Geometric and Spatial Reasoning

Today’s Agenda Symbolic and Algebraic Reasoning: Review & Sharing Geometric Reasoning Can a Picture Prove? Spatial Reasoning Baseline Assessment You might want to expand on each of these agenda items. At the beginning of the session you will ask your participants to share their experiences within their own PLCs since the last time you have met. They will share results of their assessments and discuss what they learned as they analyzed their students’ notions of mathematical proof. In Region 11 this process often takes a full hour, but if you are implementing this module on a smaller scale it may take less time. Then you will move on to today’s topic, mathematical reasoning. Defining mathematical reasoning takes a bit of work, and judging a student’s mathematical reasoning skills can be very difficult. The main focus of today will be to show teachers a framework for assessing reasoning skills, but we’ll discuss a bit of logic first, because to evaluate a student’s reasoning you often need to evaluate whether or not their logical conclusions were correct. We intentionally avoided an in-depth discussion of truth tables, converse statements, contrapositives, etc., but if you have a higher level group you might wish to add some of these concepts to your presentation.

Symbolic and Algebraic Reasoning Review Symbolic and Algebraic Reasoning is interwoven throughout the MN math standards, but specifically, students should: “Justify steps in generating equivalent expressions by identifying the properties used...” (Minnesota Math Standard 9.2.3.7, Grade 9-11, MN Dept of Ed, 2007) Discuss how the Symb and Alg Reasoning session helped students meet this standard. 3

Reasoning Review What did you learn most from the Symbolic and Algebraic Reasoning baseline and summative assessments? Describe at least two ways that you helped your students use Symbolic and Algebraic reasoning. Describe one classroom situation where you saw a student exhibit growth in Symbolic or Algebraic reasoning. Discuss Assessment Data and experience. Break up the school groups and have teachers sit with teachers from other schools, by grade level. Give teachers time to discuss in these small groups first, writing down some thoughts on the handout. Then have them share with the entire group. 4

Geometric Reasoning

Geometric Reasoning “Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations.” (Minnesota Math Standard 9.3.2.4, Grade 9-11, MN Dept of Ed, 2007)

Van Hiele Levels of Geometric Understanding: Levels provide a way to characterize student understanding Level 0: Visualization students are able to recognize and name figures based on visual characteristics students can make measurements groupings are made based on appearances and not necessarily on properties Example: Students see squares turned on their corner as “diamonds”. Some additional background and examples for the levels: Level 0. Visualization: children identify prototypes of basic geometrical figures (triangle, circle, square). They view figures holistically without analyzing their properties. At this stage children might balk at calling a thin, wedge-shaped triangle (with sides 1, 20, 20 or sides 20, 20, 39) a "triangle", because it's so different in shape from an equilateral triangle. Squares are called "diamonds" and not recognized as squares if their sides are oriented at 45° to the horizontal. Children at this level often believe something is true based on a single example.

Van Hiele levels: Level 1: Analysis students can consider all shapes within a class rather than a specific shape focus on properties Example: Students see rectangles as having right angles and parallel sides. But they may insist that a square is not a rectangle. Level 1. Analysis: children can discuss the properties of the basic figures and recognize them by these properties, but might still insist that "a square is not a rectangle." Children do not see the relationships between the properties. They might reason inductively from several examples, but not deductively.

Van Hiele levels: Level 2: Informal Deduction students can develop relationships between and among properties proofs arise here, informally focus on relationships among properties of geometric objects Students can recognize relationships between types of shapes. Example: They can recognize that all squares are rectangle, but not all rectangles are squares. Level 2. Informal Deduction (also called Abstraction): students begin to reason deductively. They understand the relationships between properties and can reason with simple arguments about geometric figures. Learners recognize relationships between types of shapes. They recognize that all squares are rectangles, but not all rectangles are squares, and they understand why. They can tell whether it is possible or not to have a rectangle that is, for example, also a rhombus.

Van Hiele levels: Level 3: Deduction students can use logic to establish conjectures made at Level 2 student is able to work with abstract statements about geometric properties and make conclusions based on logic rather than intuition focus on deductive axiomatic systems for geometry such as Euclidean Example: Students can prove that the base angles in an isosceles triangle are congruent. Level 3. Deduction: learners can construct geometric proofs at a high school level. They understand the place of undefined terms, definitions, axioms and theorems.

Van Hiele levels: Level 4: Rigor students appreciate the distinctions and relationships between different axiomatic systems focus on comparisons and contrasts among different axiomatic systems of geometry Example: Students understand the parallel postulate and its meaning in the Euclidean system compared to its meaning in the spherical geometry system. Level 4. Rigor: learners understand axiomatic systems and can study non-Euclidean geometries, such as spherical geometry.

Using Geometry Logic Can you determine the shape that satisfies all of the following clues? It is a closed figure with straight sides. It has only two diagonals. Its diagonals are perpendicular. Its diagonals are not congruent. It has a diagonal that lies on a line of symmetry. It has a diagonal that bisects the angles it joins. It has a diagonal that bisects the other diagonal. It has a diagonal that does not bisect the other diagonal. It has no parallel sides. It has two pairs of consecutive congruent sides. Answer: kite (convex kite) Have these appear one-by-one and pause for 10-20 seconds after each. The idea is that the most specific clues are given toward the end. Most teachers should have a pretty good idea of the answer before the last few clues, so that these are really checks for them. This is from a Dale Seymour publication, Logic Geometry Problems: Grades 9-12 by Wade Sherard. Dale Seymour Publications 1997

What Van Hiele Level? Discuss what Van Hiele level you think the preceding geometry logic problem was. What grade level for students? How does this problem compare to the logic puzzle from the Math Reasoning Session, where you were determining the construction sequence for city buildings? Give 5 minutes or so to discuss at tables, then discuss as a whole group. This is the geometric equivalent of the logic puzzle given on the Mathematical Reasoning Day.

Developing Reasoning Via Open-Ended Problems Open-ended problems encourage reasoning from students (NCTM Book on Open-Ended Problems) Students use geometric reasoning in making observations about a figure Students then use reasoning in formulating arguments to support their observations Their observations can indicate their level of understanding

Open-Ended Problem In the figure below, BF and CD are angle bisectors of the isosceles triangle ABC. CF is the angle bisector of exterior angle ACH. Step 1: Find as many relations as you can. Step 2: What van Hiele level is required for each? This is from the NCTM Open-Ended Approach book, p. 116. There are twenty-nine relations found by Japanese students in the article, in a variety of categories including Congruence, Similarity, Shape and Area. For example, angle ABE is congruent to angle ACD since the angles are bisected. For another example, triangle BGC is an isosceles-shaped triangle. Other relations need more justification: AD is congruent to AE – this is shown by establishing that triangles ABE and ACD are congruent and using corresponding parts.

A Construction Problem You are given two intersecting straight lines and a point P marked on one of them, as in the figure below. Show how to construct, using straightedge and compass, a circle that is tangent to both lines and that has the point P as its point of tangency. From Alan Schoenfeld’s (1985) book, Problem Solving. In his experience, given the finished diagram the students could recognize and explain why the lines were tangent to the circle, but they had great difficulty when the problem was given in this direction. After teachers work on it, discuss why that might be the case.

Can a Picture Prove Something? Discuss your answer to the question above with your colleagues. OK, why or why not? Any examples to support your claim? This is a deep philosophical question in mathematics. Many schools of thought are outlined in the Hanna and Sidoli (2007) article which is discussed on the next few slides.

Does a picture prove? Debate exists over this in the mathematics community. M. Giaquinto (noted math philosopher) noted that there is a distinction between discovery and demonstration. Discovery (of new theorems, facts, etc.) is often very visual for the expert. Demonstration (proof) can only be accomplished visually if it’s clear the order of the statements being expressed.

A Mathematical Fact Fact: 1/4 + 1/16 + 1/64 + … = 1/3 Normally justified in a calculus class using the geometric series formula , since the series has the form a + ar + ar2 + … where a is ¼ and r is also ¼. How else to show this? Work out the math here by substituting in the appropriate values. Some will have forgotten the a/1-r formula, but a is usually referred to as the first term and r is referred to as the ratio. A geometric series must have terms which are multiplied by a common ratio, in this case ¼.

Its Proof? Have teachers discuss this at their tables, decipher the diagram and decide whether it is a proof or not. This is very similar to the proofs that explain versus proofs that prove from days 1 and 3. Big question: Is the mathematical argument, step-by-step, clear in the diagram?

Visual Proof that 64 = 65? Point out to teachers that one reason we tell students “Pictures illustrate, but can’t prove” is that pictures can deceive us if we are not careful. Challenge them to find the flaw in the visual argument that 64=65. There is a very thin quadrilateral which is almost invisible in the center of the rectangle on the right. This “lozenge”, as it’s sometimes referred to, has an area of one square unit.

Visual proof necessary conditions (Hanna & Sidoli, 2007, p. 75): Reliability – that the underlying means of arriving at the proof are reliable and that the result is unvarying with each inspection Consistency – That the means and end of the proof are consistent with other known facts, beliefs, and proofs. Repeatability – That the proof may be confirmed by or demonstrated to others. Discuss the conditions above with your colleagues. Do you agree? From Hanna and Sidoli: “In arguing their position, Borwein and Jorgenson (2001) cite the many differences between the visual and the logical modes of presentation. Whereas a mathematical proof has traditionally been presented as a sequence of valid inferences, a visual representation purporting to constitute a ‘‘visual proof’’ would be presented as a static picture. They point out that such a picture may well contain the same information as the traditional sequential presentation, but would not show an explicit path through that information and thus would leave ‘‘the viewer to establish what is important (and what is not) and in what order the dependencies should be assessed.’’ For this reason these researchers believe that successful visual proofs are rare, and tend to be limited in their scope and generalizability. They nevertheless concede that a number of compelling visual proofs do exist, such as those published in the book Proofs without words (Nelsen, 1993). As one example, they present the heuristic diagram, which aims to prove that the sum of the infinite series 1/4 + 1/16 + 1/64 + ! ! ! = 1/3 (see Fig. 3).”

Let’s use the prior discussion about conditions for visual proofs to revisit an old friend…

The Pythagorean Theorem! The sum of the squares of the lengths of the legs on a right triangle is equal to the square of the length of the hypotenuse. That is, if a is the length of one leg, and b is the length of the other leg, and c is the length of the hypotenuse, then a2 + b2 = c2. It is the ultimate marriage between geometry and algebra! Let’s look at some possibilities for proving it to determine their efficacy… There are lots of stories about Pythagoras you may want to interject here. A famous one involves the society he formed, in which it was rumored that if a member believed numbers such as the square root of 2 actually existed – they were taken out in a boat and drowned! Many other interesting stories can be found on the internet and in many math history books.

Pythagorean Theorem Proof #1 Is this a correct proof of the Pythagorean Theorem? Why? Encourage teachers to work out the algebra to verify that the picture is accurate. (Shown on the next slide) Refer teachers to the Hanna and Sidoli conditions to help decide if this is a good visual proof. This appears on the interview protocol.

Pythagorean Theorem Proof #1 More (algebraic) detail added here. The area of the big square is c2. This equals the area of the four right triangles plus the area of the smaller inside square. Algebraically: Refer to the Hanna and Sidoli conditions

Pythagorean Theorem Proof #2 Is this a correct proof of the Pythagorean Theorem? Why? Not a good proof. Fails to be reliable and consistent, in the framework described by Hanna and Sidoli. Trying a few examples is good inductive reasoning, but does not constitute a proof here which is general enough. The statement “We can try others and they will be the same” is vague and unsupported. 27

Pythagorean Theorem Proof #3 Is this a correct proof of the Pythagorean Theorem? Why? Once again, not a good proof. This argument only works for a right triangle which is also isosceles, and the Pythagorean Theorem works for all right triangles, regardless of shape. So this proof fails to be general enough, though it is better than trying a few values as is Proof #2 since it works for an entire class of right triangles. Due to the lack of generalizability, the proof fails to be consistent in the framework of Hanna and Sidoli – it is inconsistent with the assumptions of the theorem in that it doesn’t work for all right triangles. Incidentally, there are over 300 known proofs of the Pythagorean Theorem. Many can be found on the internet and collected in books, especially the Proofs Without Words series by Roger Nelsen. This is a theorem that may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs. 28

Spatial Reasoning 29

Spatial Reasoning We have used spatial reasoning in several activities so far but these have been all geometric. Q: What is an activity which uses spatial reasoning but is not so dependent upon standard geometry? A: Isometric views!

Isometric Views What is an isometric view? It is a 2-d picture which portrays a 3-d shape. The standard isometric view shows three faces of the 3-d object, top, right side and left side. Example: Cube

More with Isometric Views Cubes can be arranged and stacked to form 3-d landscapes Example: This was a popular perspective for video games in the 80s and 90s as polygonal faces were fairly easy to render for the technology at that time. Some popular titles which used an isometric perspective: Qbert (of course), Populous, Diablo and the Sims. Other popular perspectives are first-person (many shooter and driving games), top-down (Pac-Man and Zelda), and the scrolling side perspective (Pitfall and Super Mario).

Spatial Reasoning with Isometric Views There are four representations at work here: Physical manipulatives: centimeter blocks Mat plans: 2-d blueprints Top, bottom and side views of the 3-d shape Isometric view Students use spatial reasoning in shifting among these representations. Today we will focus mostly on shifting from isometric views to mat plans The NCTM Navigations book, Navigating Through Geometry, Grades 6-8, has a great activity where students shift among these four types of representations.

Isometric View to Mat Plan - I The mat plan for the given isometric view is below. The value in each box refers to the number of cubes stacked on that square Notice that the values are arranged in an “L”, like the iso view Front 1 Front

Isometric View to Mat Plan - I Create the mat plan for the given isometric view Discuss your plan with your colleagues. Can there be different mat plans for one isometric view? Why? Front The key is that behind any tower of 2 cubes or more, there could be some cubes hidden from this view. After some time, teachers will hopefully discover that there can be a tower one cube less in height hidden behind any tower. In other words, if a tower is n cubes tall, then there could be a tower of height (n-1) cubes hidden behind it. For example, there could be a tower of 1 cube tall hidden behind the height 2 tower in the picture. And behind the height 3 tower there could be a tower of height 2 hidden. But there’s even more: Behind the hidden tower of height 2 (which is behind the visible tower of height 3) there could be another hidden tower of height 1. Any taller tower would have its top seen. Teachers may resist this at first, but there is a lot of logic to be used here in figuring out the locations and numbers of hidden cubes. Front

Extending Spatial Reasoning The book pictured uses Cuisenaire Rods to encourage spatial reasoning Three views are given, students then construct the shape with the rods Position, color and length are all used as clues Only include this slide if time permits. ETA Cuisenaire book which often comes with sets of the Rods. They might want to dig out those Rods from the back room and give this a try!

Baseline Assessment Before finishing the day, go over the baseline assessment with participants. Attach a copy of it to their handout and give everybody 10-15 minutes (or as needed) to complete it, before discussing it problem by problem. Information about expected student responses is contained in the Microsoft Word document with the assessment. 37

Baseline Assessment – Item 1 What is the shape described below? Clue 1: It is a closed figure with 4 straight sides. Clue 2: It has 2 long sides and 2 short sides. Clue 3: The 2 long sides are the same length. Clue 4: The 2 short sides are the same length. Clue 5: One of the angles is larger than one of the other angles. Clue 6: Two of the angles are the same size. Clue 7: The other two angles are the same size. Clue 8: The 2 long sides are parallel. Clue 9: The 2 short sides are parallel.

Baseline Assessment - Item 2 Mat Plan Front Front

Baseline Assessment – Item 3 Why is this proof of the Pythagorean Theorem incorrect?

Baseline Assessment – Item 3 Why is this proof of the Pythagorean Theorem incorrect? 41