms and linear equations are difficult to make.

51/ I would suspect any elementary teacher without 26 years of experience teaching math would also find those connections difficult to make.

The Common Core does introduce variables or algebraic expressions in grade 5, restricting its curriculum only to numerical expressions.

52/ In gr 6, algebraic expressions and substitution of values into them is introduced; while I could find nothing that specifically limits the number of terms and variables, like the Albertan draft curriculum, the examples appear to restrict expressions to terms and one variable

53/ Of course, this is grade 6 and the draft curriculum implements this in grade 5. One important difference is that the Common Core includes exponents in the substitution and evaluation.

54/ The draft curriculum does not seem to (although this is not stated by using the word linear anywhere). Lastly, the Common Core has an entire section where students are expected to “[r]epresent and analyze quantitative relationships between dependent and independent variables”

55/ In their examples, they distinctly emphasize the connection between an algebraic expression and a graph. There do not appear to be the same links to concrete and pictorial representations, but at least one for of connection is made - and that is one more than the draft offers

56/ There is so much left to discuss; about half of grade 5 and all of grade 6, but I'm not sure I will continue with this exercise. All told, I am at 27 pages on my Google Doc. Seems like enough for now. /end

57/ OK so I lied. It turns out that I missed one of the items in the Algebra section, solving for equations. I'll try to be brief:

58/ The last section, like so many before it, also has an issue with oversimplification.We are told that “[e]quality is preserved by applying inverse operations to algebraic expressions on each side of an equation”.And this is true for anything that grade 5 students will be doing

59/ But the inverse operation of multiplication is division. If students were given a question like so:

0x + 7 = 7

They would apply inverse operations and subtract 7 from both sides to get 0x = 0.

60/ Yet, if inverse operations were applied here to divide both sides by 0, the answer would be undefined (because you can’t divide by zero. Einstein supposedly famously said that “black holes are where God divided by zero” but outside of that, it can’t be done).

61/ This is a small matter but also another case of oversimplification. Too many times, this curriculum attempts to make itself look smart in its Knowledge and Understanding sections only to show exactly how little mathematical knowledge and understanding they actually possess.

62/ A broader question I have is why the only method allowed is the use of inverse operations. Again, inverse operations are efficient, effective and explainable. They are functional methods for many of the question grade 5 students will be given.

63/ But should it be the only method used? This draft tends to be prescriptive about the methods that teachers must use. That too presents a problem.

64/ It narrows creative thinking and has the potential to remove any passion or love of mathematics from children, who may find intuitive methods that work for them. This does not preclude the need to learn inverse operations; far from it.

65/ I am just saying it should not be the only method. These students are in grade 5; some concrete, or at least pictorial representation is appropriate here.

And it needs to be remembered that these students are in grade 5.

66/ They are being asked to do two-step equations (although why use that phrase when “equations with one or two operations can be used in instead”). In Ontario, Grade 5 students are limited to one operation and to numbers up to 50 (no such limit for the Albertan draft);

67/ it is not until Gr 6 that equations with two operations are introduced,again limited, but this time up to 100. In Australia,linear equations involving 2-steps are not taught until Gr 7 and involve using a variety of strategies so that students can make connections to the math

68/ The Common Core does not introduce linear equations until Gr 6, and then does so only with one-step equations. It is not until Gr 7 that students are introduced to two-step equations (but at that point are expected to deal with all types of rational numbers).

69/ In each of these curriculums, efforts are made to indicate which number systems or parameters students should be taught. Not so in the current draft. Moreover, none of these curriculums expect Grade 5 students to perform two-step equations. NONE. OF. THEM.

70/ At the earliest this started in grade 6 and in some cases it is not done until two years later. I cannot help but think that this outcome is not age appropriate based on this information. /end.